Portfolio Management - Chapter 2

8/3/2019 Portfolio Management - Chapter 2

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Chapter 2

The Two Key Concepts in Finance

1

Prof. Rushen Chahal

Prof. Rushen Chahal


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Its what we learn after we think we know it all

that counts.

- Kin Hubbard

2Prof. Rushen Chahal


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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 moneysooner

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:

7

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

! v

! v

d; and

t = time in periods

Prof. Rushen Chahal


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Present and Future Values (contd)

Apresent value is the discounted value of oneor more future cash flows

Afuture value is the compounded value of apresent value

The discountfactoris the present value of adollar invested in the future

The compoundingfactoris the future value ofa 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 interestfactorand future value

interestfactor:

12

where

1

(1 )

(1 )

t

t

PV FV PVIF

FV PV FVIF

PVIFR

FVIF R

! v

! v

!

!

Prof. Rushen Chahal


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Single Sum Factors (contd)

Example

You

ju

st invested $2,000 in a three-year bankc

ertific

ate ofdeposit (CD) with a 9 percent interest rate.

How much will you receive at maturity?



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Single Sum Factors (contd)

Example (contd)

S

olu

tion:S

olve for the fu

tu

re valu

e:

14

3$2,000 1.09

$2,000 1.2950

$2,590

FV ! v

! v

!

Prof. Rushen Chahal


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How We Get Present and Future

Valu

eT

ables 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

Valu

eT

ables (c

ontd) 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 Annu

ities Du

e An annuityis a series of payments at equal

time intervals

An ordinary annuityassumes the firstpayment occurs at the end of the first year

An annuitydue assumes the first paymentoccurs at the beginning of the first year



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Ordinary Annuities

and Annu

ities Du

e (c

ontd)Example

You

have ju

st won the lottery! You

will rec

eive $1 million in teninstallments of $100,000 each. You think youcan 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 Annu

ities Du

e (c

ontd)Example (contd)

S

olu

tion:T

hese qu

estions treat thec

ash flows as an ordinaryannuity and an annuity due, respectively:

19

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

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

PV

PV

! v !

! v !

Prof. Rushen Chahal


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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

Continu

ou

sI

ntervals 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

Continu

ou

sI

ntervals (c

ontd) Mathematical adjustment for discrete

compounding:

23

(1 / )

annual interest rate

number of compounding periods per year

time in years

mt FV PV R m

R

m

t

!

!

!

!

Prof. Rushen Chahal


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Discrete Versus

Continu

ou

sI

ntervals (c

ontd) Mathematical equation for continuous

compounding:

24

2.71828

Rt FV PVe

e

!

!

Prof. Rushen Chahal


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Discrete Versus

Continu

ou

sI

ntervals (c

ontd)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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Discrete Versus

Continu

ou

sI

ntervals (c


Solution: For quarterly compounding:

26

4

(1 / )

$100.00(1 0.03 / 4)$103.03

mt FV PV R m!

! !

Prof. Rushen Chahal


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Discrete Versus

Continu

ou

sI

ntervals (c


Solution (contd): For continuous compounding:

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0.03$100.00

$103.05

Rt FV PVe

e

!

! v!

Prof. Rushen Chahal


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Nominal Versus

Effec

tive 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

Agrowing 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

Agrowing annuityis an annuity in which the

cash flows grow at a constant rate g:

31

2

2 3 1

1

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

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

111

n

n

N

C C g C g C g PV

R R R R

C g R g R

!

!

-

Prof. Rushen Chahal


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Growing Perpetuity

Agrowing perpetuityis an annuity where the

cash flows continue indefinitely:

32

2

2 3

1

1

1

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

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

(1 )(1 )

t

t

tt

C C g C g C g PV

R R R R

C g C R R g

g

g

g

!

!

! !

Prof. Rushen Chahal


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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 (contd)

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 youchoose?



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Choosing Among

Risky Alternatives (contd)

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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

Prof. Rushen Chahal


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Choosing Among


Example (contd)Solution:

Most people would think Choice A is safe.

Choice B has an opportunity costof $90 relative toChoice A.

People who get utility from playing a game pick

Choice C.

People who cannot tolerate the chance of any losswould avoid Choice D.



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Choosing Among


Example (contd)

Solution (contd):

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

Uncertaintyinvolves a doubtful outcome

What you will get for your birthday

If a particular horse will win at the track

Riskinvolves 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

(contd)

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Investment A

Investment B

Time

Investment value

Prof. Rushen Chahal


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Types of Risk

Total riskrefers to the overall variability of the

returns of financial assets

Undiversifiable riskis 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 (contd)

Diversifiable riskcan 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 ofinterestcan be earned

without bearing any risk



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Direct Relationship (contd)

49

Risk

Expected return

Rf

0

Prof. Rushen Chahal


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Direct Relationship (contd)

The expectedreturn 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

Utilitymeasures 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 ofmoney

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 (contd)

53

$

Utility

Prof. Rushen Chahal


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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 (contd)

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Number of Tails

Before First

Head Probability Payoff

Probability

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

Total 1.00 g

Prof. Rushen Chahal


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St. Petersburg Paradox (contd)

In the limit, the expected payoff is infinite

How much would you be willing to play thegame?

Most people would only pay a couple of dollars

The marginal utility for each additional $0.50

declines



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Fair Bets

Afair betis a lottery in which the expectedpayoff 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 (contd)

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 riskrefers 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 (contd)

Convenience riskrefers to a loss of managerial

time rather than a loss of dollars

E.g., a bonds call provision

Allows the issuer to call in the debt early, meaning the

investor has to look for other investments

Documents

Portfolio Management - Chapter 2