Chapter07 Analysis of Risk and Return

CONTEMPORARY FINANCIAL MANAGEMENT

Chapter 7:

Analysis of Risk and Return

INTRODUCTION

This chapter develops the risk-return relationship for both individual projects (investments) and portfolios of projects

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RISK AND RETURN

Risk is usually defined as the actual or potential variability of returns from a project or portfolio

Risk-free returns are known with certainty Federal Government Treasury Bills are often considered the

risk-free security. The risk-free rate of return sets a floor under all other returns

in the market.

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HOLDING PERIOD RETURN

Return for holding an investment for one period (i.e. period of days, months, years, etc.)

When there is no cash flow during the holding period, then:

4

1 0

0

P - P HPR =

P

HPR = Holding Period ReturnP1 = Ending PriceP0 = Beginning Price

HOLDING PERIOD RETURN When there is a cash flow in addition to the ending price

(such as the payment of a dividend), the Holding Period Return formula is:

( )1 0

0

P +Cash Flow - P HPR =

P

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HOLDING PERIOD RETURN: EXAMPLE You bought a stock one year ago for $10. Today, it is worth $12.

Yesterday, you received a $1 dividend. What is your holding period return?

( )

( )+ −=

=

1 0

0

P +Cash Flow - P HPR =

P

12 1 10

100.30 or 30%

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RETURNS

Ex Post Returns (After the fact) Return that an investor actually realizes

Ex Ante Returns (Before the fact) Return that an investor expects to earn

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ANALYZING RETURN

Expected Return ( ): When returns are not known with certainty, there will often

exist a probability distribution of possible returns with an associated probability of occurrence.

Expected return is a weighted average of the individual possible returns (rj), with weights being the probability of occurrence (pj).

8=

= ∑n

j jj 1

r̂ rp

r̂

EXPECTED RETURN: EXAMPLE

Possible Return

Probability of Occurrence

-10% 5%

0% 10%

+5% 25%

+15% 50%

+25% 10%

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EXPECTED RETURN: SOLUTION

( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

=

=

= − + +

+ +

=

∑n

j jj 1

r̂ rp

10% .05 0% .10

5% .25 15% .50 25% .10

10.75%

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ANALYZING RISK Standard Deviation ( ): a statistical measure of the

dispersion, or variability, of outcomes around the mean or expected value ( ).

Low standard deviation means that returns are tightly clustered around the mean

High standard deviation means that returns are widely dispersed around the mean

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σ

r̂

CALCULATING STANDARD DEVIATION

Three common ways of calculating standard deviation:

Returns are known with certainty Standard deviation of a population Standard deviation of a sample

Returns are not known with certainty

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STANDARD DEVIATION OF A POPULATION

( ) ( ) ( )2 2 2

1 2 N2r - r + r - r +...+ r - r

σ =N

2σ = σ

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The standard deviation ( ) is the square root of the variance:

σ

First, calculate the variance ( 2):σ

i

r = Mean returnr = Return i

N = Number of returns

STANDARD DEVIATION OF A SAMPLE

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( ) ( ) ( )2 2 2

1 2 N2r - r + r - r +...+ r - r

S = N - 1

2s = S

First, calculate the variance (S2):

The standard deviation (s) is the square root of the variance:

STANDARD DEVIATION: EXAMPLE

You have been given the following sample of stock returns, for which you would like to calculate the standard deviation:

{12%, -4%, 0%, 22%, 5%}

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STANDARD DEVIATION: EXAMPLE

You have been given the following sample of stock returns, for which you would like to calculate the standard deviation:

{12%, -4%, 0%, 22%, 5%}

Step 1: Calculate Arithmetic Return

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+ + +=

=

1 2 Nr + r + ... + rAM =

N12 - 4 0 22 5

57%

STANDARD DEVIATION: EXAMPLE

You have been given the following sample of stock returns, for which you would like to calculate the standard deviation:

{12%, -4%, 0%, 22%, 5%}

Step 2: Calculate Variance

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( ) ( ) ( )

( ) ( ) ( ) ( ) ( )− + − − + − + − + −=

−=

2 2 2

1 2 N2

2 2 2 2 2

r - r + r - r +...+ r - rS =

N-1

12 7 4 7 0 7 22 7 5 7

5 1106

STANDARD DEVIATION: EXAMPLE

You have been given the following sample of stock returns, for which you would like to calculate the standard deviation:

{12%, -4%, 0%, 22%, 5%}

Step 3: Calculate Standard Deviation

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==

2s = S

10610.3%

STANDARD DEVIATION – RETURNS NOT CERTAIN

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( )∑ $n 2

2j j

j=1

σ = r - r p

rj = return at time period j

= expected return

pj = probability of return j occurring

r̂

STANDARD DEVIATION: EXAMPLE You have been provided with the following possible returns

and their associated probabilities. Calculate the expected return and the standard deviation of return.

State of Economy Return Probability

Boom 30% 15%

Normal 15% 60%

Recession 0% 25%

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STANDARD DEVIATION: SOLUTION

Step #1: Calculate the Expected Return

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( ) ( ) ( ) ( ) ( ) ( )=

=

= + +

=

∑n

j jj 1

r̂ rp

30% .15 15% .60 0% .25

13.5%

STANDARD DEVIATION: SOLUTION

Step #2: Calculate the Variance

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( )( ) ( ) ( ) ( ) ( ) ( )

∑ $n 2

2j j

j=1

2 2 2

σ = r - r p

= 30-13.5 .15 + 15-13.5 .60 + 0-13.5 .25

= 87.75

STANDARD DEVIATION: SOLUTION

Step #3: Calculate the Standard Deviation

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2σ = σ

= 87.75= 9.4%

NORMAL PROBABILITY DISTRIBUTION

A symmetrical, bell-like curve where 50% of possible outcomes are greater than the expected value and 50% are less than the expected value.

A normal distribution is fully described by just two statistics: Mean Standard deviation

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NORMAL PROBABILITY DISTRIBUTION

25

Mean- 1 σ

50% Probability

- 3 σ - 2 σ 1 σ 2 σ 3 σ

50% Probability

68.26%

95.44%

99.74%

NORMAL DISTRIBUTION EXAMPLE

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10%

50% Probability

5% 15%0% 20%- 5% 25%

50% Probability

68.26%

95.44%

99.74%

Mean = 10%; Standard Deviation = 5%

STANDARD NORMAL PROBABILITY

Problem: Standard deviation is correlated with size of the mean

Solution: To allow for easy comparison among distributions with different means, standardize using a Z score

Z score measures the number of standard deviations ( ) a particular rate of return (r) is from the mean or expected value ( ).

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−=σ

ˆr rz

r̂σ

STANDARD NORMAL: EXAMPLE What is the probability of a loss on an investment with an

expected return of 20% and a standard deviation of 17%?

Step #1: Calculate the Z Score for the number of standard deviations from the mean for a 0% return

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− −= = ≅ −σ

ˆr r 0% 20%z 1.18

17%

STANDARD NORMAL: EXAMPLE What is the probability of a loss on an investment with an expected

return of 20% and a standard deviation of 17%?

Step #2: Consult Table V on Page 712. Find the row with 1.10 in the left hand column. Then find the column with 0.08 in the top row. The cell where the row & column intersect is the probability of obtaining a value less than 1.18 standard deviations from the mean. The answer is 0.1190 or 11.90%

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PROBABILITY OF EARNING LESS THEN 0%

30

20%3%- 31% -14% 37% 54% 71%

1.18 st. dev.from the mean

11.9% Probability

CONCEPT OF EFFICIENT PORTFOLIOS

Has the highest possible expected return for a given level of risk (or standard deviation)

Has the lowest possible level of risk for a given expected return

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σ

r̂

A B

C A dominates B because it has the same expected return for a given risk.

C dominates B because it has a higher expected return for a given risk.

COEFFICIENT OF VARIATION (V)

The ratio of the standard deviation ( ) to the expected value ( ).

Tells us the risk per unit of return.

An appropriate measure of total risk when comparing two investment projects of different size.

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σ=vr̂

σr̂

COEFFICIENT OF VARIATION: EXAMPLE You are asked to rank the following set of investments

according to their risk per unit of return.

Security Return Standard Deviation

A 6% 7%

B 10% 13%

C 18% 21%

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COEFFICIENT OF VARIATION: SOLUTION

Security Return Standard Deviation

Coefficient of Variation

A 6% 7%

B 10% 13%

C 18% 20%

=71.17

6

=201.1

18=13

1.310

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Most Risk

Least Risk

RELATIONSHIP BETWEEN RISK AND RETURN

Required Rate of Return Discount rate used to present value a stream of expected cash

flows from an asset.

Risk-Return Relationship The riskier, or the more variable, the expected cash flow

stream, the higher the required rate of return.

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RELATIONSHIP BETWEEN RISK AND RETURN

Required Rate of Return =

Risk-free Rate of Return + Risk Premium

Risk-free Rate: rate of return on securities that are free of default risk, such as T-bills.

Risk Premium: expected “reward” the investor expects to earn for assuming risk

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RISK-FREE RATE OF RETURN

Risk-free Rate of Return (rf) =

Real Rate of Return + Exp. Inflation Premium

Real Rate of Return: the reward for deferring consumption

Expected Inflation Premium: compensates investors for the loss of purchasing power due to inflation

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TYPES OF RISK PREMIUMS

Maturity risk premium

Default risk premium

Seniority risk premium

Marketability risk premium

Business risk

Financial risk

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TERM STRUCTURE OF INTEREST RATES Term structure is a plot of the yield on securities with similar

risk but different maturities

Term structure is used to explain the maturity risk premium (why long securities tend to have higher yields than short maturity securities)

Three theories of the Term Structure: Expectations theory Liquidity premium theory Market segmentation theory

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CANADA’S TERM STRUCTURE

40

2.65%

2.69% 2.76%

2.70% 2.91%

3.28%3.57%

4.13% 4.88%

5.39%

0%

1%

2%

3%

4%

5%

6%

1Month

2Month

3Month

6Month

12Months

2 year

3 year

5 year

10 year

Longterm

November 13, 2003

EXPECTATIONS THEORY

The long interest rate is the geometric average of expected future short interest rates.

If the term structure is sloping up, future short interest rates are expected to be higher than current short interest rates.

If the term structure is sloping down, future short interest rates are expected to be lower than current short interest rates.

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LIQUIDITY PREMIUM THEORY

Investors prefer liquidity (the ability to convert to cash at or near face value)

Long securities are less liquid than short securities

Therefore, to induce investors to hold long securities, must pay a “liquidity premium”

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MARKET SEGMENTATION THEORY

The yield in each segment of the yield curve is determined by the supply and demand for funds in that maturity zone

Supply & demand driven by firms which deal primarily in a specific maturity zone Chartered banks – short maturities Trust companies – medium maturities Pension funds – long maturities

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MODERN PORTFOLIO THEORY

44Harry Markowitz William F. Sharpe Merton Miller

Modern portfolio theory was introduced by Harry Markowitz in 1952. Markowitz, Sharpe & Miller were co-recipients of the Nobel Prize in Economics in 1990 for their pioneering work in portfolio theory

EXPECTED RETURN The expected return on a portfolio is the weighted average of

the returns of each asset within the portfolio

Example: A portfolio is comprised of three securities with the following returns:

Security Return % of Portfolio

A 5% 30%

B 10% 45%

C 15% 25%

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EXPECTED RETURN

The expected return of the portfolio is the weighted average:

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( ) ( ) ( ) ( ) ( ) ( )=

=

= + +

=

∑n

j jj 1

r̂ rw

5% .30 10% .45 15% .25

9.75%

rj = return at time period j

= expected return

wj = proportion of the portfolio comprised of asset j

r̂

PORTFOLIO RISK: TWO RISKY ASSETS

Standard deviation of a two-asset portfolio is calculated as follows:

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2 2 2 2A A B B A B A,B A Bσ = w σ + w σ + 2w w ρ σ σ

A

A,B

σ = standard deviationw = the proportion of the portfolio comprised of A

ρ = the correlation coefficient between A & B

PORTFOLIO RISK: TWO RISKY ASSETS

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2 2 2 2A A B B A B A,B A Bσ = w σ + w σ + 2w w ρ σ σ

Portfolio risk is driven mainly by the correlation between the

assets!!

CORRELATION

Correlation is a measure of the linear relationship between two assets

Correlation varies between perfect negative (-1) to perfect positive (+1)

Perfect negative correlation: when the return on asset A rises, the return on Asset B falls and vice versa

Perfect positive correlation: the returns on asset A and Asset B move in perfect unison

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CORRELATION & RISK REDUCTION

50

B

ARA

σA σB

RBPerfect Negative Correlation

Perfect Positive Correlation

Less thanPerfect Correlation

To minimize portfolio risk, choose assets that have very low correlations with each other.

MOVING TOWARD MANY RISKY ASSETS

When the portfolio consists of many risky assets, they form a plot similar to a broken egg shell shape

Each dot within the broken egg shell shape represents the risk/return profile for a single risky asset or portfolio of risky assets

To maximize return per unit of risk assumed, an investor would always choose an asset or portfolio that plots along the efficient frontier

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PORTFOLIOS: MANY RISKY ASSETS

52

You would never choose Asset A, as you can earn a higher return with similar risk by choosing the asset that plots along the Efficient Frontier.

RA

σA

Return

Standard Deviation

A

CHOOSING A PORTFOLIO: SO FAR

The investor first decides how much risk to assume

The investor then chooses the portfolio that plots along the efficient frontier with that amount of risk

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INTRODUCING THE RISK FREE SECURITY

When a risk-free asset (Treasury Bill) is introduced into the set of risky assets, a new efficient frontier emerges

This new efficient frontier is known as the Capital Market Line (CML)

The CML represents all possible portfolios comprised of Treasury Bills and the Market Portfolio

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ADDING THE RISK-FREE ASSET

55

RM

σM

Return

Standard Deviation

ARf

Capital Market Line

CAPITAL MARKET LINE (CML)

To maximize return for an amount of risk, investors should hold a portion of their assets in T-bills and a portion in the market portfolio.

Linear relationship between risk and return

To earn an expected return greater than the return on the market portfolio, invest more than 100% of one’s own wealth in the market portfolio.

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( )PortfolioWell-diversified f Market fPortfolio Market

σR = r + r - r

σ

WHAT ARE WE MISSING? We know:

Investors should split their assets between Treasury bills and the market portfolio

To reduce risk, invest a greater proportion of assets in Treasury bills

To enhance expected return, invest a greater proportion of assets in the market portfolio

We do not know how to calculate the expected return (and hence the price) for a single risky asset The Capital Asset Pricing Model is needed

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THE MISSING LINK

We need to measure the Market risk that cannot be diversified away

58

Unique or Non-systematic

Risk

Market or Systematic

Risk

Diversifiable Non- Diversifiable

Total Standard Deviation (or Risk)

THE MISSING LINK

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Unique Risk Market Risk(measured with Beta)

The market will not compensate us for risk that

can be diversified away.

The market will compensate us for market risk – the risk that

cannot be diversified away

CAPM: SYSTEMATIC RISK IS RELEVANT

Systematic, or non-diversifiable, risk is caused by factors affecting the entire market interest rate changes changes in purchasing power change in business outlook

Unsystematic, or diversifiable, risk is caused by factors unique to the firm strikes regulations management’s capabilities

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PORTFOLIO DIVERSIFICATION When assets are put into a well-diversified portfolio, some of

the unique or nonsystematic risk is diversified away

The number of assets required to diversify away most of the unique risk varies with the correlation between the assets Canada’s capital markets are more highly correlated with the

natural resource sector than are the US capital markets Require more securities in Canada to diversify away most of the

unique risk

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DIVERSIFYING UNIQUE RISK

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Risk

Unique Risk

Number of Securities

Portfolio Risk

Market Risk

SYSTEMATIC RISK IS MEASURED BY BETA

Beta is a measure of the volatility of a security’s return compared to the volatility of the return on the Market Portfolio

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= Covariancej,MarketSecurity j VarianceMarket

β

CONCEPT OF BETA

64

To calculate Beta, use historical monthly

rates of return on both the security and the

market index.

Return on Stock A

Return on TSX Index

Slope equals Beta

SECURITY MARKET LINE (SML)

Shows the relationship between required rate of return and beta (ß).

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rf

ß

Security Market Line

RequiredRate ofReturn

ßj

k j

REQUIRED RATE OF RETURN The required return for any security j may be defined in terms

of systematic risk, βj, the expected market return, rm, and the expected risk free rate, rf.

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= + −j jf m fkβ ( )ˆ ˆ ˆr r r

^

^

SML: EXAMPLE

A security has a Beta of 1.25. If the yield on Treasury Bills is 5% and the return on the market portfolio is 11%, what is the expected return for holding the security?

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An investor expects a return of 12.5% to hold the security.

MARKET RISK PREMIUM

The reward for bearing risk Equal to (rm – rf)

Equal to the slope of security market line (SML)

Will increase or decrease with uncertainties about the future economic outlook the degree of risk aversion of investors

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SECURITY MARKET LINE (AGAIN)

69β

SMLReturn

βM

RM

A

B

A – return is too high; price is too low

B – return is too low; price is too high

CAPM ASSUMPTIONS

Investors hold well-diversified portfolios

Competitive markets

Borrow and lend at the risk-free rate

Investors are risk averse

No taxes

Investors are influenced by systematic risk

Freely available information

Investors have homogeneous expectations

No brokerage charges 70

CAPM DRAWBACKS

Estimating expected future market returns on historic returns.

Determining an appropriate rf

Determining the best estimate of β

Investors don’t totally ignore unsystematic risk

Betas are frequently unstable over time

Required returns are determined by macroeconomic factors

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MARKET EFFICIENCY

Capital markets are efficient if prices adjust fully and instantaneously to new information affecting a security’s prospective return.

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THREE DEGREES OF MARKET EFFICIENCY

Weak form

Semi-strong form

Strong form

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WEAK FORM MARKET EFFICIENCY

Security prices capture all of the information contained in the record of past prices and volumes

Implication: No investor can earn excess returns using historical price or volume information. Technical analysis should have no marginal value.

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SEMI-STRONG FORM MARKET EFFICIENCY

Security prices capture all of the information contained in the public domain.

Implication: No investor can earn excess returns using publicly available information. Fundamental analysis should have no marginal value.

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STRONG FORM MARKET EFFICIENCY

Security prices capture all information, both public and private.

Markets are quite efficient (but it is illegal to use private information for personal gain, when trading securities)!

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