1 Chapter 2 The Two Key Concepts in Finance Prof.S.V.MURULIDHAR....

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Chapter 2The Two Key Concepts in Finance

Prof.S.V.MURULIDHAR. M.COM.MBA.,M.Phil.,MHRD.PGDCA.,PGDMM(PhD)

Dept .of Studies in Commerce & Management

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It’s what we learn after we think we know it all that counts.

- Kin Hubbard

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Outline Introduction Time value of money Safe dollars and risky dollars Relationship between risk and return

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Introduction The occasional reading of basic material in

your chosen field is an excellent philosophical exercise• Do not be tempted to include that you “know it

all”– E.g., what is the present value of a growing

perpetuity that begins payments in five years

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Time Value of Money Introduction Present and future values Present and future value factors Compounding Growing income streams

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Introduction Time has a value

• If we owe, we would prefer to pay money later• If we are owed, we would prefer to receive

money sooner• The longer the term of a single-payment loan,

the higher the amount the borrower must repay

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Present and Future Values Basic time value of money relationships:

1/(1 )

(1 )

t

t

PV FV DF

FV PV CF

where PV = present value;

FV = future value;

DF = discount factor = R

CF = compounding factor = R

R = interest rate per perio

d; and

t = time in periods

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Present and Future Values (cont’d)

A present value is the discounted value of one or more future cash flows

A future value is the compounded value of a present value

The discount factor is the present value of a dollar invested in the future

The compounding factor is the future value of a dollar invested today

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Why is a dollar today worth more than a dollar tomorrow?• The discount factor:

– Decreases as time increases• The farther away a cash flow is, the more we discount it

– Decreases as interest rates increase• When interest rates are high, a dollar today is worth much

more than that same dollar will be in the future

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Situations:• Know the future value and the discount factor

– Like solving for the theoretical price of a bond

• Know the future value and present value– Like finding the yield to maturity on a bond

• Know the present value and the discount rate– Like solving for an account balance in the future

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Present and Future Value Factors

Single sum factors How we get present and future value tables Ordinary annuities and annuities due

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Single Sum Factors Present value interest factor and future

value interest factor:

where

1

(1 )

(1 )

t

t

PV FV PVIF

FV PV FVIF

PVIFR

FVIF R

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Single Sum Factors (cont’d)Example

You just invested $2,000 in a three-year bank certificate of deposit (CD) with a 9 percent interest rate.

How much will you receive at maturity?

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Single Sum Factors (cont’d)Example (cont’d)

Solution: Solve for the future value:

3$2,000 1.09

$2,000 1.2950

$2,590

FV

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How We Get Present and Future Value Tables

Standard time value of money tables present factors for:• Present value of a single sum• Present value of an annuity• Future value of a single sum• Future value of an annuity

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How We Get Present and Future Value Tables (cont’d)

Relationships:• You can use the present value of a single sum

to obtain:– The present value of an annuity factor (a running

total of the single sum factors)

– The future value of a single sum factor (the inverse of the present value of a single sum factor)

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Ordinary Annuities and Annuities Due

An annuity is a series of payments at equal time intervals

An ordinary annuity assumes the first payment occurs at the end of the first year

An annuity due assumes the first payment occurs at the beginning of the first year

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Ordinary Annuities and Annuities Due (cont’d)

Example

You have just won the lottery! You will receive $1 million in ten installments of $100,000 each. You think you can invest the $1 million at an 8 percent interest rate.

What is the present value of the $1 million if the first $100,000 payment occurs one year from today? What is the present value if the first payment occurs today?

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Ordinary Annuities and Annuities Due (cont’d)

Example (cont’d)

Solution: These questions treat the cash flows as an ordinary annuity and an annuity due, respectively:

of ordinary annuity $100,000 6.7100 $671,000

of annuity due $100,000 ($100,000 6.2468) $724,680

PV

PV

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Compounding Definition Discrete versus continuous intervals Nominal versus effective yields

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Definition Compounding refers to the frequency with

which interest is computed and added to the principal balance• The more frequent the compounding, the higher

the interest earned

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Discrete Versus Continuous Intervals

Discrete compounding means we can count the number of compounding periods per year• E.g., once a year, twice a year, quarterly, monthly, or

daily

Continuous compounding results when there is an infinite number of compounding periods

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Discrete Versus Continuous Intervals (cont’d)

Mathematical adjustment for discrete compounding:

(1 / )

annual interest rate

number of compounding periods per year

time in years

mtFV PV R m

R

m

t

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Mathematical equation for continuous compounding:

2.71828

RtFV PVe

e

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Example

Your bank pays you 3 percent per year on your savings account. You just deposited $100.00 in your savings account.

What is the future value of the $100.00 in one year if interest is compounded quarterly? If interest is compounded continuously?

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Example (cont’d)

Solution: For quarterly compounding:

4

(1 / )

$100.00(1 0.03/ 4)

$103.03

mtFV PV R m

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Example (cont’d)

Solution (cont’d): For continuous compounding:

0.03$100.00

$103.05

RtFV PVe

e

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Nominal Versus Effective Yields

The stated rate of interest is the simple rate or nominal rate• 3.00% in the example

The interest rate that relates present and future values is the effective rate• $3.03/$100 = 3.03% for quarterly compounding• $3.05/$100 = 3.05% for continuous

compounding

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Growing Income Streams Definition Growing annuity Growing perpetuity

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Definition A growing stream is one in which each

successive cash flow is larger than the previous one• A common problem is one in which the cash

flows grow by some fixed percentage

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Growing Annuity A growing annuity is an annuity in which

the cash flows grow at a constant rate g:

2

2 3 1

1

(1 ) (1 ) (1 )...

(1 ) (1 ) (1 ) (1 )

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1

n

n

N

C C g C g C gPV

R R R R

C g

R g R

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Growing Perpetuity A growing perpetuity is an annuity where

the cash flows continue indefinitely:

2

2 3

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1

(1 ) (1 ) (1 )...

(1 ) (1 ) (1 ) (1 )

(1 )

(1 )

tt

tt

C C g C g C gPV

R R R R

C g C

R R g

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Safe Dollars and Risky Dollars Introduction Choosing among risky alternatives Defining risk

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Introduction A safe dollar is worth more than a risky

dollar• Investing in the stock market is exchanging

bird-in-the-hand safe dollars for a chance at a higher number of dollars in the future

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Introduction (cont’d) Most investors are risk averse

• People will take a risk only if they expect to be adequately rewarded for taking it

People have different degrees of risk aversion• Some people are more willing to take a chance

than others

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Choosing Among Risky Alternatives

Example

You have won the right to spin a lottery wheel one time. The wheel contains numbers 1 through 100, and a pointer selects one number when the wheel stops. The payoff alternatives are on the next slide.

Which alternative would you choose?

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Choosing Among Risky Alternatives (cont’d)A B C D

[1-50] $110 [1-50] $200 [1-90] $50 [1-99] $1,000

[51-100] $90 [51-100] $0 [91-100] $500 [100] -$89,000

Avg.

payoff $100 $100 $100 $100

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Choosing Among Risky Alternatives (cont’d)

Example (cont’d)Solution:

Most people would think Choice A is “safe.” Choice B has an opportunity cost of $90 relative

to Choice A. People who get utility from playing a game pick

Choice C. People who cannot tolerate the chance of any

loss would avoid Choice D.

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Choosing Among Risky Alternatives (cont’d)

Example (cont’d)

Solution (cont’d): Choice A is like buying shares of a utility stock. Choice B is like purchasing a stock option. Choice C is like a convertible bond. Choice D is like writing out-of-the-money call

options.

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Defining Risk Risk versus uncertainty Dispersion and chance of loss Types of risk

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Risk Versus Uncertainty Uncertainty involves a doubtful outcome

• What you will get for your birthday• If a particular horse will win at the track

Risk involves the chance of loss• If a particular horse will win at the track if you

made a bet

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Dispersion and Chance of Loss There are two material factors we use in

judging risk:• The average outcome

• The scattering of the other possibilities around the average

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Dispersion and Chance of Loss (cont’d)

Investment A Investment B

Time

Investment value

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Dispersion and Chance of Loss (cont’d)

Investments A and B have the same arithmetic mean

Investment B is riskier than Investment A

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Types of Risk Total risk refers to the overall variability of

the returns of financial assets

Undiversifiable risk is risk that must be borne by virtue of being in the market• Arises from systematic factors that affect all

securities of a particular type

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Types of Risk (cont’d) Diversifiable risk can be removed by

proper portfolio diversification• The ups and down of individual securities due

to company-specific events will cancel each other out

• The only return variability that remains will be due to economic events affecting all stocks

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Relationship Between Risk and Return

Direct relationship Concept of utility Diminishing marginal utility of money St. Petersburg paradox Fair bets The consumption decision Other considerations

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Direct Relationship The more risk someone bears, the higher

the expected return The appropriate discount rate depends on

the risk level of the investment The risk-less rate of interest can be earned

without bearing any risk

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Direct Relationship (cont’d)

Risk

Expected return

Rf

0

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Direct Relationship (cont’d) The expected return is the weighted

average of all possible returns • The weights reflect the relative likelihood of

each possible return

The risk is undiversifiable risk• A person is not rewarded for bearing risk that

could have been diversified away

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Concept of Utility Utility measures the satisfaction people get

out of something• Different individuals get different amounts of

utility from the same source– Casino gambling

– Pizza parties

– CDs

– Etc.

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Diminishing Marginal Utility of Money

Rational people prefer more money to less• Money provides utility

• Diminishing marginal utility of money– The relationship between more money and added

utility is not linear

– “I hate to lose more than I like to win”

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Diminishing Marginal Utility of Money (cont’d)

$

Utility

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St. Petersburg Paradox Assume the following game:

• A coin is flipped until a head appears• The payoff is based on the number of tails

observed (n) before the first head• The payoff is calculated as $2n

What is the expected payoff?

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St. Petersburg Paradox (cont’d)

Number of Tails Before First

Head Probability PayoffProbability

x Payoff

0 (1/2)1 = 1/2 $1 $0.50

1 (1/2)2 = 1/4 $2 $0.50

2 (1/2)3 = 1/8 $4 $0.50

3 (1/2)4 = 1/16 $8 $0.50

4 (1/2)5 = 1/32 $16 $0.50

n (1/2)n + 1 $2n $0.50

Total 1.00

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St. Petersburg Paradox (cont’d)

In the limit, the expected payoff is infinite

How much would you be willing to play the game?• Most people would only pay a couple of dollars• The marginal utility for each additional $0.50

declines

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Fair Bets A fair bet is a lottery in which the expected

payoff is equal to the cost of playing• E.g., matching quarters• E.g., matching serial numbers on $100 bills

Most people will not take a fair bet unless the dollar amount involved is small• Utility lost is greater than utility gained

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The Consumption Decision The consumption decision is the choice to

save or to borrow• If interest rates are high, we are inclined to save

– E.g., open a new savings account

• If interest rates are low, borrowing looks attractive

– E.g., a higher home mortgage

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The Consumption Decision (cont’d)

The equilibrium interest rate causes savers to deposit a sufficient amount of money to satisfy the borrowing needs of the economy

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Other Considerations Psychic return Price risk versus convenience risk

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Psychic Return Psychic return comes from an individual

disposition about something• People get utility from more expensive things,

even if the quality is not higher than cheaper alternatives

– E.g., Rolex watches, designer jeans

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Price Risk Versus Convenience Risk

Price risk refers to the possibility of adverse changes in the value of an investment due to:• A change in market conditions• A change in the financial situation• A change in public attitude

E.g., rising interest rates and stock prices, a change in the price of gold and the value of the dollar

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Price Risk Versus Convenience Risk (cont’d)

Convenience risk refers to a loss of managerial time rather than a loss of dollars• E.g., a bond’s call provision

– Allows the issuer to call in the debt early, meaning the investor has to look for other investments

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