Present value, annuity, perpetuity

Financial Economics2012 höst

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How to Calculate Present Values

LEARNING OBJECTIVES • Time value of money• how to calculate present value of future cash flows.• To calculate the present value of perpetuities, growing

perpetuities, annuities and growing annuities.• Compound interest and simple interest• Nominal and effective interest rates.• To understand value additive property and the concept of

arbitrage.• the net present value rule and the rate of return rule.

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Time value of money

• The value of 1 € today is not the same as 1 € in a year’s time

• Suppose the interest rate on savings is 2% per year. • At the end of year 1: 1 € × (1+0,02)=1,02• At the end of year 2: 1,02€ × (1+0,02) =1,0404 year 3: 1,0404 × (1+0,02)=1,0612 i.e. FV = 1€× (1 + 0,02)3

year FV

1 1

2 1,02

3 1,0404

4 1,061208

5 1,082432

6 1,104081

7 1,126162

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Topics Covered

• Future Values and Compound Interest• Present Values• Multiple Cash Flows• Level Cash Flows Perpetuities and Annuities• Effective Annual Interest Rates• Inflation & Time Value

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Calculating future value of 1$

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 370

0.5

1

1.5

2

2.5

Value of 1 $ investment in t year’s timecompound interest rate at 2%

FV

Year

Note that the 1 $ doubled in about 37 year’s time, given interest rate 2%.

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future value and present value

• FV = PV× (1 + r)t

where FV = Future value PV = Present value r = interest rate t = number of years (Periods)

It is readily seen, PV = FV/ (1 + r)t

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Manhattan Island SaleThe Power of Compounding!

Peter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal?

trillionFV

63.140$)08.1(24$ 382

To answer, determine $24 is worth in the year 2008, compounded at 8%.

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

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Future Values

Compound Interest - Interest earned on interest.

Simple Interest - Interest earned only on the original investment.

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

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Future ValuesExample - Simple Interest

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

Interest Earned Per Year = 100 × .06 = $ 6

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Example - Simple InterestInterest earned at a rate of 6% for five years on a principal balance of $100.

Today Future Years 1 2 3 4 5

Interest EarnedValue 100

Future Values

6106

6112

6118

6124

6130

Value at the end of Year 5 = $130

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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

Future Values

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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Future Values

Future Value of $100 = FV

FV r t $100 ( )1

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Future Values

FV r t $100 ( )1

Example - FV

What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?

82.133$)06.1(100$ 5 FV

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0

200

400

600

800

1000

1200

1400

1600

1800

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Number of Years

FV o

f $10

0

0%5%10%15%

Future Values with Compounding

Interest Rates

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Present ValuesPresent Value

Value today of a future cash

flow.

Discount Rate

Interest rate used to compute

present values of future cash flows.

Discount Factor

Equals to Present value of

a $1 future payment.

DFr t

1

1( )r

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Present Values

Present Value = PV

PV = Future Value after t periods (1+r) t

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Present ValuesExample 5.2

You just bought a new computer for $3,000. The payment due in 2 years. That means you need to pay 3000$ after 2 years. 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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Present Values

Discount Factor = DF Note that discount factor is the same as PV of $1 in t year

´s time

• Discount Factors can be used to compute the present value of any cash flow.

• PV=DF*FV

DFr t

1

1( )

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• The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable.

PV FVr t

1

1( )

Time Value of Money(applications)

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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

70.703,38,000.00

2

1

)08.1(000,4

2

)08.1(000,4

1

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 Cr

Cr

11

221 1( ) ( )

....

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Perpetuities & Annuities

Perpetuity 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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Perpetuities & Annuities

PV of Perpetuity Formula

C = cash payment r = interest rate

PV Cr

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Perpetuities & Annuities

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

PV 100 00010 000 000,. $1, ,

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Perpetuities & Annuities

Example - continuedIf 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 0001 10 3 315, ,

( . )$751,

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Perpetuities & Annuities

PV 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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Perpetuities & Annuities

PV 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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Perpetuities & Annuities

Example - AnnuityYou 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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Perpetuities & Annuities

Applications• 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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Perpetuities & Annuities

Example - Future Value of annual paymentsYou 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 Rates

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

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Effective Interest Rates

exampleGiven 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

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Inflation

Inflation - 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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InflationA

nnua

l Inf

latio

n, %

Annual U.S. Inflation Rates from 1900 - 2007

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Inflation

1 real interest rate = 1+nominal interest rate1+inflation rate

approximation formula

rateinflation -rateinterest nominalrateinterest Real

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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

Savings

Bond

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Inflation

• Remember: Current dollar cash flows must be discounted by the nominal interest rate; real cash flows must be discounted by the real interest rate.