Cashman Investments
Exam II In-Class Problems
1.
There are two funds: Stock Fund(S) and Bond Fund(B) and three possible states (Boom,
Normal, Recession) with equal probability
Stocks: Boom=30%, Normal=12%, Recession= -9%
E(r) = 11%, Std Dev = 0.1595
Bonds: Boom =-6%, Normal= 6%, Recession = 12%
E(r) = 4%, Std Dev = 0.0748
Portfolio weights: ws=60% and wb=40%
a.
Calculate the Covariance of the two funds
b.
Calculate the correlation coefficient of the two funds
c.
Calculate the Expected Return, Variance and Standard Deviation of the portfolio
d.
Compare the return and risk between the portfolio and the individual components.
2.
You want to create a portfolio using IBM and Toyota that has an expected return of 19%.
You expect Toyota to return 27% and IBM 8%. What are the portfolio weights that
generate your 19% expected return?
3.
As the average investor becomes less risk averse, what do we expect to happen?
-Risk Premiums will
-The SML will
-Why
4.
The standard deviation of the market-index portfolio is 35%. If a stock has a beta of 2.5,
and a residual variance of 20% what is the stocks total variance? How important is the
market in explaining firm returns?
5.
Consider a three factor APT model. A portfolio has a beta of 2 on factor 1, 3 on factor 2
and 4 on factor 3. The risk premiums on the factor 1, 2 and 3 portfolios are 6%, 7%, and
8%, respectively. The risk-free rate of return is 7%. The expected return on portfolio A is
__________ if no arbitrage opportunities exist.
6.
Research has identified four systematic factors in the US. The factors are growth in
industrial production, changes in long-term interest rates, oil prices, and commodity
prices. Industrial production growth is expected to be 3%, long-term interest rates are
expected to increase by 1%, oil prices are expected to fall 2%, and commodity prices are
expected to rise 5%. You are analyzing a stock that has a beta of 2 on industrial
production, 0.75 on the interest rate factor, 4 on oil, and 1.5 on commodities. We
expected a return on 16% when we first invested in the stock, but over the year industrial
production actually grew at 6%, interest rates dropped 1.5%, oil rose 2%, and
commodities were flat. What is your best guess as to the stock’s actual return?
7.
You have been provided the following data on the securities of three firms, and the
market. Assume that CAPM holds.
Security
Average Return
Std Dev
Correlation
Beta
Coefficient
with Market
Firm A
0.13
0.12
(i)
0.90
Firm B
0.16
(ii)
0.40
1.10
Firm C
0.25
0.24
0.75
(iii)
The market
0.15
0.10
(iv)
(v)
Risk Free Asset
0.05
(vi)
(vii)
(viii)
a. Fill in the missing values.
b. Calculate each stocks α
c. What is your investment recommendation? Why? Graph?
8.
Draw the Capital Market Line Graph. Label the axis, relevant points, the efficient frontier,
and the CML.
a. Why do investors move along the CML?
b. What do investors hold in their portfolio?
9.
Define the three forms of market efficiency. Who CAN and CANNOT make money in the
various types of markets? Why?
10. Suppose your cousin invests in the stock market, creating a portfolio with a beta of 1, and
doubles her money in a single year while the market, on average, earned a return of only
about 15%. Is your cousin’s performance a violation of market efficiency? Why?
11.
Aerotech, an aerospace-technology research firm, announced this morning that it has hired
the world’s most knowledgeable and prolific space researcher. Before today, Aerotech’s
stock had been selling for $100.
a. What do you expect will happen to Aerotech’s stock?
b. Consider the following scenarios:
i. The stock price jumps to $118 on the day of the announcement. In subsequent
day it floats up to $123 then falls back to $116.
ii. The stock price jumps to $116 and remains there
iii. The stock price gradually climbs to $116 over the next week
Which scenario(s) indicate market efficiency? Which do not? Why?
Given Problems

There are three securities in the market. The following chart shows their possible payoffs.
State
1
2
3
4
Probability of
Return on
Return on
Return on
Outcome
Security 1
Security 2
Security 3
0.10
0.25
0.25
0.10
0.40
0.20
0.15
0.15
0.40
0.15
0.20
0.20
0.10
0.10
0.10
0.25
a. What are the co-variances and correlations between each possible pair of securities?
Covar(1,2):
Corr(1,2):
0.000625
0.384615
Covar(1,3):
Corr(1,3):
-0.001625
-1
Covar(2,3):
Corr(2,3):
-0.000625
-0.384615
b. What is the expected return and standard deviation for a portfolio with half of its funds in
stock 1, and the other half in stock 2?
Expected Return = 0.175
Std dev = 0.03354102
c. What is the expected return and standard deviation for a portfolio with half of its funds in
stock 1, and the other half in stock 3?
Expected Return = 0.175
Std dev = 0
d. What is the expected return and standard deviation for a portfolio with half of its funds in
stock 2, and the other half in stock 3?
Expected Return = 0.175
Std dev = 0.02236068
e. What do your answers above imply about diversification?
Portfolios with negatively correlated stocks can achieve a higher degree of diversification
than portfolios with positively correlated stocks, holding expected returns for each stock
constant. Applying proper weights to perfectly negatively correlated stocks can reduce
portfolio variance to 0. As long as the correlation is not 1, there is a benefit to
diversification.

Given the expected returns of stocks P & Q calculate the covariance, and correlation
coefficient between the two securities.
Probability
P
Q
0.3
-10%
8%
0.6
12%
13%
0.1
28%
24%
Stock P
E(r) = (0.3)*-0.1 + (0.6)*0.12 + (0.1)*0.28 = 0.07
Var = (0.3)*(-0.1-0.07)2 + (06)*(0.12-0.07)2 + (0.1)*(0.28-0.07)2 = 0.01458
Std Dev = 0.120748
Stock Q
E(r) = (0.3)*0.8 + (0.6)*0.13 + (0.1)*0.24 = 0.126
Var = (0.3)*(0.8-0.126)2 + (06)*(0.13-0.126)2 + (0.1)*(0.24-0.126)2 =0.001944
Std Dev = 0.044091
CoVar = (0.3)* (-0.1-0.07)*(0.8-0.126) + (0.6)*(0.12-0.07)*(0.13-0.126)
+ (0.1)*(0.28-0.07)*(0.24-0.126) = 0.00486
Correlation Coefficient = 0.00486/ (0.120748*0.044091) = 0.912871

You have a $25,000 portfolio comprised of three assets, Treasury Bills, Google, and Ford. If
you want a portfolio β of 1.3, the β’s of Google and Ford are 2.5 and 0.8, respectively, and
you have $12,000 invested in Google. How much do you have invested in each asset? What
are the portfolio weights?
1.3 = (12 / 25)*(2.5) + (Ford / 25) *(0.8) + (TB / 25)*(0)
1.3 = (12 / 25)*(2.5) + (Ford / 25) *(0.8) + 0
1.3 = (12 / 25)*(2.5) + (X) *(0.8) + 0
1.3 = 1.2 + 0.8X
0.1 = 0.8X
X = 1/8, so 1/8 of the portfolio is invested in Ford. The $ investment = 25,000/8 = 3,125
So we have 25,000 – 12,000 – 3,125 = $9,875 invested in TB
Google
Ford
TBs
$12,000
$3,125
$9,875
48%
12.5%
39.5%

Stock P and stock Q have had annual returns of -10%, 12%, 28% and 8%, 13%, 24%
respectively. Calculate the covariance, and correlation coefficient of returns between the
securities.
Expected Return P = 10%
Expected Return Q= 15%
2
Var
(P) = (1/3)*(-0.1-0.1) +(1/3)*(0.12-0.1)2+(1/3)*(0.28-0.1)2 = 0.0242667
Std Dev (P) = 0.0242667^(0.5) = 0.156
Var
(Q) = (1/3)*(0.8 -0.15)2+(1/3)*(0.13-0.15)2+(1/3)*(0.24-0.15) 2 = 0.00446667
Std Dev (Q) = 0.009933^(0.5) = 0.067
Covar(P,Q) = (1/3)*(-0.1-0.1)* (0.8 -0.15)+(1/3)*(0.12-0.1)* (0.13-0.15)+(1/3)*(0.28-0.1)*
(0.24-0.15) = 0.009933
Correlation Coefficient =

0.009933/(0.156*0.067) = 0.95
The correlation between the returns on Ceramics Craftsman, Inc. and the return on the S&P
500 is 0.675. The variance of the returns on Ceramics Craftsman Inc., is 0.004225, and the
variance of the returns on the S&P 500 is 0.001467. What is the beta of Ceramics Craftsman
stock (3 decimal places)? The risk free rate is 6%, and the market risk premium is 8.5%.
What is the expected rate of return for Ceramics Craftsman, Inc, according to CAPM?
stdev(C)
stdev(M)
0.065
0.038301436
Beta = corr(C,M) * stdev(C) / stdev(M) = 1.145518408 = 1.146
Expected Return = 0.06 + 1.146 * 0.085 = 0.1574 = 15.74%

The standard deviation of the market is 0.2828742712. The beta of a well-diversified
portfolio is 1.2. What is the variance of a well-diversified portfolio?
σi2 = βi2σm2+ σ(ei)2
σi2 = 1.22*0.28287427122+0 → σ(ei)2 = 0 because residual risk is eliminated in a well-diversified portfolio
σi2 = 0.1152

If the correlation coefficient between stock C and stock D is +1.0 and the standard deviation
of return for stock C is 15% and that for stock D is 30%, calculate the covariance between
stock C and stock D.
Correlation Coefficient = Covariance / (StdDevC * StdDevD)
1 = Covar / (0.15 * 0.3)
Covar = 0.15 * 0.3 = 0.045

An investor wants to evaluate the $2.5 million portfolio described below: The market rate of
return is 11.5 % and the risk free rate is 5.25%.
Stock's
Stock's Expected
Portfolio
Proportions
Stock
Beta
Return
Composition
I
1.3
15%
$750,000
0.30
II
0.7
8%
$250,000
0.10
III
1.25
16.25%
$500,000
0.20
IV
1.1
12.50%
$500,000
0.20
V
0.9
9%
$500,000
0.20
$2,500,000
a. Compute the expected return of the 5-asset portfolio.
Expected Return =0.3*0.15 + 0.1*0.08 + 0.2*0.1625 + 0.2*0.125 + 0.2*0.09
Expected Return: 0.1285
b. Compute the weighted-average beta and the expected return according to CAPM for
this 5-asset portfolio.
Portfolio Beta = 0.3*1.3 + 0.1*0.7 + 0.2*1.25 + 0.2*1.1 + 0.2*0.9
Portfolio Beta = 1.11
CAPM predicted return = 0.0525 + 1.11 (0.115 – 0.0525) = 0.121875
CAPM predicted return = 12.1875%
c. Based on the above calculations, would you recommend for or against investing in
this portfolio. Briefly explain your answer. Plot on the SML.
I would recommend that we invest in the portfolio, as the expected return is greater than
the return predicted by CAPM
Expected
Return
SML
Portfoli
R
f
β

The variance of Stock A is 0.0016, the variance of the market is 0.0049 and the covariance
between the two is 0.0026. What is the correlation coefficient?
Correlation coefficient = 0.0026 / (0.04*0.07) = 0.9286

Suppose the current risk free rate is 7.6%. Potpourri Inc. stock has a beta of 1.7 and an
expected return of 16.7%. (Assume CAPM is true)
a. What is the market return, and the market risk premium?
Return = 0.167 = 0.076 + 1.7 * (market return – 0.076)
Market return = 12.95%
Market Risk Premium = 5.35%
b. Magnolia Industries stock has a beta of 0.8. What is Magnolia stock’s expected return?
Expected Return = 0.076 + 0.8 * (0.0535) = 0.1188 = 11.88%
c. Suppose you have invested a total of $10,000 in both Potpourri and Magnolia, and the
beta of the portfolio is 1.07. How much did you invest in each stock? What is the
portfolio’s expected return?
Port Beta =1.07=Weight (Potpourri)*1.7+(1-Weight (Potpourri))* 0.8
1.07 = 1.7*W + 0.8 - 0.8*W
1.07 = 0.9W + 0.8
0.9W = 0.27
W = 0.27 / 0.9 = 0.3 = 30%
$3,000 invested in Potpourri, the remaining $7,000 in Magnolia
Portfolio’s Expected Return = 0.3 * 0.167 +0.7 * .1188 = 0.1333
Portfolio’s Expected Return = 13.33%
OR
Port Beta = 0.3*1.7 + 0.7*0.8 = 1.07
Return = 0.076 + 1.07 * .0535 = 13.33%

What is the expected return on this portfolio?
Stock
A
B
C
Expected
Return
8%
15%
6%
Dollar investment in each
A = 520 * 25 = 13,000
B = 300 * 48 = 14,400
C = 250 * 26 = 6,500
Number
of Shares
520
300
250
Stock
Price
$25
$48
$26
Weight
13000/33900 = 0.3835
14400/33900 = 0.4248
6500/33900 = 0.1917
Total investment
= 13000 + 14400 + 6500 = 33,900
Portfolio return = 0.3835(0.08) + 0.4248(0.15)

What is the formula for the variance of a two-stock portfolio? What do all the w terms
represent in this equation? What do all of 2 terms represent? What does the 12 represent?
What sign would you prefer the term ijij to have to maximize diversification? Why?
Portfolio Variance, σp2 = w12σ12 + w22σ22 + 2w1w2σ12
σp2 = w12σ12 + w22σ22 + 2w1w2ρ12σ1σ2
The w terms represent the weights of the assets in the portfolio
The 2 terms represent the variance of the individual assets
The 12 (correlation coefficient) represents how strongly the two assets are related
All else equal we would prefer a negative sign on the ijij t term, because this
would increase the diversification benefit

Explain why it is that in an efficient market, investments have an expected NPV of zero.
In an efficient market, prices are “fair” so that the cost of an investment is neither
higher nor lower than the investments expected NPV. Thus, on average, investments
in an efficient market will yield a zero NPV. Investors get exactly what they pay for
when they buy a security in an efficient market and firms get exactly what their stocks
and bonds are worth when they sell them.

Do you think the lessons from capital market history will hold for each year in the future?
For example, if you buy small stocks will your investment always outperform U.S. Treasury
bonds?
While on average small stocks should outperform U.S. Treasury bonds, this need not
be the case every year. As the return on small company stocks is riskier, its year to
year return is highly variable, some years it will be high and others it will be low.
While the return on the U.S. Treasury bonds less variable.