copyright © 2009 pearson prentice hall. all rights reserved. chapter 8 investor choice: risk and...

Copyright © 2009 Pearson Prentice Hall. All rights reserved.

Chapter 8

Investor Choice: Risk and Reward

Copyright © 2009 Pearson Prentice Hall. All rights reserved.8-2

Chapter 8 Outline

8.1 Measuring Risk and Reward8.2 Portfolios, Diversification, and Investor Preferences8.3 How to Measure Risk Contribution8.4 Expected Rates of Return and Market Betas for (Weighted)

Portfolios and Firms8.5 Spreadsheet Calculations for Risk and RewardAppendix: Trade-Off between Risk and Return8.6 An Investor’s Specific Trade-Off Between Risk and Reward8.7 A Shortcut Formula for the Risk of a Portfolio8.8 Graphing the Mean-Variance Efficient Frontier8.9 Adding a Risk-Free Asset


Investor Choice: Risk and RewardMeasuring Risk and Reward

• We are interested in return vs. risk.

• We consider scenarios to estimate risk.

• We may consider history to estimate risk.

• The spread of outcomes – dispersion – is important to our risk estimate.

• Low dispersion is less risky, and expected returns are lower.

• High dispersion is more risky, and expected returns are higher.

• No dispersion has no risk. It has a risk-free rate of return.


Investor Choice: Risk and Reward Measuring Reward: The Expected Rate of Return

• Expected Return is the probability-weighted average of all possible returns.

• If you had four equally likely scenarios with returns of -1%, +2%, +4%, and +11%, the expected return would be 4%.

• Calculated as a probability weighted sum:

• Calculated as a simple average also works in this case:

• Investors like assets with higher expected returns.

(r A ) 1 / 4 ( 1%) 1 / 4 (2%) 1 / 4 (4%) 1 / 4 (11%)

(r A ) 4%

(r A ) 1% 2% 4% 11%

4

(r A ) 4%


Investor Choice: Risk and Reward Standard Deviation: A=B>C>D

TABLE 8.1 Rates of Return on Five Investment Assets


Investor Choice: Risk and Reward Measuring Risk: Standard Deviation: A=B>C>D

FIGURE 8.1 Graphical Perspectives on Performance


Investor Choice: Risk and Reward Measuring Risk Standard Deviation: A=B>C>D

FIGURE 8.1 Graphical Perspectives on Performance (Cont.)


Investor Choice: Risk and Reward Measuring Risk: Standard Deviation of Returns

• Variance Calculation• Standard deviation and the variance of returns from the mean value are the

most common measures of portfolio risk.

• Variance is the difference from the mean squared, summed, and averaged. It is less useful than standard deviation because it states values in percentage squared.

• Calculating variance with mean = 4% and values of -1%, +2%, +4%, and +11% with equal likelihood:

Var(r A ) ( 1% 4%)2 (2% 4%)2 (4% 4%)2 (11% 4%)2

4

Var(r A ) ( 5%)2 ( 2%)2 (0%)2 (7%)2

419.5%%

Var(r A ) [rs1 (r)]2 [rs2 (r)]2 [rs3 (r)]2 [rs4 (r)]2

4

Var(r A ) sum overall scenarios [r s in scenario s (r)]2

number of scenarios


Investor Choice: Risk and Reward Measuring Risk: Standard Deviation of Returns

• Standard deviation calculation• Standard deviation is the square root of variance.

• Standard deviation is the most common measure of portfolio risk.

• Calculating standard deviation with mean = 4% and values of -1%, +2%, +4%, and +11% with equal likelihood:

• Investors measure reward by the expected return on the overall portfolio.Investors measure risk by the standard deviation on the overall portfolio.

Var(r A ) ( 1% 4%)2 (2% 4%)2 (4% 4%)2 (11% 4%)2

4

Var(r A ) ( 5%)2 ( 2%)2 (0%)2 (7%)2

419.5%%

Sdv(r A ) [rs1 (r)]2 [rs2 (r)]2 [rs3 (r)]2 [rs4 (r)]2

4 19.5%%

Sdv(r A ) 19.5%% 4.42%


Investor Choice: Risk and Reward Portfolio, Diversification, & Investor Preferences

• Diversification can reduce risk.

• The portfolio return formula is weight X return:

• If 50% were invested in A and 50% in B:Portfolio return = 50% • return A + 50% • return B

• Or if 25% probability in 4 Scenarios for an asset:Asset return =

25% • return 1 + 25% • return 2 + 25% • return 3 + 25% • return 4

• Formula:

rp(w1 ,w2 ,w3 ....wN ) w1 r1 ..... wN rN



FIGURE 8.2 Rate of Return Outcomes for A, B, and the 50%-50% Combination Portfolio P



FIGURE 8.2 Rate of Return Outcomes for A, B, and the 50%-50% Combination Portfolio P (Cont.)


Investor Choice: Risk and Reward Assume Investors Care Only About Risk and Reward

• More diversification has less risk.

• The market portfolio has a little of every asset.The market portfolio is a highly diversified portfolio.

• If investors hold the market portfolio, a corporate manager trying to attract investors to an alternative investment needs:

• Projects that offer higher expected rates of return• Projects that offer diversification benefits

• Diversification reduces overall portfolio risk.

• Diversification could be called imperfect correlation (non-synchronicity).

• Corporate managers should believe investors hold diversified portfolios.


Investor Choice: Risk and Reward An asset’s own risk is not a good measure for its risk contribution to a portfolio

• The important risk for an investor is an asset’s contribution to the overall risk of a diversified portfolio.

• An investor’s most important risk is an asset’s risk in the market portfolio.

• Assets with higher returns may not offer any diversification benefits compared to low-return assets once the returns are combined with the market portfolio’s returns due to high correlation with the market.

• Assets with lower correlations with the market portfolio offer the greatest amount of diversification and lower portfolio risk.

• Standard deviation is not the proper measure to determine if an asset will reduce portfolio risk.

• Beta is the better risk measure.


Investor Choice: Risk and Reward Combining the Market with Asset C or D

FIGURE 8.3 Combining M with either C or D


Investor Choice: Risk and Reward Combining the Market with Asset C or D

FIGURE 8.3 Combining M with either C or D (Cont.)


Investor Choice: Risk and Reward Beta is a good measure for portfolio risk contribution.

• Stocks with high comovement with the market have high betas.

• Stocks with low comovement with the market have low betas.

• Low and negative beta stocks offer more diversification benefits.

• Beta is the slope of the line of return plotted vs. the market return.

• Beta is the coefficient of the X term in this formula: y = + • x.

• A beta of 1 is a diagonal line, and a beta of 0 is a horizontal line.

• For C and D, the equations are:

rC .49% (1.128) rMrD 10.51% ( 2.128) rMr1 Y rM


Investor Choice: Risk and Reward Diversification and Beta

• Diversification works well if asset returns move opposite the portfolio.

• Negative beta assets have negative expected returns to the market’s positive return.

• Zero beta assets have returns that are independent of the market return.

• The market beta measures the project risk contribution to the market portfolio.

• Risk averse investors will pay more for assets with lower market betas or risk.

• Exception:

Entrepreneurs often do not hold a diversified portfolio and should care only about their project’s standard deviation.


Investor Choice: Risk and Reward Market Beta is the Slope of a Line with Market Return on the X Axis and the Asset Return on the Y Axis.

FIGURE 8.4 Possible Outcomes: Rates of Return versus Market Rate of Return


Investor Choice: Risk and Reward Computing Market Betas from Historical Returns

• Find the deviation from the mean for the market returns.

• Find the deviation from the mean for the asset returns.

• Find the variance of the market by summing the squares of the deviations and dividing by the number of scenarios.

• Find the covariance between the asset returns and the market returns by multiplying their deviations from means together and dividing by the number of scenarios.

• To find beta, divide the covariance between the asset returns and the market returns by the variance of the market.

Cov( rM , rA )

Var( rM )


Investor Choice: Risk and Reward Computing Market Betas from Scenarios


Investor Choice: Risk and Reward Computing Market Betas from Historical Returns

• Real-World Issues

• Betas are usually computed from historical rates of return.

• The use of weekly or daily returns to calculate beta results in better estimates than the use of yearly or monthly returns.

• Most researchers use the most recent 3 to 5 years of data for beta estimates.

• Historical betas are not good predictors of future betas.

• Corrections for high historical betas are to –

• either shrink the beta by averaging it with the market beta of 1, or • to sample betas of similar industries and projects.


Investor Choice: Risk and RewardWhy not Correlation as a risk measure?

• Correlation has no scale: -100% to +100%.

• Correlation is easy to interpret, but doesn’t quantify the amount of risk.

• Beta quantifies the amount of risk as well as the direction of comovement.

• Beta is the better measure of risk.


Investor Choice: Risk and RewardInterpreting Typical Stock Market Betas

• Beta = 1 Same risk as market• Beta < 1Risk-reducing asset• Beta > 1Risk-increasing asset• Beta = 0No risk asset (rare in a stock)

• Beta shows the degree by which a firm’s value tends to change with changes in the market.

• A beta of 1.9 at a firm such as Vivendi means that the firm’s value would increase 1.9 times as much as the market.

• But firm value would fall 1.9 times more than the market if the market fell.

• High betas amplify market moves and risk.

• When markets become volatile, you should avoid high beta assets.


Investor Choice: Risk and RewardInterpreting Typical Stock Market Betas

TABLE 8.2 Some Market Betas and Capitalizations on May 10, 2008


Investor Choice: Risk and RewardExpected Rates of Return and Market Betas for (Weighted) Portfolios and Firms

• Projects are often weighted by the value of their assets.

• A value-weighted portfolio is one of most common weighting methods.

• Equal-weighted portfolios give all parts equal weight regardless of value.

• Different weights on the assets in a portfolio will lead to different returns and risks for your portfolio.

• While returns can be averaged, standard deviations cannot be averaged.

• Betas can be averaged, too.


Investor Choice: Risk and RewardExpected Rates of Return Example

• Consider C and D with actual rates of return of -12% and +12%.

• If you put 1/3 in C and 2/3 in D, your portfolio return will be -4%.

• Portfolio return formula for Scenario 4:

• If C has an expected return of of +5% and D has an expected return of 2%, we can find the expected portfolio return of 3%.

• For beta, 1/3 of C’s beta of 1.128 plus 2/3 of D’s beta of -2.128 equals -1.043.

rCDD,S4

1

3 (12%)

2

3 ( 12%) 4%

rCDD,S4 wC (rC ,S4 ) wD (rD,S4 )

rCDD,S4 wC ( rC ) wD ( rD )

1

3 (5%)

2

3 (2%) 3%

CDD wC C wD D

1

3 (1.128)

2

3 ( 2.128) 1.04


Investor Choice: Risk and RewardEquity and Debt: Firm Beta

• Consider a firm with $100M in debt and $300M in equity. Value = $400M.

• Debt is ¼ of firm value with beta = 0.40• Equity is ¾ of firm value with beta = 2.00

• The overall firm beta equals 1.60.

¼ • .40 + ¾ • 2.00 = 1.60

• Firms are portfolios of debt and equity betas.• These betas average to reflect overall firm risk or beta.


Investor Choice: Risk and RewardSpreadsheet Calculations

• Average(range) computes the average of a range of numbers.

• Varp(range) computes the population variance. Uses N observations.

• Var(range) computes the variance of a data sample. Uses N-1 observations.

• Stdevp(range) computes the population standard deviation (N).

• Stdev(range) computes the standard deviation of a data sample (N-1).

• Covar(range-1, range-2) computes the population covariance between two series.

• Correl(range-1, range-2) computes the correlation between two series.

• Slope(range-Y, range-X) computes a beta. • If range-Y = asset return and range-X = market return, then is market beta.


Investor Choice: Risk and RewardSpreadsheet Example:

TABLE 8.3 The Computer Spreadsheet


Investor Choice: Risk and RewardStatistical Nuances

• When we use historical data to find variance, standard deviation, return, correlation, and covariance, we assume the future will repeat past experience.

• When comparing the output of calculators and spreadsheets, be aware that some will use N-1 and others will use N as their statistical divisor.

• Usually, it doesn’t matter because finance data often has many observations.

• If you know the scenario outcomes, you should use N observations.

• Beta is the same with either N or N-1 since both the numerator and denominator have been calculated the same way.

• If you see beta with a “hat,” then beta was estimated from historical data.


TABLE 8.4 Portfolios Used to Illustrate Mean-Variance Combinations


FIGURE 8.5 The Risk-Reward Trade-Off between H and I: More Portfolios


FIGURE 8.6 The Risk-Reward Trade-Off between H and I: Sets


FIGURE 8.7 The Risk-Reward Trade-Off between H and F


FIGURE 8.8 The Risk-Reward Trade-Off between L and F


FIGURE 8.9 The Risk-Reward Trade-Off between T and F

copyright © 2009 pearson prentice hall. all rights reserved. chapter 8 investor choice: risk and...

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