1

FUNDAMENTAL FUNDAMENTAL PRINCIPLES OF PRINCIPLES OF CORPORATE FINANCECORPORATE FINANCE

Topic #2

2

What is time value of money? (I)What is time value of money? (I) Most financial decisions involve costs and

benefits that are spread out over time Time value of money allows comparison of

cash flows from different periods Using TVM we can offer answers to

questions such as: “Would it be better for a company to invest $100,000 in a product that would return a total of $200,000 in one year, or one that would return $500,000 after two years?”

3

What is time value of money? (II)What is time value of money? (II) TVM conceptTVM concept: A dollar today is worth more : A dollar today is worth more

than a dollar tomorrow than a dollar tomorrow due to: due to: Time means means real interest ratereal interest rate and and inflationinflation Risk

PV Present Value, that is, the value today. FV Future Value, or the value at a future date. “t” the number of time periods between PV and FV “r” the discount rate

PVPV FVFV

0 1 2 3 t

4

Future values and compoundingFuture values and compounding You deposit $100 today at 10% interest. How

much will you have in 5 years?

you are interested in finding the FV for five years of the $100 today (=PV)

FV($100, 10%, 5y) = $100 (1.1)5 = $161.05

FVt,r = PV x (1+r)t

Future value factor (FVF)

5

Simple interest vs. compound Simple interest vs. compound interestinterest

Growth of $100 original amount at 10% per year

Darker shaded area represents the portion of the total that results from compounding of interest.

6

Simple versus compound Simple versus compound interestinterest

You have just won a $1 million jackpot in the lottery. You can buy a ten year certificate of deposit

which pays 6% compounded annually. Alternatively, you can give the $1 million to

your brother-in-law, who promises to pay you 6% simple interest annually over the ten year period.

Which alternative will provide you with more money at the end of ten years?

7

The power of compoundingThe power of compounding

8

Present values and discountingPresent values and discounting The process of obtaining the PV is the inverse

of the one which gives the FV Suppose you need $20,000 in three years to

pay your college tuition. If you can earn 8% on your money, how much do you need today?

PV = $20,000 1/(1.08)3 = $15,876.64

PVt,r = FV 1/(1+r)tDiscount factor or Present value factor (PVF)

9

The power of discountingThe power of discounting

10

The case of multiple cash flowsThe case of multiple cash flows FV of a set of cash flows is the sum of FVs

for the individual cash flows

11

The case of multiple cash flowsThe case of multiple cash flows PV of a set of cash flows is the sum of PVs

for the individual cash flows

12

A shortcut - perpetuitiesA shortcut - perpetuities PerpetuityPerpetuity = financial concept in which

a cash flow is received forever

ExampleExample: Suppose you receive $1,000 per year forever. Your opportunity rate is 6%. What is the value today of this set of cash flows?

rC

rflow Cashity)PV(perpetu

$16,666.660.06

$1,000ity)PV(perpetu

13

Growing perpetuitiesGrowing perpetuities Imagine an apartment building where cash

flows to the landlord after expenses will be $100,000 next year. These cash flows are expected to rise at 5% p.a. and the discount rate is 10%.

If this continues indefinitely growing growing perpetuityperpetuity

g-rC)perpetuity PV(growing

2,000,0000.050.1-

100,000)perpetuity PV(growing

14

Few points concerning growing Few points concerning growing perpetuitiesperpetuities

1. The numerator (C) is the cash flow one period hence, not at date 0

2. The discount rate (r) must be greater than the growth rate for the formula to work

3. Timing assumption formula assumes cash flows are received and disbursed at regular and discrete points in time

15

A shortcut - annuitiesA shortcut - annuities AnnuityAnnuity = a stream of regular payments that

lasts for a fixed number of periods The payments are assumed to be received at the

end of each period A good example of an annuity is a lottery, where

the winner is paid over a number of years

16

Calculating PV of an annuityCalculating PV of an annuity Suppose you need $5,000 each year for

the next three years to make your tuition payments. You need the first $5,000 in one year. You can place money in a savings account yielding 5% compounded annually. How much do you need to have in the account today?

Calculating PV of an annuityCalculating PV of an annuity

17

Calculating FV of an annuityCalculating FV of an annuity Assume you deposit $2,000 per year in

a savings account at 4% p.a., compounded annually, for 5 years. The first payment is made one year from now. How much money will you have in the account after 5 years?

18

Calculating FV of an annuityCalculating FV of an annuity

19

Annuity durationAnnuity duration Assuming that you owe $2,000 to your

bank, and the interest rate required is 2% per month. If you can make monthly payments of $50, how long will it take you to pay the debt.

20

Annuity durationAnnuity duration

21

Annuity valueAnnuity value You would like to have $3,000 in a

savings account 12 months from now. If this account pays an annual interest rate of 9%, compounded monthly. How much should you deposit monthly in order to reach your goal?

22

Annuity valueAnnuity value

23

Growing annuitiesGrowing annuities You have just been hired by a company

that offers an annual wage of $50,000. Your contract states that you annual wage will increase by 5% annually.

Suppose you intend to work for this company for 10 years, what is the current present value of your wage, considering 8% the appropriate annual discount rate?

24

Growing annuitiesGrowing annuities

25

Growing annuitiesGrowing annuities

26

Using AnnuitiesUsing Annuities Delayed annuities Special (advanced) annuities Infrequent annuities Equaling the PV of two annuities Loan amortization Valuing bonds

27

Delayed annuitiesDelayed annuities John will receive $500 annually for four

years, starting with the end of the sixth year. What is the current present value of these payments, at 10% annual discount rate?

28

Delayed annuitiesDelayed annuities

PV (annuity, today) = $984.13

0 1 2 3 10 5 6 7 8 9 4

$500$500 $500 $500

PV (annuity, 4y)= $1,584.33

Special annuitySpecial annuity Andy just won the lottery and is going

to receive $50,000 annually for 20 years. The receives the first payment today.

The lottery announced the winning of $1,000,000 ($50,000x20= $1.000.000).

If the annual interest rate Andy can get to his savings account is 8%, what is the actual value of his winnings?

30

Special annuitySpecial annuity The previous formulae of annuities (FV

and PV) assume the the first payment is one period from the present ANNUITY IN ARREARS

If the first payment is made today ADVANCED ANNUITY

31

Special annuitySpecial annuity

32

Andy’s lottery prize

Infrequent annuitiesInfrequent annuities Maria receives $500 every two years.

This annuity will be paid to her throughout 10 years. The first payment is made two years from now. What is the present value of these payments, considering 10% annual discount rate?

33

Infrequent annuitiesInfrequent annuities

PV($500, 21%, 5 periods) =$1,462.99

Two years disount rate = 1.1 1.1 - 1 = 21.0%

0 1 2 3 10 5 6 7 8 9 4

$500 $500 $500 $500 $500

10% 10% 10% 10% …………………………………………….

First determine the discount rate for two consecutive payments

Equaling the PV of two annuities Equaling the PV of two annuities Mike and Laura want to start saving for

their new born son education. They estimate spending $30,000 per year when their son begins university studies 18 years from now.

The annual interest rate is estimated at 14% for the next decades. What is the amount they should deposit each year for 17 years in order to be able to finance their son’s four years college education?

35

Equaling the PV of two annuities Equaling the PV of two annuities

36

0 1 2 21 17 18 19 20

Deposi

t 1

Last deposit

Spending $30,000

Deposi

t 2

PV today (PV17, 14%, 17 years) =9,422.91

PV17 of spendingsPV studies today

Annual deposit = 1,478.59

=

14%Spending $30,000

Spending $30,000

Spending $30,000

Loan amortizationLoan amortization Rosie borrowed $6,000, at 10% annual

interest rate, for four years. At the end of each of the following four years she will make equal payments so as the loan will be completely amortised in four years.

How much is Rosie going to pay each year?

37

Loan amortizationLoan amortization

38

Valuing bondsValuing bonds If today is Dec. 1st 2013, what is the price of the

following bond? IBM bond that pays an annual coupon rate of

11.5% (annual coupon $115) for 5 years. In 2018 the bond will also pay the nominal value $1,000.

Investors’ required return is currently at 7.5% annually.

39

PV annuity ($115, 5 years, 7.5%) PV (current price)

Valuing bondsValuing bonds Rephrasing: if the current price of the IBM bond

is $1,164.84, what is investors’ current return?

40

YIELD TO MATURITY (YTM)•Coupon rate > YTM premium bond •Coupon rate < YTM discount bond•Coupon rate = YTM bond at par

41

Compounding more frequently than Compounding more frequently than annuallyannually

Compounding more frequently than annually results in a higher effective interest rate because you are earning interest on interest more frequently

As a result, the effective interest rate is greater than then the nominal (annual) interest rate

The effective rate of interest will increase the more frequently interest is compounded

42

Compounding more frequently than Compounding more frequently than annuallyannually

What would be the difference in future value if you deposit $100 for 5 years and earn 12% annual interest compounded (a) annually, (b) semiannually, (c) quarterly, and (d) monthly?

Annually 100 x (1 + 0.12/1)51 = $176.23Semiannually 100 x (1 + 0.12/2)52 = $179.09Quarterly 100 x (1 + 0.12/4)54 = $180.61Monthly 100 x (1 + 0.12/12)512 = $181.67

43

Nominal and effective ratesNominal and effective rates The nominal interest ratenominal interest rate (sometimes denoted APR – from annual annual

percentage ratepercentage rate) is the stated or contractual rate of interest charged by a lender or promised by a borrower (r)

The effective annual rateeffective annual rate (EAR) (EAR) is the rate you actually pay or earn

EAR > APR whenever compounding occurs more than once per year

1mr1EAR

m

44

Nominal and effective ratesNominal and effective rates What is the EAR on your credit card if

the nominal rate is 18% p.a., compounded monthly?

Answer:EAR = (1 + 0.18/12)12 – 1 =

19.56% What if compounded quarterly? Answer:

EAR = (1+ 0.18/4)4 – 1 = 19.25%

45

Compounding periods, APRs and Compounding periods, APRs and EARsEARs

Compounding Compounding periodperiod

Number of Number of times times

compoundedcompounded

Effective Effective annual rate (%)annual rate (%)

Year 1 10.000000Quarter 4 10.381289Month 12 10.471307Week 52 10.506479Day 365 10.515578Hour 8,760 10.517029MinuteSecond

525,60031,536,000

10.51709110.517092

46

Continuous compoundingContinuous compounding The number of compounding periods per year

approaches infinity The FV equation becomes:

What is the future value of a $100 deposit after 5 years if interest of 12% is compounded continuously?

FV(5,12) = $100 e0.125 = $182.22

2.7183

FV(t,r) = PV ert