Download - Time value of money
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Corporate Finance
Time Value of Money
Session One
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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.
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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.”
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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.
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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
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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
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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
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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
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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 ?
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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
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Manhattan Island Sale
Peter Minuit bought Manhattan Island for $24 in 1629. Was this a good deal? Suppose the interest rate is 8%.
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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.
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Present Values
Present Value = PV
PV = Future Value after t periods
(1+r) t
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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
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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
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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
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PV of Multiple Cash Flows
• PVs can be added together to evaluate multiple cash flows.
PV C
r
C
r
1
12
21 1( ) ( )....
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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.
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PerpetuitiesPV of Perpetuity Formula
C = cash payment r = interest rate
PV Cr
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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, ,
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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,
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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( )
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AnnuitiesPV Annuity Factor (PVAF) - The present value of
$1 a year for each of t years.
PVAF r r r t
1 11( )
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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, .
. . ( . )
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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
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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,
. . ( . )
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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.
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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
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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.
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Inflation1 real interest rate = 1+nominal interest rate
1+inflation rate
approximation formula
Real int. rate nominal int. rate - inflation rate
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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
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The End