Chapter 11 Return and Risk the Capital Asset Pricing Model.Pdf

University of Science and Technology Beijing

Dongling School of Economics and management

ChapterChapter 1111 ReturnReturn andand RiskRisk TheThe CapitalCapital AssetAsset PricingPricing ModelModel

Nov. 2012

Dr. Xiao Ming USTB 1 Key Concepts and Skills

• Know how to calculate the return on an investment • Know how to calculate the standard deviation of an investment’s returns • Understand the historical returns and risks on various types of investments • Understand the importance of the normal distribution • Understand the difference between arithmetic and geometric average returns

Dr. Xiao Ming USTB 2 Key Concepts and Skills

• Know how to calculate expected returns • Know how to calculate covariances, correlations, and betas • Understand the impact of diversification • Understand the systematic risk principle • Understand the security market line • Understand the risk-return tradeoff • Be able to use the Capital Asset Pricing Model

Dr. Xiao Ming USTB 3 Chapter Outline

11.1 Individual Securities 11.2 Expected Return, Variance, and Covariance 11.3 The Return and Risk for Portfolios 11.4 The Efficient Set for Two Assets 11.5 The Efficient Set for Many Assets 11.6 Diversification 11.7 Riskless Borrowing and Lending 11.8 Market Equilibrium 11.9 Relationship between Risk and Expected Return (CAPM)

Dr. Xiao Ming USTB 4 11.1 Individual Securities

• The characteristics of individual securities that are of interest are the: – Expected Return – Variance and Standard Deviation – Covariance and Correlation (to another security or index)

Dr. Xiao Ming USTB 5 11.2 Expected Return, Variance, and Covariance

Consider the following two risky asset world. There is a 1/3 chance of each state of the economy, and the only assets are a stock fund and a bond fund.

Rate of Return Scenario Probability Stock Fund Bond Fund Recession 33.3% -7% 17% Normal 33.3% 12% 7% Boom 33.3% 28% -3%

Dr. Xiao Ming USTB 6 Expected Return

Stock Fund Bond Fund Rate of Squared Rate of Squared Scenario Return Deviation Return Deviation Recession -7% 0.0324 17% 0.0100 Normal 12% 0.0001 7% 0.0000 Boom 28% 0.0289 -3% 0.0100 Expected return 11.00% 7.00% Variance 0.0205 0.0067 Standard Deviation 14.3% 8.2%

Dr. Xiao Ming USTB 7 Expected Return

rE )( 1 %)7( 1 %)12( 1 ×+×+−×= %)28( S 3 3 3

rE S = %11)(

Dr. Xiao Ming USTB 8 Variance

2 =−− 0324.%)11%7(

Dr. Xiao Ming USTB 9 Variance

Dr. Xiao Ming USTB 10 Standard Deviation

= 0205.0%3.14

Dr. Xiao Ming USTB 11 Covariance

Stock Bond Scenario Deviation Deviation Product Weighted Recession -18% 10% -0.0180 -0.0060 Normal 1% 0% 0.0000 0.0000 Boom 17% -10% -0.0170 -0.0057 Sum -0.0117 Covariance -0.0117

“Deviation” compares return in each state to the expected return. “Weighted” takes the product of the deviations multiplied by the probability of that state.

Dr. Xiao Ming USTB 12 Correlation

baCov ),( ρ = σσba − 0117. ρ = −= 998.0 )082)(.143(.

Dr. Xiao Ming USTB 13 11.3 The Return and Risk for Portfolios

Note that stocks have a higher expected return than bonds and higher risk. Let us turn now to the risk-return tradeoff of a portfolio that is 50% invested in bonds and 50% invested in stocks.

Dr. Xiao Ming USTB 14 Portfolios

Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.0016 Normal 12% 7% 9.5% 0.0000 Boom 28% -3% 12.5% 0.0012

Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08%

The rate of return on the portfolio is a weighted average of the returns on the stocks and bonds in the portfolio:

= + rwrwr SSBBP = × − + × %)17(%50%)7(%50%5

Dr. Xiao Ming USTB 15 Portfolios

Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.0016 Normal 12% 7% 9.5% 0.0000 Boom 28% -3% 12.5% 0.0012

Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08% The expected rate of return on the portfolio is a weighted average of the expected returns on the securities in the portfolio. P = + rEwrEwrE SSBB )()()(

= × + × %)7(%50%)11(%50%9

Dr. Xiao Ming USTB 16 Portfolios

Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.0016 Normal 12% 7% 9.5% 0.0000 Boom 28% -3% 12.5% 0.0012

Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08% The variance of the rate of return on the two risky assets portfolio is 2 2 2 σ P = (w σ BB (w) σ SS ++ 2(w) σ )(w )ρσ BSSSBB

where ρBS is the correlation coefficient between the returns on the stock and bond funds.

Dr. Xiao Ming USTB 17 Portfolios

Rate of Return Scenario Stock fund Bond fund Portfolio squared deviation Recession -7% 17% 5.0% 0.0016 Normal 12% 7% 9.5% 0.0000 Boom 28% -3% 12.5% 0.0012

Expected return 11.00% 7.00% 9.0% Variance 0.0205 0.0067 0.0010 Standard Deviation 14.31% 8.16% 3.08% Observe the decrease in risk that diversification offers. An equally weighted portfolio (50% in stocks and 50% in bonds) has less risk than either stocks or bonds held in isolation.

Dr. Xiao Ming USTB 18 11.4 The Efficient Set for Two Assets

% in stocks Risk Return 0% 8.2% 7.0% Portfolo Risk and Return Combinations 5% 7.0% 7.2% 10% 5.9% 7.4% 100%

15% 4.8% 7.6% 12.0% stocks 20% 3.7% 7.8% 11.0% 10.0%

25% 2.6% 8.0% Return 30% 1.4% 8.2% Portfolio 9.0% 100% 8.0% 35% 0.4% 8.4% 7.0% bonds 40% 0.9% 8.6% 6.0% 5.0% 45% 2.0% 8.8% 50.00% 3.08% 9.00% 55% 4.2% 9.2% 0.0%Portfolio 5.0% Risk (standard 10.0% deviation) 15.0% 20.0% 60% 5.3% 9.4% 65% 6.4% 9.6% 70% 7.6% 9.8% 75% 8.7% 10.0% We can consider other 80% 9.8% 10.2% 85% 10.9% 10.4% portfolio weights besides 90% 12.1% 10.6% 50% in stocks and 50% in 95% 13.2% 10.8% 100% 14.3% 11.0% bonds.

Dr. Xiao Ming USTB 19 The Efficient Set for Two Assets

% in stocks Risk Return Portfolo Risk and Return Combinations 0% 8.2% 7.0% 5% 7.0% 7.2% 10% 5.9% 7.4% 15% 4.8% 7.6% 12.0% 100% 20% 3.7% 7.8% 11.0% stocks 25% 2.6% 8.0% 10.0%

30% 1.4% 8.2% 9.0% 35% 0.4% 8.4% 8.0% 100% 40% 0.9% 8.6% Portfolio Return7.0% 6.0% bonds 45% 2.0% 8.8% 5.0% 50% 3.1% 9.0% 55% 4.2% 9.2% 60% 5.3% 9.4% Portfolio Risk (standard deviation) 65% 6.4% 9.6% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% 70% 7.6% 9.8% Note that some portfolios are 75% 8.7% 10.0% 80% 9.8% 10.2% “better” than others. They have 85% 10.9% 10.4% 90% 12.1% 10.6% higher returns for the same level of 95% 13.2% 10.8% risk or less. 100% 14.3% 11.0%

Dr. Xiao Ming USTB 20 Portfolios with Various Correlations

ρ = -1.0 100% stocks return

ρ = 1.0 = 0.2 100% ρ bonds σ • Relationship depends on correlation coefficient -1.0 < ρ < +1.0 • If ρ = +1.0, no risk reduction is possible • If ρ = –1.0, complete risk reduction is possible

Dr. Xiao Ming USTB 21 11.5 The Efficient Set for Many Securities return

Individual Assets

σP Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios.

Dr. Xiao Ming USTB 22 The Efficient Set for Many Securities

r ntie fro ent return ci effi minimum variance portfolio

Individual Assets

σP The section of the opportunity set above the minimum variance portfolio is the efficient frontier.

Dr. Xiao Ming USTB 23 Announcements, Surprises, and Expected Return

• The return on any security consists of two parts. – First, the expected returns – Second, the unexpected or risky returns • A way to write the return on a stock in the coming month is:

+= URR where R theis expected theofpart return U theis unexpected theofpart return

Dr. Xiao Ming USTB 24 Announcements, Surprises, and Expected Returns

• Any announcement can be broken down into two parts, the anticipated (or expected) part and the surprise (or innovation): – Announcement = Expected part + Surprise.

• The expected part of any announcement is the part of the information the market uses to form the expectation, R, of the return on the stock. • The surprise is the news that influences the unanticipated return on the stock, U .

Dr. Xiao Ming USTB 25 Diversification and Portfolio Risk

• Diversification can substantially reduce the variability of returns without an equivalent reduction in expected returns. • This reduction in risk arises because worse than expected returns from one asset are offset by better than expected returns from another. • However, there is a minimum level of risk that cannot be diversified away, and that is the systematic portion.

Dr. Xiao Ming USTB 26 Portfolio Risk and Number of Stocks

In a large portfolio the variance terms are σ effectively diversified away, but the covariance terms are not. Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk Portfolio risk Nondiversifiable risk; Systematic Risk; Market Risk n

Dr. Xiao Ming USTB 27 Risk: Systematic and Unsystematic

A systematic risk is any risk that affects a large number of assets, each to a greater or lesser degree. An unsystematic risk is a risk that specifically affects a single asset or small group of assets. Unsystematic risk can be diversified away. Examples of systematic risk include uncertainty about general economic conditions, such as GNP, interest rates or inflation. On the other hand, announcements specific to a single company are examples of unsystematic risk.

Dr. Xiao Ming USTB 28 Total Risk

• Total risk = systematic risk + unsystematic risk • The standard deviation of returns is a measure of total risk. • For well-diversified portfolios, unsystematic risk is very small. • Consequently, the total risk for a diversified portfolio is essentially equivalent to the systematic risk.

Dr. Xiao Ming USTB 29 Optimal Portfolio with a Risk-Free Asset

100% return stocks

rf 100% bonds σ In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills.

Dr. Xiao Ming USTB 30 11.7 Riskless Borrowing and Lending

L M C 100% return stocks Balanced fund

rf 100% bonds σ Now investors can allocate their money across the T-bills and a balanced mutual fund.

Dr. Xiao Ming USTB 31 Riskless Borrowing and Lending

L M C efficient frontier return

σP

With a risk-free asset available and the efficient frontier identified, we choose the capital allocation line with the steepest slope.

Dr. Xiao Ming USTB 32 11.8 Market Equilibrium

L M efficient frontier

return C

σP With the capital allocation line identified, all investors choose a point along the line—some combination of the risk-free asset and the market portfolio M. In a world with homogeneous expectations, M is the same for all investors.

Dr. Xiao Ming USTB 33 Market Equilibrium

L M C 100% return stocks Balanced fund

rf 100% bonds σ Where the investor chooses along the Capital Market Line depends on her risk tolerance. The big point is that all investors have the same CML.

Dr. Xiao Ming USTB 34 Risk When Holding the Market Portfolio

• Researchers have shown that the best measure of the risk of a security in a large portfolio is the beta (β ) of the security. • Beta measures the responsiveness of a security to movements in the market portfolio (i.e., systematic risk).

, RRCov Mi )( βi = 2 σ RM )(

Dr. Xiao Ming USTB 35 Estimatingβwith Regression

ee iinn LL tiicc iisst eerr cctt rraa hhaa Security Returns Security Returns CC SlopeSlope == ββi

Return on market %

RRi == αα i ++ ββi RRm ++ eei

Dr. Xiao Ming USTB 36 The Formula for Beta

, RRCov Mi )( σ Ri )( βi = 2 = ρ σ RM )( σ RM )(

Clearly, your estimate of beta will depend upon your choice of a proxy for the market portfolio.

Dr. Xiao Ming USTB 37 11.9 Relationship between Risk and Expected Return (CAPM)

• Expected Return on the Market:

M RR F += Market Risk Premium • Expected return on an individual security:

i βiF M −×+= RRRR F )(

Market Risk Premium

This applies to individual securities held within well-diversified portfolios.

Dr. Xiao Ming USTB 38 Expected Return on a Security

• This formula is called the Capital Asset Pricing Model (CAPM):

i βiF M −×+= RRRR F )(

Expected Risk- Beta of the Market risk return on = + × free rate security premium a security

• Assume βi = 0, then the expected return is RF. • Assume βi = 1, then = RR Mi

Dr. Xiao Ming USTB 39 Relationship Between Risk & Return

i βiF M −×+= RRRR F )(

RM Expected return

1.0 β

Dr. Xiao Ming USTB 40 Relationship Between Risk & Return

%5.13

%3 Expected return

1.5 β

βi = 5.1 RF = %3 R M = %10

Ri =−×+= %5.13%)3%10(5.1%3

Dr. Xiao Ming USTB 41 Quick Quiz

• How do you compute the expected return and standard deviation for an individual asset? For a portfolio? • What is the difference between systematic and unsystematic risk? • What type of risk is relevant for determining the expected return? • Consider an asset with a beta of 1.2, a risk-free rate of 5%, and a market return of 13%. – What is the expected return on the asset?

Dr. Xiao Ming USTB 42 Dr. Xiao Ming USTB 43