Time value of money

Corporate Finance

Time Value of Money

Session One

Decision Dilemma—Take a Lump Sum or Annual Installments

A suburban Chicago couple won the Lottery.

They had to choose between a single lump sum $104 million, or $198 million paid out over 25 years (or $7.92 million per year).

The winning couple opted for the lump

sum.

Did they make the right choice? What basis do we make such an economic comparison?

To make such comparisons (the lottery decision problem), we must be able to compare the value of money at different point in time.

Time Value of Money

“The idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity.”

Future Values and Compound Interest

Future Value – Amount to which an investment will grow after earning interest.

Compound Interest – Interest earned on interest.

Simple Interest – Interest earned only on original investment; no interest is earned on interest.

Future ValuesYou have $100 invested in a bank account. Suppose banks are currently paying an interest rate of 6 percent per year on deposits. So after a year, your account will earn interest of $6:

FV = PV (1 + r) �

FV = 100 (1+06)¹

Value of investment after 1 year = $106

Future ValuesExample - Simple Interest

Interest earned at a rate of 6% for five years on a principal balance of $100.

Today Future Years 1 2 3 4 5

Interest Earned

Value 100

6106

6112

6118

6124

6130

Value at the end of Year 5 = $130

Future Values

Example - Compound InterestInterest earned at a rate of 6% for five years on the previous year’s balance.

Interest Earned Per Year =Prior Year Balance x .06

Future Values

Example - Compound InterestInterest earned at a rate of 6% for five years on the previous year’s balance.

Today Future Years 1 2 3 4 5

Interest EarnedValue 100

6106

6.36112.36

6.74119.10

7.15126.25

7.57133.82

Value at the end of Year 5 = $133.82

Practice Problem: Warren Buffett’s Berkshire Hathaway

• Went public in 1965: $18 per share

• Worth today (August 22, 2003): $76,200

• Annual compound growth: 24.58%

• Current market value: $100.36 Billion

• If he lives till 100 (current age: 73 years as of 2003), his company’s total market value will be ?

Market Value

• Assume that the company’s stock will continue to appreciate at an annual rate of 24.58% for the next 27 years.

27$100.36(1 0.2458)

$37.902 trillions

F

Manhattan Island Sale

Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal? Suppose the interest rate is 8%.

Manhattan Island SalePeter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal?

trillion

FV

979.75$

)08.1(24$ 374

To answer, determine $24 is worth in the year 2003, compounded at 8%.

FYI - The value of Manhattan Island land is well below this figure.

Present Values

Present Value = PV

PV = Future Value after t periods

(1+r) t

Present ValuesExample

You just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?

572,2$2)08.1(3000 PV

PV of Multiple Cash FlowsExample

Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?

$15,133.06 PVTotal

36.429,3

70.703,3

8,000.00

2

1

)08.1(

000,42

)08.1(

000,41

payment Immediate

PV

PV

Present Values

Present Value

Year 0

4000/1.08

4000/1.082

Total

= $3,703.70

= $3,429.36

= $15,133.06

$4,000

$8,000

Year0 1 2

$ 4,000

$8,000

PV of Multiple Cash Flows

• PVs can be added together to evaluate multiple cash flows.

PV C

r

C

r

1

12

21 1( ) ( )....

Perpetuities & AnnuitiesPerpetuity A stream of level cash payments that

never ends.

Annuity Equally spaced level stream of cash

flows for a limited period of time.

PerpetuitiesPV of Perpetuity Formula

C = cash payment r = interest rate

PV Cr

PerpetuitiesExample - Perpetuity

In order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today if the rate of interest is 10%?

PV 100 00010 000 000,. $1, ,

PerpetuitiesExample - continued

If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

PV

1 000 000

1 10 3 315, ,

( . )$751,

AnnuitiesPV of Annuity Formula

C = cash payment r = interest rate t = Number of years cash payment is received

PV C r r r t

1 11( )

AnnuitiesPV Annuity Factor (PVAF) - The present value of

$1 a year for each of t years.

PVAF r r r t

1 11( )

AnnuitiesExample - Annuity

You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?

PV

PV

4 000

947 41

110

110 1 10 3,

$9, .

. . ( . )

AnnuitiesApplications• Value of payments• Implied interest rate for an annuity• Calculation of periodic payments – Mortgage payment– Annual income from an investment payout– Future Value of annual payments

FV C PVAF r t ( )1

AnnuitiesExample - Future Value of annual payments

You plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?

FV

FV

4 000 1 10

100

110

110 1 10

2020, ( . )

$229,

. . ( . )

Effective Interest Rates

Annual Percentage Rate - Interest rate that is annualized using simple interest.

Effective Annual Interest Rate - Interest rate that is annualized using compound interest.

Effective Interest Ratesexample

Given a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?

12.00%or .12=12 x .01=APR

12.68%or .1268=1-.01)+(1=EAR

r=1-.01)+(1=EAR12

12

InflationInflation - Rate at which prices as a whole are

increasing.

Nominal Interest Rate - Rate at which money invested grows.

Real Interest Rate - Rate at which the purchasing power of an investment increases.

Inflation1 real interest rate = 1+nominal interest rate

1+inflation rate

approximation formula

Real int. rate nominal int. rate - inflation rate

InflationExample

If the interest rate on one year govt. bonds is 6.0% and the inflation rate is 2.0%, what is the real interest rate?

4.0%or .04=.02-.06=ionApproximat

3.9%or .039 = rateinterest real

1.039 =rateinterest real1

=rateinterest real1 +.021+.061

The End