Chapter 5The Trade-off between Risk

and Return

© 2007 Thomson South-Western

2

Introduction to Risk and Return

Valuing risky assets - a task fundamental to financial management

Three-step procedure for valuing a risky asset

1. Determine the asset’s expected cash flows2. Choose discount rate that reflects asset’s risk3. Calculate present value (PV cash inflows - PV

outflows)

The three-step procedure is called discounted cash flow (DCF) analysis.

3

Historical vs. Expected Returns

Decisions Must Be Based On Expected Returns

There Are Many Ways to Estimate Expected Returns

Assume That Expected Return Going Forward Equals the Average Return in the Past

Simple Way to Estimate Expected Return

4

Risk and Return Fundamentals

Equity risk premium: the difference in equity returns and returns on safe investments

• implies that stocks are riskier than bonds or bills

• trade-off always arises between expected risk and expected return

5

Risk Aversion

Risk Neutral• Investors Seek the Highest Return

Without Regard to Risk

Risk Seeking• Investors Have a Taste for Risk and Will

Take Risk Even If They Cannot Expect a Reward for Doing So

Risk Averse • Investors Do Not Like Risk and Must Be Compensated For Taking It

Historical Returns on Financial Assets Are Consistent with a Population of Risk-Averse Investors

6

Probability Distribution

Probability distribution tells us what outcomes are possible and associates a probability with each outcome.Normal distribution

7

Two Assets With Same Expected Return But Different Distributions

8

Return on an Asset

t

tttt P

CPPR 11

1

Return - The Total Gain or Loss Experienced on an Investment Over a Given Period of Time.

An example....

Investor Bought Utilyco for $60/share

Dividend = $6/share

Sold for $66/share

+- = R util

%2060$

12$60$

6$60$66$

9

Arithmetic Versus Geometric Returns

• Arithmetic return the simple average of annual returns: best estimate of expected return each year.

• Geometric average return the compound annual return to an investor who bought and held a stock t years:

• Geometric avg return= (1+R1)(1+R2)(1+R3)….(1+Rt)]

1/t – 1

The Difference Between Arithmetic Returns and Geometric Returns Gets Bigger the More Volatile the Returns Are

10

Arithmetic Versus Geometric Returns

The Difference Between Arithmetic Returns and Geometric Returns Gets Bigger the More Volatile the Returns Are

AAR = 6.25%

GAR = 5.78%

An example....

Year Return

5 -10%

7 +12%

9 +15%

11 + 8%

11

Distribution of Historical Stock Returns, 1900 - 2003

Histogram of Nominal Returns on Equities 1900-2003

<-30 -30 to -20 to -10 to 0 to 10 to 20 to 30 to 40 to >50-20 -10 0 10 20 30 40 50

Percent return in a given year

Probability distribution for future stock returns is unknown. We can approximate the unknown

distribution by assuming a normal distribution.

12

Variance

A reasonable way to define risk is to focus on the dispersion of returns

• most common measure of dispersion used as a proxy for risk in finance is variance, or its square root, the standard deviation.

• distribution’s variance equals the expected value of squared deviations from the mean.

13

Expected Return For A Portfolio

Most Investors Hold Multiple Asset Portfolios

Key Insight of Portfolio Theory: Asset Return Adds Linearly, But Risk Is (Almost Always) Reduced in a Portfolio

)NE(RNw...)3E(R3w)2E(R2w)1E(R1w)pE(R

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Two-Asset Portfolio Standard Deviation

2112212

22

22

12

12 2 wwwwp

2Deviation Standard p

Correlation Between Stocks Influences Portfolio Volatility

15

Correlation Coefficients And Risk Reduction For Two-Asset Portfolios

10%

15%

20%

25%

0% 5% 10% 15% 20% 25%

Standard Deviation of Portfolio Returns

Exp

ecte

d R

etu

rn o

n t

he

Po

rtfo

lio

is +1.0

-1.0 < <1.0

is -1.0

16

Portfolios of More Than Two Assets

Five-Asset Portfolio

)()(

)()()()(

5544

332211

REwREw

REwREwREwRE p

Expected Return of Portfolio Is Still The Average Of Expected Returns Of The Two Stocks

How Is The Variance of Portfolio Influenced By Number Of Assets in Portfolio?

17

5

4

3

2

154321Asset

The Covariance Terms Determine To A Large Extent The Variance Of The Portfolio

5

4

3

2

154321Asset

5

4

3

2

154321Asset

Variance of Individual Assets Account Only for 1/25th of the Portfolio Variance

Variance – Covariance Matrix

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2

5

1

12

2

5

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1

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2

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1

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1

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1

18

Effect of Diversification on Portfolio Variance

19

Portfolio Risk

variance cannot fall below the average covariance of securities in the portfolioUndiversifiable risk (systematic risk, market

risk)Only systematic risk is priced in the market. Beta is one way to measure the systematic risk of

an asset.

Diversifiable risk (unsystematic risk, idiosyncratic risk, or unique risk)

20

What Is a Stock’s Beta?

Beta Is a Measure of Systematic Risk

m

imi

What If Beta > 1 or Beta <1?

• The Stock Moves More Than 1% on Average When the Market Moves 1% (Beta > 1)

• The Stock Moves Less Than 1% on Average When the Market Moves 1% (Beta < 1)

What If Beta = 1?

• The Stock Moves 1% on Average When the Market Moves 1%

• An “Average” Level of Risk

21

Diversifiable And Non-Diversifiable Risk

As Number of Assets Increases, Diversification Reduces the Importance of a Stock’s Own VarianceDiversifiable risk, unsystematic risk

Only an Asset’s Covariance With All Other Assets Contributes Measurably to Overall Portfolio Return VarianceNon-diversifiable risk, systematic risk

22

How Risky Is an Individual Asset?

First Approach – Asset’s Variance or Standard Deviation

What Really Matters Is Systematic Risk….How an Asset Covaries With Everything Else

Use Asset’s Beta

23

The Impact Of Additional Assets On The Risk Of A Portfolio

Number of Securities (Assets) in PortfolioNumber of Securities (Assets) in Portfolio

Po

rtfo

lio R

isk,

k p

Nondiversifiable RiskNondiversifiable Risk

Diversifiable RiskDiversifiable Risk

Total riskTotal risk

1 5 10 15 20 25 1 5 10 15 20 25