2-1 CHAPTER 5 Time Value of Money Future value Present value Annuities Rates of return Amortization.

2-1

CHAPTER 5Time Value of Money

Future value Present value Annuities Rates of return Amortization

2-2

Last week

Objective of the firm Business forms Agency conflicts Capital budgeting decision and

capital structure decision

2-3

The plan of the lecture Time value of money concepts

present value (PV) discount rate/interest rate (r)

Formulae for calculating PV of perpetuity annuity

Interest compounding How to use a financial calculator

2-4

Financial choices with time

Which would you rather receive?

$1000 today $1040 in one year

Both payments have no risk, that is, there is 100% probability that you will

be paid

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Financial choices with time Why is it hard to compare ?

$1000 today $1040 in one year

This is not an “apples to apples” comparison. They have different units

$1000 today is different from $1000 in one year

Why? A cash flow is time-dated money

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Present value

To have an “apple to apple” comparison, we

convert future payments to the present values

or convert present payments to the future values

This is like converting money in Canadian $ to money in US $.

2-7

Some terms

Finding the present value of some future cash flows is called discounting.

Finding the future value of some current cash flows is called compounding.

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What is the future value (FV) of an initial $100 after 3 years, if i = 10%?

Finding the FV of a cash flow or series of cash flows is called compounding.

FV can be solved by using the arithmetic, financial calculator, and spreadsheet methods.

FV = ?

0 1 2 3

10%

100

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Solving for FV:The arithmetic method

After 1 year: FV1 = c ( 1 + i ) = $100 (1.10)

= $110.00 After 2 years:

FV2 = c (1+i)(1+i) = $100 (1.10)2

=$121.00 After 3 years:

FV3 = c ( 1 + i )3 = $100 (1.10)3

=$133.10 After n years (general case):

FVn = C ( 1 + i )n

2-10

Set up the Texas instrument

2nd, “FORMAT”, set “DEC=9”, ENTER 2nd, “FORMAT”, move “↓” several

times, make sure you see “AOS”, not “Chn”.

2nd, “P/Y”, set to “P/Y=1” 2nd, “BGN”, set to “END” P/Y=periods per year, END=cashflow happens end of periods

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Solving for FV:The calculator method

Solves the general FV equation. Requires 4 inputs into calculator, and

it will solve for the fifth.

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10 0

133.10

-100

2-12PV = ? 100

What is the present value (PV) of $100 received in 3 years, if i = 10%?

Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding).

The PV shows the value of cash flows in terms of today’s worth.

0 1 2 3

10%

2-13

Solving for PV:The arithmetic method

i: interest rate, or discount rate

PV = C / ( 1 + i )n

PV = C / ( 1 + i )3

= $100 / ( 1.10 )3

= $75.13

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Solving for PV:The calculator method

Exactly like solving for FV, except we have different input information and are solving for a different variable.

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10 0 100

-75.13

2-15

Solving for N:If your investment earns interest of 20% per year, how long before your investments double?

INPUTS

OUTPUT

N I/YR PMTPV FV

3.8

20 0 2-1

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Solving for i:What interest rate would cause $100 to grow to $125.97 in 3 years?

INPUTS

OUTPUT

N I/YR PMTPV FV

3

8

0 125.97-100

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Now let’s study some interesting patterns of cash flows…

Annuity Perpetuity

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ordinary annuity and annuity due

Ordinary Annuity

PMT PMTPMT

0 1 2 3i%

PMT PMT

0 1 2 3i%

PMT

Annuity Due

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Value an ordinary annuity

Here C is each cash payment n is number of payments If you’d like to know how to get the

formula below (not required), see me after class.

1 1

(1 )nPV C

i i i

2-20

Solving for FV:3-year ordinary annuity of $100 at 10%

$100 payments occur at the end of each period. Note that PV is set to 0 when you try to get FV.

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10 -100

331

0

2-21

Solving for PV:3-year ordinary annuity of $100 at 10%

$100 payments still occur at the end of each period. FV is now set to 0.

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10 100 0

-248.69

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Example

you win the $1million dollar lottery! but wait, you will actually get paid $50,000 per year for the next 20 years if the discount rate is a constant 7% and the first payment will be in one year, how much have you actually won?

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Solving for FV:3-year annuity due of $100 at 10%

$100 payments occur at the beginning of each period.

FVAdue= FVAord(1+i) = $331(1.10) = $364.10. Alternatively, set calculator to “BEGIN”

mode and solve for the FV of the annuity:

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10 -100

364.10

0BEGI

N

2-24

Solving for PV:3-year annuity due of $100 at 10%

$100 payments occur at the beginning of each period.

PVAdue= PVAord(1+I) = $248.69(1.10) = $273.55. Alternatively, set calculator to “BEGIN” mode

and solve for the PV of the annuity:

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10 100 0

-273.55

BEGIN

2-25

What is the present value of a 5-year $100 ordinary annuity at 10%?

Be sure your financial calculator is set back to END mode and solve for PV: N = 5, I/YR = 10, PMT = 100, FV = 0. PV = $379.08

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What if it were a 10-year annuity? A 25-year annuity? A perpetuity?

10-year annuity N = 10, I/YR = 10, PMT = 100, FV = 0;

solve for PV = $614.46. 25-year annuity

N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $907.70.

Perpetuity (N=infinite) PV = PMT / i = $100/0.1 = $1,000.

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What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%?

22.297$09.1

97.323$0

PV

0 1 2 3 4 5

$100 $100 $100 $100$323.97$297.22

1 1 2 3 4

$100 $100 $100 $100$323.97

(1.09) (1.09) (1.09) (1.09)PV

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What is the PV of this uneven cash flow stream?

0

100

1

300

2

300

310%

-50

4

90.91247.93225.39 -34.15530.08 = PV

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Solving for PV:Uneven cash flow stream

Input cash flows in the calculator’s “CF” register: CF0 = 0 CF1 = 100 CF2 = 300 CF3 = 300 CF4 = -50

Enter I/YR = 10, press NPV button to get NPV = $530.09. (Here NPV = PV.)

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Detailed steps (Texas Instrument calculator)

To clear historical data: CF, 2nd ,CE/C To get PV: CF , ↓,100 , Enter , ↓,↓ ,300 , Enter, ↓,2,

Enter, ↓, 50, +/-,Enter, ↓,NPV,10,Enter, ↓,CPT

“NPV=530.0867427”

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The Power of Compound Interest

A 20-year-old student wants to start saving for retirement. She plans to save $3 a day. Every day, she puts $3 in her drawer. At the end of the year, she invests the accumulated savings ($1,095=$3*365) in an online stock account. The stock account has an expected annual return of 12%.

How much money will she have when she is 65 years old?

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Solving for FV:Savings problem If she begins saving today, and

sticks to her plan, she will have $1,487,261.89 when she is 65.

INPUTS

OUTPUT

N I/YR PMTPV FV

45 12 -1095

1,487,262

0

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Solving for FV:Savings problem, if you wait until you are 40 years old to start If a 40-year-old investor begins saving

today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20.

Lesson: It pays to start saving early.

INPUTS

OUTPUT

N I/YR PMTPV FV

25 12 -1095

146,001

0

2-34

Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated i% constant? LARGER, as the more frequently

compounding occurs, interest is earned on interest more often.

Annually: FV3 = $100(1.10)3 = $133.10

0 1 2 310%

100 133.10

Semiannually: FV6 = $100(1.05)6 = $134.01

0 1 2 35%

4 5 6

134.01

1 2 30

100

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What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

2 33S

63S

123Q

0.10FV $100 ( 1 )

2

FV $100 (1.05) $134.01

FV $100 (1.025) $134.49

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Classifications of interest rates

1. Nominal rate (iNOM) – also called the APR, quoted rate, or stated rate. An annual rate that ignores compounding effects. Periods must also be given, e.g. 8% Quarterly.

2. Periodic rate (iPER) – amount of interest charged each period, e.g. monthly or quarterly. iPER = iNOM / m, where m is the number of

compounding periods per year. e.g., m = 12 for monthly compounding.

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Classifications of interest rates 3. Effective (or equivalent) annual

rate (EAR, also called EFF, APY) : the annual rate of interest actually being earned, taking into account compounding.

If the interest rate is compounded m times in a year, the effective annual interest rate is

1 1m

nomi

m

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Example, EAR for 10% semiannual investment

EAR= ( 1 + 0.10 / 2 )2 – 1 = 10.25%

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

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EAR on a Financial Calculator

keys: description:

[2nd] [ICONV] Opens interest rate conversion menu

[↓] [EFF=] [CPT] 10.25

Texas Instruments BAII Plus

[↓][NOM=] 10 [ENTER] Sets 10 APR.

[↑] [C/Y=] 2 [ENTER] Sets 2 payments per year

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Why is it important to consider effective rates of return? An investment with monthly

payments is different from one with quarterly payments.

Must use EAR for comparisons. If iNOM=10%, then EAR for different

compounding frequency:Annual 10.00%Quarterly 10.38%Monthly 10.47%Daily 10.52%

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If interest is compounded more than once a year

EAR (EFF, APY) will be greater than the nominal rate (APR).

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2-44

What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually?

Payments occur annually, but compounding occurs every 6 months.

Cannot use normal annuity valuation techniques.

0 1

100

2 35%

4 5

100 100

61 2 3

2-45

Method 1:Compound each cash flow

110.25121.55331.80

FV3 = $100(1.05)4 + $100(1.05)2 + $100

FV3 = $331.80

0 1

100

2 35%

4 5

100

61 2 3

100

2-46

Method 2:Financial calculator

Find the EAR and treat as an annuity.

EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%.

INPUTS

OUTPUT

N I/YR PMTPV FV

3 10.25 -100

331.80

0

2-47

When is periodic rate used?

iPER is often useful if cash flows occur several times in a year.