Risk and Return - Part 2
• Efficient Frontier
• Capital Market Line
• Security Market Line
Risk and Return
C h an g e s in Po rtf o lio St an d ard D ev iat io n s w it h Ad d itio n s o f S ec u rit ies to th e Po rtf o lio fo r
V ary in g L e vels o f C o rr ela tio n s A m o n g Se c u ritie s
0.4 0
P o rtfo lio Sta n d a r d D e via tio n s
0.3 5
0.3 0
0.2 5
C orr =
0.7 0
0.2 0
C orr =
0.5 0
0.1 5
C orr =
0.0 0
0.1 0
0.0 5
0.0 0
1
2
3
4
5
6
7
8
9
10 11
12 13 14 15
16 17 18 19
N u m b e r o f S e c u ritie s
20 21 22 23
24 25 26 27
28 29 30
Risk and Return
Expected Returns and Standard Deviations for Various Levels of Correlation
16.00
14.00
12.00
Expected Returns
Correlation = -1.0
10.00
Correlation = 1.0
8.00
6.00
4.00
-1.0 < Correlation < 1.0
2.00
0.00
0.00
5.00
10.00
15.00
Standard Deviations
20.00
25.00
Risk and Return: Feasible Investments
What does the risk-return tradeoff look like if we combine many securities into
a portfolio?
Expected Returns and Standard Deviations
20.00
18.00
16.00
14.00
Expected Returns
•
12.00
10.00
8.00
6.00
4.00
2.00
0.00
0.00
5.00
10.00
15.00
Standard Deviations
20.00
25.00
30.00
Risk and Return (Continued)
•
Why do we care what the risk-return tradeoff looks like?
– Efficient Investment
• For a given level of risk, choose those investments that provide the highest expected
return.
• For a given level of expected return, choose those investments with the lowest level of
risk.
– Which investments on the graph are efficient?
•
With what decisions might variations of the above analysis help managers
and/or investors?
•
The graph above includes only risky assets. What happens if we also include a
risk-free asset?
Risk and Return: Efficient Frontier
Expected Returns and Standard Deviations
20.00
18.00
16.00
Expected Returns
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
0.00
5.00
10.00
15.00
Standard Deviations
20.00
25.00
30.00
Risk and Return with a Risk-Free Asset
Expected Returns and Standard Deviations
20.00
18.00
16.00
Expected Returns
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
0.00
5.00
10.00
15.00
Standard Deviations
20.00
25.00
30.00
Risk and Return: Capital Market Line
• What investmentsExpected
are efficient
if we include the risk-free asset?
Returns and Standard Deviations
20.00
18.00
16.00
Expected Returns
14.00
12.00
10.00
8.00
6.00
4.00
2.00
0.00
0.00
5.00
10.00
15.00
Standard Deviations
20.00
25.00
30.00
Risk and Return: Capital Market Line
• Important conclusions so far:
– When securities’ returns are less than perfectly positively correlated,
diversifying enables investors
• to increase their investment opportunity set and
• to invest efficiently
– When investors have the same expectations about what opportunities for
risk and return they have, they will invest in two sets of assets that include
• positive (lending) or negative (borrowing) amounts of the risk-free asset and
• the market portfolio that has been completely diversified across all risky
assets.
– What are the risk/return tradeoffs in this type of environment?
• Risk is measured as the standard deviation of portfolio returns.
• Expected return is measured as the expected return on the portfolio.
Risk and Return: Capital Market Line
• Numerical Example: Suppose the return on the risk-free asset is 4%,
the expected return on the market portfolio is 11%, and the standard
deviation of the return on the market is 15%. What are the expected
return and standard deviation of the returns on a portfolio in which
50% of our wealth is invested in both the market portfolio and the riskfree asset?
• The expected return is
E ( Rp )  wRf * ( Rf )  wM * E ( RM )

• The standard deviation is
 ( Rp )  wRf 2 *  Rf 2  wM 2 *  M 2  2 * wRf * wM *  Rf *  M *  Rf ,M

Risk and Return: Capital Market Line
• Numerical Example: Suppose investors want to earn more than the
market rate of return, say 15%. What proportions of their investment
must they invest in the risk-free asset and the market portfolio,
respectively?
E ( Rp)  15%  wRf * (4%)  wM * E (11%)
• How much risk would investors be exposed to?
 ( Rp )  wM 2 * M 2  wM * M
Risk and Return: Capital Market Line
• For well diversified portfolios, the required return can be determined
by what is known as the Capital Market Line. That is,
p
E( Rp )  R f 
* [ E ( Rm )  R f ]
m
• Where,
–
–
–
–
Rf is the rate of return on a risk-free asset.
E(Rm) is the expected return on the market portfolio
p is the standard deviation of the return on the portfolio in question, and
m is the standard deviation of the return on the market portfolio.
Risk and Return: Capital Market Line
• Thought questions about the Capital Market Line:
– For the numbers in the example above, what are the probabilities that
portfolios with expected returns of 7.5% and 15.0% will be below the riskfree rate of return? Hint 1:
Z-score = [Rf - E(Rp)]/p
Sharpe’s Ratio = [E(Rp) – Rf]/p = [E(Rm) – Rf]/ M
– How do returns on the portfolios with expected returns of 7.5% and 15%
correlate with returns on the Market Portfolio? Hint 2: All investors in
this model invest in only two assets: the market portfolio and the riskfree asset.
– Could we use the Capital Market Line to find the expected and required
rate of return for individual securities? Why or why not? Hint 3: See
Hint 2.
Risk and Return: Capital Market Line for
Individual Securities? Why or why not?
Risk and Return
Ch a ng e s in P o rtfo lio S tan d a rd D e viation s w ith Ad d itio n s o f Se cu r itie s to th e Po rtfo lio fo r
Va ry ing L e v els o f C o rr ela tio n s A m o n g S ec u rities
0.40
P ortf ol io S t an da rd D ev i atio n s
0.35
0.30
0.25
C or r =
0.70
0.20
C or r =
0.50
0.15
C or r =
0.00
0.10
0.05
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
N um be r of Se cu rit ie s
19
20
21
22
23
24
25
26
27
28
29
30
Risk and Return: The Risk of Individual
Securities in an N-stock Portfolio
Securities
1
2
3
1
W12*12
W1*W2*1
*2*1,2
W1*W3*1
*3*1,3
2
W1*W2*1
*2*1,2
W22*22
W2*W3*2
*3*2,3
3
W1*W3*1
*3*1,3
W2*W3*2
*3*2,3
W32*32
N
N
WN2*N2
Risk and Return: The Risk of Individual
Securities in an N-stock Portfolio
• Suppose N=100, how many terms in the portfolio variance are
influenced by correlations between returns on the different stocks?
• How many terms in the portfolio variance are not influenced by the
correlations?
• How much risk does stock 1 contribute to the portfolio? How do we
measure risk?
Risk and Return: The Risk of Individual
Securities in an N-stock Portfolio
• The contribution to portfolio risk made by an individual stock is called
beta, .
• Do we have to calculate the correlations between returns on each pair
of stocks in the portfolio to calculate ?
– No, there is a simpler approach: Regression Analysis.
– How does this regression work?
Risk and Return: Calculating 
Regression of Returns on Security i against Returns on the Market
30
25
Returns on Security i
20
15
Rit = 0.723*Rmt + 2.6253
10
5
0
-15
-10
-5
0
5
10
15
-5
-10
Returns on the Market
20
25
30
35
Risk and Return: What does  mean?
• As we noted earlier,  is the contribution to portfolio risk made by a
security (or group of securities) in a specific portfolio. Another way
of saying this is that the portfolio p is the weighted average of the
individual s in the portfolio. That is,
N
 p   wi * i
i 1
• We also said that  is the slope from a regression that explains the
relationship between returns on an individual security (or group of
securities) and returns on the market. What does the slope tell us?
Risk and Return: What does  mean?
• Because  is the slope of the regression line, it tells us two things: 1)
the relative direction of the stock’s movements when the market moves
up or down, and 2) the relative amplitude of the stock’s movements
compared to the movements in the market. This can be shown by
decomposing i as follows:
 i  i , M
i
*
M
• What economic significance does  have? Could you estimate it
without running a regression? What are the likely values of these
components of ? What determines the values of i,M and i /M?
Risk and Return: Can you explain these s?
Company
Microsoft
AT&T
Novell
Ford Motor Company
Union Pacific Corp
Pacificorp
Delta Air Lines
American Express
Geneva Steel
Iomega
Beta
1.45
0.86
1.70
0.96
0.63
0.19
0.75
1.36
0.24
2.07
Risk and Return: How do we use  ? The
Security Market Line
• One way to use  is to calculate the cost of equity for individual
firms. This is done by using the Security Market Line as follows
E ( Ri )  R f  i * [ E ( Rm )  R f ]
• Where
–
–
–
–
E(Ri) is the expected and required return on stock i
Rf is the rate of return on the risk-free asset
i is the beta for stock i, and
E(Rm) is the expected and required rate of return on the market.
• What determines the values of these factors?
Risk and Return: How do we use  ? The
Security Market Line
Required Returns on Stock Using the Security Market Line
20.00%
15.00%
Returns
10.00%
5.00%
-1.50
-1.00
-0.50
0.00%
0.00
0.50
-5.00%
Beta
1.00
1.50
2.00
2.50
Risk and Return: How do we use  ?
The Security Market Line
• Numerical Example: Suppose three stocks have s of 1.5, .75, and .30, respectively. We plan to invest 30% of our wealth in the first two
stocks and 40% of our wealth in the third stock. If the risk -free rate is
4% and the expected (required) return on the market is 11%, calculate
the following:
– The  for the portfolio of three stocks.
– The contribution to the portfolio risk made by each stock.
– The required return for the portfolio.
– The required return for the individual stocks.
Risk and Return: Assignment for Next Time
• Estimate the cost of equity for Star Appliance
– Convert price and earnings per share data to returns for Star
– Estimate the 
• Estimate  without the numbers, using what you know about Star’s product
and industry.
• Estimate  with the numbers, using regression analysis.
• Determine Star’s cost of capital for capital budgeting.