1. Chapter 6
Risk and
Rates of
Return

2. Copyright © 2014 by Nelson Education Ltd.

3. Investment Returns: An Example
Dollar return:
$Received – $Invested
$1,100 – $1,000 = $100
Percentage return:
$Return / $Invested
$100/$1,000 = 0.10 = 10%
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4. Stand-Alone Risk
Typically, investment returns are not known with certainty.
An asset’s stand-alone risk pertains to the probability of earning a return on one asset less than that expected.
The greater the chance of a return far below the expected return, the greater the risk.
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5. Probability Distributions
Probability distributions are used to describe the certainty of returns by listing all possible returns and their probabilities.
Graphically, the tighter (i.e., more peaked) the probability distribution, the more likely it is that the actual returns will be close to the expected value.
The tighter the probability distribution, the lower the risk assigned to a stock.
Copyright © 2014 by Nelson Education Ltd.

6. Probability Distributions
Discrete Probability Distributions:
A limited or finite number of outcomes
Continuous Probability Distributions:
Unlimited or infinite number of outcomes

7. Expected Rate of Return
The weighted average of various possible outcomes; it is based on the probability that each outcome will occur where the outcomes’ probabilities as weights are used. It is the most likely return on a asset. + = n 2 1 k ˆ
return of
rate
Expected Pr L å = n 1 i k Pr
ki = the result of outcome i
Pri = the probability that outcome i will occur

8. Expected Rate of Return - Example

9. Measuring Total Risk: The standard deviation
The standard deviation, σk, measures the dispersion around the expected value of an asset’s risk.
Variance, 2—measures the variability of outcomes n 2 1 ) k ˆ ( = Pr - + s L n i 2 1 = ) k ˆ ( Pr - å
Standard deviation, 

10. The standard deviation - Example

11. The coefficient of variation
The coefficient of variation, CV, is a measure of relative dispersion that is useful in comparing various assets with differing risks and expected returns.
Coefficient of variation = Risk = σ
Return k

12. Risk aversion and Required Returns
Assuming that all investors are risk averse, that investor will ALWAYS choose to invest in portfolios with lower returns but with lower risk as well.
Risk averse investors will demand higher expected returns for riskier investments
Investors will hold a diversified portfolio of assets because the investor will diversify away a portion of the risk that is inherent in “putting all your eggs in one basket.”

13. Relationship between required rates of return and Risk for Risk Averse Investors
Risk
Return
k = kRF + RP
Risk Premium = RP
kRF
Risk-Free Return = kRF

14. Risk in a Portfolio Context
Investors often hold portfolios, not the asset of only one kind.
A particular asset going up or down is important, but what matters the most is the return on the portfolio and its risk.
Therefore, risk/return of an asset should be analyzed in terms of how that asset affects the overall risk/return of the portfolio in which it is held.
Copyright © 2014 by Nelson Education Ltd.

15. Portfolio Returns
The expected return on a portfolio is the weighted average of the expected returns on the individual assets forming the basket, with the weights being the fraction of the total portfolio invested in each asset.
Copyright © 2014 by Nelson Education Ltd.

16. Portfolio Risk and Return

17. Rate of Return distributions for perfectly positively and negatively correlated stock

18. Correlation Coefficient

19. Two-Asset Portfolios With Various Correlations
-1.0 < ρ < +1.0
The smaller the correlation, the greater the risk reduction potential.
If ρ = –1.0, complete risk reduction is possible.
If ρ = +1.0, no risk reduction is possible.
E(RP)
 = -1.0
 = 1.0
 = 0.2 σP
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20. Efficient Portfolios Portfolio is a collection of assets.
In a mean-variance (۸R – σ) space, a set of portfolios maximize expected return at each level of portfolio risk.
Equally, a set of portfolios minimize risk for each expected return.
Investors choose along the efficient set for the best mix of risk and return with their own risk attitudes.
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21. The Efficient Set for Two Assets
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22. Effects of Portfolio Size on Portfolio Risk for Average Stocks

23. Relevant – Irrelevant Risk
Relevant risk is the risk that cannot be reduced or diversified away (systematic, or market risk)
“Irrelevant” risk is the portion of total risk can be reduced through diversification (firm-specific, or unsystematic risk)

24. Relevant Risk Return kRF Risk Risk-Free Return Risk Premium based on
Risk
Return
kRF
Risk-Free Return
Risk Premium based on
systematic risk

25. The Capital Asset Pricing Model (CAPM)
The CAPM is a model developed to determine the required rate of return for an investment that considers the fact that some of the total risk associated with the investment can be diversified away; in essence, the model suggests that the risk premium associated with an investment should only be based on the risk that cannot be diversified away rather than the total risk; investors should not be rewarded for not diversifying—that is, they should not be paid for taking on risk that can be eliminated through diversification.

26. The concept of Beta
The market, or systematic, risk can be measured by comparing the return on an investment with the return on the market in general, or an average stock; the resulting measure is called the beta coefficient, and is identified using the Greek symbol β; graphically, β can be determined as follows:
Return on the
Market, kM
Stock, kj .
b =
slope

27. The concept of Beta
The beta coefficient shows how the returns associated an investment move with respect to the returns associated the market; because the market is very well diversified, its returns should be affected by systematic risk only—unsystematic risk should be completely diversified away in a portfolio that contains all investments in the market; thus, the beta coefficient is a measure of systematic risk because it gives an indication of the degree of movement in returns associated with an investment relative to the market, which contains only systematic risk; for example, an investment with β = 2.0 generally is considered twice as risky as the market, such that the risk premium associated with the investment should be twice the risk premium on the market.

28. Relative volatility of Assets S, R

29. Beta Coefficients for Selected Stocks

30. Interpreting Beta
The beta value of a company j stock is an index of the amount of company j’s systematic risk relative to that of the market portfolio
The beta value of a company j stock indicates the degree of responsiveness of expected return on the stock relative to movements in expected return of the market
The beta of a the stock j indicates the relative magnitude of the change in the stock’s risk premium as a result of a change in risk premium of the market portfolio
Beware: Beta does not indicate the degree of total volatility that can be expected on an investment’s return but only the extent to which expected return is likely to react to overall market movements

31. Portfolio Beta Coefficients
A portfolio’s beta, p is a function of the betas of the individual investments in the portfolio;
A portfolio beta is the weighted average of the betas associated with the individual investments contained in the portfolio
wj = % of total funds invested in asset j
j = asset j’s beta coefficient

32. Relationship between Risk and Rates of Return
Return = Risk-free rate + Risk Premium
kj = kRF RPInvest
= kRF (RPM)βj
= kRF (kM - kRF)βj
Capital Asset Pricing Model (CAPM)

33. Relationship between Risk and Rates of Return
Market risk premium = RPM = kM - kRF
where RPM is the return associated with the riskiness of a portfolio that contains all the investments in the market. RPM is based on how risk averse investors are on average. Because an investment’s beta coefficient indicates volatility relative to the market, we can use β to determine the risk premium for an investment. Investment risk premium = RPInvest = RPM x βInvest A more volatile investment—that is, an investment with a high β—will earn a higher risk than a less volatile investment

34. The Security Market Line (SML)
The CAPM Graph
Risk—Measured by 
Return, %
kRF
Risk-Free
Return
Risk Premium based on 
1.0 kM
Security Market Line, SML
RPM

35. CAPM - Example
Calculate the required return for Federal Express assuming it has a beta of 1.25, the rate on US T-bills is 5. %, and the expected return for the S&P 500 is 15%.
ki = 5% [15% - 5%]
ki = 17.5%

36. asset’s risk premium (12.5%) market risk premium (10%)
Sensitivity to risk-aversion / betas
ki%
SML
17.5%
15.0%
asset’s risk premium (12.5%)
market risk premium (10%)
RF = 5% bi
1.0
1.25