HOW TO CALCULATE PRESENT VALUES

Chapter

Brealey, Myers, and Allen Principles of Corporate Finance 11th Edition

HOW TO CALCULATE PRESENT VALUES

2

2-1 FUTURE VALUES AND PRESENT VALUES

•

Calculating Future Values

•

Future Value

• Amount to which investment will grow after earning interest

•

Present Value

• Value today of future cash flow

2-1 FUTURE VALUES AND PRESENT VALUES

• Future Value of $100 =

• Example: FV

• What is the future value of $100 if interest is compounded annually at a rate of 7% for two years?

r)

1 ( 100

$

FV   

49 . 114

$ )

07 . 1 ( ) 07 . 1 ( 100

$

FV    

FİGURE 2.1 FUTURE VALUES WİTH

COMPOUNDİNG

2-1 FUTURE VALUES AND PRESENT VALUES

factor

discount

= PV

PV

= lue

Present va

 C

2-1 FUTURE VALUES AND PRESENT VALUES

•

Discount factor = DF = PV of $1

• Discount factors can be used to compute present value of any cash flow

2-1 FUTURE VALUES AND PRESENT VALUES

• Given any variables in the equation, one can solve for the remaining variable

• Prior example can be reversed

100 49

. 114 PV

DF PV

)2

07 . 1 (

1

2 2













C

FİGURE 2.2 PRESENT VALUES WİTH

COMPOUNDİNG

2-1 FUTURE VALUES AND PRESENT VALUES

•

Valuing an Office Building

• Step 1: Forecast Cash Flows

• Cost of building = C₀ = $700,000

• Sale price in year 1 = C₁ = $800,000

• Step 2: Estimate Opportunity Cost of Capital

• If equally risky investments in the capital market offer a return of 7%, then cost of

2-1 FUTURE VALUES AND PRESENT VALUES

• Valuing an Office Building

• Step 3: Discount future cash flows

• Step 4: Go ahead if PV of payoff exceeds investment

664 ,

747

$ PV 





664 ,

47

$

000 700

$ 664

, 747

$ NPV





 ,

2-1 FUTURE VALUES AND PRESENT VALUES

r C C

 



= 1 NPV

investment required

PV

= NPV

1 0

•

Net Present Value

2-1 FUTURE VALUES AND PRESENT VALUES

• Risk and Present Value

• Higher risk projects require a higher rate of return

• Higher required rates of return cause lower PVs

664 ,

747 .07 $

1

$800,000 PV

7%

at

$800,000 of

PV ₁

 



 C

2-1 FUTURE VALUES AND PRESENT VALUES

•

Risk and Net Present Value

$47,664

$700,000 47,664

7

$

= NPV

investment required

PV

= NPV







NPV at 5%= $761,000 - $700,00 = $61,904 NPV at 12%= $714,286 – 4700,00 = $14,286

2-1 FUTURE VALUES AND PRESENT VALUES

• Net Present Value Rule

• Accept investments that have positive net present value

• Using the original example: Should one accept the project given a 12% opportunity cost?

286 ,

14 1.12 $

$800,000 +

$700,000

=

NPV  

2-1 FUTURE VALUES AND PRESENT VALUES

• Rate of Return Rule

• Accept investments that offer rates of return in excess of their opportunity cost of capital

• In the project listed below, the opportunity cost of capital is 12%. Is the project a wise

investment?

14.3%

or .143,

$700,000

0,000 70

$ 000

, 00

$8 investment

profit

Return  





FİNDİNG THE DİSCOUNT RATE

• Often we will want to know what the implied interest rate is in an investment

•

• FV = PV(1 + r)^t => FV / PV = (1 + r)^t

• r = (FV / PV)^1/t – 1

• Ex.: You are looking at an investment that will pay

$1.200 in 5 years if you invest $1.000 today.

What is the implied rate of interest?

• r = (1200 / 1000)^1/5 – 1 = 0.03714 = 3.714%

FİNDİNG THE NUMBER OF PERİODS

• Solve for t:

FV = PV(1 + r)^t FV / PV = (1 + r)^t

ln(FV / PV) = t ln(1 + r) t = ln(FV / PV) / ln(1 + r)

18

NUMBER OF PERİODS – EXAMPLE 1

• You want to purchase a new car and you

are willing to pay $20.000. If you can invest at 10% per year and you currently have

$15.000, how long will it be before you

have enough money to pay cash for the car - without additional savings?

t = ln(FV / PV) / ln(1 + r)

t = ln(20,000 / 15,000) / ln(1.1) =

2-1 FUTURE VALUES AND PRESENT VALUES

• Multiple Cash Flows

• Discounted Cash Flow (DCF) formula:

t t

r C r

C r

C

) 1

( )

1 ( )

1

0 (

....

PV 



 

FİGURE 2.5 NET PRESENT VALUES

2-2 PERPETUİTİES AND ANNUİTİES

•

Perpetuity

• Financial concept in which cash flow is theoretically received forever

r C

PV

rate discount

flow cash

flow cash

of PV





2-2 PERPETUİTİES AND ANNUİTİES

•

Perpetuity

•

PV

lue present va

flow cash

Return r  C



2-2 PERPETUİTİES AND ANNUİTİES

•

Present Value of Perpetuities

• What is the present value of $1 billion every year, for eternity, if the perpetual discount rate is 10%?

2-2 PERPETUİTİES AND ANNUİTİES

•

Present Value of Perpetuities

• What if the investment does not start making money for 3 years?

2-2 PERPETUİTİES AND ANNUİTİES

• Annuity

• Asset that pays fixed sum each year for specified number of years

Perpetuity (first payment in year 1)

Perpetuity (first payment in year t + 1)

Annuity from

Asset Year of Payment

1 2…..t t + 1

Present Value









C C 1

2-2 PERPETUİTİES AND ANNUİTİES

  ^ _

 





 





r r r

C 1

1 annuity 1

of

PV

2-2 PERPETUİTİES AND ANNUİTİES

• Example: Tiburon Autos offers payments of $5,000 per year, at the end of each year for 5 years. If interest rates are 7%, per year, what is the cost of the car?

2-2 PERPETUİTİES AND ANNUİTİES

• Annuity

• Example: The state lottery advertises a jackpot prize of $365 million, paid in 30 yearly

installments of $12.167 million, at the end of each year. Find the true value of the lottery prize if interest rates are 6%.

 

000 ,

500 ,

167

$

06 . 1 06 .

1 06

. 167 1

. 12 Value

Lottery ₃₀





 





 



 Value

2-2 PERPETUİTİES AND ANNUİTİES

• Suppose you take a 4-year car loan from bank. You

borrow $20,000 and will make monthly payments. Bank charges you 8% per year, compounded monthly.

• What is your monthly payment?

• What is monthly interest rate?

• 8%/12 = 0.667% per month or 0.08/12=0.00667

00923 .

0

37785 .

0 )

0067 .

1 ( 0067 .

0

1 )

0067 .

1 20000 (

) 1

(

1 )

1 20000 (

48 48





 



 



 





 









 

C C

r r

C r _T

T

2-2 PERPETUİTİES AND ANNUİTİES

•

Future Value of an Annuity

 

 

 



  



 r

C r

t

1

annuity 1 of

FV

2-2 PERPETUİTİES AND ANNUİTİES

• Future Value of an Annuity

• What is the future value of $20,000 paid at the end of each of the following 5 years, assuming investment returns of 8% per year?

 

332 ,

117

$

08 .

1 08

. 000 1

, 20 FV

5





 



  





2-3 GROWİNG PERPETUİTİES AND ANNUİTİES

• Constant Growth Perpetuity

• This formula can be used to value a perpetuity at any point in time

g r

C

 ¹

PV0 g = the annual growth rate of the cash flow

g r

C_t

t  ^1

PV

2-3 GROWİNG PERPETUİTİES AND ANNUİTİES

• Constant Growth Perpetuity

• What is the present value of $1 billion paid at the end of every year in perpetuity, assuming a rate of return of 10% and constant growth rate of 4%?

billion 667

. 16

$

04 . 10 . PV₀ 1



 

2-3 GROWİNG PERPETUİTİES AND ANNUİTİES

• Growing Annuities

• Golf club membership is $5,000 for 1 year, or

$12,750 for three years. Find the better deal given payment due at the end of the year and 6% expected annual price increase, discount rate 10%.

ANNUITY EXAMPLE:

• Jacqui wants to retire twenty years from

now. She wants to save enough so that she can have a pension of $10,000 a year for

ten years. How much must she save each year, denoted S furing the the first 20 years to achieve her goal if the interest rate is

5%?

• Assume all cash flows ocur at the end of the year. The first date at which S is saved

2-4 HOW INTEREST İS PAİD AND QUOTED

•

Effective Annual Interest Rate (EAR)

• Interest rate annualized using compound interest

•

Annual Percentage Rate (APR)

•

Also called: Quoted rate

• Interest rate annualized using simple interest

2-4 HOW INTEREST İS PAİD AND QUOTED

• Invest $1000 for 1 year.

• Bank pays 12% interest rate ‘compounded semiannually’.

• What is APR ?

• Semiannual interest rate= 6%

• APR = 2 x 6% = 12%

• What is EAR?

2-4 HOW INTEREST İS PAİD AND QUOTED

• EAR:

• Bank pays 12% int rate ‘compounded semiannually’.

• Semi-annual rate = 0.10/2 = 0.06.

• FV = $1000(1.06)(1.06)=$1,123.6

• Rate of return = (1123.6-1000) / 1000= 12.36 %

1236 .

0 1

6 . 123 .

1

6 . 123 .

2 1 12 . 1 0

1 )

1 (

2







 



 

 

 



 

 



 EAR

m EAR APR

m

2-4 HOW INTEREST İS PAİD AND OUTLAİD

• Given a monthly rate of 1%, what is the (EAR)? What is the (APR)?

12.00%

or .12,

= 12 .01

= APR

12.68%

or .1268,

= 1 .01)

+ (1

= EAR

= 1 .01)

+ (1

= EAR

12 12





 r

2-4 How Interest is Paid and Quoted

Quoted Rate %

Compoundin g Freq.,

(m)

Periodica l Interest Rate %,

Invest ($)

FV, after a year: Rate of Return

%

12 Annual, 1 12 1000 1000(1.12)=1120 12

12 Semiannual, 2

6 1000 1000(1.06)²= 1123.60

12.36

12 Quarterly, 4 3 1000 1000(1.03)⁴= 1125.50

12.55

12 Monthly, 12 1 1000 1000(1.01)¹²= 1126.83

12.68

12 Daily, 365 0,032876 1000 1000(1.00329)³⁶⁵= 1127.57

12.76

12  ?

AMORTİZATİON LOANS & SCHEDULES

• Amortized loans: Borrower repays the principal over the life of the loan (ex. Home mortgages, car loans, etc).

• Two methods:

• Fixed amount of principal to be repaid each period, that results in uneven payments.

• Fixed payments, results in uneven principal reduction.

• Tables showing how do you amortize your loan to a bank are called amortization schedules.

Amortization

Construct an amortization schedule

for a car loan of 20,000 TL, at 1.2% monthly rate, (APR= 1.2x12= 14.4%per year). Loan to be paid back in 3 years – monthly payments.

0 1.2% 1 t 36

-20,000 PMT ... PMT

DEVELOPİNG THE AMORTİZATİON SCHEDULE – FİXED PAYMENTS

1. We start by finding the required payments.

20,000 = PMT (1/0.012 – 1/0.012(1.012)³⁶) PMT = 20,000 / (83.33 – 54.24) = 687.44 TL

2. Find interest charge for month 1:

Int = Beginning balance * periodical int. rate Int = 20,000 * 0.012 = 240 TL

Developing the amortization schedule

3. Find repayment of principal in month 1:

Repmt = PMT – Interest

Repmt = 687.44 – 240 = 447.44 TL

4. Find ending balance after month 1:

Ending Balance = Beginning Balance – Repmt End. Balance = 20,000 – 447.44 = 19,552.56 TL Let us see the rest of it on excel.

AMORTİZED LOAN WİTH FİXED PRİNCİPAL PAYMENT - EXAMPLE

• Consider a $50,000, 10 year loan at 8% interest.

The loan agreement requires the firm to pay

$5,000 in principal each year plus interest for that year.

• Note that pmt period is annual in this example.

• Click on the Excel icon to see the amortization table