TIME VALUE OF MONEY

FACULTY OF BUSINESS AND ACCOUNTANCY

Week 5

Based on positive time preference~ a ringgit today is worth more than a ringgit expected in the future

TVM tools are used to; Calculate deposits required to accumulate a future sum

Amortize loans by calculating loan payments schedules

Determine interest or growth rates of money streams

Evaluate perpetuities

Find the required rate of return

Basic Concepts Time Lines

Future Values

Present Values

Perpetuities

Single Sum

Annuity

Nominal Rate Periodic Rate

Effective Annual Rate

Required Rate of Return

Compounding Periods

Amortization

Types of Interest Rates

TMV Solution Methods

1. Numerical – using regular calculator w/o financial functions

2. Interest Tables - given with the text book a. Present Value Interest Factor b. Present Value Interest Factor for Annuity c. Future Value Interest Factor d. Future Value Interest Factor for Annuity

3. Financial Calculator

4. Worksheet

Time Lines

Show the timing of cash flows. Tick marks occur at the end of periods, so

Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.

CF0 CF1 CF3CF2

Time 0 1 2 3i%

End of period 2 & beginning

of period 3

Time line illustrations

100 100100

0 1 2 3i%

0 1 25%

RM100 lump sum due in 2 years

End of period 2PV

3 year RM100 ordinary annuity (fixed pmts)

100 50 75

0 1 2 3 i%

Uneven cash flow stream

CF0 CF1 CF2 CF3

Key Description

Clear all data

No of payment per year

No of year

Annual interest rate

Present value

Future value

(No keyed in) x (P/YR)

Begin End

Calculates amortization table

BEG/END

Future and Present Values

Future Value & Present Value

Future Value - Ending amount of your account at the end of n periods

Present Value – Beginning amount in your account

In determining the final value of a cash flow or series of cash flows, compound interest will be applied.

What is compound interest? Is it the same thing as simple interest?

The process of going from today’s values, or PV to future value is called compounding.

Simple interest;

105 110

2nd period; (Principal) [(2 x 0.05) + 1 ] = RM110 or, [100 (1.05) - 100] + 100(1.05) = RM110

0 1 25%

Compounding interest;

100 105 110.25

2nd period; (Principal + interest)(1 + i) (RM100 + RM5) (1 + 0.05) = RM110.25

Future Value : One period caseCase 1: Given that a car dealer offers a car for

RM20,000 in cash or RM25,000 on credit for one year. Given an annual i.r. of 10%, which payment is better for you & which for the car dealer?

0 1i% = ?

20,000 25,000

Given RM20,000 now, in 1 year at ir of 10%, the money deposited will be ?

20,000

0 110%

FV = ?

FV1 = PV + INT = PV + PV (i)

= PV ( 1 + i )

Numerical Solution (N/S) ;

FV1 = PV ( 1 + i ) = 20,000 ( 1 + 0.1) = RM22,000

Future Value Interest Factor for i & n (FVIFi,n)~ the future value of RM1 left on deposit for n periods at a rate of i percent per period

~ where ( 1 + i)n = FVIFi,n

Tabular Solution (T/B) ;

FV1 = PV (FVIFi,n) = 20,000 ( 1.1000) = RM22,000

Find the FV of RM20,000 given an interest rate of 10% in one year.

Data Key

Clear all data

1 No of payment per year

1 No of year

10 Annual interest rate

20,000 Present value

22,000.00

Present Value : One period case

FV1 = PV ( 1 + i ) , so PV = FV1

(1 + i)

C2 : Given the annual ir of 10%, at what amount of cash would the car dealer be indifferent to receiving RM25,000 at time 1?

PV = ?

25,000

nn,i )i1(1)PVIF(

Numerical solution ;

PV = FV1 / (1 + i) = 25,000 = RM22,727.27 1 + 0.01

Tabular Solution ;Present Value Interest Factor for i & n (PVIFi,n)~ the present value of RM1 due n periods in the future discounted at i percent per period

PV = FV (PVIFi,n) = 25,000 (0.9091) = RM 22,727.50

~ where;

Find the PV of RM25,000 given an interest rate of 10% in one year.

Data KeyClear all data

1 Payment per year

10 Annual interest rate

25,000

-22727.27273

Future Value & Present Value : Multi – period case

Important terms;Compound interest – interest earned on the principal & on the accumulated interest

Discount interest rate – the rate that will make the future value equivalent to the present value

Fair (Equilibrium) Value – the price at which investors are indifferent btw buying or selling a security

discounting

compounding

The discount rate is often also referred to as the opportunity cost, the required return, and the cost of capital.

C3 : Find the FV o RM100 left for 3 years in an account paying 10 percent, annual compounding;

FV = ?

0 1 2 310%

FV1 = PV + INT = PV + PV (i)

= PV ( 1 + i )

FV2 = FV1 (1 + i) = PV ( 1 + i ) (1 + i) = PV (1 + i)2

FV3 = PV (1 + i)3

FV3 = 100 (1.10)3 = 133.10

Data Key

133.10

FVn = PV (1 + i)n = PV (FVIFi,n) = 100 (1.10)3 = 100 (1.3310) = RM133.10

C4 : Find the PV of RM100 to be received in 3 years if the appropriate ir is 10 percent, compounded annually:

0 1 2 310%

PV = ?

PVn = FV (1 + i)n

PV3 = FV (1 + i)3

PVn = FVn/ (1 + i)n = FVn 1 n = FVn(PVIFi,n) 1 + i

= 100 (1/1.10)3 = 100 (0.7513) = RM75.13

Data Key

-75.13

n (periods) and

i (interest rate)

Solving for n in TVM problems

C5: How long will it take a firm’s sales to double, if sales are growing at a 15% rate?

0 15% n = ?n - 1

RM1 RM2

FVn = PV (1 + i)n

2 = 1 (1.15)n

2 = (1.15)n

FVn = PV (FVIFi,n)(FVIFi,n) = FV / PV = 2 / 1 = 2

Look in FVIF Table for (FVIF15%,n) = 2 n 5 periods

0 15% n = ?n - 1

RM1 RM2

Financial Calculator Solution ;

Data Key

Solving for interest rateC6: What annual ir will cause RM100 to grow to RM125.97 in 3

years?

125.97

0 1 2 3

100 (1 + i) 100 (1 + i)2 100 (1 + i)3

100 (1 + i)3 = 125.97100 (FVIFi,3) = 125.97 FVIFi,3 = 1.2597

Look at Row 3 of FVIF Table.1.2597 is in the 8% column

Annuities & Perpetuities

Annuities

An annuity is a series of equal payments made at fixed intervals for a specific number of periods

Ordinary annuity - payments occur at the end of each period - eg. Students loan

Annuity due – payments are made at the beginning of each period - eg. Mthly rentals, insurance premiums

What is the difference between an ordinary annuity and an annuity due? Both are 3-yr annuities ( 3 pmts)

Ordinary Annuity

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

Annuity Due

Future Value of an Ordinary Annuity

C7: What is the future value of an ordinary annuity of RM100 per period for 3 yrs if the ir is 10 percent, compounded annually?

Time line approach;

0 1 2 310%100 100 100

331100(1 + i)2

100(1 + i) +

Twice compounding

N/S; FVA3 = PMT (1 + i) + PMT (1 + i)1 + PMT (1 + i)2

= 100 (1) + 100 (1.10) + 100 (1.21) = RM331

T/S; FVAn = PMT (FVIFAi,n)FVA3 = 100 (FVIFA10%,3) = 100 (3.3100) = RM331

= 331.0010 3-100

Make sure no BGN sign

PMT I/YR N FVP/YR

Future Value of an Annuity DueC8: What is the future value of RM100 payments made at

beginning of each year for 3 yrs in a saving account that pays 10 percent, compounded annually?

Time line approach;

0 1 2 310%

100 100 0

133.10100(1 + i)2

100(1 + i) +

Triple compounding

100 (1 + i)3364.10

N/S; FVAD3 = PMT (1 + i)3 + PMT (1 + i)2 + PMT (1 + i)1 = 100 (1.331) + 100 (1.21) + 100 (1.10) = RM364.10

T/S; FVADn = FVA3 (1 + i) or = PMT (FVIFA10%,3) (1 + i) = 331 (1.10) = 100 (3.3100) (1.10) = RM364.10 = 364.10

= 364.1010 3-100

Make sure BGN sign

P/YR PMT I/YR N

BEG/END

Present Value of an Ordinary AnnuityC9: What is the PV of an annuity of RM100 per period for 3

years if the ir is 10 percent annually?

Time line approach;

0 1 2 310%

100 100

100 /(1 + i)2

100 / (1 + i)

Triple discounting

100 / (1 + i)3

75.13248.68

Present Value of an Annuity DueC10: How much lump sum today to make it equivalent with a 3

year annuity paying RM100 at beginning of each year?

Time line approach;0 1 2 310%

100 100

100 /(1 + i)2

100 / (1 + i)Double

discounting

273.55

Make sure BGN sign

An annuity due will always be greater than an otherwise equivalent ordinary annuity because interest will compound for an additional period.

Perpetuities- is a stream of equal payments expected to

continue forever- a type of annuity

PV (Perpetuity) = payment = PMT interest rate i

the current price

C11: A perpetual bond promised to pay RM100 per year in perpetuity. What would the bond’s worth today if the opportunity cost, or discount rate was 5 percent

PV (Perpetuity) = RM100 = RM2000 0.05

As the interest rate increases, the perpetuity’s value will drop.

When ir = 10%;

PVp = 100 = RM1000 0.1

Uneven Cash Flow Stream

Payment (PMT) - equal cash flows at regular intervals

Cash flow (CF) - uneven cash flows

Examples of uneven cash flows;

- common stock’s dividend - returns from fixed asset investments ~ production income ~ rentals

Present Value of an Uneven Cash Flow Stream

C12: Find PV of the following cash flows stream, discounted at 10%

0 1 2 3 410% 0 100 300 300 -50

PV = CF0 1 0 + CF1 1 1 + CF2 1 2

1 + i 1 + i 1 + i

+ CF3 1 3 + CF4 1 4 1 + i 1 + i

CF0 CF1 CF2 CF3 CF4

For cash flow calculation ;

Clear all

No of periods per year

Cash flow j

No of consecutive times CFj occurs

Internal rate of return per year

Net present value

IRR/YR

530.09

+/-CFj CFj

0 1 2 3 4 510% 0 100 50 200 200 200

C13: Find the PV of the following c/f discounted at 10%

497.38 200 (PVIFA10%,3)= 200 (2.4869)

100(1/1 + i)1

50(1/1 + i)2

497.38(1/1 + i)2 or

200 (PVIFA10%,3) or 200 (1/1 + i)3 200 (PVIFA10%,4) or 200 (1/1 + i)4

200 (PVIFA10%,5) or 200 (1/1 + i)5

411.03543.26150.26

136.60

124.18

Future Value of an Uneven Cash Flow StreamC14: Find FV of the following cash flows stream,compounded at 10%

0 1 2 3 4 510%

0 100 50 200 200 200

FV = CF5 (1 + i)0 + CF4 (1 + i)1 + CF3 (1 + i)2

+ CF2 (1 + i)3 + CF1 (1 + i)

4 + CF0 (1 + i)5

CF0 CF1 CF2 CF3 CF4 CF5

0 1 2 3 4 510%

0 100 50 200 200 200

420 ( 1 + i)200 (FVIFA10%,2) = 200 (2.1)

200 (1 + i)2

200 (1 + i)

50 (1 + i)3 = 50 (1.331)

100 (1 + i)4 = 100 (1.4641)

146.41NFV = 874.96

0 1 2 3 4 510%

0 100 50 200 200 200

Data Key

CF0 CF1 CF2 CF3 CF4 CF5 NFV=?

Data Key

543.28

847.93

Using formula; NFV = NPV ( 1 + i)n = 543.26 (1 + 0.1)5

= 874.93

Different Compounding Periods

- annually / semiannually / quarterly / monthly / daily compounding

- the quoted interest rate is normally the annual one.

- If bank promised 10% annual interest rate semiannually

what does it means ? ~ interest will be added each 6 months but will the interest be 10% as quoted or more/less?

Say that you want to deposit your money in the bank which offer you the highest return. As you shopped around, you come up with these rates:

Bank A : 15 percent, compounded daily

Bank B : 15.5 percent, compounded quarterly

Bank C : 16 percent, compounded annually

Which one has the best rates for deposits?

C15: A bank declares that it pays a 6% annual ir semiannually & you want to deposit RM 100. What is FV at the end of 3rd year?

P/YR 1

PV (-)100 n 3

FV 119.1

P/YR 1 2PV (-)100 (-)10

N 6 6 I% 6/2 6FV 119.4

1119.41

0 1 2 36%Annual compounding

Semiannual compounding0 1 2 3 4 5 6i%

FV3 = PV (1 + i)3

= 100 (1 + 0.06)3 = RM119.10

-100 FV = ?

-100 FV = ?FV6 = PV (1 + i)6

= 100 (1 + 0.03)6

= RM119.41

i% = 6% /2 = 3%

n = 3 x 2 = 6

Different compounding periods are used for different types of investment

In order to compare securities with different compounding periods, need to put them on a common basis.

Types of interest rates; Nominal or quoted interest rates

Annual percentage rates (APR)

Effective annual rates (EAR)

1. Nominal or Quoted interest rates , inom

Is the contracted, or stated, or declared ir.

The rate which is given by the bank or issuer.

Annual Percentage Rate (APR)The interest rate charged per period multiplied by the number of periods per year.

C16: If a bank is charging 1.2% per month on car loans, what is the APR?

APR = 1.2% x 12 = 14.4%

It is the nominal rates for loan that some government requires the bank to display to customers.

Periodic rate is the nominal rate at each period; where m is the no of compounding periods per year

Eg: 6% compounded quarterly. periodic rate, iper = inom / m = 6% / 4 = 1.5%

Periodic Rate

mii nom

So periodic rate is the rate charged by a lender or paid by a borrower in each period.

C17: A bank charges 18% annual interest rates monthly on credit card loans, what is the periodic rate?

iper = inom / m = 18% / 12 = 1.5%

Bank will charge 1.5% of interest monthly or per month

So if we delayed paying our credit card debt for a year, will the debt be the same as we take a loan of the same amount at 18% annual interest?

Notice that iper is the rate that is shown on time lines and used in certain calculations, not the annual rate. C18: How much would you have at the end of the 2nd year

when you make RM100 deposit in an account that pays 12% interest rate semiannually.

0 1/2 1 1/2 2 (x2) = N 6%

-100 FV = ?

For calculation of FV given only PV, must use this rate not the annual rate given in this case. AND maintain P/YR = 1.

2. Effective Annual Rates (EAR) or (EFF)The rate which would produce the same ending (future) value if annual compounding has been used. ~ (the interest rate expressed as if it were compounded once per year)

0 1 2 36%Annual compounding

Semiannual compounding0 1 2 3 4 5 63%

-100 119.41

-100 119.10

0 3i = ?-100 119.41

EAR ;The annual rate that produces the same FV as if we had compounded at a given periodic rate m times per year

An investor would be indifferent between an investment offering a 10.25% annual return and one offering a 10% annual return, compounded semiannually.

C19: EFF% or EAR for 10% semiannual investment.

10.25%

120.101

1mi1 EAR or EFF%

C20: What is the FV of RM100 compounded semiannually for 3 years if inom = 10%? Would it be different if it were compounded quarterly?

Quarterly compounding

Semiannual compounding0 1 2 3 5

-100 FV = ?

0 1 2 3

-100 FV = ?

134.01

134.49

FVn = PV 1 + inom mn

m FV3 = 100 1 + 0.1 2(3)

2 = 100(1.05)6

= 134.01

EAR = ( 1 + inom / m )m - 1 = ( 1 + 0.10 / 2 )2 – 1

= 10.25%

Semiannual compounding

Quarterly compoundingFVn = PV 1 + inom mn

m FV3 = 100 1 + 0.1 4(3)

4 = 100(1.025)12

= 134.49

EAR = ( 1 + inom / m )m - 1 = ( 1 + 0.10 / 4 )4 – 1

= 10.38%

F/C ;Given annual i.r. of 10%, compounded semiannually;

To find APR, given the EAR of 10.25%;

The APR formula;

APR = 1 + EFF 1/m - 1 x m x 100 100

NOM%102 P/YR

EFF% EARNOM%

EFF%10.252

Compounding More Frequently than Annually

• Compounding more frequently than once a year results in a higher effective interest rate because you are earning on interest on interest more frequently.

• As a result, the effective interest rate is greater than the nominal (annual) interest rate.

• Furthermore, the effective rate of interest will increase the more frequently interest is compounded.

Nominal & Effective Rates The nominal interest rate is the stated or contractual

rate of interest charged by a lender or promised by a borrower.

The effective interest rate is the rate actually paid or earned.

The effective rate > nominal rate whenever compounding occurs more than once per yearEAR > inom

EAR = inom = iper

If compounding occurs only once a year, then;

Nominal & Effective Rates

C21: What is the effective rate of interest on your credit card if the nominal rate is 18% per year, compounded monthly?

EAR = (1 + .18/12) 12 -1EAR = 19.56%

Why is it important to consider effective rates of return?

An investment with monthly payments is different from one with quarterly payments. Must put each return on an EFF% basis to compare rates of return. Must use EFF% for comparisons. See the following values of EFF% rates at various compounding levels.

EARANNUAL 10.00% EARQUARTERLY 10.38% EARMONTHLY 10.47% EARDAILY (365)

10.52%

Fractional Time Periods

Before, payments only occur at beginning or end of periods.

What if, payments occur at some date within a period?

0 1st 2nd month 20th day month

1%-100 FV = ?

C22: Deposits RM100 in a bank that pays 12% ir compounded monthly. How much the amount will be then in 1 month and 20 days?

N/S; FVn = PV (1 + i)n

= PV (1 + i)1+ 20/30

= 100(1 + 0.1)1.67

= RM101.68

N 20÷30+1

I/YR 1PV (-)100P/YR 1FV 101.67

If use 360-day year;iper = 0.12 /360 = 0.00033333 per day

No of days deposited; = 50 days

FVn = PV (1 + iper)n

= 100 ( 1.0003333)50

= RM 101.68

Note of Caution;

Rule 1: Money saved/deposited/invested should be in negative sign. Money withdrawn/received should be in positive sign.

C23: RM1000 is deposited today for a semiannual payment of RM300 for 3 years. Given an interest rate of 10% semiannually, how much would be left in the account in 3 years time?

0 1 2 3 5%

-1000 300 300 300 300 300 300 FV = ?

Rule 2 : If there is only PV & no PMT, either;a. If use periodic ir, keep P/YR = 1. b. If use nominal rate, change P/YR

accordingly. C24: RM1000 is deposited today. Given an interest rate of 10%

semiannually, how much would be in the account in 3 years time?0 1 2 3 5

-100 FV = ? N 3x2 = 6I/YR 10÷2 = 5PV (-)100PMT 0P/YR 1FV 134.01

N 3x2 = 6I/YR 10PV (-)100PMT 0P/YR 2FV 134.01

Rule 3: For case with PMT or PMT and PV, N = no of payment made, I/YR = annual interest rate, P/YR = no of payments made per year.

0 1 2 3 8%

-100 -150 -150 -150 -150 -150 -150 FV = ?

C25: Interest is 8% compounded quarterly. Initial deposit is RM100, and regular payments of RM150 will be made every semiannually.

2 P/YR3x2 =

8.08 I/YR(-)100 PV(-)150 PMT1122.7

8 NOM%4 P/YR

8.24 I/YR8.24 EFF%

2 P/YR8.08 NOM%

C26: Someone offers to sell you a note calling for the pmt of RM1000, 15 months from today for RM850. You have RM850 in the bank, which pays a 7% nominal rate with daily compounding. Should you buy the note or leave your money in the bank.

An Example of Everything

0 456 days-850 1,000

iper = 7%/365 = 0.0192%

How to solve this? Have to compare both investments on similar grounds;Fvnote vs. FVbank PVnote vs. PVbank EARnote vs. EARbank

Fvnote vs. Fvbank

Bank : FV = 850 (1.000192)456 = 927.67Note : FV = 1,000 Buy note (more value in future)

PVnote vs. Pvbank

Bank : PV = RM850Note : PV = 1,000 (1.000192)-456 = 916.27 Buy note (more value now) EARnote vs. EARbank

Bank : iper = 0.0192%Note : 1,000 = 850 (1 + i)456, solving i = 0.0356% Buy note (higher iper means higher EAR)

C27: Cost of note = RM850 PMT = RM190 quarterly for 5 quarters inom = 7% compounded daily Is this a good investment?

0 91 182 274 366 456 days

-850 190 190 190 190 190

iper for bank = 7% / 365 = 0.0192%

PVAnote = 190 (1.000192)-91 + 190 (1.000192)-182 + … + 190 (1.000192)-456

= 901.68PVApocket = 850 Buy note (more value now)

EARnote ; finding iper;

inom = (iper) (m) = (3.83) (4) = 15.3%

So for daily rate = inom / 365 = 0.0419% Buy note coz iper,note > iper, bank = 0.0192%

-850 CFj

190 CFj

IRR 3.82586

Quarterly iper

Loan Types

1. Pure Discount Loans - the borrower receives money today & repays a single sum at some time in the future - eg. A 1-year, 10% RM100 pure discount loan, would require the borrower to repay RM110 in one year.

2. Interest-Only Loans - a loan that has a repayment plan that calls for the borrower to pay interest each period & repay the entire principal (original loan amount) at some point in the future - eg. With a 3-year, 10%, interest-only loan of RM1000, the borrower would pay RM1000(0.1) = 100 in interest at the end of 1st & 2nd years. At the end of 3rd year, the borrower would return the principal along with RM100 in interest for that year.

3. Amortized Loans- a loan that is repaid in equal payments over its life.- eg. Car & home loans

C28: Say, borrow RM1,000 at 10% interest and have to pay equally at the end of each of the next 3 years.

0 1 2 3-1,000 PMT PMT PMT

T/S ;PVAn = PMT (PVIFAi,n)1,000 = PMT(PVIFA10%,3) PMT = 1,000 / (2.4869) = RM402.11

F/C ; 3 N 10 I/YR-1000 PV 0 FV PMT 402.1

Constructing an amortization table:Repeat steps 1 – 4 until end of loan

Interest paid declines with each payment as the balance declines.

Year Beginning Balance

PMT(2)

Interest(3)

PrincipRepmt

EndBalance

1 RM1,000 RM402 RM100 RM302 RM6982 698 402 70 332 3663 366 402 37 366 0Total 1,206.34 206.34 1,000 -

PVn(1 + i)(2) – (3) (1) – (4)

1 P/YR 3 N 10 I/YR-1000 PV 0 FV PMT 402.1

1 INPUT AMORT 1-1 = 302.11

PRIN= 100.00 INT

= -697.89 BAL

1 INPUT 2 AMORT 1-2 = 634.43

PRIN= 169.79 INT

= -365.57 BAL

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