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Chapter 2

Valuation, Risk, Return, and Uncertainty

Portfolio Construction, Management, & Protection, 5e, Robert A. StrongCopyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved.

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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 Valuation Safe Dollars and Risky Dollars Relationship Between Risk and Return The Concept of Return Some Statistical Facts of Life

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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 conclude 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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Valuation Valuation may be the most important part

of the study of investments• Security analysts make a career of estimating

“what you get” for “what you pay”• The time value of money is one of the two key

concepts in finance and is very useful in valuation

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Growing Income Streams 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 )

11

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

11

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 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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Safe Dollars and Risky Dollars (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)

$100$100$100$100

Average

payoff

–$89,000[100]$550[91–100]$0[51–100]$90[51–100]

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

DCBA

Number on lottery wheel appears in brackets.

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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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Risk Versus Uncertainty Uncertainty involves a doubtful outcome

• What birthday gift you will receive• 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 riskless 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

× Payoff

0 (1/2) = 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 bigger 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 influence stock prices, and a change in the price of gold can affect 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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The Concept of Return “Return” can mean various things, and it is

important to be clear when discussing an investment

A general definition of return is “the benefit associated with an investment”• In most cases, return is measurable

• e.g., a $100 investment at 8 percent, compounded continuously is worth $108.33 after one year

– The return is $8.33, or 8.33 percent

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Holding Period Return The calculation of a holding period return

is independent of the passage of time• e.g., you buy a bond for $950, receive $80 in

interest, and later sell the bond for $980– The return is ($80 + $30)/$950 = 11.58 percent

– The 11.58 percent could have been earned over one year or one week

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Arithmetic Mean Return The arithmetic mean return is the

arithmetic average of several holding period returns measured over the same holding period:

iR

n

R

i

n

i

i

periodin return of rate the~

~mean Arithmetic

1

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Arithmetic Mean Return (cont’d)

Arithmetic means are a useful proxy for expected returns

Arithmetic means are not especially useful for describing historical return data• It is unclear what the number means once it is

determined

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Geometric Mean Return The geometric mean return is the nth root

of the product of n values:

1)~

1(mean Geometric/1

1

nn

iiR

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Arithmetic and Geometric Mean Returns

Example

Assume the following sample of weekly stock returns:

Week Return Return Relative

1 0.0084 1.0084

2 –0.0045 0.9955

3 0.0021 1.0021

4 0.0000 1.000

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Arithmetic and Geometric Mean Returns (cont’d)

Example (cont’d)

What is the arithmetic mean return?

Solution:

0015.04

0000.00021.00045.00084.0

~mean Arithmetic

1

n

i

i

n

R

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Arithmetic and Geometric Mean Returns (cont’d)

Example (cont’d)

What is the geometric mean return?

Solution:

1/

1

1/ 4

Geometric mean (1 ) 1

1.0084 0.9955 1.0021 1.0000 1

0.001489

nn

ii

R

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Comparison of Arithmetic andGeometric Mean Returns

The geometric mean reduces the likelihood of nonsense answers• Assume a $100 investment falls by 50 percent

in period 1 and rises by 50 percent in period 2

• The investor has $75 at the end of period 2– Arithmetic mean = [(0.50) + (–0.50)]/2 = 0%

– Geometric mean = (0.50 × 1.50)1/2 – 1 = –13.40%

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Comparison of Arithmetic andGeometric Mean Returns (Cont’d) The geometric mean must be used to

determine the rate of return that equates a present value with a series of future values

The greater the dispersion in a series of numbers, the wider the gap between the arithmetic mean and geometric mean

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Expected Return Expected return refers to the future

• In finance, what happened in the past is not as important as what happens in the future

• We can use past information to make estimates about the future

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Return on Investment Return on investment (ROI) is a term that

must be clearly defined• Return on assets (ROA)

– Return ÷ Total Assets

• Return on equity (ROE)– Return ÷ Total stockholder’s Equity

– ROE is a leveraged version of ROA

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Standard Deviation and Variance

Standard deviation and variance are the most common measures of total risk

They measure the dispersion of a set of observations around the mean observation

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Standard Deviation and Variance (cont’d)

General equation for variance:

If all outcomes are equally likely:

2

2

1

Variance prob( )n

i ii

x x x

2

2

1

1 n

ii

x xn

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Standard Deviation and Variance (cont’d)

Equation for standard deviation:

2

2

1

Standard deviation prob( )n

i ii

x x x

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Semi-Variance Semi-variance considers the dispersion only

on the adverse side• Ignores all observations greater than the mean• Calculates variance using only “bad” returns

that are less than average• Since risk means “chance of loss,” positive

dispersion can distort the variance or standard deviation statistic as a measure of risk

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Some Statistical Facts of Life One has to understand key terms:

• “Constants,” “Variables”, “Populations,” “Samples,” and “Sample statistics”

Properties of Random Variables Linear Regression R Squared and Standard Errors

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Constants A constant is a value that does not change

• e.g., the number of sides of a cube• e.g., the sum of the interior angles of a triangle

A constant can be represented by a numeral or by a symbol

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Variables A variable has no fixed value

• It is useful only when it is considered in the context of other possible values it might assume

In finance, variables are called random variables• Designated by a tilde

– e.g., x

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Variables (cont’d) Discrete random variables are countable

• e.g., the number of trout you catch

Continuous random variables are measurable• e.g., the length of a trout

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Variables (cont’d) Quantitative variables are measured by

real numbers• e.g., numerical measurement

Qualitative variables are categorical• e.g., hair color

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Variables (cont’d) Independent variables are measured

directly• e.g., the height of a box

Dependent variables can only be measured once other independent variables are measured• e.g., the volume of a box (requires length,

width, and height)

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Populations A population is the entire collection of a

particular set of random variables The nature of a population is described by

its distribution• The median of a distribution is the point where

half the observations lie on either side• The mode is the value in a distribution that

occurs most frequently

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Populations (cont’d) A distribution can have skewness

• There is more dispersion on one side of the distribution

• Positive skewness means the mean is greater than the median

– Stock returns are positively skewed

• Negative skewness means the mean is less than the median

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Populations (cont’d)Positive Skewness Negative Skewness

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Populations (cont’d) A binomial distribution contains only two

random variables• e.g., the toss of a coin (heads or tails)

A finite population is one in which each possible outcome is known• e.g., a card drawn from a deck of cards

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Populations (cont’d) An infinite population is one where not all

observations can be counted• e.g., the microorganisms in a cubic mile of

ocean water

A univariate population has one variable of interest

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Populations (cont’d) A bivariate population has two variables

of interest• e.g., weight and size

A multivariate population has more than two variables of interest• e.g., weight, size, and color

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Samples A sample is any subset of a population

• e.g., a sample of past monthly stock returns of a particular stock

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Sample Statistics Sample statistics are characteristics of

samples• A true population statistic is usually

unobservable and must be estimated with a sample statistic

– Expensive

– Statistically unnecessary

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Properties of Random Variables

Example Central Tendency Dispersion Logarithms Expectations Correlation and Covariance

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Example

Assume the following monthly stock returns for Stocks A and B:

Month Stock A Stock B

1 2% 3%

2 –1% 0%

3 4% 5%

4 1% 4%

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Central Tendency Central tendency is what a random variable

looks like, on average The usual measure of central tendency is the

population’s expected value (the mean)• The average value of all elements of the population

n

iii R

nRE

1

~1)

~(

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Example (cont’d)

The expected returns for Stocks A and B are:

n

iAA R

nRE

1

%50.1%)1%4%1%2(4

1~1)

~(

n

iBB R

nRE

1

%00.3%)4%5%0%3(4

1~1)

~(

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Dispersion Investors are interested in the variation of

actual values around the average A common measure of dispersion is the

variance or standard deviation

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Example (cont’d)

The variance and standard deviation for Stock A are:

%8.1018.0000325.0

000325.0)0013.0(4

1

%)5.1%1(%)5.1%4(%)5.1%1(%)5.1%2(4

1

)~(

2

2222

22

xxE i

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Example (cont’d)

The variance and standard deviation for Stock B are:

%87.10187.000035.0

00035.0)0014.0(4

1

%)0.3%4(%)0.3%5(%)0.3%0(%)0.3%3(4

1

)~(

2

2222

22

xxE i

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Logarithms Logarithms reduce the impact of extreme

values• e.g., takeover rumors may cause huge price

swings• A logreturn is the logarithm of a return relative

Logarithms make other statistical tools more appropriate• e.g., linear regression

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Logarithms (cont’d) Using logreturns on stock return

distributions:• Take the raw returns

• Convert the raw returns to return relatives

• Take the natural logarithm of the return relatives

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Expectations The expected value of a constant is a

constant:

The expected value of a constant times a random variable is the constant times the expected value of the random variable:

( )E a a

( ) ( )E ax aE x

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Expectations (cont’d) The expected value of a combination of

random variables is equal to the sum of the expected value of each element of the combination:

( ) ( ) ( )E x y E x E y

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Correlations and Covariance Correlation is the degree of association

between two variables

Covariance is the product moment of two random variables about their means

Correlation and covariance are related and generally measure the same phenomenon

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Correlations and Covariance (cont’d)

( , ) ( )( )ABCOV A B E A A B B

( , )AB

A B

COV A B

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Example (cont’d)

The covariance and correlation for Stocks A and B are:

1(0.5% 0.0%) ( 2.5% 3.0%) (2.5% 2.0%) ( 0.5% 1.0%)

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(0.001225)40.000306

AB

( , ) 0.0003060.909

(0.018)(0.0187)ABA B

COV A B

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Correlations and Covariance (cont’d)

Correlation ranges from –1.0 to +1.0. • Two random variables that are perfectly

positively correlated have a correlation coefficient of +1.0

• Two random variables that are perfectly negatively correlated have a correlation coefficient of –1.0

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Linear Regression Linear regression is a mathematical

technique used to predict the value of one variable from a series of values of other variables• e.g., predict the return of an individual stock

using a stock market index Linear regression finds the equation of a

line through the points that gives the best possible fit

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Linear Regression (cont’d)Example

Assume the following sample of weekly stock and stock index returns:

Week Stock Return Index Return

1 0.0084 0.0088

2 –0.0045 –0.0048

3 0.0021 0.0019

4 0.0000 0.0005

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Linear Regression (cont’d)Example (cont’d)

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

-0.01 -0.005 0 0.005 0.01

Return (Market)

Re

turn

(S

tock

)

Intercept = 0Slope = 0.96R squared = 0.99

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R Squared and Standard Errors R squared and the standard error are used to assess the

accuracy of calculated securities

R squared is a measure of how good a fit we get with the regression line• If every data point lies exactly on the line, R squared is 100%

R squared is the square of the correlation coefficient between the security returns and the market returns• It measures the portion of a security’s variability that is due to the

market variability

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Standard Errors The standard error is equal to the standard

deviation divided by the square root of the number of observations:

Standard errorn

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Standard Errors (cont’d) The standard error enables us to determine

the likelihood that the coefficient is statistically different from zero• About 68 percent of the elements of the

distribution lie within one standard error of the mean

• About 95 percent lie within 1.96 standard errors• About 99 percent lie within 3.00 standard errors