Cfa examination quantitative study

© Knowledge Varsity – 2011 Page 1

Concept Notes For Reading 5 - Time value of money

Reading Summary

This reading is the most important reading for the CFA Level 1 Examination as it covers the basics of

present value (PV) of the expected future cash flows. The PV concept is used throughout the curriculum

with major emphasis in Equity Valuation, Fixed Income valuation and corporate finance. A mastery of

time value is paramount to be successful in the examination.

The topic covers calculation of present value and future value for a single cash flow, a series of cash

flow (known as annuity), computation of effective rate, problems related to mortgages, retirement.

Understanding of time line is critical and it is advised that the candidates should draw the timeline while

solving any problems. For easy problems it might look that drawing time line is waste of time but

timeline is important when you are solving complex problems with multiple cash flows as the timeline

will give a perfect picture of the cash inflows and outflows.

Basic Idea and Concepts

Time Value of Money (TVM) links cash flow, interest rate, compounding frequency, time period and the

present value or the future value. TVM reflects that a $1 amount with a person today is more valuable

than $1 at a future date because $1 can be invested now and interest can be earned on the investment.

TVM would require calculation of Future Value (FV), which is result of compounding the investment at a

certain interest rate. TVM also entails finding the present value (PV), which is reverse of compounding

and is referred as discounting. Comparison of PV or FV is very useful in analyzing investments and then

making an investment decision. As an investor you would be more interested to invest in those

products which will fetch you more money in the future for the same amount invested today.

Usage of Financial Calculator

Solving TVM problems becomes very easy when you are using financial calculator. There are 2 types of

calculator permitted in the CFA Exam, one from Hewlett Packard and another from Texas Instruments.

We recommend Texas Instrument’s calculators because they are similar to the calculators that you have

used in the past and are easy to master.

In TI BA II Plus calculator you have a set of TVM Calculation Key. The following are the function keys

available for TVM calculation

N = Number of compounding periods FV = Future value

I/Y = Interest rate per compounding period PMT = Annuity payments, or constant periodic cash

flow

PV = Present value CPT = Compute

We will provide detail in some of the questions on the operation of the calculator.

Timeline

Timeline is a representation of the cash flows with respect to time. We follow a notation when drawing

a timeline

Cash Outflow – Any cash outflow is treated as Negative

Cash Inflow – Any cash outflow is treated as Positive

-$500 $200 $200 $200 $200

T=0 T=1 T=2 T=3 T=4

© Knowledge Varsity – 2011

In the timeline above, we have an initial investment of $500 being done and hence it is an outflow, that

is why we have taken the amount as -$500 in the timeline. Since the initial investment is done now, the

time has been put as T=0 in the timeline. Likewise we are getting a series of return every period from

the investment, since the amount of $200 is cash inflow to us, we have taken a positive sign for it. There

are four payments of $200 each received and hence we are displaying four payments in the time line.

Drawing a time line is a good idea to solve any TVM problem.

LOS 5.a. Interpret interest rates as required rate of return, discount rate or opportunity cost

There are three ways in which we can interpret the interest rates:-

1. Required rate of return is the return required by the investors to postpone their current

consumption to a future period.

2. Discount rate is the rate used to discount the future cash flows (that is the money to be received in

the future) to the current period. In essence discount rate and required rate are one and the same. We

use the term “discount rate” to bring a future cash flow to present period and the term “required rate”

to compound a current cash flow to future period.

3. Opportunity cost is the benefit that investors would have received if they had invested their money

in some other investment and they are investing in any other product of the same risk they should be

able to meet the opportunity cost. This can also be said as the value that investors forego by choosing a

particular investment. This would be clear from the following example:-

Suppose an investor has taken loan at the rate of 10% pa from a bank and is offered an opportunity to

invest in a product A which is promising a yield of 12%. Now the investor’s opportunity cost is 12%, he

would invest in any other product having same risk if the return from the other product is more than

12%.

LOS 5.b. Explain an interest rate as the sum of a real risk-free rate, expected inflation, and premiums

that compensate investors for distinct types of risk;

When there are no uncertainty and no inflation then the interest rate received by the investor is known

as Real Risk Free Rate. The risk free rate is observed in only the government bonds (also known as

treasury securities), however these are not real rate as there is some amount of inflation present. The

interest rate given by the government securities is Nominal risk free rate.

Here we bring the concept of inflation premium. When an inflation premium is added to real risk free

rate, the resulting rate is the nominal risk free rate. Following equation will make things clear

Or we can approximate the equation to;

Since we are in an uncertain world there are risks present in any investment, an investor should be

compensated for these risks, as a result the required return need to be increased by what is termed as

risk premium. There are three types of risk premium we will discuss in this topic, these are:-

Default Risk Premium is the premium added for the likelihood or the probability that the

borrower will not meet interest or principal payment on time and is likely to default on the

loan. When there is higher uncertainty the premium would be more.

Maturity Risk Premium is the premium that one obtains for parting away with his money for

longer period of time. Typically as the no of days of an investment increases, the investor is

being deprived of his wealth for longer time hence the investor should be compensated by way

of higher return. This higher return is attributed as maturity risk premium.

Liquidity Risk Premium is the premium provided to the investor for investing in securities

having low liquidity. Liquidity here implies how fast the asset can be sold in the market and

money received. Typically real estate investments give higher return because the investment is

not liquid, it may take months to sell a property and hence liquidity premium should be more.


So, for an investor the required return on equity should be the approximately the sum of real risk free

rate and the various premiums.

Required rate = Real Risk Free Rate + Inflation Premium + Default Risk premium

+ Liquidity Risk premium + Maturity Risk premium

In Fixed Income study session 15, reading 61, we will come across other risk premiums that an investor

should consider when investing.

LOS 5.c. Calculate and interpret the effective annual rate, given the stated annual interest rate and the

frequency of compounding

Compounding is a concept in which interest accumulate over a period of time. A compounding period is

a period in which interest is accrued or accumulated; there can be different compounding period in a

year.

There are three ways in which we can quote the interest rate in a year

Periodic Interest rate – This is the interest rate that is applicable for a period, the period can be

1 month, 1 day, 1 year or anything. For example if the 6 months periodic rate is 5%, it means

that we will get return of 5% in 6 month.

Stated annual interest rate – This is also known as quoted interest rate. Here the interest rate

is stated annually, so for the example of the periodic rate given above, the stated annual rate

will be 5% multiplied by 2 or 10%.

Effective annual rate (EAR) takes the effect of compounding within a year. For a stated annual

rate, the higher the number of compounding the higher is the effective annual rate. EAR can be

stated as per the following formula

Where, periodic rate = Stated annual rate / Number of compounding period in a year

Continuous compounding is a concept in which there is infinite number of compounding period in a

year. The interest rate thus is called as continuous compounding interest rate.

For a continuous compounding rate, the EAR is given as

Where r is the stated annual rate

Concept Builder – Effective Annual Rate Computation - Single Computation

1. Find Effective Annual rate for a bank deposit, in which the stated interest rate is 8% per annum and

the frequency of compounding is semi-annually

Answer

Remember whenever we are computing effective annual rate, we need to find first the periodic rate.

Periodic rate = stated rate / number of periods in a year

Periodic rate = 8%/2 = 4%

EAR = (1.04)2 -1 = .0816 = 8.16%

Concept Builder – Effective Annual Rate Computation – Understanding increasing frequency

2. For a bank deposit, the stated interest rate is 12% per annum compounded monthly?

Find the Effective Annual Rate (EAR), for the following compounding frequency

a) Semi-annually b) Quarterly c) Monthly d) Daily e) Continuously

Answer

a) When the compounding is semi-annual => EAR = (1.06)2 – 1 = 12.36%

b) When the compounding is quarterly => EAR = (1.03)4 – 1 = 12.5509%

Calculator TIP: Press 1.03 ; then press yx function key (present above 9), now press 4

You have computed 1.034 ; now you press -; then press 1; then multiply by 100 to get the rate


c) When the compounding is monthly=> EAR = (1.01)12 – 1 = 12.6825%

d) When the compounding is daily=> periodic rate = 12%/365 = 0.032877%

EAR = (1.000329)365 – 1 = 12.7475%

e) The EAR for a continuously compounding rate is – ; please note that r is the stated rate

EAR = –

Calculator TIP: Press 0.12; then Press 2nd; then press LN (present on the left side of 7); this will

compute ; now subtract 1; after this multiply by 100.

Now, the stated rate was 12%, as we increase the number of compounding, you can see that the EAR is

increasing, this is primarily because as the compounding period increases, the more interest is earned

on the interest, so it results in higher effective rate.

The highest possible effective interest rate would be achieved when continuous compounding is done.

LOS 5.d. Solve time value of money problems when compounding periods are other than annual;

When compounding periods are not annually then we must take into consideration that the number of

period will be more than 1. Also we need to adjust the periodic rate to reflect this.

As a thumb-rule, if there are m periods in a year and there are n years, then we should divide the

interest rate by m and multiply the number of years by m.

For example, if we are required to solve for a problem in which the stated rate is 8% pa, with quarterly

compounding and 5 years, we should have the periodic rate as 2% and the number of periods as 20.

Concept Builder – Finding Future Value when compounding period is more than annual

3. Canara bank is offering interest rate of 9.15% for fixed deposits for 2 years. The compounding is

quarterly, how much amount would you receive at the end of the period, if you deposit $1000.

Answer

There are 2 approach to solve problems of this kind

1. Find EAR and then find the FV using the compounding for multiple years

Periodic rate = 9.15%/4 = 2.2875%

EAR = (1.022875)4 – 1 = 9.4688%

FV = PV( 1 + EAR)N

So the amount after 2 years will be equal to = 1000 * 1.0946882 = $1,198.3412

2. Solve directly using the calculator

Here FV = ?

PV = -$1000 (Note –ve sign) ; PMT = 0 ; I/Y = 9.15/4 = 2.2875; N = 2 * 4 = 8

FV = $1,198.34

Calculator TIP: First Press 2nd ; Press FV; this will clear the previous TVM calculation

For PV: Press 1000; Press +|- (placed right of decimal); Press PV => PV = -1000

For PMT: Press 0; Press PMT => PMT = 0

For I/Y : Press 2.2875 ; Press I/Y => I/Y = 2.2875

For N: Press 8; Press N => N = 8

Now to find out FV; Press CPT (top left corner); Press FV => You will get FV = 1198.3412

Concept Builder – Finding Present Value when compounding period is more than annual

4. How much money should you deposit now in your investment account to get $1000 after 3 years.

Assuming that the investment account offers a return of 10%, compounded semi-annually?

Answer

As we have seen in the above example, its easier to use the TVM function of the calculator to solve

these kind of problems, we will use the TVM keys to solve it.

First we need to identify the values associated with the TVM Keys.


Here, FV = 1000; PV = ? ; N = 2 * 3 = 6; I/Y = 10/2 = 5; PMT = 0

So we need to input these values in the calculator to find out the answer

The answer would be -$746.21, the negative sign implies that an amount of 746 goes out from you

(outflow) in the investment account.

Calculator TIP: First Press 2nd ; Press FV; this will clear the previous TVM calculation

For FV: Press 1000; Press FV => FV = 1000

For PMT: Press 0; Press PMT => PMT =0

For I/Y : Press 10; Press ÷ ; Press 2; Press =; Press I/Y ; Press I/Y => I/Y = 5

For N: Press 8; Press N=> N = 8

Now to find out PV; Press CPT (top left corner); Press PV => You will get PV = -746.21

LOS 5.e. calculate and interpret the future value (FV) and present value (PV) of a single sum of money,

an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows;

Future value is the money to be received at a later date. We have the following formula for the Future

Value for a single cash investment. We assume that the interest generated is also invested at the given

rate.

Concept Builder – Future Value of a Single Sum

5. How much would my term deposit of $1000 will become in 7 years, if the bank is offering me 10%

interest rate compounded annually?

Answer

This is similar question like Concept Builder # 3; only thing here is that the compounding period is

annual.

We can solve this using the calculator as shown in the #3. But whenever you see a single cash flow, its

much simpler to use the formula directly than using calculator.

FV = PV (1 + I/Y) N => FV = 1000 * (1.1)7 = $1948.7171

So, if you invest $1000 now, you will get $1948.71 after 7 years.

Present Value is the money that you can assume to be equivalent of the future cash flows. The present

value of the expected single future cash flows is given by

• FV = future value at time n

• PV = present value

• I/Y = interest rate per period

• N = number of periods

Concept Builder – Present Value of a Single Sum

6. How much would be the present value of an investment which promises to return $1000 in 7 years,

the required rate of return is 10%?

Answer

This is similar question like Concept Builder # 4; only thing here is that the compounding period is

annual.

We can solve this using the calculator as shown in concept builder #4. But whenever you see a single

cash flow, its much simpler to use the formula directly than using calculator.

PV = FV/ (1 + I/Y) N => PV = 1000 / (1.1)7 = 513.158

Calculator TIP:

First Find 1.17 => Press 1.1; Press YX; Press 7; => you will get 1.17; which is 1.948717

Now, Press button (above YX) => you will get 0.513158

Now, press X; press 1000; press =; You will get $513.158 (which is the present value)


Perpetuity is a concept in which same cash flow is received for an indefinite (infinite) number of periods

in the future. Present value of perpetuity is what the infinite cash flow should be worth now.

PV of Perpetuity = Cash flow / Required rate of return

Or

PV of perpetuity is used in the valuation of preferred stock.

Preferred stock is a financial product which gives a constant dividend till perpetuity. Value of preferred

stock is given by

Concept Builder – Present Value of a Perpetuity

7. Tata motor has issued a preferred stock, which pays an annual dividend of $10. If an investor’s

required rate of return is 10%, how much the preferred stock would be valued by that investor?

Answer

Price of preferred stock would be the same as the present value of the dividends that it is paying.

Since, preferred stock is a perpetuity, we will be calculating the present value from its formula.

PV = D/r

PV = $10/0.1 = $100

So, the price at which the preferred stock should sell will be equal to $100.

Concept Builder – Required rate of return of a perpetuity when PV is given

8. A preferred stock is trading at a price of $120 per share. The preferred stock pays annual dividend

of $8. What is the required rate of return for the preferred stock?

Answer

In this problem, we know the price and the dividend that it is paying, the required rate of return can be

calculated using the formula of PV of preferred stock.

PV = D/r => r = D/PV => required rate = $8/$120 = 6.67%

Annuities – Sometimes we do not invest the money in lump-sum only. We invest regularly, so this is the

concept of annuity. The regular period can be annually, quarterly, monthly or daily.

Present Value of Annuity is the sum that you can receive now in lieu of the future cash flows.

Future Value of Annuity is the sum that you will receive in the future for the investments that

you are doing periodically.

You need not remember the formula of the perpetuity, we should use financial calculator to solve

perpetuity problems. You will be explained in the class and the video as to how to apply the concept.

There are 2 types of annuities

Ordinary Annuity – In ordinary annuity the investment is done at the end of the period. So if you

are buying a product on 1st January 2010 and the product is annual pay then first payment will be

done on 31st December 2010. If the product matures in 5 years then you will make the last payment

on 31st December 2014 and also receive the maturity value on 31st December 2014.

Annuity due – In annuity due the investment is done at the beginning of the period. So if you are

buying a product on 1st January 2010 and the product is annual pay then first payment will be done

on 1st January 2010. If the product matures in 5 years then you will make the last payment on 1st

January 2014 and receive the maturity value on 31st December 2014.

So, in both the case we are receiving maturity value on 31st December 2014, but in ordinary annuity

the last payment is not earning any interest, hence the Future Value will be lesser than that of the

annuity due.


Concept Builder – Present Value and Future Value of Ordinary Annuity

9. I have invested in an ordinary annuity product; the annual payment is $1000 for a total of 10

payments. First installment will start after 1 year from today, how much is the PV of this? How

much it will become at the end of 10 years? Interest rate on this product is 8%.

Answer

We would be solving all the problems in which there are intermediate cash flows using the financial

calculator. You should be able to identify the value of any 4 of the 5 functions of TVM. One unknown

value can be easily found out from calculator.

NOTE: For the Ordinary Annuity; we use the END Mode in the calculator.

For Computing PV – PV is unknown

Here, PMT = -$1000; N = 10; I/Y = 8; FV = 0; PV =?

Why is FV zero here? – Please note that when you are calculating PV, assume that FV is zero and vice

versa. The PMT and N is taking care of the amount of cash that is being deposited.

Please see the timeline for this below

So, using financial calculator, we will input the values for the various parameters and then compute FV.

CPT -> PV will give the present value; PV = $6,710.08

For Computing FV – FV is unknown

Here, PMT = -$1000; N = 10; I/Y = 8; PV = 0; FV =?

CPT -> FV will give the future value; FV = $14,486.56

Concept Builder – Using Annuity Concept to Value a Bond

10. A bond having face value of $100 is paying annual coupon, the coupon rate is 10%. The bond has 5

years to maturity. If the current market interest rate (required rate) is 6%, then at what price the

bond should sell in the market?

Answer

Face value of a bond is the amount that we receive at the maturity of the bond. So face value equals the

future value that is received from an investment in bond.

The coupon payment is calculated on the face value. So a bond having coupon rate of 10% will given

coupon equal to 10% * Face value = 10% * $100 = $10, coupon can be regarded as the PMT.

The bond pays the first coupon at the end of the period, so a bond valuation can be thought of as

Ordinary Annuity.

For an annual pay bond, the years to maturity will be the number of payment made. For bonds with

semi-annual coupon, the number of period will be twice the number of years to maturity and coupon

would be half of the annual coupon.

The market interest rate can be thought of as required rate or I/Y.

So, for this bond valuation, we have the following parameters

N = 5; I/Y = 6; PV = ? ; PMT = $10; FV = $100


Please note that as an investor, we will receive the coupon and also the face value, and hence as per

our sign convention, both of these should be Positive.

CPT -> PV = -$116.849

The negative sign means, that you will have to pay $116.849 to buy the bond.

This is how the bonds are valued and this would be used in Financial Reporting and Fixed Income

Valuation.

Concept Builder – Present Value of Ordinary Annuity, when the first payment is at a later date

11. My grandfather will give me $1000 for 6 years when I become 18 years old. I have just completed

15 years. My required rate of return is 10% p.a. How much is the present value of my grand pa’s

gift?

Answer

Please see the timeline below, I will receive the first payment of $1000, when I will become 18 years.

Since there are total of 6 payments, I will get the payment till age of 23 (and not 24)

For ordinary annuity or In END Mode, the following point need to be imprint in your mind

1. The PV is one period before the first payment day

2. The FV is on the last payment day

If we calculate the present value from the calculator, it will give us the present value at T = 17 and not

at T = 15, which is our requirement.

So, this problem involves 2 steps. First find out the present value using the calculator, it would come at

T =17 and then discount this for 2 more periods to arrive at the present value at T=15.

1st Step: N = 6; I/Y =10; PV = ? ; PMT = 1000; FV = 0

CPT -> PV = -$4,355.26 ; Please note that this is PV at T =17

2nd Step: PV (at T = 15) = PV (at T =17) / (1+r)2

PV15 = PV17/(1+r)2 => 4355/1.12 => $3,599.38

So, the present value of the payment is $3,599.38

Concept Builder – Present and Future Value of Annuity due

12. I have invested in annuity due product of installment $1000 per year for 10 payments. First

installment started today, how much is the PV of this? How much it will become at the end of 10

years? Interest rate on this product is 8%.

Answer

COMPUTATION OF Present Value.

See the timeline of this problem below, pay attention here, the first payment is being done at T=0 and

there is no payment done at T=10, the last payment was done at T= 9.

There are 2 approach to solve such type of problems


A. Using our financial calculator in Begin Mode

B. Using our financial calculator in End Mode (Ordinary annuity mode) & then making adjustment

First using the Begin Mode.

Please change the mode in your calculator by performing the following functions

Press 2nd; Press PMT; Press 2nd ; Press ENTER – You will begin seeing BGN in your calculator, it shows

that your calculator is now in begin mode.

In BEGIN Mode, the following point need to be imprint in your mind

1. The PV is on the first payment day

Now, with begin mode set, it is a simple problem. You just need to enter the parameters like the way

you had entered in concept builder #9

Here, PMT = -$1000; N = 10; I/Y = 8; FV = 0; PV =?

CPT-> PV => PV = $7246.88

So, the present value of the investment is $7246.88

Using the other approach : The problem can be solved using the end mode also. Since your calculator is

in Begin Mode, you need to change it to END Mode. The steps to do this is same as the way we

changed to begin mode

Press 2nd; Press PMT; Press 2nd ; Press ENTER – You will begin seeing END in your calculator, it shows

that your calculator is now in END mode.

When the calculator is in END mode, we had observed that the present value is 1 period before the

first payment.

So here the PV will come at T =-1. See the timeline below.

But we need to find out the value at T =0 and not at T = -1, so what we need to do is to find out the PV

at T =0 from the value we obtained at T = -1

PV (At T = 0) = PV (At T = -1) * (1+r)

Or, we can say that PV(Annuity Due) = PV(Ordinary Annuity) * (1 + r )

Here, PMT = -$1000; N = 10; I/Y = 8; FV = 0; PV =?

CPT-> PV => PV = $6710.08

This PV is at T=-1

=> PV0 = PV-1 * (1+r) => PV0 = 6710.08 * 1.08 = $7246.88

So, the value of PV is same in both the case.

COMPUTATION OF Future Value.

Here also, we have the same 2 approach to solve the problem.

First we will solve using the BEGIN Mode.

In BEGIN Mode, the following point need to be imprint in your mind

1. The FV is one period after the last payment day

We will change the mode from END To BEGIN, using the steps highlighted earlier.

Please change the mode in your calculator by performing the following functions

Press 2nd; Press PMT; Press 2nd ; Press ENTER – You will begin seeing BGN in your calculator, it shows

that your calculator is now in begin mode.


In the Begin mode, the timeline looks like below.

So, we will enter the value and find out the future value.

Here, PMT = -$1000; N = 10; I/Y = 8; FV = ?; PV =0

CPT-> FV => FV = $15,645.48

Secondly, we will solve the problem using the END Mode

Since your calculator is in Begin Mode, you need to change it to END Mode. The steps to do this is same

as the way we changed to begin mode

Press 2nd; Press PMT; Press 2nd ; Press ENTER – You will begin seeing END in your calculator, it shows

that your calculator is now in END mode.

When the calculator is in END mode, we had observed that the Future value is on the last payment.

So here the FV will come at T =9. See the timeline below.

But, we want the FV at T=10

So, we can arrive at the FV at 10, by compounding the FV at 9

FV10 = FV9 * (1+r)

So, we have a new rule => FV(Annuity Due) = FV(Ordinary Annuity) * (1 + r )

So, we will enter the value and find out the future value.

Here, PMT = -$1000; N = 10; I/Y = 8; FV = ?; PV =0

CPT-> FV => FV9 = $14,486.56

We know, FV10 = FV9 * (1+r) => FV10 = $14,486.56 * 1.08 = $15,645.48

So, the answer is the same as it was obtained from the Begin Mode.

IMPORTANT

So, whenever we are asked to calculate the PV or FV of annuity due, we can solve using the default END

Mode. We advise you to follow this approach, because in exam, if you forget to change the mode then

it would be problematic for you, since most of the questions would be end mode questions.

Things to Learn

In END Mode

1. The PV is one period before the first payment day

2. The FV is on the last payment day

In BEGIN Mode

1. The PV is on the first payment day

2. The FV is one period after the last payment day


Concept Builder – Future Value at a later date of Annuity due

13. I am making an investment of $100 every year, starting from now for a total 3 payments. How much

money I will receive at the end of 6 years? The interest rate of the investment is 10%.

Answer

First you need to make timeline for this particular problem, which is given below

We would be using our calculator in END MODE

In the END Mode, if we are computing the FV, it would come at T=2

Now, we should compound the value received at T=2 to get the FV at T=6. Since there are 4 periods in

between, we can write

FV6 = FV2 * (1+r)4

Plugging in the values in the financial calculator

Here, PMT = -$100; N = 3; I/Y =10; PV =0; FV = ?

CPT-> FV => FV2 = $331

FV6 = $331 * (1.1)4 = $484.617

So, I will receive $484.6 at the end of 6 years.

Concept Builder – Present and Future value of cash flows when the cash flow is not same

14. Find the present value and the future value for the following cash flows. Required rate of return is

10%.

Answer

Since, the TVM functions assume that the PMT remains the same, we can’t use TVM here. We should

use the Cash Flow function here.

You will find, a button named CF in the 2nd row, 2nd Column.

Whenever we are calculating using CF, we should first clear the memory.

To Clear Memory – Press CF; Press 2nd; Press CE|C

You will observe CF0 on the screen, it is asking for the Cash flow at T=0

For CF0: Press 1000; Press +|- ; Press ENTER; Press ↓

You See CF1: Press 2000; Press +|- ; Press ENTER; Press ↓

You see F01 => By default its value is 1 in the calculator. This is the concept of frequency, it is asking you

how many times -2000 is coming consecutively in the problem. Since -2000 is coming consecutively

only once, we will leave F01 at 1 only and press the down arrow ↓ to move to C02.

You see C02: Press 3000; Press +|- ; Press ENTER; Press ↓↓ (Yes 2 times, as the frequency of -3000 is

also 1)

You see C03: Press 2000; Press +|- ; Press ENTER; Press ↓

Now, the data is entered, we will calculate the PV.

Note that, for CF function, there is a button to calculate NPV.

NPV is Net Present Value and is equal to PV of Inflows – Outflows , but for this NPV will be same as the

PV.


Press NPV; You will see I in the screen, Press 10; Press ENTER; Press ↓; You will see NPV on the screen;

Press CPT

NPV = -6800.15

For finding out the future value, you can assume that you are investing 6,800 for a period of 4 years.

FV = 6800 * 1.14 => FV = $9,956.1

LOS 5.f. Draw a time line and solve time value of money applications (for example, mortgages and

savings for college tuition or retirement).

As mentioned, we can solve various type of problems by employing time value of money calculation and it is better to draw a timeline before solving the problem. This LOS covers the application of TVM concepts in real life. We will cover this LOS through examples and understand the various applications. Concept Builder – Calculation of I/Y

15. You have deposited $100 in the account today, after 7 years, the amount would become $200, what

is the stated annual rate if the compounding is annual?

Answer

Here we need to calculate the rate or I/Y

The parameters of TVM are :

FV = $200; PV = -$100; N = 7; PMT = 0; I/Y = ?

CPT -> I/Y => I/Y = 10.4%

Concept Builder – Calculation of PMT

16. You are required to deposit a fixed amount every year in your investment account starting from the

end of this year. If the stated rate is 10% p.a. and you are depositing for a total of 10 years, you will

receive an amount of $100,000. How much amount you should deposit every year?

Answer

Here we need to calculate the amount deposited every year or PMT

Note that there is no money in the account today, it is a case of ordinary annuity since the money is

deposited at the end of the period.


FV = $100,000; PV = $0; N = 10; I/Y =10; PMT = ?

CPT -> PMT => PMT = -$6,274.5

So, you need to deposit 6,274.5 every year to get $100,000 at the end of 10 years.

Concept Builder – Calculation of Number of Periods

17. You are shown an investment plan that will require depositing every year $10,000 starting from the

end of this year; it is promised that at the end you will get a sum of $115,000. If the interest rate

offered by the investment plan is 10.15%, then find out for how many years you will have to deposit

the money in the plan?

Answer

Here we need to calculate N


FV = $115,000; PV = $0; PMT = -$10,000; I/Y =10; N= ?

CPT -> N => N = 8.03

So, you need to deposit the amount for 8 years.


Concept Builder – Compounded Annual Growth Rate

18. Tata Steel’s EPS was $5 in 2002, In year 2008 the EPS was $14. Find the compounded annual growth

rate (CAGR) of the EPS.

Answer

CAGR is the rate at which the EPS has grown.

Please see the timeline below

Easily we can solve this with the financial calculator

So, the number of periods from 2002 to 2008 is equal to 6, therefore N = 6


FV = -$14; PV = $5; PMT = 0; N=6; I/Y =?

CPT -> I/Y => I/Y = 18.72%

So, the EPS has grown at an average rate of 18.72% for the 6 years.

Concept Builder – Loan Calculation

19. I took a housing of $200,000 for 15 years; the rate of the loan is 10%. Calculate

A. How much monthly payment I am doing.

B. How much is the principal payment done in 1st month?

C. How much is the principal remaining after 60 installments.

Answer

This is another set of problem, where you will find financial calculator handy.

Please note that housing loan is an amortizing loan, where both principal and interest is paid in the

monthly payment that is being done. At the end of the loan term, there is no money that is required to

be paid to the bank, so the future value is 0.

For Part A – It is just asking for the EMI that I am paying and hence we need to find the PMT


FV = 0; PV = $200,000; I/Y =10/12 = 0.833; N=15 *12 = 180; PMT =?

CPT ->PMT => PMT = -2,149.21

So, every month, you should pay $2,149.2 towards your housing loan

For Part B –So, for the 1st month, we can calculate the interest that is accrued on $200,000 for 1 month

using the simple interest formula

Interest for the 1st month = PRT/100 => ($200,000 * 10 * 1 )/(12 *100) = $1,666.67

Out of the EMI of $2,149.21 ; $1667.67 is towards the interest charge

The principal payment = $2,149.21 - $1667.67 = $482.54

For Part C – This is a difficult question if you think conventionally, here you need to think Out Of Box, to

answer it in a simple manner. Think, if everything was as planned, you were paying the EMI and now 60

months are left. Can I say, that if you approach the bank after 60 months for a loan, they should give

you an amount which will be paid off by the end of 120 months, So we can say that whatever the

principal is remaining, should be equal to the loan amount that you would be given after 60 months.

To solve, this now your N = 120

FV = 0; I/Y =10/12 = 0.833; N=120; PMT =-2,149.21 ; PV = ?

CPT-> PV =>PV = $162,633.24

Since you have financial, these things are much easier to solve. Your calculator has AMORT Function,

which can be accessed by pressing 2nd and then Pressing PV.

Once you do that, you will find P1 on the screen.


P1 is the point from which we want to calculate the principal payment/interest payment etc.

If you press ↓; you will find P2.

P2 is the point till which we want to calculate the principal payment/interest payment etc.

For Part B – we can solve using P1 = 1 and P2 = 1 , since we are interest for the principal payment for

the 1st month only.

Calculator TIP: Press 2nd; Press PV; You will see P1; Press 1; Press ENTER; Press ↓; You will see P2;

Press 1; Press ENTER;

Press ↓; You will See BAL => this is the Principal Remaining after P2

Press ↓; You will See PRN => this is the Principal paid between P1 and P2 => $482.54

Press ↓; You will See INT => this is the Interest paid between P1 and P2

For PART C – Using the financial calculator

P1 = 1; P2 = 60 (as we are interested in finding the principal remaining after 60 months)

Calculator TIP: Press 2nd; Press PV; You will see P1; Press 1; Press ENTER; Press ↓; You will see P2;

Press 60; Press ENTER;

Press ↓; You will See BAL => this is the Principal Remaining after P2 = $162,633

Press ↓; You will See PRN => this is the Principal paid between P1 and P2 => $37,366.76

Press ↓; You will See INT => this is the Interest paid between P1 and P2 => $91,585.85

So, in first 60 months, I have paid $91,585.85 as the interest and $37,366.76 as the principal.

Concept Builder – Retirement Calculation

20. An investor, plan to retire at the age of 60. He expects to live till the age of 90. He is currently aged

25. He invests the first payment in the account today and invest for 35 years (that is total of 35

investment done). The retirement account earns 12% per annum. Assume that he would like to

withdraw $30,000 per year starting from the point when he turns 60 for 30 years, find out the

amount he should deposit in his retirement account every year? Here assume, that the investor

doesn’t leave any money for his heirs.

Answer

This is a complex problem involving 2 series of cash flows. It’s good to draw a timeline to solve the

problem

In these type of problems, we need to come up with a common point.

If we take the middle point as common point, that will be the best.

So, we can find out the PV of the cash inflows at T = 59; Using that, we can find out the PV at T=60; and

then we can compute the PMT for the series of cash outflows.

Lets do step by step calculation

Step 1: Find out the PV at T = 59

Keep the calculator in the END mode, the 2nd series of cash flows from T=60 to T=89 looks like ordinary

annuity and the PV will come one period before the first cash flow.

FV = 0; I/Y =12; N=30; PMT =$30,000; PV = ?

CPT-> PV => PV = $241,655.19

Step 2: Find out the PMT. Here you have to note that the FV of the investment would be equal to the


present value that we have found out.

FV = PV59 = $241,655.19; PV = $0; N = 35; I/Y =12; PMT = ?

CPT -> PMT => PMT = -$559.82

So, the investor need to deposit $559.82 every year for 35 years, in order to get $30,000 every year

when he retires.

Experience, the power of Compounding, the investor is depositing only $560 and is able to withdraw

$30,000 per year.

If the investor, delays his investment by 5 years, that is the first payment , he does at the age of 30, then

he would be required to deposit $1,000 to experience the same inflow during retirement.

So, it’s always advisable to start the investment early.


Concept Notes For Reading 6 - Discounted Cash Flow Applications

Reading Summary

This topic is another one which is relevant across the CFA Curriculum. In recent times, CFA Institute has

started asking questions on the concepts in this chapter and it might happen that you may be getting

questions on some of these topics. The calculation of cash flow using financial calculator will be covered

in this topic, however the same would be used in Corporate Finance and Equity Valuation. You should

be thorough with the Cash Flow function of your calculator. The return concepts of Time Weighted and

Money Weighted Returns may be intimidating for some to start with, but these are essential concepts

for any person in the investment industry. Lastly you are expected to understand some other return

measures which are kind of industry convention and are popular because of their simplicity. You should

solve all the questions that are given in this book as well as the problems that are there in the institute

book then only a clear-cut understanding of this chapter will emerge.

LOS 6.a. Calculate and interpret the net present value (NPV) and the internal rate of return (IRR) of an

investment

Companies take up projects as part of their business activity, typically a project requires upfront

investment (a cash outflow) and then subsequently companies benefit in the form of cash inflow. For a

company to undertake a project, not only the amount of return it is getting should be more than the

investment being done but also the company should be able to earn a decent rate of return on the

investment. To make investment decisions companies evaluate projects on the basis of NPV and IRR.

NPV or Net Present Value of an investment is its present value of the expected cash inflows minus the

present value of its expected cash outflows.

IRR or Internal Rate of Return is the discount rate which would make the NPV of a project equal to

Zero.

NPV Calculation Process – NPV of any project involves the following steps

1. Identify all the cash inflows and outflows

2. Determine the discount rate that should be used to discount these cash flows. In this section we

need not calculate the discount rate ourselves, when we cover the corporate finance section,

we will understand how to compute the rate.

3. Using the discount rate computed in step 2, find out the present value of cash outflows and

cash inflows. We as per our convention treat outflows as negative (hence they decrease NPV)

and treat inflow as positive (hence they increase NPV)

4. NPV is the summation of all the present value of the cash flows computed above. Please note

that since we are abiding by the sign convention, hence we need to just do the addition.

NPV Formula:

CFt : Cash flow in time period t

N : the number of periods for which investment is made

r : the discount rate used for finding out the present value


Concept Builder – NPV and IRR Computation

1. Compute the NPV of a project which requires initial investment of $10 million. The cash flows are

expected to be $6 million in year 1, $5 million in year 2, $4 million in year 3 and $3 million in year 4.

The discount rate for the investment is 15%.

Answer

NPV:The NPV of the project can be calculated by discounting the cash flows.

NPV = -$10 + $5.217 + $3.780 + $2.630 + $1.715

NPV = $3.343 million

It is better to compute the NPV using financial calculator, the usage of TI BA II Plus will be discussed in

the classroom and also in the Pre-class content.

IRR: Computation of IRR is difficult manually and should be done using financial calculator only. There is

IRR function in the calculator which will be able to derive the IRR, please note that when discounted at

the IRR rate, the NPV will be equal to Zero.

The value of IRR = 32.9783%

LOS 6.b. Contrast the NPV rule to the IRR rule, and identify problems associated with the IRR Rule

In any investment the idea is to generate more wealth than what you have invested or produce a return

which is more than the cost of capital or required rate of return.

Typically the projects can be classified under the following

1. Independent Projects – When acceptance or rejection of a project does not affect the acceptance

or rejection of other projects then the projects are considered as independent projects. As an

example suppose a firm is considering investment in a new printer and investment in a vehicle,

these can be considered as two independent projects if the company has fund to buy both and the

company is evaluating these two investments as per their cash flow.

2. Mutually Exclusive Projects – Typically companies do not have high amount of capital and most of

the times it becomes necessary to take up only one project out of two, due to capital constraints. In

such case we say that the projects are mutually exclusive. For example if the firm has money either

to install printer or to buy the vehicle but not for both then it becomes important for the firm to

identify the project which will be more beneficial.

To evaluate any investment decision we have the following two rules

1. NPV Decision Rule: If a project is yielding positive value to the firm then the project is a good

project and it should be accepted. We can summarize this decision rule

a. Accept projects which have positive NPV as these projects are adding to the shareholder

wealth

b. Reject project having negative NPV as these projects are destroying the shareholder wealth

c. In case of mutually exclusive projects, select the one which has higher NPV. Mutually

exclusive projects are those projects in which only one can be selected, naturally the higher

NPV project is increasing the shareholder wealth more and hence we should accept that

project

2. IRR Decision Rule: The decision rule based on IRR helps in evaluation of how much return is being

generated.

a. Accept projects whose IRR is more than the required rate of return

b. Reject those whose IRR is less than the required rate

c. Unlike the NPV decision rule here we can’t say directly that in case of mutually exclusive

project we should accept project having higher IRR. This is because of certain problems with

the IRR method


Problems Associated with the IRR Rule

Before we take up the problems with the IRR rule, first we need to understand that NPV and IRR would

give the same result for the independent projects. This is because

If NPV > 0 then IRR should be more than the required rate of return

If NPV < 0 then IRR should be less than the required rate of return

However for mutually exclusive projects (let’s say projects A &B) there can be conflicting results, that is

the NPV rule may result in selection of project A whereas IRR may result in selection of project B.

This conflict happens because of the following problems

1. Reinvestment–The calculation of IRR assumes that all the cash flows are reinvested at the IRR rate.

For example if a project has IRR of 20%, then it is assumed that the cash inflows which are expected

would be re-invested at 20% rate. Now many times it might happen that the firm is not able to find

lucrative investments and hence it might not be able to re-invest those cash flows at IRR rate, let’s

say that the firm is reinvesting those cash flows at a rate of 15% then the actual return from the

investment would be lower than the 20% which was estimated earlier. We will be covering this

concept in the Fixed Income in detail.

2. Timing – Another reason for conflict is the timing of the cash flows, one project may produce higher

cash flows during the early periods and another project may have higher cash flows in the later part

of the project.

3. Initial Investment –Conflicting results can be observed in cases where one project requires smaller

investment and produce smaller amount of cash flows compared to another project which require

more investment and produces higher cash flows.

You can remember these through the acronym RTI (Right to Information)

Concept Builder –NPV and IRR Rule Conflict

2.There are two mutually exclusive projects which have the following cash flows

Project A: CF0 = -$10,000; CF1 = $20,000

Project B: CF0 = -$40,000; CF1 = $60,000

The required rate of return is 10%. Which project should be selected and why?

Answer

Compute the NPV and the IRR of each of the project using the financial calculator

Project NPV IRR

A $8,181 100%

B $14,545 50%

If these were independent projects then both the project should have been selected because the NPV is

greater than 0 or IRR is more than the required rate of return.

However since the projects are mutually exclusive, we should select one of these projects. In case of

mutually exclusive projects we should compare the NPV and the project having higher NPV should be

selected, hence project B should be selected.

LOS 6.c. Define, calculate, and interpret a holding period return (total return)

Holding Period Return (HPR) is the return that is generated over the holding period. The holding period

can be any period; it can be a minute, several hours, many days or years, please do not confuse holding

period as yearly period. Here we are interested in determining the total return which means that the

return does not only include the change in the investment value but also include any cash flow that is

generated by the investment within the holding period. These cash flows which are generated are

called as intermediate cash flows. Stock investment has intermediate cash flow in the form of dividend

and bond investment results in coupon payment.


Where for an equity investment

P1 is the price at the end of the period

P0 is the price at the end of the period

D is the dividend received

Concept Builder –Holding Period Return

3. An investor purchased a share for $20 and sold it at $22 after 6 months, also he received a dividend

of $0.5 just before he sold the share. Find his holding period return?

Answer

Ending Value of investment = $22

Beginning Value of investment = $20

Dividend received = $0.5

HPR = (22- 20 + 0.5) / 20 = 2.5/20

HPR = 12.5%

Concept Builder –Holding Period Return

4. A t bill priced at $98 with face value of $100 and 180 days until maturity. Find the holding period

return for the T bill?

Answer

Ending Value of investment = $100

Beginning Value of investment = $98

HPR = (100- 98) / 98 = 2/98

HPR = 2.0408%

LOS 6.d. Calculate, interpret, and distinguish between the money-weighted and time-weighted rates of

return of a portfolio, and appraise the performance of portfolios based on these measures

Money Weighted Return(MWR) – It is actually the internal rate of return for an investment. All the

deposits in the investment account are treated as cash inflow and all the withdrawals are treated as

cash outflow. After determining the inflow and outflow, IRR is calculated. This calculation is similar to

the calculation that we had done in Example 1.

Time Weighted Return(TWR)– It is actually the Geometric mean return of an investment. For

calculation of TWR we need to divide the investment horizon into several periods. The period for the

return calculation is computed whenever there is a major inflow or outflow from the portfolio. The

geometric mean of all the periodic returns is then calculated to find the time weighted return.

If there are N holding periods, the TWR can be calculated as follows

(1 + TWR)N = (1+ HPR1)*(1+ HPR2)*(1+ HPR3)* …….. (1+ HPRN)

Where HPRi is the Holding Period Return in the ith period

Concept Builder –Money Weighted Return and Time Weighted Return

5. An investor purchased one share of Goldman Sachs (GS) at $100 at T = 0, he again purchased one

more share of GS at $120 at T =1. He received total dividend of $2 at T= 1 and a dividend of $3 per share

at T =2, the investor sold the shares at T=2 for a total consideration of $260. Find money weighted and

time weighted return?

Answer

Important assumption : We assume that the dividend is paid just before the period ends.

Money Weighted Return Calculation:

Step 1 : We should identify the cash flows that are happening at the various intervals. As per our sign

convention, money going out from investor should be treated as negative and money coming to an


investor should be treated as positive.

At T =0 : Purchase of 1st share was done => Cash outflow of $100 => CF0 = -$100

At T =1 : Purchase of 2nd share was done => Cash outflow of $120

Dividend is received => Cash Inflow of $2

Sum of these two results in cash outflow of $118 => CF1 = -$118

At T =2 : 2 shares are sold => Cash inflow of $260

Dividend is received => Cash Inflow of $3 * 2 = $6

Sum of these two results in cash inflow of $266 => CF1 = $266

Step 2: Plug the cash flow values in the financial calculator to compute the IRR (because MWR is nothing

but IRR)

IRR = 14.438% or Money Weighted Return is 14.438%

Time Weighted return Calculation:

Step 1: We should first find out the number of periods, here significant cash in-flow is happening at T=1

and cash outflow is happening at T=2. So we have one period from T=0 to T=1 and another period from

T=1 to T=2

Step 2: Compute the portfolio value just before the significant cash inflow or outflow.

For Period 1: Beginning Value = $100; Ending Value = $120; Dividend Received = $2

For Period 2: Beginning Value = $120 (for the 1st share) + $120 (amount that is invested) = $240

Ending Value = $260; Dividend Received = $6 (for 2 shares)

Note: Many candidates have confusion regarding why the beginning value of the portfolio should be

$240, why not it should be $220, the point here is that at the beginning of period 2, the first share could

be sold for $120 and there is an investment of $120 more, hence the portfolio value is $240. Also we

should consider the dividend in this period for both the shares.

Step 3: Find out the holding period return of the periods

For Period 1: HPR = (120 – 100 + 2)/100 = 22%

For Period 2: HPR = (260 – 240 + 6)/240 = 10.83%

Step 4: Find out the geometric mean return or TWR from the holding period returns

(1 + TWR)2 = (1+ HPR1)*(1+ HPR2)

=> (1 + TWR)2 = (1.22) * (1.1083) = 1.3522

=> (1 + TWR) = 1.1628 => TWR = 0.1628

Or Time Weighted return is 16.28%

Is it always the cash that Time Weighted return is less than money weighted return?

Please note that this is not the case and we got into this situation for this problem only. The next

example will clear this concept.

Concept Builder –Money Weighted Return and Time Weighted Return

6. An investor purchased one share of Goldman Sachs (GS) at $100 at T = 0, he again purchased one

more share of GS at $120 at T =1. He received total dividend of $2 at T= 1 and a dividend of $3 per share

at T =2, the investor sold the shares at T=2 for a total consideration of $360. Find money weighted and

time weighted return?

Answer

Money Weighted Return Calculation:

Step 1 : Identify the cash flows that are happening at the various intervals.

At T =0 : Purchase of 1st share was done => Cash outflow of $100 => CF0 = -$100

At T =1 : Purchase of 2nd share was done => Cash outflow of $120

Dividend is received => Cash Inflow of $2

Sum of these two results in cash outflow of $118 => CF1 = -$118


At T =2 : 2 shares are sold => Cash inflow of $360

Dividend is received => Cash Inflow of $3 * 2 = $6

Sum of these two results in cash inflow of $366 => CF1 = $366

Step 2: Plug the cash flow values in the financial calculator to compute the IRR (because MWR is nothing

but IRR)

IRR = 41.2% or Money Weighted Return is 41.2%

Time Weighted return Calculation:

Step 1: Here significant cash in-flow is happening at T=1 and cash outflow is happening at T=2. So we

have one period from T=0 to T=1 and another period from T=1 to T=2

Step 2: Compute the portfolio value just before the significant cash inflow or outflow.

For Period 1: Beginning Value = $100; Ending Value = $120; Dividend Received = $2

For Period 2: Beginning Value = $120 (for the 1st share) + $120 (amount that is invested) = $240

Ending Value = $360; Dividend Received = $6 (for 2 shares)

Step 3: Find out the holding period return of the periods

For Period 1: HPR = (120 – 100 + 2)/100 = 22%

For Period 2: HPR = (360 – 240 + 6)/240 = 52.5%

Step 4: Find out the geometric mean return or TWR from the holding period returns

(1 + TWR)2 = (1+ HPR1)*(1+ HPR2)

=> (1 + TWR)2 = (1.22) * (1.525) = 1.8605

=> (1 + TWR) = 1.3640 => TWR = 0.364

Or Time Weighted return is 36.4%

So for this example the MWR is more than that of the TWR. So which return is more or less depends on

the amount and the timing of the cash flows.

What can we observe from the above 2 examples, either TWR or MWR can be more or less, it depends

on how the investment is done and the periodic return that is generated.

Practical Understanding of the differences in the two return measures

1. Money Weighted Return gives weightage to the money that is deposited in the account. If the

portfolio performance is good when more money is in the portfolio then the overall return

increases, where as if the portfolio performance is bad with more money the overall return

decreases.

2. In time weighted return, the return is not affected by the timing of the cash flows and the amount

of money invested, it is purely depended on the periodic return generated. For purpose of simplicity

we will assume that the holding period will be the same. In reality, for the calculation of mutual

fund returns, we define the holding period as 1 day, because typically everyday there are cash

inflows and outflows.

You need to remember the following table because many problems will not ask you to calculate the

Money weighted or Time weighted return but an understanding of the impact on the return measure

when one of these is choses.

Amount of Money compared to

previous period

Return compared to

previous period

Interpretation

More Money More Return MWR > TWR

More Money Less Return MWR < TWR

Less Money More Return MWR < TWR

Less Money Less Return MWR > TWR


Important: TWR is the preferred method for return calculation in most of the investments. Only in one

case we prefer the MWR over the time weighted return and that is when the portfolio manager has the

complete control on the cash flow of the portfolio that is the portfolio manager decides on when to

increase or when to decrease the size of the portfolio.

LOS 6.e. Calculate and interpret the bank discount yield, holding period yield, effective annual yield, and

money market yield for a U.S. Treasury bill

Money market instruments are highly liquid, low risk instruments having maturity of less than 1 year.

These instruments are of two types:-

1. Discounted instruments – These instruments do not give any coupon or interest during the holding

period. These are priced at less than the face value. Face value is the amount of money that the

bondholder will receive when the bond matures. The difference between the face value and the

selling price is called as the bond discount. The investor will earn the discount value if he holds the

instrument till maturity. The best example of this type of instrument is a treasury bill.

For example a T bill having face value of $100 at maturing after 90 days will be selling now at a price

which is less than $100, let’s say the price is $99. Then $100 minus $99 or $1 is the discount.

2. Interest Bearing Instruments - These instruments will produce intermediate cash flows in the form

of interest. You can think of a money market (or income) mutual fund as an example.

Yield Measures of Treasury bill

There are various yield measures of Treasury Bills; here we will go through the formula and

understanding of each of the yield measures. You might fear that how you will remember so many

formulas, in reality there are not so many formulas; the idea is to understand the bank discount yield

really well.

1. Bank Discount Yield – Here the yield is calculated on the basis of the face value and not on the

actual investment done. The formula for bank discount yield is

Where; rBD : Bank Discount yield D: Discount of the T Bill

F: Face value of the T bill t : The time to maturity

Please note that Bank discount yield is an annualized yield, however it is not a good measure. Think

about Bank discount yield as BaD yield, the problems with it are:-

a. It is calculated on face value and not the actual price

b. The number of days in one year is treated as 360 days instead of 365 or 366

c. The calculation is based on simple interest concept rather than the compound interest

concept

2. Money market yield – This is another yield measure, here we are correcting one of the problems of

the bank discount yield, which is the return is calculated on the actual investment and not on the

face value. The formula for money market yield is given as

or

So money market yield is also not a good measure, but it is better than bank discount yield. The

problems are:-

a. The number of days in one year is treated as 360 days instead of 365 or 366

b. The calculation is based on simple interest concept rather than the compound interest

concept

3. Holding Period Yield: We have covered this concept in earlier LOS, here the yield is calculated on

the actual investment but only for the period the investment is made and it is not annualized.

For a T bill, the HPY is given as;


4. Effective Annual Yield: This is the best yield measure for the Treasury bill. It corrects the problem of

money market yield by considering

a. Compounding

b. Considering one year as 365 days

The formula is given by: -1

EAY is the annualized yield of the holding period and can be used to compare across various

investments

Based on the above, we can say that for any period

Effective Annual Yield > Money Market Yield > Bank Discount Yield

Concept Builder –Various Yield Measures

6. A T bill priced at $98 with face value of $100 and 180 days until maturity.

a. Calculate the bank discount yield

b. Find the Holding period yield

c. Find the effective annual yield

d. Find the money market yield

Answer

First calculate the discount. Discount = Face value – Price => discount = $100 - $98 = $2

a. Bank Discount Yield (The BaD Yield) = (D/F) * (360/t) = (2/100) * (360/180) = 4%

b. Holding Period Yield = D/P = 2/98 = 2.0408%

c. Effective Annual Yield = (1+ HPY)365/t – 1 => (1.020408)365/180 -1 => 1.0204082.0278 -1

=> EAY = 4.1817%

d. Money Market Yield = (D/P) * (360/t) = (2/98) * (360/180) = 4.0816%

LOS 6.f. Convert among holding period yields, money market yields, effective annual yields, and bond

equivalent yields.

There is relationship among all the yield measures, if you have understood the underlying concept of

each of the yield then you need not remember any formula.

1. Conversion from bank discount yield to Money Market yield – There is a very complex formula for

the conversion; we will not follow the formula, but the logic for this conversion.

During the conversion we will assume that the face value of the T bill is $100. The process is best

described using the below example

Concept Builder –Conversion from Bank Discount Yield to Money Market Yield

7. A T bill having 120 days to maturity is yielding 3.5% on bank discount basis, how much will be its

money market yield?

Answer

Step 1: Assume the face value to be $100, we would like to find out the discount at which the T bill is

trading

3.5 % = (D/100) * (360/120) => 3.5% = (D/100) * 3

=> D = 3.5% * 100 / 3 = $1.1667

Step 2: Once discount is calculated, Find out the price at which the bond is trading

Price = F – D => $100 - $1.667 = $98.8333

Step 3: Since now we know the discount, the price and the days to maturity, calculate MM yield

Money Market yield = (D/P) * (360/t) => (1.667/98.333) * (360/120)

=>Money Market yield = 3.5413%

2. Conversion from holding period yield to Money Market yield to Effective Annual Yield –

Please note that the

Money market yield is annualized version of HPY based on simple interest and 360 days in a year.


EAY is annualized version of HPY based on compound interest and 365 days in a year.

- 1

Concept Builder –Conversion from Money Market Yield to HPY and EAY

8. A T bill having 120 days to maturity has money market yield of 3.5413%, calculate the HPY and EAY?

Answer

Holding Period Yield = 3.5413 * 120/360 = 1.1804%

EAY = (1.011804)365/120 – 1 = 3.6339%

Bond Equivalent Yield (BEY) – In US, most of the bonds pay coupon semi-annually, based on the price

of the bond, the market participants calculate the semi-annual yield of the bond. The bond equivalent

yield is based on the simple interest concept of the semi-annual yield, therefore

3. Conversion from HPY to Bond Equivalent Yield: For this conversion we first need to find the effective

semi-annual yield. Please note that when we are referring to semi-annual yield; you should convert the

semi-annual into 6 months and the holding period should also be in month. Please note that here we

need to assume compounding for the 6 months rather than simple interest. You can take 30 days as one

month period over here. We can see an example of this to understand the concept better.

Concept Builder –Conversion from HPY to BEY

9. A T bill having 120 days to maturity has HPYof 2%, calculate BEY?

Answer

First Step: Find the Holding period in months by taking 30 days as 1 month

=> holding period = 120/30 = 4 months

Second Step: Find the effective semi-annual yield

=>effective semi-annual yield = (1.02)1.5 -1 = 3.01495%

BEY = 2 * 3.01495% = 6.0299%

4. Conversion from EAY to Bond Equivalent Yield: Here the idea is to convert the EAY to effective semi-

annual yield and then find the BEY.

- 1 => BEY = 2 *{ – 1}

Concept Builder –Conversion from EAY to BEY

10. A T bill having 120 days to maturity has EAY of 5%, calculate the BEY?

Answer

First Step: Find the semi-annual yield

=> Semi-annual yield = (1.05)0.5 – 1 = 2.4695%

Second Step: Multiply the semi-annual yield by 2 to get the BEY

=> BEY = 2 * 2.4695% = 4.939%


Concept Notes For Reading #7-Statistical Concepts and Market Returns

Reading Summary

This topic introduces the concept of descriptive statistics. The focus in this topic is on Measures of

central tendency and dispersion. You need to understand and calculate these measures. From

investment point of view, important measure of central tendency is mean and standard deviation is the

important dispersion measure. Percentiles are definitely a testable topic in the examination. A

candidate should understand the deviation of a data set from what is known as Normal distribution.

Some may find Chebysev’s inequality to be a difficult concept, but an understanding of it would make

life easier in Readings 10 and 11. Problems related to Skewness and Kurtosis are favorites of the

Institute. Sharpe Ratio is a concept which is found in portfolio management also.

LOS 7.a.Differentiate between descriptive statistics and inferential statistics, between a population and a sample, and among the types of measurement scales;

Statistics can be used to refer to numerical data (as an example the net profit of a firm over the last 10

years). It can also be used to refer to the methods of collecting, classifying, interpreting or analyzing the

data. Statistics are widely used in finance and its applications should be well understood by the finance

professionals.

Following are the two categories of statistics:-

Descriptive Statistics –This is the branch of statistics where we describe the important aspects or

characteristics of the data set that have been collected. The description can consists of

o Measure of Central Tendency -Where the data is centered, like mean

o Measure of dispersion - How the data is spread out, like range

In this reading we will focus on the descriptive statistics, we would understand the concepts

through which we can describe huge amount of data sets, thus trying to make sense of the data.

Inferential Statistics –This branch of statistics deals with prediction or inference of the large set of

data (population) from small set of data (a sample). As an example suppose we want to know the

average height of the men aged between 20 and 25, then it would be impossible to measure the

height of each and every man. What we can do is to take a sample of the men across the various

cities/town/villages and then find out the average. Using this simple average we can try to estimate

the average height of the men population. Please note that since we are trying to predict, there will

always be an error associated with the prediction. We will focus on this part of statistics in Reading

10 and Reading 11.

We have used the word population and sample in the paragraphs above; let’s see what is meant by

these terms

Population –A population can be thought of as a universal set. It consists of the entire data set or

has each and every member of the specified group. As an example, all the men of India aged

between 20 and 25 would form a population. An important point to note is that the population

doesn’t have anything to do with the human. We can have a population of living, non-living or even

virtual objects.

Sample – A sample is a subset of the population or in other words we can say that a sample

contains some of the member of the population. It is impractical to conduct analysis on a

population and hence typically we select a sample from the population. The best is to get a random

sample from the population but sometimes we may be interested in something more specific. We

will cover Sampling later in Reading 10.


Measurement Scales

When dealing with data, it becomes important to organize the data. The ways in which data can be

organized are listed below

Nominal (Grouping): This is the weakest level of measurement scale. Here we only categorize or

group the data. However in no way we would be able to find out which group of data is better as

there is no ranking done.

Example of Nominal Scale would be categorizing the stocks in Large Cap; Small Cap etc.

• Ordinal(Grouping and Ranking): This is better measurement than the nominal scale. In this scale

the data is grouped as well as ranked. In Ordinal scale the data is ordered data (or ranked data)

which means assigning some rating or value to a group with a view that one group is better than the

other based on the value. However it provides no information on differences in performance

between groups.

Example: Categorizing the stocks based on performance; like assigning 5 star rating to the top 20%

stocks. Another example is of bond rating, where the bonds are rated as AAA, AA, A etc., a bond

with a rating of AAA will be safer than that of a bond with rating of AA, but we don’t have

information on how much better the AAA bond is as compared to AA bond.

• Interval(Grouping, Ranking, includes exact distance): This scale is one step further, here we work

with ranked data but we make sure that the difference between scale values are equal. Here the

values can be added or subtracted. However a value of zero here does not mean the absence of

what is being measured, in simpler words this scale lacks “True Zero”. We will try to explain this

concept using the below two examples

Example 1:Temperature is measured in Celsius scale which is an interval scale. Water freezes at a

temperature of 0 but Zero 0Cdoes not mean absence of temperature because we have temperature

value in negative also. In reality the true absence of temperature is observed at -273 0C which is also

Zero Kelvin. Now in the absence of true zero, we can’t say that 50 0C is twice that of 25 0C, so a

meaningful comparison is not possible in this scale.

Example 2:In GMAT examination, it is not possible for a candidate to score zero marks. The

candidates are assigned marks in multiples of 10. So a person scoring 800 in GMAT has not score

two times that person who has scored 400 in GMAT.

• Ratio: Ratio scale is the strongest measurement scale, it has all the qualities of interval scale plus it

also provides a true zero. Since we have a true zero, meaningful comparison can be made among

values. This is the most important scale, here the datasets can be evaluated on the basis of ratio.

Example 1: The amount of money a person has can be considered as Ratio scale, a person having

$10,000 has twice as much money as a person having $5,000.

Example 2: In a typical traditional examination, having 100 marks, a person can score zero marks

also, here we can say that a person scoring 100 marks has scored 2 times of a person scoring 50

marks.

Note: For remembering the measurement scale you can remember NOIR (meaning black in French).

Concept Builder – Identification of the scales

1. In the cases given below, identify what type of scale it represents

A. Ranking of mutual funds by Morning Star

B. Runs scored by a player in a cricket match

C. Measurement of temperature in Fahrenheit scale

D. Classification of students in a class on the basis of sex

Explanation

A. Morning Star, assigns a one star to five star ranking. Five star ranking is given to the best funds

as determined by Morning Star’s evaluation criterion. Here we can say that a 5 star mutual fund


is better than a 2 star mutual fund, but we can’t say by exactly how much, hence it is an

example of ordinal scale.

B. The run scored can’t be negative. A batsman who scores 100 runs has scored twice as much a

batsman who has scored 50 runs. Hence, it is an example of Ratio Scale.

C. Fahrenheit scale like Celsius scale doesn’t have a true zero, but we can identify the difference

between the temperatures and hence it is a measure of interval scale.

D. Classification on the basis of sex into male or female is just grouping and hence it is an example

of Nominal Scale.

LOS 7.b. Define a parameter, a sample statistic, and a frequency distribution

Parameter – A parameter is a numerical quantity measuring some characteristics of the population. In

statistics we have to deal with many parameters, but from our point of view, we are mainly interested

in mean standard deviation of the returns. Parameters are usually denoted by Greek letters. For

example, population mean is denoted by µ (pronounced as mu) and population standard deviation is

denoted by σ (called as sigma).

Sample Statistics – A sample statistics is a numerical quantity measuring the characteristics of the

sample. A sample mean is denoted by , whereas sample standard deviation is denoted by s. As

mentioned earlier, most of the times population parameters are not known and they have to be

estimated from sample, so we find out sample statistics to make prediction of the population

parameter.

Mostly we deal with large data sets and it becomes difficult to deal with individual member of the set,

frequency distribution is a simpler way of dealing with the data.

The process of creation of Frequency distribution is outlined in the following steps

Step 1 - We first create an interval (also known as class), which is basically a group. Following

points are important

A. An observation has to fall in one of the interval.

B. The intervals should not overlap (that is they are mutually exclusive)

C. Also an interval has a lower limit and an upper limit.

Please note that an interval need not be a range always, it can be classified as one of the scale

measure. For example we can have a Class as

1. Just grouping (Nominal) – Type of bond, Govt., private etc.

2. Ranking (Ordinal) - AAA bonds, AA bonds etc.

3. Interval scale – Have a range of return from these bonds

4. Ratio Scale – Have the bond grouped as per the coupon rate

Step 2 –Assign the observation in the data set to the class that you have identified in Step 1.This

process is called as tallying the data.

Step 3 – Count the number of observation in each of the class, this number is called as the class

frequency or frequency or absolute frequency.

Note: Please note that in the examination, you will NOT be asked to construct a frequency distribution

table, the example below is an illustration of the process. However there can be a question (very low

probability) where you would be asked to find the frequency of a class interval.


Concept Builder – Frequency Distribution

2. Given the following returns data for a stock in the month of January 20X1, construct a frequency

distribution table

-3.60 3.23 5.32 -5.84 -17.99

-7.14 2.26 6.42 6.70 17.51

-9.26 7.89 0.89 -2.85 -10.50

3.17 -3.25 2.09 -13.98 6.23

3.87 -6.70 1.20 -18.22 18.57 Explanation

For each observation, identify which class interval it should belong to and draw a line in the Tally

column. Once done, count the number to identify the absolute frequency.

Class Interval Tally Absolute Frequency

-20% ≤ R < 15% // 2

-15% ≤ R < 10% // 2

-10% ≤ R < 5% /// 3

-5% ≤ R < 0% //// 4

0% ≤ R < 5% //// // 7

5% ≤ R < 10% //// 5

10% ≤ R < 15% 0

15% ≤ R < 20% // 2

Total 25

LOS 7.c. Calculate and interpret relative frequencies and cumulative relative frequencies, given a frequency distribution

Lets now define the following types of frequency that we observe in a frequency distribution

Absolute Frequency – This is the actual number of frequency of a given interval. The minimum

value of absolute frequency is ZERO, It can’t be negative.

Relative Frequency –This measures how much time one of the class has occurred as relative to

the entire class. Numerically,

Cumulative Absolute Frequency and Cumulative Relative Frequency – Starting from the first

class and moving down the classes, we can add the absolute and relative frequencies to get the

cumulative absolute and cumulative relative frequency. Since the frequency will always be

greater than or equal to ZERO, the cumulative frequencies are always increasing or remaining

the same.

The below example will build upon the example #2 to explain the concepts of this LOS.


Concept Builder – Relative and Cumulative Frequencies

3. Given the following returns data for a stock in the month of January 20X1, construct a frequency

distribution table

Explanation

We will use the formula given in the text above to calculate each of the measure.

Class Interval Absolute

Frequency Relative

Frequency

Cumulative Absolute

Frequency

Cumulative Relative

Frequency

-20% ≤ R < 15% 2 0.08 2 0.08

-15% ≤ R < 10% 2 0.08 4 0.16

-10% ≤ R < 5% 3 0.12 7 0.28

-5% ≤ R < 0% 4 0.16 11 0.44

0% ≤ R < 5% 7 0.28 18 0.72

5% ≤ R < 10% 5 0.2 23 0.92

10% ≤ R < 15% 0 0 23 0.92

15% ≤ R < 20% 2 0.08 25 1

Total 25 1

LOS 7.d. describe the properties of a data set presented as a histogram or a frequency polygon

Histogram – It is a bar chart (without space between the bars) that is constructed using the frequency

distribution. On the X (horizontal) Axis we plot the Class and on the Y Axis (Vertical Axis) we plot the

frequency. Looking at the histogram, you can easily identify some distribution characteristic, that is

which class has the highest, lowest frequency, how the data is dispersed etc. Hence histogram is a very

useful tool for an analyst.

Frequency Polygon – This is a similar representation like the histogram, but here we don’t display the

bars. The mid-points of the intervals are joined with a line to create a polygon structure.


LOS 7.e. define, calculate, and interpret measures of central tendency, including the population mean, sample mean, arithmetic mean, weighted average or mean (including a portfolio return viewed as a weighted mean), geometric mean,harmonic mean, median, and mode

Measures of Central Tendency - Central tendency refers to the presence of center in a data set. There

are various measures through which we try to identify the central (or middle) point of a population or a

sample. Please note that practically it is expensive to find out the central tendency measures for a

population.

Population Mean is the average value of a population, there is only one mean for one population. The

formula for a population mean is given as

Where are the individual members or data point of a population. N is the population size (that is the

number of data point in the population).

Sample Mean is the average value of a sample, there is only one mean foronesample. But from a

population we can create many samples; hence we can have many sample means for one population.

We compute sample mean and estimate the population mean from the sample means.

The formula for a population mean is given as

Note that we are using N (big) to denote the population size and n (small) to denote the sample size.

Arithmetic Mean is the simple average for any data set. The population and the sample mean which we

have explained above fall in the category of the arithmetic mean. Arithmetic mean is the most popular

measure of central tendency because of its calculation simplicity.

Properties of Arithmetic Mean:

1. Arithmetic mean is unique

2. Arithmetic mean is only applicable to interval and ratio scale. Arithmetic mean is meaningless for

nominal and ordinal scale.

3. Needless to say that arithmetic mean can only be computed by taking the sum of all the data points

in the set, its value will be inaccurate if you exclude some data purposely.

4. Sum of the deviations from the mean is ZERO. First let’s define what deviation is – A deviation is the

distance between the mean and one data point.

Concept Builder – Deviation from the Mean

4. As an example consider the following data set - 10, 20, 25, 5

The mean of this data set is (10+20+25+5)/4 = 15. Please note that there is no data point which has

value of 15, so it’s not necessary that mean would be equal to one of the data point.

Now let’s find the deviation:

Data Point Mean Deviation

10 15 (10-15) = -5

20 15 (20-15) =5

25 15 (25-15) = 10

5 15 (5 -15) = -10

Sum Of Deviation -5 + 5 + 10 + -10 = 0

Deviation is important information for a data set, as it is a measure of risk. We will cover this point

when we are covering the dispersion topic.

Advantages of Arithmetic Mean:

1. Easy to calculate

2. Mean uses all the information about the size and magnitude of the observation (or data points)

Disadvantages of Arithmetic Mean:


1. Mean is highly sensitive to the extreme values, a higher value will be able to attract the mean

towards its side. For example in MBA colleges, average salary is not a good measure because some

students get very high salary and as a result, the mean is high. For the following data set - 10, 20, 25,

965, the mean is 250. See the impact on the mean because of the presence of a high number (as

compared to the previous example). In such cases median is a better measure.

2. Mean cannot be calculated for a data set which is not finite, that is the number of data points is not

known.

Median: Median divides a data set into two equal parts; it means that median is the Mid-Point of a

sorted data set. The data set can be sorted either in ascending or descending order.

For an odd number of data set, median is easier to identify as we have a data point which is the middle

value. For odd number, the median is {(n+1)/2}th data point

Concept Builder – Calculation of median for odd number of observation

5. A stock has generated the following returns over the past 5 days

-2%, 5%, 4%, 2%, 8%

Compute the median of the stock return?

Explanation

For computing the median, first we need to arrange the data in either ascending or descending order.

Lets arrange in ascending order

-2%, 2%, 4%, 5%, 8%

There are total of 5 data points, since this is an odd number the median will be {(5+1)/2}th number or

(6/2)th or 3rd number.

The 3rd number here is 4%. Hence the median is 4%.

For an even number of data set, the median needs to be calculated. Median is the average of the

(n/2)th and ((n/2) + 1)th data point (or observation)

Concept Builder – Calculation of median for odd number of observation

6. A stock has generated the following returns over the past 4 days

-2%, 5%, 4%, 2%

Compute the median of the stock return?

Explanation

For computing the median, first we need to arrange the data in either ascending or descending order.

Lets arrange in ascending order

-2%, 2%, 4%, 5%

There are total of 4 data points, since this is an even number the median will be the average of the

(4/)2th and {(4/2)+1}th number or Average of 2nd and 3rd number.

2nd Number is 2%, 3rd number is 4%. Hence the average is (2% + 4%)/2 = 3%

Hence the median is 3%.

Mode: The mode is the most frequently occurring data point (or observation) in a data set (population

or a sample).

Unlike Mean and Median, It is not necessary that a data set should have a mode. If all the observation in

a data set is different, then it is said to have No Mode.

Unimodal – When a data set has only one observation which has the highest frequency (or most

occurrences) then the data set is said to have One Mode (or Unimodal)

Bimodal -When a data set has only two observations having the highest frequency (or most

occurrences) then the data set is said to have Two mode (or Bimodal)


Modal Interval – In a frequency distribution with class intervals, we can’t classify as single data point

and hence finding mode is not possible, However, we can find out the class interval which has the

highest frequency, that class interval is said to be the modal interval.

Concept Builder – Mode

7. Find the mode for the following data set.

a. 5%, 7%, 9%, 11%, 13%

b. 5%, 7%, 9%, 5%, 11%, 13%,

c. 5%, 7%, 9%, 5%, 11%, 13%,13%

Explanation

a. There is no observation which is occurring more than once, and hence no mode exist for

this data set

b. The observation 5% is coming twice in the data set, and hence the mode is 5%

c. The observations 5% and 13% are coming twice in the data set, so this is an example of

bimodal distribution

Other Concepts of Mean

Earlier we have covered arithmetic mean which is an important concept, but in investment we use

other concepts of mean also, and it’s worthwhile to cover those concepts.

The Weighted Mean The concept of weighted mean is used in portfolio analysis. The idea behind weighted mean is that not all observation has the same weight and hence if we take a simple average, then the resulting answer would not be accurate. Before moving to this concept, let’s see the example given below

Concept Builder – Why We Need Weighted Mean

8. Consider that you have made investment in 3 stocks at T=0 and after one year (i.e. at T=1) you sell

the stock. You would want to know the return that has been generated. The purchase price (at T=0)

and the price at T=1 (the price at which you sold the stock) is given in the table below. Assuming

that you have purchased one share of each stock.

Stock Price at T=0 Price at T=1 Return

A $100 $120 20%

B $60 $66 10%

C $40 $80 100%

What should be the return as per your understanding, Should it be the arithmetic mean of the return?

Let’s compute the arithmetic mean

It would be equal to (20%+10%+100%)/3 = 43.33%

Now, let’s think in term of the money that is invested at T=0 and the money that you got when you sold

the stocks at T=1.

Money Invested = $100 + $60 + $40 = $200

Money Received from selling the stocks = $120 + $66 + $80 = $266

So the return that you generated = ($266 - $200)/$200 = 33%

Now, compare the return that your portfolio has generated with the arithmetic mean return, they are

not same. Hence there should be a way in which we should be able to find out the mean return.

Weighted mean is the concept through which we would be able to find out that.

In weighted mean calculation, we assign a weight to each observation. For stocks in the portfolio, the

weight is the amount of money invested in them.


Where, X1 , X2 , …. Xn are the observations

W1 , W2 , …. Wn are the weights assigned to these observations

Please note that, it is must to have the sum of the weights equal to 1, that is

In case of a portfolio of stocks, the Weight is given as

Now let’s solve the example above using the weighted mean formula, we should be getting the same

answer

Concept Builder –Weighted Mean

9. Consider that you have made investment in 3 stocks at T=0 and after one year (i.e. at T=1) you sell

the stock. You would want to know the return that has been generated. The purchase price (at T=0)

and the price at T=1 (the price at which you sold the stock) is given in the table below. Assuming

that you have purchased one share of each stock.

Stock Price at T=0 Price at T=1 Return

A $100 $120 20%

B $60 $66 10%

C $40 $80 100%

Explanation

First We need to calculate the weight of each of the stocks in the portfolio

WA = Amount invested in A / Total amount invested => $100/$200 = 0.5

WB = $60/$200 = 0.3

WC = $40/$200 = 0.2

Please check that the sum of the weight is equal to 1 => 0.5+0.3+0.2 = 1

Now, apply the weighted mean formula on the returns that we are seeing

Weighted Mean Return = WA RA+ WB RB+ Wc Rc = 0.5*20% + 0.3 * 10% + 0.2 * 100% = 10% + 3% + 20%

Weighted Mean Return =33%

This is the same return which we got in the previous example.

Hence, for portfolio return, weighted mean is the appropriate return measure.

The Geometric Mean

Geometric mean is used to calculate investment returns over multiple time periods. It is also used to

compute the average growth rate over a period, for example, the average sales growth of a firm over

last 5 years. When calculating the average growth rate it is called as Compounded Annual Growth Rate

of CAGR.

The formula for Geometric mean is given below

Given that Xi ≥ 0 for i=1 , 2, 3 …… n

Where, GM is the Geometric Mean.

X1 ,X2,…… Xn are the observation and n is total number of observation

However this formula can’t be used to calculate the Geometric mean of the returns, because the

returns can be negative and we can’t have negative number inside the root.

So, a solution for that is to replace each Xi with (1 + Return in decimal form). Now for any traditional

investment the return will not be less than -100% (that is you can’t lose more than what you have

invested), so we will not have any negative value inside the root.

So, for returns the Geometric Mean is given by the following formula


Where, RG is the Geometric Mean.

R1 ,R2,…… Rn are the returns and n is total number of returns

Concept Builder –Geometric Mean Return

10. Consider that you have made investment in a stock at T=0, it provided a return of 10% in the first

year, 20% in the second year and -30% in the third year. How much is the return you have received

from the stock over the past 3 years?

Explanation

R1 = 10% or 0.1; R2 = 0.2; R3 = -0.3

(1+R1) = 1.1 ; (1+R2) = 1.2 and (1+R3) = 0.7

(1+ RG) = => (1+RG) = => (1+ RG) = 0.974

RG = -0.026 or -2.6%

So, this geometric mean return means that you have lost your money at a rate of 2.6% every

year.

And if you had invested $100 at T=0, it would have become $92.4 (Think How!!)

This is the most appropriate place to cover the last LOS of this reading. The LOS is given below

LOS 7.l. Discuss the use of arithmetic mean or geometric mean when determining investment returns

Now to calculate the performance of any portfolio over a time period, should we use Arithmetic Mean

return or Geometric Mean Return?

Let’s take an example to explain this scenario.

Concept Builder –Geometric Mean Return Versus Arithmetic Mean Return

11. Consider that you have made investment in a stock and it provided a return of 100% in the first year

and -50% in the second year. How much is the return you have received from the stock?

Explanation

Rather than getting into complication, let’s work from basic reasoning. Let’s assume that you had

invested $100 in the stock at T=0.

Since the first year return is 100%, the value of your stock will be $200 at T=1.

Now, in the 2nd year, the return is -50%, therefore the value of your stock will be $100 at T=2.

So, you started with $100 and have $100 in the end.

It implies that over the 2 years, the return is 0%, so it means that the average return should also be 0%.

Suppose, we use Arithmetic Mean return to explain the return over the 2 year period. The arithmetic

mean return will be (100% + -50%)/2 = 25%

But, we see that the return is 0% over the 2 years. This implies that the Arithmetic mean return is not a

correct measure to find the average stock return.

Now Let’s calculate the Geometric mean return over the 2 years.

R1 = 100% or 1; R2 = -0.5

(1+R1) = 2and (1+R2) = 0.5

(1+ RG) = => (1+RG) = => (1+ RG) = 1

RG = 0 or 0%

So, the geometric mean return is 0%, which is the actual average return that we have calculated.

So, from the above example it is clear that the Geometric Mean return is the correct measure of the

average return over a period of Time.


The Harmonic Mean

As compared to the arithmetic and geometric mean, harmonic mean is used lesser in the investment

industry. One scenario, where harmonic mean is used is to find out the average purchase price of shares

when somebody is depositing equal dollar amount in the stock every period. BTW when somebody is

depositing equal amount every month or every period, the concept is called as Dollar Cost Averaging

and in India, we also know this by the term Systematic Investment Plan.

Let’s first try to find out the average purchase price of the stock when dollar cost averaging scheme is

adopted and in the process we will derive the formula of the harmonic mean.

Concept Builder –Harmonic Mean Return

12. Suppose you as an investor have invested in a Systematic Investment Plan of HDFC Tax Saver Fund.

You are making contribution of $1000 every month in the fund. The below table outlines the price

of the fund at which you have made the investment. What is the average purchase price at which

you bought the fund?

Note: You will learn in the later chapters that for Mutual funds, the price at which we buy is not

called as price but it is known as NAV (Net Asset Value)

Time Investment Price (NAV)

T=0 $1000 $100

T=1 $1000 $125

T=2 $1000 $80

Explanation

Will the answer be the arithmetic mean of the price for the 3 periods?

Let’s find the arithmetic mean, which will be ($100+$125+$80) = $305/3 = $101.67

Let’s try to build the following table, where we identify the number of units of the mutual fund that we

have bought.

Time Investment Price (NAV) Number of Units

T=0 $1000 $100 =$1000/$100 = 10

T=1 $1000 $125 =$1000/$125 = 8

T=2 $1000 $80 =$1000/$80 = 12.5

So, the total investment is 3 * $1000 = $3000

Total Number of units that is bought is 10 + 8 + 12.5 = 30.5

We know that logically, the average purchase price should be equal to total investment divided by the

total number of units

Average price = $3000 / 30.5 = $98.3606

So, it implies that the arithmetic mean is not the correct approach to find out the average purchase

price in Dollar Cost Averaging method.

Now, what exactly we have calculated which we are saying as the correct answer?

We have calculated the harmonic mean of the purchase price, and that is the correct mean.

Let’s try to rewrite the Average Price Equation, which we have in the above

Now, $1000 will get cancelled in the numerator and the denominator

Or,

This is the formula of the Harmonic Mean, which is given in detail in the text below


Harmonic mean for a set of observation X1 ,X2 …… Xnis given by the following formula

Or in more general format

LOS 7.f. Describe, calculate, and interpret quartiles, quintiles, deciles, and percentiles

Other Measures of Location – Quantiles

Median divides a distribution into half. The concept of Quantile divides the distribution into smaller

sizes. Median and quantiles are known as measure of location. Following are the examples of Quantiles

Percentiles – The distribution is divided into 100 parts.

Deciles – The distribution is divided into 10 parts. (Think Deca)

Quintiles – The distribution is divided into 5 parts.

Quartiles – The distribution is divided into 4 parts. (Think Quarter)

Any quantile can be expressed as a percentile. So we use a standard formula to represent the quantiles.

As an example, 3rd quintile (or 3/5th) is same as 60 percentile. This is because when we mention 3rd

quintile, it means that 60% of the data are below the particular value or observation.

For a data set having n members and arranged in ascending order, the formula for finding out the

position for a given percentile y is

Here Ly is the position that we will get in the sorted data set.

Y is the given percentile.

So, if we are asked to find out the 2nd quartile, then y will be 50. If we are asked to find 6thdecile then y

will be 60. If we are asked to find out 90 percentile then y will be 90.

Also note that when we are calculating using the formula, we are essentially finding out the position (or

level) at which y percent of the data set will be below that particular position.

100 percentile means that 100% of the data is below the particular observation, which is not possible.

Hence the formula has (n+1) in it.

Concept Builder –Quantiles

13. Suppose there are 9 stocks in a portfolio, whose returns are given below. Find out the 4th quintile

of the stock returns.

-5%, 4%, 3%, 10%, 2%, 8%, -10%, 7% and 12%

Explanation

First arrange the returns in ascending order

-10%, -5%, 2%, 3%, 4%, 7%, 8%, 10%, 12%

Now we need to find the position at which we will get the 4th quintile.

4th quintile is same as 80 percentile.

=> Ly = 8

Therefore, the 4th quintile is 8th data from the left.

The 8th data is 10%, and hence the 4th quintile is 10%. Or we can say that 80% of the distribution is

below 10%.

Now, let’s see what happens when we do not get a whole number as the position. Whenever this

happens, we need to apply interpolation to get the value. This is a very simple concept related to

unitary method, it would be clear from the following example.


Concept Builder – Quantiles

14. Suppose you have added one more stock in the portfolio and now there are 10 stocks in the

portfolio, whose returns are given below. Find out the 4th quintile of the stock returns.

-5%, 4%, 3%, 10%, 2%, 8%, -10%, 7% , 12% and 18%

Explanation

First arrange the returns in ascending order

-10%, -5%, 2%, 3%, 4%, 7%, 8%, 10%, 12%, 18%

Now we need to find the position at which we will get the 4th quintile.

4th quintile is same as 80 percentile.

=> Ly = 8.8

Therefore, the 4th quintile is 8.8th data from the left. Now 9th data is 12% and 8th data is 10%, this

means that the value would be between 10% and 12%. What we can say is that the value will be 10% +

0.8 times the difference between 12% and 10%.

This is because for 1 unit, the difference is 12% minus 10% or 2%, therefore for 0.8 units, the difference

will be 0.8 times 2%, this concept is Interpolation.

So the value will be = 8th Value + 0.8 * (9th Value – 8th Value)

10% + 0.8 * (12% - 10%) => 10% + 0.8* 2% = 11.6%

So the 4th quintile is 11.6%, so we can say that 80% of the distribution is below 11.6%

IMPORTANT: We will apply interpolation only when the data set is population. When the data set is a

sample, then interpolation should not be used. So, if we had mentioned that in this example, it was a

sample then the interpolation would not have been used. The Ly value was 8.8, so we would just take

the integer part of it, which in this case is 8, so the value would be the observation in the 8th position.

Hence it would still be 10%.

LOS 7.g. Define, calculate, and interpret 1) a range and a mean absolute deviation and 2) the variance

and standard deviation of a population and of a sample

Measures of Dispersion

This LOS covers the measures of dispersion, that is, how the data is dispersed around the central

tendency or simply mean. We can think of mean as the reward for investing and dispersion as the risk

associated with investing.

There are two types of dispersion

Absolute Dispersion – Absolute dispersion is the amount of variability present in a distribution

without considering any benchmark or any reference point. In this LOS we will discuss about

Absolute dispersion. The measures of absolute dispersion are range, mean absolute deviation,

variance and standard deviation.

Relative Dispersion - Relative dispersion isthe amount of variability present in a distribution relative

to a benchmark or a reference value. Coefficient of variation is an example of relative dispersion

and is covered in LOS i.

Range is the simplest measure of dispersion. Here we are just interested in finding out the distance

between the maximum and the minimum value in a distribution. It is useful when we are comparing

two data set (or distribution) and it is very simple to compute.

Range has the following disadvantages

It uses only two values from the distribution

It is very sensitive to the extreme values.

Does not give any idea about the shape of the distribution

As a result, range is not used on a stand-alone basis but is used to supplement other measures of

dispersion.


Mean Absolute Deviation – While covering arithmetic mean, we calculated the mean deviation and we

found that it was always ZERO. Using the deviation we can get information about the dispersion but we

need to address the problem of the sum coming zero, a solution is to take only the absolute value of the

deviation, that is, when the deviation is negative, we ignore the negative sign and take the value. Hence

Mean absolute deviation is the average of the absolute deviation around the mean. The formula is

Here is the sample mean, is the observation and n is the total number of observations.

MAD is a better measure than the range, but mathematically it has been found that it’s not the most

superior measure of dispersion and hence we move on to the next measure which is Variance.

Variance – In MAD, we had taken absolute measure to get the positive deviation. Another way to get

positive deviation is to square the deviations. Mathematically this is a better measure. By taking the

average of the squared deviations we get Variance. There are differences in the variance calculation

when we are calculating for population and when it is calculated for sample. The formula for Population

Variance is given below

Here, is the population mean and is the size of the population

Sample Variance –Statistically, it has been found that if we use the population variance formula to find

out the sample variance, the variance thus obtained was lesser than what is actually observed for a

sample. To remove this bias, it has been suggested to divide the squared deviations by (n-1) rather than

n. The sample variance thus obtained was closer to what was actually observed. The formula for sample

variance is given below

Here, is the sample mean and n is the size of the sample

Standard Deviation - Variance is the best measure for dispersion, however if you observe, variance is a

squared number and hence its unit is also squared. For example if we are measuring the variance of

height of a group of people, the unit of variance will be height squared. We as human would like to

think in linear measures and also all the calculation that we perform on the distributions are linear and

hence it became important to have a measure which is linear along with a superior measure of

dispersion. The solution is to take the square root of the variance. The value obtained after taking the

root is known as standard deviation. So, variance is essentially square of the standard deviation.

The population standard deviation is denoted by σ (called as sigma), therefore the population variance

is denoted by σ2. So, the formula for population standard deviation will be

The sample standard deviation is denoted by s, therefore the sample variance is denoted by s2. The

formula for sample standard deviation is given as


Concept Builder – Measures of Dispersion

15. A stock has the following returns over the past 7 years

7%, 6%, 9%, 10%,11%,2%, 4%

You are interested in finding out the measures of dispersion.

a. What is the range?

b. What is the Mean Absolute Deviation (MAD)?

c. What is the sample and population variance?

d. What is the sample and population standard deviation?

Explanation

Range : Range is the difference between the highest and the lowest return values

Range = 11% - 2% = 9%

MAD: Before we find MAD, we will have to find the mean of the returns

Mean = (7+6+9+10+11+2+4)/7 = 7%

MAD = 2.57%

For the rest of the calculations, it is better to use the financial calculator.

Using our TI BA II Plus Financial Calculator, we can easily find the variance and the deviation.

Press 2nd ; Press 7 ; It means you are pressing DATA.

Press 2nd Press CE|C ; It means you are pressing Clear Work

You will find in your calculator X01 , the calculator is asking you to input the value here.

Please note that since we have only one set of data, we will work with only the X data series.

Now Press 7 and then Press ENTER. You will see a down arrow, you will have to Press it Twice ↓↓

You will get X02, press 6 and then Press ENTER, again press ↓↓.

The below has the subsequent steps that you need to perform

Press 9 Press ENTER Press ↓↓




Press 4 ENTER

So, you have entered total of 7 data.

Now we need to go to the Statistics function in the calculator

Press 2nd ; Press 8, it means that you are pressing STAT.

You will see LIN in the display, it means that you are working on linear data.

Now Press ↓=> You will see n = 7, double check that your total data points were 7

Now Press ↓=> You will see , = 7, which is the mean of the data (we calculated the same earlier)

Now Press ↓=> You will see ,Sx = 3.26598, which is the sample standard deviation

Now Press ↓=> You will see , σx = 3.023716, which is the population standard deviation

For this exercise, these were the only calculation that you had to perform in the calculator

Population Variance can be found out by squaring the population standard deviation.

Population Variance = 9.14%2

Sample Variance can be found out by squaring the sample standard deviation.

Sample Variance = 10.67%2

Before moving to other LOS, we will visit LOS i as it is closely tied with the concepts which we have

discussed in this LOS.


LOS 7.i. Define, calculate, and interpret the coefficient of variation and the Sharpe ratio

When we are comparing two distributions which have large differences in the mean and the variance, it

becomes difficult to compare with the absolute dispersion measure, In such cases we employ relative

dispersion measure.

Coefficient of Variation (CV) is a relative dispersion measure and is used to standardize the absolute

dispersion measure. It is given by the following formula

For portfolios, CV measures the amount of Risk (deviation) per unit of return of the portfolio. We would

like to have lesser risk per unit of return and hence the lower the coefficient of variation, the better it

is.

Concept Builder – Coefficient Of Variation

16. You are Fund Of Fund Investment manager and would like to invest some money in one of the fund.

You are evaluating two fund managers and would like to invest in one of them. The following table

has the return and the standard deviation data for the two fund managers. Based on the concept of

Coefficient of Variation, with which fund manager would you invest and why?

Fund Manager Return Standard Deviation (Risk)

A 10% 20%

B 15% 25%

Explanation

You might think it is better to invest with A because he has lower variability of the return. But note that

here we are seeing that the funds have different return and risk measure, and hence we are unable to

make a decision based on the absolute measure. We will have to employ a relative measure to find out

the best manager.

Coefficient of Variation (CV) which measures the risk per unit of return would be ideal here. So we will

compute the CV for both the manager. Computation is shown below

Fund Manager Return Standard Deviation (Risk) CV

A 10% 20% =20/10 = 2

B 15% 25% = 25/15 = 1.67

Since the fund manager B has lower Coefficient of Variation (risk per unit of return) we should invest in

B’s fund.

Sharpe Ratio – This ratio is attributed to Prof. William Sharpe and he received Nobel Prize for coming up

with this concept. This ratio is basically a reward to risk ratio and has prominent place in the

investment industry. Using this ratio we are able to compare the portfolios on risk and return

characteristics. You may find it similar to the coefficient of variation, but this ratio is far more reaching

that coefficient of variation because it measures the excess return over the risk free rate. The idea on

which it is built is that, Since there is no risk when one invest in risk free asset, one should not look at

the absolute return from any investment but should look at the excess return that the investment is

generating over the risk free rate and then compare investments on the excess return versus the risk.

Sharpe ratio is opposite to the coefficient of variation, here we measure the excess return generated

per unit of risk taken. It would become clearer from its formula

Here, is the portfolio return, RFR is the risk free rate of return, is the standard deviation (or risk)

of the portfolio.

Since higher excess return per unit of risk is better, hence higher the Sharpe ratio, the better it is.


Concept Builder –Sharpe Ratio

17. The data is same as that of the previous problem. Additionally risk free rate of return is given as 4%.

Based on Sharpe Ratio, with which fund manager would you invest and why?

Explanation

The following table has the calculation of Sharpe Ratio in the fourth column.

Fund Manager Return Standard Deviation (Risk) Sharpe Ratio

A 10% 20% =(10-4)/20 = 0.3

B 15% 25% = (15-4)/25 = 0.44

Since the fund manager B has higher Sharpe Ratio (excess return per unit of risk) than fund manager A,

we should invest in B’s fund.

IMPORTANT: Compare, this example with the previous example, in both the case we have got the same

answer, which is to invest in B’s fund. Is it always the case that the result in which portfolio to invest is

same in Sharpe Ratio and Coefficient of Variation?

The example, here are peculiar and it is by chance that we are getting the same return, It is not always

the case. Let’s see the next example.

Concept Builder – Coefficient of Variation Versus Sharpe Ratio

18. You are Fund Of Fund Investment manager and would like to invest some money in one of the fund.

You are evaluating two fund managers and would like to invest in one of them. The following table

has the return and the standard deviation data for the two fund managers. The risk free rate of

return is 7%. Based on the concept of Coefficient of Variation and Sharpe Ratio, find out with which

fund manager would you invest and why?

Fund Manager Return Standard Deviation (Risk)

A 12% 15%

B 14% 20%

Explanation

The following table has the calculation of CV and the Sharpe Ratio.

Fund Manager Return Standard Deviation CV Sharpe Ratio

A 12% 15% =15/12 = 1.25 =(12-7)/15 = 0.33

B 14% 20% = 20/14 = 1.43 = (14-7)/20 = 0.35

As per the CV, Fund manager A is better, as CV is lower for A.

As per the Sharpe Ratio, Fund manager B is better, as Sharpe Ratio is higher for B.

So, we can have conflicting result from Sharpe Ratio and CV comparison. Since Sharpe Ratio, measures

the excess return relative to Risk, Sharpe Ratio is Preferred over CV.

LOS 7.h. Calculate and interpret the proportion of observations falling within a specified number of

standard deviations of the mean using Chebyshev’s inequality

Ideally, this LOS should have been covered in reading no -9, where we are covering the distribution.

Covering the concept over here would entail some unnecessary hard-ship on you. Leaving this LOS, at

this point of time will not impact the flow of study. So, we will cover this LOS in Reading 9.

LOS 7.j. Define and interpret Skewness, explain the meaning of a positively or negatively skewed return distribution, and describe the relative locations of the mean, median, and mode for a nonsymmetrical distribution

Skewness -Ideally the distribution should be symmetrical around the mean, that is if we put a mirror

along the Y Axis on the mean, the reflection of the left side should be identical to the shape of the right

side. However in real life, we usually do not find symmetrical distribution.


In everyday language, the terms “skewed” and “askew” are used to refer to something that is out of line

or distorted on one side. When referring to the shape of frequency or probability distributions,

“skewness” refers to asymmetry of the distribution.

Skewness is of two types

Positively Skewed –In this case the distribution is skewed to the right side and hence it is also

called as Right Skewed. If you see the diagram given below, you would see that in case of positive

skewed distribution there is a long tail in the right side. The long tail denotes that there are

outliers in the positive side of the distribution, that is, there are some observations which have

large positive value and hence they are able to shift the mean to the right side.

As an example consider that in a class, there is a brilliant student and he scores 100 out of 100 in

the examination, there are 4 other student in the class, who are average and they score 60 marks

each in the examination. The mean marks of the students in the class would be 68, but the mode

and the median are equal to 60. Hence due to the presence of above average student the mean has

shifted to the right.

For this kind of distribution, you will find that the mean is more than median and median is more

than the mode.

Negatively Skewed –In this case the distribution is skewed to the left side and hence it is also

called as Left Skewed. If you see the diagram given below, you would see that in case of negative

skewed distribution there is a long tail in the left side. The long tail denotes that there are outliers

in the negative side of the distribution, that is, there are some observations which have large

negative value and hence they are able to shift the mean to the left side.

An example, here could be the returns in a stock market, many times we observe that there are

some returns which are highly negative and there are more returns which are positive , because of

large negative returns the mean return shift to the left and hence we can see a left skewed

distribution.

Mnemonic: As per dictionary, the order of the words is Mean, Median and Mode. Hence Negative

skewed distribution follows the dictionary order that is, mean is less than median which in turn is

less than mode. Please note that median will always be in the middle.

Important Exam Points for Skeweness

Mean always get shifted in the direction of skewness, for positively skewed distribution, mean is

shifted to the positive side (or right side)

For a symmetrical distribution, the distance from the mean to the highest observation would be

same as that of distance from the mean to the lowest observation.

Concept Builder – Skeweness

19. Suppose the mean is given as 50% and the highest return value is given as 100% and lowest return

value is given as 0%. State whether the distribution is symmetrical or not?

Explanation

Since, here the distance from the mean to the highest value and the distance to the lowest value is

same, we would say that the distribution is symmetrical.


20. Suppose the mean is given as 50% and the highest return value is given as 80% and lowest return

value is given as 0%. State whether the distribution is symmetrical or not? If it is not symmetrical

then state whether it is positively skewed or negatively skewed?

Explanation

Since, here the distance from the mean to the highest value is 30 and the distance to the lowest value is

50, then we would say that the distribution is skewed to the left as we have long tail there.

LOS 7.k. Define and interpret measures of sample Skewness and kurtosis

This LOS asks us to define the measure of sample skeweness and kurtosis. In the CFA text book, you

would find that formula for sample skeweness and kurtosis are given, but you don’t have to remember

the formula, since the LOS doesn’t ask you to calculate. However you require interpretation of these.

Skeweness is measure of 3rd moment, that is, we are taking the cube of the deviation from the mean

and summing them up. This result in the following interpretation

o Skewness can range from minus infinity to positive infinity.

o For a symmetrical distribution, skeweness = 0

o For positively skewed distribution, skeweness>0

o For negatively skewed distribution, skeweness< 0

From examination point of view, you just need to know the above and nothing more.

Kurtosis - Kurtosis is a measure of the Peakness of a symmetric distribution as compared to a normal

distribution of the same variance. Here, before covering anything lets cover normal distribution.

Normal Distribution – As of now, just understand that this is one type of distribution which is

symmetrical in nature. It has a kurtosis of 3. We would cover this distribution in detail in the Reading

Number 9.

Kurtosis is of three types:-

Leptokurtic

o This type of distribution is more peaked than the normal distribution.

o The kurtosis of Leptokurtic distribution is more than 3

o In this type of distribution, there is large number of observation which is near the mean and

also large number of observation away from the mean.

o It has fatter and long tails

o Mnemonic -It has a shape like that of small L and L stands for Lepto

Mesokurtic

o A mesokurtic distribution has kurtosis of 3 which is same as normal distribution and hence

mesokurtic distribution is actually a normal distribution. We compare kurtosis with respect

to mesokurtic distribution only.

Platykurtic

o This type of distribution is less peaked than the normal distribution.

o Platykurtic distribution is Flat.

o The kurtosis of Platykurtic distribution is less than 3

o It has thinner and shorter tails

o Mnemonic – Platy can be thought of as “FLAT”, add P in FLAT to get PLAT


Excess Kurtosis – Excess kurtosis tries to find out the kurtosis which is more than that of a mesokurtic

(normal) distribution. Since the Kurtosis for a normal distribution is equal to 3. The excess kurtosis

formula is given by

Excess Kurtosis = Kurtosis - 3

Therefore, Excess Kurtosis for

Leptokurtic distribution is more than 0

Mesokurtic distribution is 0.

Platykurtic distribution is less than 0

Kurtosis Measure: We don’t need to know the formula of Kurtosis; we just need to know that Kurtosis

is the 4th moment. That is kurtosis is calculated by taking the sum of the fourth power of the

deviations. As a result Kurtosis is always positive.


Concept Notes For Reading 8- Probability Concepts

Reading Summary

This reading covers important probability concepts. Here we will discuss about random variables,

properties of probability. We will also understand how probability theory is used in betting. From

examination point of view, this is an important reading, because the concepts would be used in

portfolio management and also in the next readings where without understanding of probability it

would be difficult to understand the distributions. Quite a few questions from this reading are expected

in the examination.

From our experience, we have seen that to do well in this topic, you should understand how to

structure a probability problem using a tree diagram. Bayes theorem is a very important topic which

you need to master. However permutation and combination topic can be taken lightly.

LOS 8.a.Define a random variable, an outcome, an event, mutually exclusive events, and exhaustive events

A random variable is a quantity whose observed value is uncertain.

Outcome is the observed value of a random variable.

Event is a specified set of outcomes. It can be a single outcome also.

Mutually exclusive events are those events in which only one of the events would happen at a

particular time. So, in mutually exclusive event it is not possible to observe all the events at one time.

Exhaustive means that the outcome of the events covers the entire possible scenario.

An example of mutually exclusive and exhaustive event would be the pattern of stock price. At any

point of time, the share price would

Increase

Decrease

Stay same

So, the three states given above are mutually exclusive but note that they are collectively exhaustive

because, these are the only states possible. You can’t have a stock’s price changing to some other state.

Mnemonic: MECE – Mutually Exclusive and Collectively Exhaustive

LOS 8.b.Explain the two defining properties of probability and distinguish among empirical, subjective, and a priori probabilities

The two defining properties of probability are as follows:-

1. The probability of any event (E) is a number between 0 and 1, that is,

2. The sum of the probabilities of any set of mutually exclusive and collectively exhaustive event should

be always equal to 1. That is,

Going back to the stock price example of last LOS. It’s not possible to have the probability of any of the

state as negative or more than 1. Let’s say that the probability of increase is 0.4, decrease is 0.5 and

remaining the same is 0.1.

Here we are seeing that the sum of the probabilities of these mutually exclusive and exhaustive events

should be equal to 1.

How do we determine the probability of any set of events?

Estimation of Probability: There are 3 approaches in which we can estimate the probability. The

approach is divided into Objective and Subjective. Objective is further divided into Empirical and Priori

Probability. The following diagram will make the concept clearer.


Let’s cover what these probability mean

Empirical Probability – Empirical probability is estimated using the past data.

Example – Suppose we have seen that over the past 250 trading days. The number of days the stock

price increased was 100. The number of days the stock price decreased was 125 and the number of

days the stock price remained the same was 25.

Now probability is also defined as

So with the historical observation, we can say that the probability that the stock price will increase

will be equal to 100 (no of favorable days) divided by 250 (total number of days), which will be 0.4.

Likewise we can estimate the probability of decrease and remaining same also.

A Priori Probability–A priori probability is estimation of probability using formal reasoning or logical

method.

Example – Suppose, we estimate that the inflation of the economy is high and it is expected that

the central bank will increase the interest rate. In this scenario with formal reasoning we expect

that the probability that the stock price will decrease, we do some analysis and come up with a

probability value of 0.7, and similarly other probabilities.

Subjective Probability–Subjective probability is estimation of probability based on gut feeling. It is

the least formal method and involves lot of subjectivity by the estimator. In investment, its usage is

frequent in nature.

Example – A person may say that in his personal opinion the probability of stock rising the next day

is 0.7.

LOS 8.c.State the probability of an event in terms of odds for or against the event

This LOS covers the concept related to betting. In betting we use the term odds for a team winning etc.

Terminologies

Odds for an event–Out of the total how much is in favor of a particular event.

Example : Odds for India winning against Pakistan is 5 to 3, it implies that out of (5+3 = 8) times,

India will win the match 5 times and Pakistan will win the match 3 times.

Odds against an event - Out of the total how much is against a particular event.

Example : Odds against India winning in a match with Pakistan is 3 to 5, it implies that out of (5+3 =

8) times, Pakistan will match 3 times and India will win the match 5 times .

Note: Odds for an event is just the reciprocal of the odds against the event

In examination, there are 2 types of questions that would be asked

a) Find out the probability, given odds for an event or against an event.

a. If odds for an event is given as A:B then the probability that event A will occur is

b. If odds against an event is given as A:B then the probability that event A will occur is

b) Given probability for an event, find out the odds for or against the event

Probability

Estimation

Objective

Empirical A Priori

Subjective


a. If Probability of an event is given as P(E) then the odds for the event is

b. If Probability of an event is given as P(E) then the odds against the event is

Even though we have given the formula, dealing these concepts in formula is not the correct approach,

rather than that you should understand the underlying concept. It would become clear by seeing the

below example.

Concept Builder – Finding probability when Odds For and Odds Against is given

1. The bookies are quoting the following for the final match between Liverpool and ManU

Odds For Liverpool : 7 to 11

What is the probability of Liverpool winning the match? What is the probability of ManU winning

the match?

Explanation

The probability that Liverpool will win the match is its favorable case divided by the total case

P(Liverpool winning) = 7/18 = 0.389

The probability that ManU will win the match is its favorable case divided by the total case

P(ManU winning) = 11/18 = 0.611

Since, This could also be obtained by subtracting P(Liverpool winning) from 1

P(ManU winning) = 1 - P(Liverpool winning) = 1 – 0.389 = 0.611

Needless to say, we are ManU fans!!

Concept Builder – Finding Odds For and Odds Against when probability is given

2. Suppose you determine that the probability of Chennai Super Kings winning a match against

Mumbai Indian is 0.6, then what would be the

a. Odds that Chennai Super Kings will win the match

b. Odds that Chennai Super Kings will lose the match (Note that here we are asking odds

against in an indirect way)

Explanation

a. For finding the odds for; without following the formula; we should think of odds for as

Probability the event will occur : Probability the event will NOT occur

Probability that Chennai will win : Probability that Chennai will lose

0.6 : 0.4 Or 6:4 Or Simplifying it a bit, 3:2

So the odds that Chennai Super Kings will win the match is 3 : 2

b. For finding the odds against; without following the formula; we should think of odds against as

Probability the event will NOT occur : Probability the event will occur

Probability that Chennai will lose : Probability that Chennai will win

0.4 : 0.6 Or 4:6 Or Simplifying it a bit, 2:3

So the odds that Chennai Super Kings will lose the match is 2 : 3

LOS 8.d.Distinguish between unconditional and conditional probabilities

Unconditional Probability: It is also referred as marginal probability. It is the probability that an event

will occur regardless of any past or future occurrence of any other event.

Conditional Probability: The probability that an event will occur, given that one or more other events

have already occurred. More precisely the probability that B will occur given that A has occurred.

The symbol P (B|A) represents Conditional Probability and is read as Probability of B given A has

occurred.


LOS 8.g.Distinguish between dependent and independent events

We will be covering this LOS before we cover LOS 8.e and LOS 8.f

1. Independent Events – These are events in which the occurrence of one event has no impact on the

occurrence of the other events. As an example, your score in an exam is independent of the score

of your friend. In terms of conditional probabilities, two events are independent, if and only if

P(A | B ) = P(A) or P(B | A ) = P(B)

If the condition above is not satisfied then the events are dependent events

2. Dependent Events – These are the events in which occurrence of one event has an impact on other

events. As an example, your score in an exam is dependent on the amount of effort you have put in

to study for the exam.

Here in this section, we will be covering both LOS 8.e and LOS 8.f

LOS 8.e.Define and explain the multiplication, addition, and total probability rules LOS 8.f.Calculate and interpret 1) the joint probability of two events, 2) the probability that at least one of two events will occur, given the probability of each and the joint probability of the two events, and 3)

a joint probability of any number of independent events

a) Multiplication Rule of probability – It is used to find out the Joint Probability of 2 events. Joint

probability is the probability that 2 events would happen together (or simultaneously).

We would see the following type of joint probability in the curriculum.

1. Joint Probability of conditional events

2. Joint Probability of independent events

Let’s cover these two concepts

1. Joint Probability of conditional events –The joint probability of conditional event is given as

Where, is the conditional probability that A will happen, given that B has happened.

And is the unconditional probability of B.

2. Joint Probability of independent events – In case of independent events A and B, the joint

probability is given as

So, the joint probability is the multiplication of the individual probabilities and the multiplication

rule is used to denote the joint probability.

This concept can be extended for any number of independent events

Let’s say that A1, A2, ……, AN are independent events, the probability that all of these events occur

simultaneously is given by

Note that when you see AND , this means that you have to use multiplication.

b) Addition Rule of probability – It is used to find out the probability that at least one of the event will

occur. In case of two events A and B.

Addition rule will say that Event A OR B will occur.

Note that when you see OR, this means that you have to use addition.

You can extend the addition rule concept to determine the probability that at least one of the event

would occur. If you notice the below formula, here we are saying that Either A occurs or B occurs, it

implies that sometimes A occurs, sometimes B occurs or sometime both will occur, it will not happen

anytime that none of the event is occurring.

You can understand this better from the Venn diagram, which is shown below. The shaded region

represent that both A and B will occur together. Note that when we add P(A) and P(B), the shaded


region will come twice, hence we need to subtract it from the addition to come up with the true

probability that at least A or B will occur.

c) Total Probability is the probability of occurrence of an event, given conditional probability.

Let’s say there are N mutually exclusive events denoted as A1, A2, ……, AN. Each of these events are

conditional on mutually exclusive and exhaustive events B1, B2, ……, BN.

We can obtain the unconditional probability of event A (which we say is the sum of the probabilities of

each event Ai)

Since A1 is conditional on B1, We can write

Replacing this for all Ai’s we have

The probability of A that we have obtained here is un-conditional probability, that is, it doesn’t depend

on whether B1 is occurring or B2 is occurring.

Hence we can say that Total Probability is used to find out the unconditional probability of an event

when the conditional probabilities are given.

2 Scenario Case: The total probability rule is stated below for a special case involving 2 scenarios. If we

have an event denoted by B, then we will denote the event which is NOT B (or complement of B) with

(spelt as B bar). Here B and are mutually exclusive and collectively exhaustive event.

Let’s say we have an event A, which will occur when B occurs and also when B doesn’t occur. The total

probability for this two scenario case can then be written as

Below is a representation of the total probability concept using tree diagram. Tree diagram is

considered in a separate LOS, but this is ideal place to introduce the concept of tree diagram.


So, from the above diagram, it becomes clear that Node 1 and Node 3, are representative of occurrence

of A, whereas node 2 and node 4 are representative of occurrence of NOT A (or A bar).

If you add the value obtained in Node 1 and Node 3, you will get the probability of occurrence of A.

If you add the value obtained in Node 2 and Node 4, you will get the probability of occurrence of A bar

(or Not A).

The tree diagram will become clearer as and when we solve more problems. We would suggest thinking

about tree diagram when you are solving problems which involve conditional probability because it

gives a better understanding.

Concept Builder – Joint Probability Of Conditional Events and Addition Rule

3. Suppose probability that RBI will decrease the interest rate is 0.6 and the probability that recession will

happen if the interest rate is decreased is 0.3.

A. Find the joint probability of decrease in interest rate and recession?

B. If the probability that either interest rate will decrease or recession will occur is 0.82, find the

unconditional probability of recession

Explanation

A. In notation term we can write ; P(Decrease Interest) = 0.6; P(Recession | Decrease Interest) = 0.3

The joint probability is P(Decrease Interest AND Recession); as per the joint probability for conditional

probabilities we have

P(Decrease Interest AND Recession) = P(Recession | Decrease Interest) * P(Decrease Interest)

P(Decrease Interest AND Recession) = 0.6 *0.3 = 0.18

B. Unconditional probability of recession = P(Recession)

Here we have OR condition

=> P(Recession Or Decrease Interest) = P(Recession) + P(Decrease Interest) – P(Recession AND Decrease

Interest)

=> 0.82 = P(Recession) + 0.6 – 0.18 => P(Recession) = 0.82 -0.6 +0.18 = 0.4

Therefore the unconditional probability that recession will happen is 0.4

Concept Builder – Joint Probability Of Independent Events

4. Suppose that you are tossing a coin 5 times in a row. What is the probability that Head will come in

each of the time?

Explanation

Here , the tossing of coins are independent events . For independent events, we have the following

P(A and B) = P(A) * P(B)

Probability of Head coming in 1 toss = ½ = 0.5

Probability of Head coming in 5 tosses = ½ * ½ * ½* ½* ½ = 0.0312

LOS 8.h.Calculate and interpret, using the total probability rule, an unconditional probability

As we had stated in the last LOS that total probability is also known as unconditional probability. This

LOS specifically asks us to calculate the unconditional probability using the total probability rule.

Many of us are not able to appreciate fact that the total probability is the unconditional probability.

Let’s solve one example to calculate the total probability and understand the concept of unconditional

probability.

Concept Builder – Calculating the Total Probability

5. If you are following the recent events, Reserve Bank of India is increasing the interest rates to

contain the inflation. One of the major drawback of raising interest rate is that the chances that the

economy will go into recession increases. However, it’s not the case that an economy can’t be in

recession if the interest rates are decreased (see USA, even though the interest rates are decreased

in US, the state of the economy is not good).

Coming to the question. Let’s say the probability that the Central Bank would increase the interest

rate is 0.7 and the conditional probability of Recession happening when the interest rate is


increased is 0.9. Also, the conditional probability that the recession will happen if the interest rate is

decreased is 0.2.

We are interested in finding out

a) The unconditional probability that the Recession will happen

b) The unconditional probability that the Recession will NOT happen

Explanation

You will find that in book, there is notation to the unconditional probabilities. We will go through this

problem by following those notations, and then try to solve the problem through the formula of the

total probability. In the next step we will try to solve the problem using the TREE DIAGRAM. We advise

the candidates to follow the tree diagram.

Probability of Increase in Interest Rate = = 0.7

Probability of decrease in Interest Rate = = 1 – 0.7 = 0.3

Probability of Recession happening Given Interest Rate Increases =

Probability of Recession happening Given Interest Rate Decreases = = 0.2

Probability of Recession NOT happening Given Interest Rate Increases = = 1 -

= 1 -0.9 = 0.1

Probability of Recession NOT happening Given Interest Rate Decreases = = 1- 0.2 = 0.8

The unconditional probability that the recession will happen is same as the total probability that the

recession will happen.

As per the total probability formula, we can write

=> P(R) = 0.9 *0.7 + 0.2 * 0.3 =0.63 +0.06 = 0.69

So, the unconditional probability that recession will happen is 0.69

The probabilities that you had seen earlier or were conditional probabilities, that is, they

were dependent on whether the interest rate is increased or decreased. But the total probability is not

depended on increase/decrease of interest rate, it doesn’t matter whether interest rate increases or

decreases, if the interest rate increases, then the probability of recession will increase and if the

interest rate decreases then the probability of recession will decreases, but overall the probability will

be the same and hence un-conditional.

We can find the un-conditional probability of recession NOT happening – it will simply be 1 minus the

un-conditional probability of Recession happening.

The unconditional probability that recession will NOT happen = 1 - 0.69 = 0.31

We can also find out the above value using the total probability formula

=> 0.1 *0.7 + 0.8 * 0.3 = 0.07+0.24 = 0.31

Let’s see, how we can solve this using Tree Diagram, You will find it intuitive


While constructing tree diagram,

1st STEP - We start with first the un-conditional probability, which we have here for the Interest rate. So

the first branch will be for the interest rate increase and the other branch is for the interest rate NOT

Increasing.

2nd STEP - Then we start with the conditional probabilities and draw branches for them. Throughout the

process, we will keep in mind that the branches are Mutually exclusive and Collectively Exhaustive.

3rd STEP – We highlight the node is related to the respective probabilities. Here Node 1 and Node 3 are

the nodes for Recession happening and Node 2 and Node 4 are the nodes for Recession NOT

happening.

4th STEP - To find out the Total Probability of recession, we add up the value at the nodes were

recession happens. Therefore Recession probability = 0.63 + 0.06 = 0.69

To find out the Total Probability of recession NOT happening, we add up the value at the nodes were

recession does not happen. Therefore Recession probability = 0.07 + 0.24 = 0.31

This is the way we will be drawing the TREE Diagram and hence finding out the total or un-conditional

probabilities.

LOS 8.i.Explain the use of conditional expectation in investment applications

In previous reading we had covered the concept of Weighted Mean return in a portfolio, which was

based on the weight of the individual asset. We had obtained the weighted mean return by multiplying

the weight of asset with the return of the particular asset. If you think about the Total Probability, it can

also be considered like a weighted mean; here the weight is the conditional probability. We are going to

introduce the concept of Expected value which can be viewed like weighted mean.

Expected Value –Expected value of a random variable is the probability weighted average of the

possible outcomes of that random variable.

Expected value is denoted by E(X) where X is the random variable.

Expected value is a futuristic value and is based on our expectation about the future, it is not

necessary that whatever we are expecting about the future will happen.

The formula for the expected value can be written as

Where, is the outcome of the random variable with a probability of


Concept Builder – Finding Expected Value for throw of Dice

6. Find the Value that you would expect when you throw a Six faced dice having numbers from 1 to 6

Explanation

When we throw a dice, we can get any number from 1 to 6. By Expected value we imply what is the

value on an average we will get when we throw a dice.

The random variable can take values from 1 to 6.

The probability that either 1,2,3,4,5 or 6 would come when we throw a dice is 1/6

Hence Expected Value is calculated as,

Therefore we should expect that in one throw we should get on an average 3.5

But 3.5 is not a number that you will observe, we get sometimes 1 and sometimes 6, hence there is

variance in our observation.

We will take up the variance concept after next example.

Concept Builder – Finding Expected Earnings Per Share (EPS) in an investment Setting

7. You are analyzing, Infosys Technologies (NASDAQ: INFY) stock and based on your evaluation you

have come up with the following estimates of EPS for the year 20X2 and the probabilities that the

firm would be able to achieve the EPS. See the table below for the estimates

Probability EPS

0.35 $2.8

0.3 $2.9

0.25 $3.1

0.1 $4

What is the expected EPS of INFY in 20X2?

Explanation

Expected Value is probability weighted

Hence the expected value of EPS is $3.03, so you have estimated that INFY will have EPS of $3.03 in the

year 20X2.

Since, expected value can be considered as mean; hence we will also get variance because there are

times when the value thus obtained will not be equal to the expected value.

Variance –The variance of a random variable is the probability weighted average of the squared

deviations from the expected value. The formula for the variance is given below

Standard deviation is square root of variance.

Concept Builder – Finding Variance and Standard Deviation of an Expected value

8. You are analyzing, Infosys Technologies (NASDAQ: INFY) stock and based on your evaluation you

have come up with the following estimates of EPS for the year 20X2 and the probabilities that the

firm would be able to achieve the EPS. See the table below for the estimates

Probability EPS

0.35 $2.8

0.3 $2.9

0.25 $3.1

0.1 $4


What is the variance and standard deviation of the expected EPS of INFY in 20X2?

Explanation

We have calculated the expected value in the previous example as $3.025

For simplicity, we will calculate the variance using the following table

Probability EPS EPS – Expected Value {EPS – E(X)}2 P * {EPS – E(X)}2

0.35 $2.8 (2.8 – 3.025) = -0.225 0.050625 0.01771875

0.3 $2.9 (2.9 – 3.025) = -0.125 0.015625 0.0046875

0.25 $3.1 (3.1 – 3.025) = 0.075 0.005625 0.00140625

0.1 $4 (4 – 3.025) = 0.975 0.950625 0.0950625

Sum 0.118875

So, the variance (σ2)is 0.118875 ($2)

The standard deviation (σ )= 0.1188750.5 = $0.34478

This is a simple problem and there is a high probability that this would be coming in the exam.

There are lots of pitfalls here, please avoid them

1. Rather than first subtracting the expected value from the observation and then multiplying, many

students they first multiply and then they subtract

2. Many students do not multiply with probability but take the square of the difference and then divide

by the number of occurrence

Conditional Expectation – In investments, many times we expect outcome which is based on some

event (hence conditional), the expected value based on conditional probability is called as conditional

expectation. Let’s say that the expected value of a random variable X is based on a scenario S, then the

expected value is denoted by E(X|S) , that is, X is conditional on S.

Let’s say that X takes on values X1, X2, ……….Xn then the Expected Value for X is given by

If you see, the above formula looks equivalent to the Total Probability Rule. So, the above can be

viewed as the Total Probability rule for Expected Value, when there are N scenarios.

LOS 8.k.Diagram an investment problem using a tree diagram

An investment problem is when we are calculating the EPS, return or any other measure related to

investment. We associate probability to those measures. So, essentially in an investment problem we

are interested in finding out the expected value. Tree diagram is used to represent these problems as it

becomes very easy to solve the problems.


Concept Builder – Tree diagram for Investment Problem

9. There is a probability of 0.6 for decrease in interest rate, the probability of stable interest rate is 0.3

and probability of increase in interest rate is 0.1.

A firm may perform well or poorly in all these scenarios.

The probability that the firm will do well when interest rate increases is 0.3 and the EPS is $2.5.

The probability that the firm will do poorly when interest rate decreases is 0.7 and the EPS is $1.5

The probability that the firm will do well when interest rate is stable is 0.5 and the EPS is $3.

The probability that the firm will do poorly when interest rate decreases is 0.5 and the EPS is $2

The probability that the firm will do well when interest rate decreases is 0.8 and the EPS is $4.

The probability that the firm will do poorly when interest rate decreases is 0.2 and the EPS is $3.

A. Draw a tree diagram to represent the above problem.

B. Find out the expected EPS conditional on the firm doing well

C. Find out the expected EPS conditional on the firm doing poor

D. Find out the expected EPS (that is unconditional EPS)

Explanation

B. The Node 1, 3 and 5 represent the Expected EPS conditional on the firm doing well .

The Expected EPS will be the probability weighted of the EPS

C. The Node 2, 4 and 6 represent the Expected EPS conditional on the firm doing well .

The Expected EPS will be the probability weighted of the EPS

D. The expected EPS or the unconditional EPS is given as

LOS 8.n.Calculate and interpret an updated probability using Bayes’ formula

We will cover this LOS before we cover the other LOSs.

Till now we have covered probabilities which are based on our expectation or historical data and were

static. In reality, the probabilities keep on changing as and when new information becomes available.

Bayes’ formula addresses the concept of updated probability. Following is the formula given for the

updated probability


The following is the probability notation

The updated probability is also known as posterior probability.

Rather than focusing on the formula, we will be covering the Bayes’ theorem using the tree diagram.

Concept Builder – Bayes’ Theorem

10. Going back to one of the previous problem. The probability that the Central Bank would increase

the interest rate is 0.7 and the conditional probability of Recession happening when the interest

rate is 0.9. Also, the conditional probability that the recession will happen if the interest rate is

decreased is 0.2.

You want to update your Prior probabilities with some new information

a) Given that recession has happened, what is the updated probability that interest rate was

increased?

b) Given that recession has NOT happened, what is the updated probability that interest rate was

increased?

c) Given that recession has happened, what is the updated probability that interest rate was

decreased?

d) Given that recession has NOT happened, what is the updated probability that interest rate was

decreased?

Explanation

The Tree Diagram is given below.

a) Here, we know that the recession has happened; it means that either Node 1 or Node 3 has

happened. Now since we are interested in finding out the updated probability of increase in interest

rate given that recession has happened. So, if you see the tree diagram, our favorable case is Node 1

and the total case is Node 1 and Node 3.

The probability is given by : favorable case / total case

=> Updated probability = 0.63/(0.63+0.06) = 0.9130

You will appreciate the fact that recession is mainly happening due to increase in interest rate and

hence the updated probability is very high as compared to the prior probability.

b) Here our favorable case is node 2 and total case is node 2 and node 4



c) Here our favorable case is node 3 and total case is node 1 and node 3


d) Here our favorable case is node 4 and total case is node 2 and node 4


If you take the sum of probabilities found in a) and c) it would come to 1. This implies that the

probability of recession happening is 1 (As we already know that recession has happened)

LOS 8.k.Calculate and interpret covariance and correlation

Variance measures the dispersion of a single random variable. Many times during investment we are

interested in knowing how two variables move with respect to each other. For example, we might be

interested in knowing if the S&P 500 is increasing what would be the behavior of Microsoft stock,

whether it would increase or decrease.

To understand the movement of 2 variables we have Covariance and Correlation as the measures.

Covariance – Covariance measures how 2 variables move with respect to each other. The formula for

the covariance is given below

So, covariance is the expected value of the product of deviation of the two random variables from their

respective expected value (can be thought of as mean).

Properties of Covariance

Covariance values ranges from negative infinity to positive infinity

The covariance of a variable X with itself is the variance of X

If there is no relation between the assets, then the covariance is 0.

The unit of covariance is squared units

Drawback – Covariance is not able to measure the strength of the relationship. Higher Covariance does

not necessarily means that the strength of the relationship is high. As a result we have another measure

known as Correlation.

Correlation Coefficient – It measures the strength of the relationship. Higher correlation value implies

higher strength. The formula for correlation coefficient is given by

Properties of Correlation Coefficient

Correlation coefficient measures only strength of Linear Relationship

Correlation coefficient is unit less

Correlation ranges between -1 and +1

If there is no linear relation between the assets, then the correlation coefficient is 0.

If the correlation coefficient is +1, then the variables are Perfectly Positively Correlated

If the correlation coefficient is +1, then the variables are Perfectly Negatively Correlated

Most of the time we would be using covariance and correlation from investment point of view, so the

returns are used as the random variable.


Concept Builder – Covariance and Correlation

11. You are analyzing two stocks and are trying to find out the expected return. You are predicting that

the economy can be in either good state or bad state. Following table illustrate the return and the

probabilities that you are expecting from the two stocks.

State of the economy Probability Return of A Return of B

Good 0.7 15% 20%

Bad 0.3 10% 5%

Explanation

First we will have to find out the expected returns of A and B

E(RA) = 0.7 *15% + 0.3*10% = 13.5%

E(RB) = 0.7 *20% + 0.3*5% = 15.5%

The following table has the calculation for Covariance

Probability Return of A Return of B RA – E(RA) RB – E(RB) P * {RA-E(RA)} * {RB-E(RB)}

0.7 15% 20% 1.5% 4.5% 4.725

0.3 10% 5% -3.5% -10.5% 11.025

Sum 15.75

So, the covariance is 15.75 %2 or 0.001575

For finding out the Correlation we need to find the standard deviation of A and B

The standard deviation is computed in the following tables

Probability Return of A RA – E(RA) P * {RA-E(RA)} * {RA-E(RA)}

0.7 15% 1.5% 1.575

0.3 10% -3.5% 3.675

Sum 5.25

So, the variance of A = 5.25%2 => standard deviation = 2.2913 %

Probability Return of B RB – E(RB) P * {RB-E(RB)} * {RB-E(RB)}

0.7 20% 4.5% 14.175

0.3 5% -10.5% 33.075

Sum 47.25

So, the variance of B = 47.25%2 => standard deviation = 6.8739 %

So, the correlation coefficient will be given by

So, here we see that the correlation coefficient is the maximum.

LOS 8.l.Calculate and interpret the expected value, variance, and standard deviation of a random variable and of returns on a portfolio

A portfolio of assets will also have expected return and variance similar to the assets itself. The only

thing is that we observe the benefit of diversification in case of portfolio. By diversification we mean

that we are including more assets in the portfolio. If there are 2 assets in a portfolio and the correlation

coefficient between them is less than 1, then the standard deviation of the portfolio is lesser than the

weighted standard deviation of the individual assets.

Asset Weight in a portfolio - The weight of any asset in the portfolio is given by the following formula

Expected Return of Portfolio – The expected return of the portfolio is the weighted average of the

expected return of the individual asset.


Variance of the Portfolio – There is a complex formula for the variance of a portfolio having n assets.

For level 1 we are required to remember the variance of 2 asset portfolio.

The variance of a 2 asset portfolio is given by

In terms of covariance, the formula can be written as

Concept Builder – Portfolio Variance

12. A portfolio is made up of 2 assets A and B. The current market value of investment in A is $600 and

current market value of investment in A is $400. Correlation between A & B is 0.3. The standard

deviation of A is 20% and standard deviation of B is 30%. Find out the portfolio standard deviation?

Explanation

First find out the weight of each asset in the portfolio.

Weight of asset A = 600/(600+400) = 0.6 => Weight of B = 0.4

We will use the 2 asset, standard deviation formula.

=> = 374.4

The variance of the portfolio is 374.4 %2

The standard deviation will be square root of the variance = 19.3494%

If, you calculate the weighted average standard deviation it will be 24%

So, you are seeing that the standard deviation of the portfolio is lesser than the weighted average, this

is the benefit of diversification, which we had discussed.

LOS 8.m.Calculate and interpret covariance given a joint probability function

Joint Probability Table: This is a table in which we mention the joint probabilities of the two variables

for a specified set of outcomes. The Joint probability is very neat way to denote the probabilities of

occurrence of return of 2 investments.

Below is an example of a Joint Probability table

Asset RB = x% RB = y% RB = z%

RA = a% A1 A2 A3

RA = b% A4 A5 A6

RA = c% A7 A8 A9

Some Observations from the joint probability table

Here P1 means the probability of A returning a% and B returning x%

The total probability of A returning a% will be the some across the row which will be A1 + A2 + A3

The total probability of B returning x% will be the some across the column which will be A1 + A4 +

A7

The sum of all the Ai should be equal to 1; that comes from the probability property.

Correlation Matrix – In a correlation matrix the correlation between the variables is mentioned. The

usage of matrix is a very clear representation of the correlation and is widely used.

Below is an example of correlation matrix

Asset A B C

A 1

B 1

C 1


You will find that the lower part of the matrix is redundant, because the correlation coefficient between

A and B is same as the correlation coefficient between B and A.

So, in many places you will find that only the top part is mentioned.

Covariance Matrix – In covariance matrix, the covariance between the random variables is mentioned.

Below is an example of covariance matrix.

Asset A B C

A Cov (RA, RA) = Var(A) Cov (RA, RB) Cov (RA, RC)

B Cov (RB, RA) = Cov (RA, RB) Cov (RB , RB) = Var(B) Cov (RB, RC)

C Cov (RC, RA) Cov (RC , RB) Cov (RC, RC) = Var(C)

Like correlation matrix, here also you will find that the lower part of the matrix is redundant.

LOS 8.o.Identify the most appropriate method to solve a particular counting problem, and solve counting problems using the factorial, combination, and permutation notations

There is a very low probability of this section coming in the examination

We will be covering 3 type of formula here

1. Labeling Formula – The number of ways in which n object can be labeled with k different labels,

with n1 of first type, n2 of second type and so on. n1 + n2 + ….. +nk = 1

The formula is given by

2. Selection Or Combination formula – This is used when we are selecting from a group. It is a special

case of labeling formula. Here the order of the selection doesn’t matter. For example if we are

selecting 6 players from 9 players, it doesn’t matter who gets selected first.

The formula is given as or =

Where n is the total number of object and we need to select r object from it.

So, in the example above, n = 6 , r = 9, so the number of ways = 84

3. Permutation – This formula is used when arrangement of the items selected is also important. For

example in a cricket match, let’s say that the player who gets selected first will be the first one to

bat, hence here the order of selection is important.

The formula is given as =

As an example, if there are 9 players and we need to select 6 players and the order of selection is

important, then the number of ways in which it can happen = 60,480

Concept Builder – Counting Problem

13. Your portfolio has 10 mutual funds,

a. You met an investment advisor and he advised to hold 6 mutual funds and sell 4 mutual

funds. In how many ways this can be achieved?

b. In how many ways you can select 3 mutual funds from the 10 funds

c. In how many ways you can sell 3 mutual funds, where the order of selling is important

Explanation

a. The total number of ways in which we can arrange 10 mutual funds is 10!. The total no of ways in which we

can arrange 6 and 4 mutual fund is 6! And 4! respectively.

Here the sequence doesn’t matter, so we are doing double counting by arranging the 6 and 4

mutual funds, which have to be removed.


Hence, the number of ways = 10!/(6! * 4!) = 210

Alternatively, we can directly use the labeling formula, because we are labeling 4 mutual funds as

sell and 6 funds as hold.

the number of ways = 10!/(6! * 4!) = 210

b. This is a selection problem, the number of ways in which we can select 3 mutual funds is given by

10C3 = 120

c. Here the order is important. So we will use permutation

No of ways = 10P3 = 720

Cfa examination quantitative study

Business

required rate of return

future period

term discount rate

cash outflow

essence discount rate

expected future cash

calculation of future

time period