Chapter 5The Trade-off between Risk
and Return
© 2007 Thomson South-Western
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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
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Probability Distribution
Probability distribution tells us what outcomes are possible and associates a probability with each outcome.Normal distribution
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Two Assets With Same Expected Return But Different Distributions
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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$66$
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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
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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.
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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.
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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
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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
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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?
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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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Effect of Diversification on Portfolio Variance
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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)
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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
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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
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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
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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