Expected returns
• As we talk about annual expected returns, keep in mind what they are:
E(D1 ) + E ( P1 )
E(ri ) =
-1
P0
E(D1 ) E(P1 ) - P0
=
+
P0
P0
Risk
• As time passes, realized stock prices and dividends may differ from what you expected.
• Such future deviations from expectations represent, from today’s perspective, risk.
• Standard deviation measures this risk
(“average deviation from expectation”).
Portfolios
• When you form portfolios of securities, you combine the “expected returns” and “risks” of the individual securities in a particular way.
• There are two ways to calculate the portfolio’s expected return and standard deviation from information about the individual securities.
Method 1
STEP 1: Compute the return distribution of the portfolio. STEP 2: Then compute the expected value and the standard deviation of that distribution.
Method 1 Example
• Consider a “50-50” portfolio of two securities.
• You are provided with the individual return distributions of the two securities:
State
Probability
Return Security A
Return Security B Portfolio Return
1
20%
50%
30%
.5*50%+.5*30%
2
60%
0%
0%
0%
3
20%
-50%
-30%
-.5*50%-.5*30%
• STEP 1: Compute the portfolio return distribution.
Method 1 Example
State
Probability
Portfolio Return
1
20%
40%
2
60%
0%
3
20%
-40%
• STEP 2: Compute the expected value and standard deviation of the portfolio return distribution.
– Expected portfolio return: 0%
– Portfolio return variance: (.4-0)^2*20%+(0-0)^2*60%+(-.40)^2*20%=6.4%
– Portfolio standard deviation: (.064)^(1/2)=29.3%
(When squaring or taking the root of percentages, first convert to decimal numbers.)
Method 1 Example
• What if I change the original distribution a bit?
State
Probability
Return Security A
Return Security B Portfolio Return
1
20%
50%
-30%
10%
2
60%
0%
0%
0%
3
20%
-50%
30%
-10%
• In this case, we get
– Expected portfolio return:
0%
– Portfolio standard deviation: 6.3%
Method 2
• STEP 1: Compute the expected return and return variance of each security. Also compute the covariance between all pairs of securities.
• STEP 2: Use the “portfolio formulas” to compute the expected portfolio return and portfolio standard deviation.
(Particularly good method if STEP 1 is already given.)
Method 2 Example
• Consider a “50-50” portfolio of two securities.
• You are provided with the individual return distributions of the two securities:
State
Probability
Return Security A
Return Security B
1
20%
50%
30%
2
60%
0%
0%
3
20%
-50%
-30%
• STEP 1: Compute the expected return and return variance for each security. Also compute the covariance between all pairs of securities.
Method 2 Example
State
Probability
Return Security A
Return Security B
1
20%
50%
30%
2
60%
0%
0%
3
20%
-50%
-30%
Expected Return
0%
0%
Return Variance
10%
3.6%
Covariance
6%
• Covariance between A and B: (.5-0)(.3-0)*20%+(00)(0-0)*60%+(-.5-0)(-.3-0)*20%=6%
Method 2 Example
Return Security A
Return Security B
Expected Return
0%
0%
Return Variance
10%
3.6%
Covariance
6%
• STEP 2: Use the “portfolio formulas.”
– Expected portfolio return: .5*0%+.5*0%=0%
ri = rf + bi ( rm - rf ) s im bi º 2 sm • Understand what the components stand for, and what drives them.
Using the formula
• Internalize three basic ways of using the CAPM formula (there are more):
1. Finding the required return of a security
2. Assessing whether a security is mis-valued
3. Understand how changes in economic fundamentals affect expected returns
Use #1
• You have the following information:
Risk-free rate (rf)
2.5%
Market risk premium (rm-rf)
7.2%
Market standard deviation (σm)
Covariance of return of security i with market return (σxm)
6%
0.64%
• The required (expected) return – or equivalently, the discount rate – for security i is:
Beta_i=.64%/(6%)^2=1.78
R_i=2.5%+1.78*7.2%=10.87%
Use #2
• Suppose you have the following information…
Risk-free rate (rf)
2.5%
Market risk premium (rm-rf)
7.2%
Market standard deviation (σm)
Covariance of return of security i with market return (σxm)
6%
0.64%
…and believe, for your own reasons, that the expected return on security i will be Es(ri) = 6% if you buy the security at the current market price. (The subscript “s” denotes your subjective beliefs.)
Use #2
• Given security i’s beta, the expected return required from a security with this systematic risk (according to
CAPM) is 10.87%
• Since you expect the return to be lowe than 10.87%
(based on your subjective information), the security is in your opinion currently overvalued. You should therefore sell the security.
Use #3
ri = rf + bi ( rm - rf ) s im bi º 2 sm • Suppose company i moves its business away from
“utilities” and into “information technology.”
• Suppose uncertainty about future interest rates
(interest rate risk) decreases.
