Chapter 2: Valuation of Stocks and Bonds

2.1 Time Value of Money

2

A Rupee today is more worthy than a Rupee a year hence. Why ?

• Individuals, in general prefer current consumption to future consumption

• Reinvestment opportunity with rate of return ‘r’.

• In an inflationary period, a rupee today represents a greater real purchasing power than a rupee year hence.

3

Importance:

• Valuing securities• Analyzing investment projects• Determining lease rentals• Choosing right financing instruments• Setting up loan amortization schedules• Valuing companies• Setting up sinking fund etc….

4

Time lines and notations

• Difference between period of time and point in time.

• End of the year cash flow vs Beginning of the year cash flow

• Positive cash flow = cash inflow• Negative cash flow = cash outflow

5

3.1.1 Future value and compounding

• The phenomenon whereby the principle along with interest are reinvested is called compounding.

nn,r

nn,rn

)r1(FVIF

FactorInterest Value Future FVIF

periods ofnumber n

rate discountr

,where

)r1(PVFVIFPVFV

6

Example:

• Suppose you deposit Rs 1000 today in a bank that pays 10 % interest compounded annually, how much will the deposit grow after 8 years ?

Ans: Rs 2,144

7

Variation of FVIF with n and r

• Higher the interest rate, faster the growth rate.• Higher the period, higher the FVIF

8

Compound and Simple Interest

rnPVPVrn1PVFV

Simple Interest:

9

Power of compounding:

“ I don’t know what the seven wonders of the world are, but I know the eighth – THE COMPOUND INTEREST”

- Albert Einstein

10

Doubling period

How long would it take to double the amount at a given rate of interest ?

RateInterest

72 Period Doubling

:72 of Rule

RateInterest

690.35 Period Doubling

:rule accurate More

11

Finding the growth rate

ABC ltd had revenues of $ 100 million in 1990 which increased to Rs 1000 million in the year 2000. What was the compound growth rate in revenues ?

Ans: g = 26 %

Future Value of streams of cash flow

n1n

2n

1n C.......)r1(C)r1(CFV

13

3.1.2 Present Value and Discounting

nnr

nnnrn

rPVIF

discountr

where

rFVPVIFFVPV

)1(

1

FactorInterest ValuePresent PVIF

periods ofnumber n

rate

,

)1(

1

,

,

14

Example:

What is the present value of $ 1,000 receivable 20 years hence if the discount rate is 8 % ?

Ans: $ 214

15

Variation of PVIF with r and n

• The PVIF declines as the interest rate rises and as the length of time increases.

16

Present Value of uneven series of cash flow

year t of end at the occuring flow cashC

,where

)r1(

C

)r1(

C........

)r1(

C

)r1(

CPV

t

n

1tt

t

nn

221

n

17

3.1.3 Future Value of an Annuity

• An annuity is a stream of constant cash flow occurring at the regular intervals of time.

• When the cash flows occur at the end of each period, the annuity is called an ordinary annuity or a deferred annuity.

• When the cash flows occur at the beginning of each period, the annuity is called an annuity due.

18

Formula

r

1r)(1

Annuityfor Factor Interest Value FutureFVIFA

flowcash annuityA

,wherer

1r)(1AFVIFAAFVA

n

nr,

n

nr,

19

Applications

1. Knowing what lies in store for you.

Suppose you have decided to deposit Rs 30,000 per year in your PPF Account for 30 years. What will be accumulated amount in your PPF Account at the end of 30 years if the interest rate is 11 % ?

Ans: Rs 5,970,600

20

Applications (contd…)

2. How much should you save annually ?

You want to buy a house after 5 years when it is expected to cost Rs 2 million. How much should you save annually if your savings earn a compound return of 12 % ?

Ans: Rs 314,812

21

Applications (contd…)

3. Annual Deposit in Sinking Fund

ABC ltd has an obligation to redeem Rs 500 million bonds 6 years hence. How much should the company deposit annually in a sinking fund account wherein it earns 14 % interest ?

Ans: Rs 58.575 million

22

Applications (contd…)

4. Finding the Interest Rate

A finance company advertises that it will pay a lump sum of Rs 8,000 at the end of 6 years to investors who deposit annually Rs 1,000 for 6 years. What interest rate is implicit in this offer ?

Ans: 8.115 %

23

Applications (contd…)5. How long should you wait ?

You want to take up a trip to the moon which costs Rs 1 million – the cost is expected to remain unchanged in nominal terms. You can save annually Rs 50,000 to fulfill your desire. How long will you have to wait if your savings earn an interest of 12 % ?

Ans: 10.8 years

24

Present Value of an Annuity

n

n

nr,

n

n

nr,

r)(1r

1r)(1

Annuityfor Factor Interest Value resentPPVIFA

flowcash annuityA

,where

r)(1r

1r)(1APVIFAAPVA

25

Applications1. How much can you borrow for future need ?

After reviewing your budget, you have determined that you can afford to pay Rs 12,000 per month for 3 years towards a new car. You call a finance company and learn that the going rate of interest on car finance is 1.5 % per month for 36 months. How much can you borrow ?

Ans: Rs 332,400

26

Application (contd…)2. Period of Loan amortization

You want to borrow Rs 1,080,000 to buy a flat. You approach a housing finance company which charges 12.5 % interest. You can pay Rs 180,000 per year toward loan amortization. What should be the maturity period of the loan ?

Ans: 11.76 years

27

Application (contd….)

3. Determining the Loan Amortization Schedule• Most loans are repaid in equal periodic

installments (monthly, quarterly, or annually), which cover interest as well as principal repayment. Such loans are referred to as amortized loans.

• For amortized loans, we would like to know:a) The periodic installment payment and b) The loan amortization schedule showing the

breakup of installment between the interest component and the principal repayment component.

28

Loan Amortization Schedule

A firm borrows Rs 1,000,000 at an interest rate of 15 % and the loan is to be repaid in 5 equal installments payable at the end of each of the next 5 years.

What is the annual installment payment ?

Ans: Rs 298,312

29

Loan Amortization Schedule

5     Year

Beginning Amount

Annual Installment

InterestPrincipal

RepaymentRemaining

Balance

(1) (2) (3) (2)-(3)=(4) (1)-(4)=(5)

1          

2          

3          

4

5          

30

Loan Amortization Schedule

Year

Beginning Amount

Annual Installment

InterestPrincipal

RepaymentRemaining

Balance

(1) (2) (3) (2)-(3)=(4) (1)-(4)=(5)

1 1,000,000 298,312 150000 148312 851688

2 851,688 298,312 127753 170559 681129

3 681,129 298,312 102169 196143 484987

4 484,987 298,312 72748 225564 259423

5 259,423 298,312 38913 259399 24*

* Rounding off error

31

Applications (contd….)

4. Determining the Periodic Withdrawal

A father deposits Rs 300,000 on retirement in a bank which pays 10 % annual interest. How much can be withdrawn annually for a period of 10 years ?

Ans: Rs 48,819

32

Applications (contd…)

5. Finding the Interest Rate

Someone offers you the following financial contract: If you deposit Rs 10,000 with him to pay Rs 2,500 annually for 6 years. What interest rate do you earn on this deposit ?

Ans: 13 %

33

Present Value of a Growing annuity

nA.only be shall PV which,of case

in ther gfor not but r g andr gfor trueis This

r)(1)gr(

)g1(r)(1g)A(1annuity growing of PV

yearnth of end at the flowcash g)A(1

year 2nd of end at the flowcash g)A(1

year1st of end at the flowcash g)A(1

,If

n

nn

n

2

34

Example:Suppose you have the right to harvest a teak plantation for the next 20 years over which you expect to get 100,000 cubic feet of teak per year. The current price per cubic feet of teak is Rs 500, but it is expected to increase at a rate of 8 % per year. The discount rate is 15%. What is the present value of the teak that you can harvest ?

3551,736,68 Rs

0.15)(1)08.015.0(

)08.01(0.15)(10.08)100,000(1500 Rs teak of PV

20

2020

35

Annuity Due

• Annuity which occur at the beginning of the period are called annuity due.

• Eg: monthly lease rentals in apartments

Annuity due value = Ordinary annuity value * (1+r)

• This applies to both, present and future value.• Two steps are involved:

– Calculate the PV or FV as though it were an ordinary annuity

– Multiply your answer by (1+r)

36

Present Value of a Perpetuity

A perpetuity is an annuity of infinite duration.

PV = A * (1/r)

37

3.1.4 Intra-Year Compounding and Discounting

• So far we assumed that compounding is done annually.

• Now we shall consider the case, where compounding is done more frequently within a year.

Suppose you deposit Rs 1,000 with a finance company which advertises that it pays 12 % interest semi-annually – this means that the interest is paid every six months.

38

Semi-Annual Compounding (Example)

• First Six months:– Principal at the beginning = Rs 1,000– Interest for 6 months = Rs 1,000*0.06 = Rs 60– Principal at the end = Rs 1,060

• Second Six months:– Principal at the beginning = Rs 1,060– Interest for 6 months = Rs 1,060 * 0.06 = Rs 63.6– Principal at the end = Rs 1,123.6

• Note: If compounding is done annually, the principal at the end of one year would be Rs 1,120

39

Intra-Year Compounding

ratediscount (annual) nominal r

yearsin periods ofnumber n

yearper gcompoundin offrequncy m

,where

m

r1PVFV

:isyear a timesm

done is gcompoundin when yearsn after flowcash

single a of valuefuture for the formula general The

nm

n

40

Example

Suppose you deposit Rs 5,000 in a bank for 6 years. If the interest rate is 12 % and the compounding is done quaterly, then you deposit after 6 years will be ……….. ?

Rs 10,164

41

Effective versus Nominal Interest Rate

• Note the example of semiannual compounding with 12 % interest rate for Rs 1000.

• At the end of a year, it grew to Rs 1,123.6• That means Rs 1,000 grows at the rate of 12.36

% per annum.• This figure of 12.36 % is called effective

interest rate. • And 12 % interest rate is called nominal

interest rate. • 12.36 % under annual compounding produces

the same result as that produced by an interest rate of 12 % under semi-annual compounding.

42

Relationship:

12.36% i.e. 1236.012

0.121 RateInterest Effective

example,our For

yearper gcompoundin offrequency m

,Where

1m

RateInterest Nominal1 RateInterest Effective

2

m

43

Comparing Rates: The effect of compounding.

• Interest Rates are quoted in different ways.• Sometimes the way a rate is quoted is the

result of tradition.• Sometimes it’s the result of legislation.• At time, they are quoted deliberately in

deceptive ways to mislead borrowers and investors.

• Lets make sure that we never fall victim of such deception.

44

Effective Annual Rates (EAR) and compounding• A rate is quoted as 12% compounded semi-

annually.• What it means is that the investment actually

pays 6 % every six months. • Is 6 % every six months the same thing as 10

% a year ?NO

• If you invest $ 1 at 12 % per year, you’ll have $ 1.12 at the end of the year.

• If you invest at 6 % every six months, then you’ll have $ 1.1236 at the end of the year.

45

EAR and the effect of compounding

• 12 % compounded semi-annually is actually equivalent to 12.36 % per year.

• In other words, 12% compounded semiannually is equivalent to 12.36 % compounded annually.

• In this example, 12 % is called STATED, OR QUOTED OR NOMINAL INTEREST RATE.

• 12.36 % is EFFECTIVE ANNUAL RATE (EAR)

46

Lets not get decieved…

• You’ve researched and come up with following three rates:– Bank A : 15 % compounded daily.– Bank B: 15.5 % compounded quarterly.– Bank C: 16 % compounded annually.

• Which of these is the best if you are thinking of opening a savings account ? Which of these is best if they represent loan rates ?

• Find out EAR for each.

47

Answer:

• Bank A – 16.18 %• Bank B – 16.42 % - Good for savers• Bank C – 16 % - Good for borrowers

• Inference:– The highest quoted rate is not necessarily the best.– Compounding during a year can lead to a significant

difference between the quoted rate and the effective rate.

– Remember, EAR is what you get or what you pay.

48

Annual Percentage Rate (APR)

• It is the interest rate charged per period multiplied by the number of periods per year.

• If a bank quotes a car loan as 1.2 % per month, then the APR that must be reported is 1.2 % * 12 = 14.4 %.

• If a bank quotes a car loan at 12 % APR, is the consumer actually paying 12 % interest ?

• i.e. IS APR and EAR ?NO.

• APR of 12 % is actually 1 % per month.• EAR on such loan is 12.68 %.• Hence APR is actually Stated or quoted or nominal rate

in the sense we’ve been discussing.

49

Continuous compounding

yearper interest stated r

logarithm natural of base e

where,

1e RateInterest Effective r

50

Compounding Frequency and Effective Interest RateFrequency Nominal Int rate

%m Effective Int rate

%

Annual 12 1 12.00

Semi-annual

12 2 12.36

Quarterly 12 4 12.55

Monthly 12 12 12.68

Weekly 12 52 12.73

Daily 12 365 12.75

Continuous 12 inf 12.75

The effect of increasing the frequency of compounding is not as dramatic as some would believe it to be – the additional gains dwindle as the frequency of compounding increase

52

Intra-Year Discounting

ratediscount (annual) nominal r

yearsin periods ofnumber n

yearper gdiscountin offrequncy m

,wheremr

1

1FVPV

shorter is period

gdiscountin when luepresent va for the formula general The

nm

n

53

3.1.5 Loan Types and Loan Amortization

• There might be unlimited number of possibilities to the way the principal and interest of loan are repaid.

• Three basic types of loans are :

1. Pure Discount Loans

2. Interest-Only Loans

3. Amortized Loans

54

1. Pure Discount Loans

• Borrower receives money today and repays a single lump sum at some time in the future.

• Very common when the loan term is short.• However, they’ve become increasingly

common for much longer period recently.• Eg. Treasury Bill (T-bills)• If a T-bill promises to repay $ 10,000 in 12

months and the market interest rate is 7 %, how much will the bill sell for in the market ?

Ans: $ 9,345.79

55

2. Interest-Only Loans

• Borrower pays interest each period and repays the entire principal at some point in the future.

• If there’s only one period, a pure discount loan and an interest-only loan are the same thing.

• For eg, a 50- year interest-only loan would call for the borrower to pay interest every year for next 50 years and then repay the principal.

• Most corporate bonds are interest-only loan.

56

3. Amortized Loan

• The process of providing for a loan to be paid off my making regular principal reductions is called amortizing the loan.

• A simple way of amortizing a loan is to have the borrower pay the interest each period plus some fixed amount as the principal repayment.

• This approach is common with medium-term business loans.

• Almost all consumer loans and mortgages work this way.

57

Partial Amortization or “Bite the Bullet”

• A common arrangement in real state lending might call for a 5-year loan, with say 15-year amortization.

• What this means is that the borrower makes a payment every month of a fixed amount based on a 15 year amortization.

• However, after 60 months, the borrower makes a single, much larger payment called a “balloon” or “bullet” to pay off the loan.

• Because the monthly payments don’t fully pay off the loan, the loan is said to be partially amortized.

58

Example

Suppose we have a $ 100,000 commercial mortgage with a 12 % annual percentage rate and a 20-year amortization (240 months). Further, suppose the mortgage has a five-year balloon. What will the monthly payment be ? How big will the balloon payment be ?

• Here, monthly interest = 12 % / 12 = 1 % per month.

• The monthly payment can be calculated based on an ordinary annuity with a PV = $ 100,000.

59

Solution:

09.101,1$A

8194.09A

0.01)(101.0

10.01)(1APVIFAA000,100$

240

240

1%,240

• That means, for 60 months i.e. 5 years we have to pay $ 1,101.09

• Remaining amount shall be paid in lump-sum balloon. What shall be that balloon payment ?

60

Solution (Contd….)

91,744.69 $

0.01)(101.0

10.01)(11,101.09 $PVIFA1,101.09 $Balance Loan

:payments remaining theof PV the thusis balanceloan The

month.per % 1 still is

rateinterest and month,per 1,101.09 $ still isPayment

loan.month 180 60-240 have wemonths, 60After

180

180

1%,180

The balloon payment is $ 91,744.

Why is it so large ?

End of section:

2.1: Time value of Money