Sunil

Risk and Return

Holding Period Return Three month ago, Peter Lynch purchased 100 shares of Iomega Corp. at $50 per share. Last month, he received dividends of $0.25 per share from Iomega. These shares are worth $56 each today. •Compute Peter’s holding period return from his investment in Iomega common shares.

Probability Concept •Random variable ⇒Something whose value in the future is subject to uncertainty.

•Probability ⇒The relative likelihood of each possible outcome (or value) of a random variable ⇒Probabilities of individual outcomes cannot be negative nor greater than 1.0 ⇒Sum of the probabilities of all possible outcomes must equal 1.0

•Moments ⇒Mean, Variance (or Standard deviation), covariance

Computing the Basic Statistics A security analyst has prepared the following probability distribution of the possible returns on the common stock shares of two companies: Compu-Graphics Inc. (CGI) and Data Switch Corp. (DSC). Probability Return on Return on CGI DSC 0.30 10% 40% 0.50 14% 16% 0.20 20% 20%

The Mean

For CGI, the mean (or expected) return is: 3

µ =∑pn x n CGI

n =1

=0.30(10%)+0.50(14%)+0.20( 20%) =14.00%

Similarly, the mean return for DSC is 24.00%

The Variance and Standard Deviation

The variance of CGI’s returns is: N

σ x2 =∑pn ( x n −µx )

2

n=1

=0.30(10−14)2 +0.50(14−14 ) +0.20( 20−14)2 =12.00 2

The Standard Deviation of CGI’s return is: σ x = σ = 12.00 =3.46% 2 x

The Covariance

The covariance of the returns on CGI and DSC is: σ =∑pn ( x n −µx )( y n −µy ) N

CD

n=1

=0.30(10−14)( 40−24)+0.50(14−14)(16−24) +0.20( 20−14)( 20−24) =− 24.00

The Correlation Coefficient

The correlation coefficient between CGI and DSC is:

σxy ρxy = σ x σy − 24.00 = (3.46 )(10.58 ) =−0.655

Summary of Results for CGI and DSC

CGI Mean Standard Deviation Correlation Coefficient

14.00% 3.46%

DSC 24.00% 10.58% -0.655

Portfolio Securities • A portfolio is a combination of two or more securities. • Combining securities into a portfolio reduces risk. • An efficient portfolio is one that has the highest expected return for a given level of risk. • We will look at two-asset portfolios in fair detail.

Portfolio Expected Return and Risk Expected Return

The Expected Returns of the Securities

Risk

The

The Risk

The

The

Portfolio

of the

Portfolio

Correlation

Weights

Securities

Weights

Coefficients

Portfolio Weights and Expected Return Portfolio Weights CGI

DSC

1.00 0.75 0.67 0.50 0.25 0.00

0.00 0.25 0.33 0.50 0.75 1.00

Portfolio’s Expected Return 14.00% 16.50% 17.33% 19.00% 21.50% 24.00%

Standard Deviation 3.46% 2.18% 2.64% 4.36% 7.40% 10.58%

Portfolio Expected Return and Risk Home

Expected Return

25%

DSC

20%

15%

CGI

10% 0%

5% Standard Deviation

10%

Diversification of Risk •Note that while the expected return of the portfolio is between those of CGI and DSC, its risk is less than either of the two individual securities. •Combining CGI and DSC results in a substantial reduction of risk diversification! •This benefit of diversification stems primarily from the fact that CGI and DSC’s returns are not perfectly correlated.

Correlation Coefficient and Portfolio Risk • All else being the same, lower the correlation coefficient, lower is the risk of the portfolio. – Recall that the expected return of the portfolio is not affected by the correlation coefficient.

• Thus, lower the correlation coefficient, greater is the diversification of risk.

Correlation Coefficient and Portfolio Risk Consider stocks of two companies, X and Y. The table below gives their expected returns and standard deviations. Stock X

Stock Y

10% 12%

25% 30%

Expected Return Standard Deviation

Plot the risk and expected return of portfolios of these two stocks for the following (assumed) correlation coefficients:

-1.0 0.5 0.0

+0.5 +1.0

Correlation Coefficient and Portfolio Risk Y

Expected Return

25%

Correlation Coefficient -1.0

15%

-0.5 0.0

X

+0.5 +1.0

5% 0%

5%

10%

15% 20% Standard Deviation

25%

30%

35%

Portfolios with Many Assets •The above framework can be expanded to the case of portfolios with a large number of stocks. •In forming each portfolio, we can vary ⇒the number of stocks that make up the portfolio, ⇒the identity of the stocks in the portfolio, and ⇒the weights assigned to each stock.

•Look at the plot of the expected returns versus the risk of these portfolios

All Combinations of Risky Assets

Efficient Frontier •A portfolio is an efficient portfolio if ⇒no other portfolio with the same expected return has lower risk, or ⇒no other portfolio with the same risk has a higher expected return.

•Investors prefer efficient portfolios over inefficient ones. •The collection of efficient portfolio is called an efficient frontier.

Efficient Frontier

µ (expected return)

F

E

σ (risk)

Choosing the Best Risky Asset •Investors prefer efficient portfolios over inefficient ones. •Which one of the efficient portfolios is best? •We can answer this by introducing a riskless asset. ⇒There is no uncertainty about the future value of this asset (i.e. the standard deviation of returns is zero). Let the return on this asset be rf. ⇒For practical purposes, 90-day U.S. Treasury Bills are (almost) risk free.

Combinations of a Risk Free and a Risky Asset

µ (expected return)

F

N

E rf

σ (risk)

Best Risky Asset

µ (expected return)

F M

E rf

σ (risk)

The Capital Market Line •Assume investors can lend and borrow at the risk free rate of interest. ⇒borrowing entails a negative investment in the riskless asset.

•Since every investor hold a part of the “best” risky asset M, M is the market portfolio. •The Market portfolio consists of all risky assets. ⇒Each asset weight is proportional to its market value.

The Capital Market Line

 µm − rf µp =rf +  σm

 σ p 

µ (expected return)

The Capital Market Line

F M

E

rf

σ (risk)

Next Coverage • Understand the determination of the expected rate of return  Capital Asset Pricing Model • Decomposition of Risk: Systematic Vs. Unsystematic.

• Explain the importance of asset pricing models. • Demonstrate choice of an investment position on the Capital Market Line (CML). • Understand the Capital Asset Pricing Model (CAPM), Security Market Line (SML) and its uses.

Asset Pricing Models •These models provide a relationship between an asset’s required rate of return and its risk. •The required return can be used for: ⇒computing the NPV of your investment.

Individual’s Choice on the CML

σσ (risk) (risk)

The Capital Asset Pricing Model (CAPM) •It allows us to determine the required rate of return (=expected return) for an individual security. ⇒Individual securities may not lie on the CML. ⇒Only efficient portfolios lie on the CML

•The Security Market Line (SML) can be applied to any securities or portfolios including inefficient ones.

The Security Market Line (SML)

(

µ j =r f +β j µ −r m

f

)

•where COV ( r , r ) ρ σ σ ρ σ j M j, m j m j, m j β = = = j σ σ2 σ2 m m

m

What does the SML tell us • The required rate of return on a security depends on: ⇒ the risk free rate ⇒ the “beta” of the security, and ⇒ the market price of risk.

• The required return is a linear function of the beta coefficient. ⇒ All else being the same, higher the beta coefficient, higher is the required return on the security.

Graphical Representation of the SML

Computing Required Rates of Return Common stock shares of Gator Sprinkler Systems (GSS) have a correlation coefficient of 0.80 with the market portfolio, and a standard deviation of 28%. The expected return on the market portfolio is 14%, and its standard deviation is 20%. The risk free rate is 5%. • What is the required rate of return on GSS?

Required Return on GSS

First compute the beta of GSS: ρ GSS,mσ GSS β GSS = σm ( 0.80 )( 28 ) = =1.12 20

Next, apply the SML: μ GSS =rf + β GSS (μ m −rf ) =5%+1.12(14%−5% )= 15.08%

Required Rate of Return on GSS •What would be the required rate of return on GSS if it had a correlation of 0.50 with the market? (All else is the same) ⇒Beta = 0.70 and µGSS = 11.30%

•What would be the required rate of return on GSS if it had a standard deviation of 36%, and a correlation of 0.80? (All else is the same) ⇒Beta = 1.44 and µGSS = 17.96%

Estimating the Beta Coefficient If we know the security’s correlation with the market, its standard deviation, and the standard deviation of the market, we can use the definition of beta: ρ j,mσ j β j=

σm

•Generally, these quantities are not known.

Interpreting the Beta Coefficient

The beta of the market portfolio is always equal to 1.0. ρm,m σm βm = = 1 sin ce ρm,m = 10 . σm

The beta of the risk free asset is always equal to 0.0 ρ f ,m σ f βf = =1 since σ f =0.0 σm

Interpreting the Beta Coefficient Beta indicates how sensitive a security’s returns are to changes in the market portfolio’s return. ⇒It is a measure of the asset’s risk.

Suppose the market portfolio’s risk premium is +10% during a given period. ⇒if β = 1.50, the security’s risk premium will be +15%. ⇒if β = 1.00, the security’s risk premium will be +10% ⇒if β = 0.50, the security’s risk premium will be

Beta Coefficients for Selected Firms

Common Stock Alex Brown Nike Inc. (Class B) Microsoft PepsiCo. Inc. McDonald’s Corporation Boeing Co. AT&T Corp. Exxon Corp.

Beta 1.90 1.50 1.40 1.10 1.05 1.00 0.85 0.60

Beta of a Portfolio •The beta of a portfolio is the weighted average of the beta values of the individual securities in the portfolio.

β p = w1 β 1 + w 2 β 2 + w 3 β 3 +  + w n β n where wi is the proportion of value invested in security i, and βi is the beta of the security i.

Applying the CAPM • The CML prescribes that investors should invest in the riskless asset and the market portfolio. • The true market portfolio, which consists of all risky assets, cannot be constructed. • How much diversification is necessary to get substantially “all” of the benefits of diversification? ⇒About 25 to 30 stocks!

Next Class

•Capital Budgeting ⇒Investment Decision ⇒Determination of Cost of Capital ⇒Determination of Cash Flows