CHAPTER 9 Capital Market
I. Review of Basic Concepts
1. Variance [or STDEV] are appropriate measures of risk for assets held by themselves.
2. Variance and standard deviation are not good measures of an asset’s risk once it is a part of a
portfolio since the individual asset’s risks may offset one another.
3. Beta is the appropriate measure of an asset’s risk if held as part of a portfolio
4. Investors will only invest in riskier assets (as measured by beta) if compensated for taking
this additional risk as follows:
E(ri) = rf + [E(rM) - rf]
NOTE: this is the basic result of the Capital Asset Pricing Model (CAPM)
known as the security market line (SML)
II. Estimating Risk and Return using Historical Data
KEY  past return and risk on an asset may indicate return and risk will face in future
NOTE: danger  future return and risk may be unlike past returns and risk
Ex. Suppose have collected returns for New Honda Dealership Inc. and for Harry’s Automotive
Repair Inc. for past 4 years.
Year
1991
1992
1993
1994
Return on:
Dealership
Repair
19
14
39
8
15
11
0
19
A. Average historical return
 best estimate of what will earn in any one year (if past returns representative)
 add up historical returns and divide by number of observations
r
 rt
T
where:
r = average return
rt = return for period t
T = number of observations
Ex.
rDealership 
rRepair  13
19  39  15  0
 18.25
4
1
B. Variance (2) and standard deviation ()
NOTES:
1) measures dispersion around the average return
2) measures the risk of an asset when held by itself
 the higher the std. deviation, the greater the uncertainty about the return
3) usually work with standard deviation since same unit of measurement as average return
KEY  Add up squared deviations from average and divide by T-1
 the financial calculator ?
 rt  r 

2

2
T 1
  2
Year
1991
1992
1993
1994
average
Ex.
 2Dealership
Return on:
Dealership
Repair
19
14
39
8
15
11
0
19
18.25
13
(19 -18.25) 2 + (38 -18.25) 2 + (15 -18.25) 2 + (0 -18.25) 2
 258.25
4 -1
 Dealership  258.25  16.07%
 Repair  4.69%
 more uncertainty about the return on the dealership
2
C. Covariance
Year
1991
1992
1993
1994
average
 1,2 
Return on:
Dealership
Repair
19
14
39
8
15
11
0
19
18.25
13
r1  t   r1 r2  t   r2 
t
T 1
where:
ri(t) = return on asset i at time t
ri = average return on asset i
 positive covariance
 when one asset’s return is above average, other asset’s return tends to be above average also
 negative covariance : Interpretation ?
zero covariance : Interpretation
 D, R  ?
D. Correlation
 1,2
 1,2 
 1 2
Ex. 
D,R
 ?
1) Interpretation : same as covariance
2) Correlation ranges between -1 and +1, unlike covariance.
 = +1  perfectly positively corelated
 = -1  perfectly negatively corelated
0 <  < 1  positively correlated
-1 <  < 0  negatively correlated
3
III. Estimating Risk and Return using Forecasted Data
Key
 based on forecasts of possible returns and associated probabilities
 invest in the future, not in the past
danger : difficult to forecast the future
Ex. Suppose have estimated possible returns for New Honda Dealership Inc. and Harry’s
Automotive Repair for the coming year based on how the economy does
Economy
Boom
Average
Bust
Prob.
.25
.55
.20
Return on:
Dealership
Repair
40%
6%
15%
15%
-1%
17%
E. Expected Return
 best estimate of future return on the asset
 probability weighted average of asset returns across all states
 actual return may be higher or lower than expected return
E r    Pr sr s
s
where:
Pr(s) = probability of scenario s
r(s) = return in scenario s
Example.
E(rDealership) = .25(40) + .55(15) + .2(-1) = 18.05%
E(rRepair) = 13.15%
 expect to earn a higher return on dealership than repair shop for the coming year
F. Variance and Standard Deviation
1) measures dispersion around the average return
2) measures the risk of an asset when held by itself
 the higher the standard deviation, the greater the uncertainty about the return
3) usually work with standard deviation since same unit of measurement as average return
 2   Pr  sr s  E r 
2
  2
s
Example.
 2Dealership .25 40  18.05 2 .5515  18.05 2 .2 1  18.05 2  1981475
.
Dealership = 14.08%
Repair = 4.20%
 return is more uncertain for the dealership
4
IV. Risk and Return for Portfolios using Forecasted Data
 Portfolio  collection of assets
A. Expected Return
 probability weighted avg. of portfolio returns across all states : just like individual assets
 determining portfolio return for each state
rp  s   X i ri  s
s
where: Xi = % of portfolio value invested in asset i
Ex. Assume invest $8000 in dealership and $12,000 in the repair shop
Return on:
Economy Prob.
Dealership
Repair
Boom
.25
40%
6%
Average .55
15%
15%
Bust
.20
-1%
17%
rP(boom) = .4(40) + .6(6) = 19.6%
rP(average) = .4(15) + .6(15) = 15%
rP(bust) = 9.8%
E(rP) = .25(19.6) + .55(15) + .2(9.8) = 15.11%
5
B. Variance and Standard Deviation
Economy
Boom
Average
Bust
Prob.
.25
.55
.20
Return on:
Dealership
Repair
40%
6%
15%
15%
-1%
17%
1. weighted average of squared deviations from E(r) same as individual assets
 P  .2519.6  1511
.  2 .5515  1511
.  2 .29.8  1511
.  2  3.27%
NOTE: when combined dealership and repair shop, std. deviation of portfolio (3.27) less
than either the dealership (14.08) or the repair shop (4.19)
2. what happened to the risk?
 diversification : risk reduction from offsetting variability
 key to diversification : more closely related returns, less diversification possible
NOTE: The relationship between asset returns and diversification benefits will be more
obvious with a following portfolio example.
C. Covariance
 1,2   Pr sr1  s  Er1  r2  s  Er2 
s
where:
Pr(s) = probability of scenario s
ri(s) = return on asset i in scenario s
E(ri) = expected return on asset i
Ex. D,R = ?
6
THE RELATIONSHIP BETWEEN THE RISK AND #. OF STOCKS IN PORTFOLIO
1. Single Stock
Year
91
92
93
94
95
Avg. Ret.
Stdev.
Stock W
40.00%
-10.00%
35.00%
-5.00%
15.00%
15.00%
22.64%
2. Two-Stock Portfolio [ The Weight of Each Stock : 50% ]
A. Perfectively positive correlated case : Coefficient = 1
Stock W
40.00%
-10.00%
35.00%
-5.00%
15.00%
15.00%
22.64%
Stock X
40.00%
-10.00%
35.00%
-5.00%
15.00%
15.00%
22.64%
Portfolio WX
40.00%
-10.00%
35.00%
-5.00%
15.00%
15.00%
22.64%
Returns on Two Stocks
50%
Stock X
Year
91
92
93
94
95
Avg. Ret.
Stdev.
10%
-30.00%
10.00%
50.00%
-30%
Stock W
B. positively correlated case : coefficient = 0.69
Stock W
40.00%
-10.00%
35.00%
-5.00%
15.00%
15.00%
22.64%
Stock Y
24.00%
17.00%
43.00%
-19.00%
10.00%
15.00%
22.64%
Portfolio WY
32.00%
3.50%
39.00%
-12.00%
12.50%
15.00%
20.81%
Returns on Two Stocks
50%
Stock Y
Year
91
92
93
94
95
Avg. Ret.
Stdev.
10%
-30.00%
10.00%
50.00%
-30%
Stock W
7
THE RELATIONSHIP BETWEEN THE RISK AND #. OF STOCKS IN PORTFOLIO
(Continued)
C. Negatively correlated case : coefficient = -0.72
Stock W
40.00%
-10.00%
35.00%
-5.00%
15.00%
15.00%
22.64%
Stock Z
11.41%
32.49%
-4.00%
43.60%
-8.50%
15.00%
22.64%
Portfolio WZ
25.71%
11.25%
15.50%
19.30%
3.25%
15.00%
8.45%
Returns on Two Stocks
50%
Stock Y
Year
91
92
93
94
95
Avg. Ret.
Stdev.
10%
-30.00%
10.00%
50.00%
-30%
Stock W
D. Perfectively negative correlated case : Coefficient = -1
Stock W
40.00%
-10.00%
35.00%
-5.00%
15.00%
15.00%
22.64%
Stock P
-10.00%
40.00%
-5.00%
35.00%
15.00%
15.00%
22.64%
Portfolio WP
15.00%
15.00%
15.00%
15.00%
15.00%
15.00%
0.00%
Returns on Two Stocks
50%
Stock Y
Year
91
92
93
94
95
Avg. Ret.
Stdev.
10%
-30.00%
10.00%
50.00%
-30%
Stock W
8
V. Relationship Between Risk and Return
A. Systematic vs. Unsystematic Risk
systematic risk - risk that affects all assets in the market portfolio
unsystematic risk - risk that only affects a single asset or a small group of assets
B. Impact of Diversification
1. unsystematic risk
 as combine assets, unsystematic risk begins to cancel out
 after combine 20 or 30 randomly chosen assets, almost all unsystematic risk gone
reason: very unlikely that all 20 have positive surprises at same time
 once part of market portfolio, all is gone
2. systematic risk
 not affected by diversification
 affects all assets
 should only be compensated for systematic risk since only it remains once part
of market portfolio
C. Measuring Systematic Risk
KEY  systematic risk of an asset measured by its beta

 i ,m
 2m
where:
 i,m = covariance between asset and the market
 2m = variance of returns on the market
NOTE: discussion of how to calculate covariance will be discussed in class.
Ex. Suppose that the variance of returns on the market is 124.3275. Assume also that the
covariance between New Honda Dealership and the market is 144.1825 and that the
covariance between Harry’s Automotive Repair and the market is -36.6025. What are the
betas for New Honda Dealership and Harry’s Automotive Repair?
144.1825
 11597
.
124.3275
36.6025

 0.2944
124.3275
  Dealership 
  Repair
9
NOTE : beta is the slope of least squares regression line of the return on the asset
against the return on the market
 called characteristic line
. .
.
.
.
.
.
ri
rM
Graph #6
 dots represent period returns
 slope of line is the beta for asset i
 slope = 1.5 ( = 1.5)  stock tends to be 1.5 times as volatile as the market
 slope = .8 (= 0.8)  stock tends to be 80% as volatile as the market
 deviations from line = unique risk
NOTE: an easier approach  look up  in value line or S&P
10
D. Portfolio Betas
KEY  weighted average of betas in portfolio
 P   Xi  i
i
Ex. Invest $8000 in the the dealership with a beta of 1.1597 and $12,000 in the repair
shop with a beta of -0.2944.
 P = .4(1.1597) + .6(-.2944) = 0.2872
E. The Capital Asset Pricing Model (CAPM)
KEY  since beta is relevant measure of risk, there should be a relationship
between beta and return


E ri   r f  E rM   r f  i
NOTE:
1) this equation is known as the security market line
2) the model has very strong intuition
If  = 2, the asset has two “units” of market risk
 twice as volatile as the market (as part of a portfolio)
 should earn twice as much of a premium (above the risk free rate)
If  = .5, the asset has one-half of a “unit” of market risk
 one-half as volatile as the market (as part of a portfolio)
 should earn one-half of a premium
Ex. The return on T-bills (risk-free rate) is 5% and the market risk premium is 8%.
What does the CAPM tell us is the required return on New Honda and Harry’s
Automotive Repair?
E(rDealership) = .05 + (.08)(1.1597) = .1428
E(rRepair) = .05 + (.08)(-.2944) = .0265
 Solve the following problems in the textbook.
1, 2, 3, 4, 7, 8, 10, 11, 15
11
EXERCISE PROBLEMS FOR CHAPTER 9.
1. Find the expected return on Bio-Pharma stock when there are two possible states of economy. The
probability that the high state will occur is 32% and that the low state will occur is 68%. The return
on Bio-Pharma stock is 16.83% in the high state and 7.91% in the low state. [[10.76%]]
2. With two possible states of the economy, find the variance of returns on Texas Eastern stock. The
probability that State 1 will occur is 28% and that State 2 will occur is 72%. The return in State 1 is
16.49% and in State 2 is 7.04%. [[18.0034]]
3. Your budget manager demands new calculations of covariance of returns based on the best case and
worst case possibilities. You estimate that your firm will receive returns given in the following table
in the two states. You also estimate returns to the Dow Jones 30 in these two states. Find the
covariance of returns. [[0.2855]]
State of Nature
The best state
The worst state
Probability
52%
48%
Texas Eastern Return
16.79%
5.35%
Index Return
16.02%
6.02%
4. You are considering investing in a portfolio of two stocks, with 57% of your money in the first stock
and 43% in the second stock. You calculate that the expected returns for the two stocks are 9.60%
and 16.25%, respectively. Find the portfolio expected return.[[12.46%]]
5. Find the variance of the portfolio when the variance of expected returns is 0.10262 for Asset 1 and
0.11397 for Asset 2. The covariance of returns is 0.01925. Funds are invested with 53% of the
money in Asset 1 and 47% in Asset 2.[[0.0636]]
6. The stock analyst for US East has determined that if the best state occurs in the economy, US East
stock will yield a return of 16.17%. If the worst state occurs, the return will be 6.96%. If the normal
state occurs, the return will be 12.56%. With probabilities of 17% on the best state, 31% on the worst
state, and 52% on the normal state, what is the expected return on the stock? [[11.44%]]
7. Before approving acquisition of new machinery, the treasurer of Sip-n-Smoke wants to know the
standard deviation of returns on it. You tell him that the firm should receive a return of 15.15% with
a probability of 49%, and a return of 5.21% with a probability of 51%. What is the standard deviation
of returns on the machinery?[[4.97%]]
8. The project manager at O'Reilly Corp figures the states of nature and returns for O'Reilly Corp from a
new project and also the returns for the value weighted stock index. These appear in the following
table. What is the covariance of returns of the project with the value weighted stock index?
[[0.1348]]
State of Nature
The boom state
The recession
Probability
73%
27%
O'Reilly Corp Return
15.47%
6.15%
Index Return
14.64%
7.30%
9. You have invested 29% of your money in Stock X, 50% in Stock Y, and the rest in Stock Z. The
portfolio expected return is 7.48%, and the expected returns of X and Y are 7.83% and 6.83%. What
is the expected return of Stock Z? [[8.54%]]
10. You are considering a portfolio of two stocks, with 90% of your money in the first stock, and 10% in
the second stock. The variance of returns of Stock 1 is 0.10065 and the variance of returns of Stock 2
is 0.13753. The correlation coefficient of returns is 0.598. Find the variance of the portfolio.
[[0.0956]]
12
CHAPTER 10. RISK AND RETURN [CAPM]
I. Examples from Chapter 9.
Ex. Historical Returns for Dealership Inc. and for Repair Inc. for past 4 years.
- How to compute by using financial calculator ?
Return on:
Year
Dealership
Repair
1992
19
14
1993
39
8
1994
15
11
1995
0
19
r

18.25
16.07
13.00
4.69
Ex. Estimated possible future returns for Dealership Inc. and Repair Inc for the coming year
based on how the economy does
Economy
Boom
Average
Bust
Prob.
.25
.55
.2
E(r)

Return on:
Dealership
Repair
40%
6%
15%
15%
-1%
17%
18.05
14.08
13.15
4.20
II. Risk and Return for Portfolios using Forecasted Data
 weighted average of variances and covariances
 2p  X 12 12  2 X 1 X 2 1,2  X 22 22
where:
 p2  variance of returns on portfolio
Xi = % of portfolio invested in asset i [=weight or proportion]
i2  variance of returns on asset i
 1, 2  covariance between asset 1 and asset 2
13
III. Estimating Risk and Return for Portfolios using Historical Data
 essentially the same as w/ forecasted data except use historical data except for
calculation of covariance
IV. The relationship between risk and return
[ A.] Combinations of risky assets
1. Portfolios of two risky assets
Ex. Combinations of New Honda Dealership and Harry’s Auto Repair based on forecasted
data.
XD
.0
.2
.4
.6
.8
1.0
E(rP)
13.15%
14.13
15.11
16.09
17.07
18.05
P
4.20%
0.98
3.27
6.84
10.45
14.08
Note: Curve due to diversification : Correlation negative.
What if perfect positive correlation ?
Return and STDEV
RETURN for PF
18
17
16
15
14
13
0
5STDEV for PF10
15
14
2. Portfolios of many risky assets
a. Feasible set – the set of all portfolios that could be formed from a group of N securities.
b. The efficient set – An investor will choose his or her optimal portfolio from the set of
portfolios that
1. Offer maximum expected return for varying levels of risk and
2. Offer minimum risk for varying levels of return.
note: optimal portfolio depends on risk preferences
15
[ B.] Combinations of a risky asset and risk-free borrowing or lending
1. relationship between risk and return
note: std. deviation of risk-free borrowing/lending is zero and covariance between risk-free
borrowing/lending and any other asset is zero
a. E(rp) = XAE(rA) + Xrf rf
b. P =
x 2A  2A
+ 2 x A xrf  rf ; A + xrf2  2rf
= XAA
 second and third terms drop out since zero
 can combine 1st & 2nd equations to obtain relationship between risk and return
 solve for XA in eq. b, substitute into a, and simplify
 E(r ) - rf 
 E(rp) = rf +  A
 P
A


Ex. Assume combine risky asset “A” from the efficient set with riskfree borrowing/lending
E(rA) = 14%; A = 10%; rf = 9%
 E(rp) =
Assume initial wealth is $5000 and invest $3500 in “A” and lend the remainder at riskfree rate.
 E(rp) = ?
Assume initial wealth is $5000 and that invest $7500 in “A” by borrowing
 E(rp) = ?
16
2. The optimal risky asset
A
Q: Is “A” the best risky asset to combine with riskfree-borrowing and lending?

 Point M deserves some attention. Why ?
Because there is no other portfolio consisting purely of risky assets that, when connected by
a straight line to the riskfree asset, lies northwest of it. In other words, of all the lines that
can be drawn emanating from the riskfree asset and connecting with either risky asset or
risky portfolio, none has a greater slope than the line that goes to M.
 Above statement is important. Why ?
: Efficient set has been changed when we introduce the riskfree lending and borrowing.
 The crucial point is that every investor holds the optimal risky portfolio P, regardless of
his/her risk aversion.
 E(r ) - rf 
3) The equation of the capital market line is E(rp) = rf +  M
 P
M


[ C.] Homogeneous expectations
 assume investors have homogeneous expectations
17
[ D.] The Capital Asset Pricing Model (CAPM)
1. relationship between risk and return
 assume combine some risky asset “A” with risk-free borrowing or lending
 beta of risk-free borrowing/lending is zero
a. E(rp) = XAE(rA) + Xrf rf
b. P = XAA + Xrf(0) = XA A
 can combine a and b to obtain relationship between risk and return
 solve for XA in eq. b, substitute into a, and simplify
 E(r ) - rf 
 E(rp) = rf +  A
P
A


 E(r ) - rf 
1) slope of line =  A

A


 called the reward to risk ratio
Ex. E(rA) = 14%, rf = 9%, A = 0.5
 reward to risk ratio = ?
E(R)
slope = 10
14
9
.5

Graph #7
2. Market portfolio
RTRM =


 E ri   r f  E rM   r f  i
 called security market line (SML)
Ex. Suppose asset A has a beta of 0.5. How can we combine the market with risk-free
borrowing and lending to achieve the same risk?
Ex. Suppose asset B has a beta of 1.3. How can we combine the market with risk-free borrowing
and lending to achieve the same risk?
18
TAKE-HOME EXERCISE [CHAPTER 10]

1.
John Beere's economists have determined that the best state in the market will occur with a 20% probability,
that the worst state will occur with a 25% probability, and that the normal state will occur with a
55%probability. If the return on John Beere will be 14.73% in the best state, 7.46% in the worst state, and
12.57% in the normal state, what is the expected return on John Beere stock?
2.
With three possible states of the economy, find the standard deviation of returns on Quicken stock. The
probability that the best state will occur is 17%, that the worst state will occur is 26% and that the normal state
will occur is 57%. The return in the best state is 14.40%, with 5.70% in the worst state and 10.31% in the
normal state.
3.
The project manager at Dell figures the states of nature and returns for Dell from a new project and also the
returns for for the S&P500. These appear in the floating table. What is the covariance of returns of the project
with the S&P500?
State of Nature
Probability
Dell Return
Index Return
Boom
76%
17.24%
17.20%
Recession
24%
7.42%
7.23%
4.
Suppose the expected return of a three-stock portfolio is 13.95%. The expected returns of the first and second
stocks are 17.83% and 6.84%. With investment proportions of 37%, 27%, and 36%, find the expected return of
the third stock.
5.
Find the variance of returns of the two-asset portfolio where 67% is invested in Asset 1 and 33% is invested in
Asset 2. The standard deviation of Asset 1 is 30.41%and the standard deviation of Asset 2 is 24.36%. The
covariance of returns is -0.00126.
6.
You are considering a portfolio of two stocks, with 87% of your money in the first stock, and 13% in the second
stock. The variance of returns of Stock 1 is 0.05653 and the variance of returns of Stock 2 is 0.04421. The
correlation coefficient of returns is 0.188. Find the variance of the portfolio.
7.
Find the variance of returns of the two-asset portfolio where 64% is invested in Asset 1 and 36% is invested in
Asset 2. The standard deviation of Asset 1 is 31.94%and the standard deviation of Asset 2 is 29.24%. The
covariance of returns is -0.03306.
8.
You've been told that the beta of a three-stock portfolio is 1.465. The betas of the first and second stocks are
2.161 and 2.071. With investment proportions of 24%, 37%, and 39%, what is the beta of the third stock?
9.
Your investment analyst has determined the expected return for Wacom Tech stock is 17.66%. The stock's beta
is 1.525 and the current T-bill rate is 3.92%. What is the expected return on the market portfolio?
10. The expected market risk premium is 9.81% and the expected return on a stock with a beta of 1.460 is 17.94%.
What is the risk-free rate?
ANSWERS
1.)
4.)
7.)
10.)
11.72%
15.29%
0.0376
3.62%
2.) 2.85%
5.) 0.0474
8.) 0.463
3.) 0.1786
6.) 0.0457
9.) 12.93%
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