Tutorial 5 - Unit III/Part III Portfolio Theory II/II
SELF-STUDY PROBLEM SET 1
When adding real estate to an asset allocation program that currently includes only stocks,
bonds, and cash, which of the properties of real estate returns affect portfolio risk? (You can
choose more than one answer in this problem set.)
a. Standard deviation.
b. Expected return.
c. Correlation with returns of the other asset classes.
SELF-STUDY PROBLEM SET 2
Which statement about portfolio diversification is correct?
a. Proper diversification can reduce or eliminate systematic risk.
b. Diversification reduces the portfolio’s expected return because it reduces a portfolio’s
total risk.
c. As more securities are added to a portfolio, total risk typically would be expected to
fall at a decreasing rate.
d. The risk-reducing benefits of diversification do not occur meaningfully until at least 30
individual securities are included in the portfolio.
Explanations:
b. Diversification does not necessarily reduce portfolio’s expected return if each security
has similar E(r). However, the correlation coefficient between each security’s return must
be lower than +1 in reality. As a result, portfolio standard deviation must fall when adding
more asset while expected return may not change.
SELF-STUDY PROBLEM SET 3
Which of the following statements about the minimum variance portfolio of all risky
securities are valid?
a. Its variance must be lower than those of all other risky securities or portfolios.
b. Its expected return can be lower than the risk-free rate.
c. It may be the optimal risky portfolio.
d. It must include all individual securities including the risk-free asset.
Explanations:
b. MVP E(r) cannot be lower than the risk-free rate because MVP is a risky portfolio.
c. MVP cannot be the optimal risky portfolio because optimal risky portfolio is always the
tangency point between CAL and Minimum Variance Frontier. If MVP has to be a tangency
point between CAL and Minimum Variance Frontier, then CAL has to originate from the
horizontal axis (measure of standard deviation) instead of vertical axis (measure of expected
return). CAL can only originate from vertical axis.
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
SELF-STUDY PROBLEM SET 4
Which one of the following portfolios cannot lie on the efficient frontier as described by
Markowitz?
Portfolio
Expected Return (%) Standard Deviation (%)
a.
W
15
36
b.
X
12
15
c.
Z
5
7
d.
Y
9
21
You can plot portfolios W, X, Y, Z in an expected return-standard deviation diagram. You
will then discover portfolio X is dominating Y as portfolio X has higher expected return but
lower standard deviation than portfolio Y. In other words, portfolio W, X, and Z can be on
the efficient frontier but portfolio Y must be inside the efficient frontier.
SELF-STUDY PROBLEM SET 5
Stocks A, B, and C have the same expected return and standard deviation. The following
table shows the correlations between the returns on these stocks.
Given these correlations, the portfolio constructed from these stocks having the lowest risk
is a portfolio:
a. Equally invested in stocks A and B.
b. Equally invested in stocks A and C.
c. Equally invested in stocks B and C.
d. Totally invested in stock C.
SELF-STUDY PROBLEM SET 6
The correlation coefficients between pairs of stocks are as follows: ρAB = 0.85; ρAC = 0.60;
ρAD = 0.45. Each stock has an expected return of 8% and a standard deviation of 20%.
i. If your entire portfolio is now composed of stock A and you can add some of only one
stock to your portfolio, you would choose:
a. B.
b. C.
c. D.
d. Need more data.
ii. Suppose that in addition to investing in one more stock you can invest in T-bills as well.
Would you change your answers to Part i if the T-bill rate is 8%?
The risky portfolio is at 8%, so the optimal CAL runs from the risk-free rate
through MVP. This implies risk-averse investors will just hold T-Bills.
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
SELF-STUDY PROBLEM SET 7
The standard deviation of the portfolio is always equal to the weighted average of the
standard deviations of the assets in the portfolio. True or False?
False. The portfolio standard deviation equals the weighted average of the
component-asset standard deviations only in the special case that all assets are
perfectly positively correlated. Otherwise, as the formula for portfolio
standard deviation shows, the portfolio standard deviation is less than the
weighted average of the component-asset standard deviations. The portfolio
variance is a weighted sum of the elements in the covariance matrix, with the
products of the portfolio proportions as weights.
SELF-STUDY PROBLEM SET 8
A pension fund manager is considering three mutual funds. The first is a stock fund, the
second is a long-term government and corporate bond fund, and the third is a T-bill money
market fund that yields a rate of 8%. The information of the risky funds is as follows:
Expected Return
Standard Deviation
Stock fund (S)
20%
30%
Bond fund (B)
12%
15%
The correlation between the fund returns is 0.1.
a. Tabulate and draw the investment opportunity set of the two risky funds. Use
investment proportions for the stock fund of zero to 100% in increments of 20%.
Proportion in Proportion in Expected Standard
stock fund
bond fund
return Deviation
0.00%
20.00%
40.00%
60.00%
80.00%
100.00%
100.00%
80.00%
60.00%
40.00%
20.00%
0.00%
12.00%
13.60%
15.20%
16.80%
18.40%
20.00%
Graph shown below.
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
15.00%
13.94%
15.70%
19.53%
24.48%
30.00%
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
25.00
INVESTMENT OPPORTUNITY SET
CML
20.00
Tangency
Portfolio
Efficient frontier
of risky assets
15.00
Minimum
Variance
Portfolio
10.00
rf = 8.00
5.00
0.00
0.00
5.00
10.00
15.00
20.00
25.00
30.00
b. Solve numerically for the proportions of each asset and for the expected return and
standard deviation of the optimal risky portfolio.
Covariance Matrix: Cov(rS , rB ) = ρSB σS σB
Bonds
Stocks
Bonds
Stocks
0.0225
0.0045
0.0045
0.09
[E(rS ) − rf ]σ2B − [E(rB ) − rf ]Cov(rB , rS )
wS =
[E(rS ) − rf ]σ2B + [E(rB ) − rf ]σ2S − [E(rS ) + E(rB ) − 2rf ]Cov(rB , rS )
[0.2 − 0.08]0.0225 − [0.12 − 0.08]0.0045
=
[0.2 − 0.08]0.0225 + [0.12 − 0.08]0.09 − [0.2 + 0.12 − 2 × 0.0.8]0.0045
= 𝟎. 𝟒𝟓𝟏𝟔
wB = 1 − 0.4516
= 𝟎. 𝟓𝟒𝟖𝟒
E(rp ) = wS E(rS ) + wB E(rB )
= 0.4516 × .20 + 0.5484 × .12
= 𝟎. 𝟏𝟓𝟔𝟏
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
σP = √wS2 σ2S + wB2 σ2B + 2wS wB Cov(rS , rB )
= √0.45162 × 0.09 + 0.54842 × 0.0225 + 2 × 0.4516 × 0.5484 × 0.0045
= 𝟏𝟔. 𝟓𝟒%
c. What is the reward-to-volatility ratio of the best feasible CAL?
E(rP ) − rf
S=
σP
0.1561 − 0.08
0.1654
= 𝟎. 𝟒𝟔𝟎𝟏
=
d. You require that your portfolio yield an expected return of 14%, and that it be efficient,
on the best feasible CAL.
i.
What is the proportion invested in the T-bill fund and each of the two risky
funds?
E(rc ) = y E(rp ) + (1 − y)rf
E(rc ) = rf + y[E(rp ) − rf ]
14% = 0.08 + y[0.1561 − 0.08]
y = 0.7884
Proportion invested in T − bills
= (1 − y)
= 1 − 0.7884
= 𝟎. 𝟐𝟏𝟏𝟔
Proportion invested in stock
= 0.7884 × 0.4516
= 𝟎. 𝟑𝟓𝟔𝟎
Proportion invested in bond
= 0.7884 × 0.5484
= 𝟎. 𝟒𝟑𝟐𝟒
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
ii.
What is the standard deviation of your portfolio?
σC = yσp
= 0.7884 × 0.1654
= 𝟎. 𝟏𝟑𝟎𝟒
e. If you were to use only the two risky funds, and still require an expected return of 14%,
what would be the investment proportions of your portfolio? Compare its standard
deviation to that of the optimized portfolio in d) ii, what do you conclude?
14% = 0.2 × wS + 0.12 × (1 − wS )
wS = 𝟎. 𝟐𝟓
wD = (1 − wS ) = 𝟎. 𝟕𝟓
σP = √wS2 σ2S + wB2 σ2B + 2wS wB Cov(rS , rB )
= √0.252 × 0.09 + 0.752 × 0.0225 + 2 × 0.25 × 0.75 × 0.0045
= 𝟏𝟒. 𝟏𝟑%
This is considerably greater than the standard deviation of 13.04% achieved using Tbills and the optimal portfolio.
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
SELF-STUDY PROBLEM SET 9
Suppose that there are many stocks in the security market and that the characteristics of
Stock A and B are given as follows:
Stock Expected Return Standard Deviation
A
10%
5%
B
15
10
Correlation = –1
What must be the value of the risk-free rate? (Hint: Think about constructing a risk-free
portfolio from stock A and B.)
σB
wA =
σA + σB
10%
=
5% + 10%
= 66.67%
σA
5%
=
= 33.33%
σA + σB 5% + 10%
= (1 − wA ) = 33.33%
wB =
E(rMin ) = rf = 66.67% × 10% + 33.33% × 15%
= 𝟏𝟏. 𝟔𝟕%
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
SELF-STUDY PROBLEM SET 10
Abigail Grace has a $900,000 fully diversified portfolio. She subsequently inherits ABC
Company common stock worth $100,000. Her financial adviser provided her with the
following forecast information:
The correlation coefficient of ABC stock returns with the original portfolio returns is 0.40.
a. The inheritance changes Grace’s overall portfolio and she is deciding whether to keep the
ABC stock. Assuming Grace keeps the ABC stock, calculate the:
i. Expected return of her new portfolio which includes the ABC stock.
ii. Covariance of ABC stock returns with the original portfolio returns.
iii. Standard deviation of her new portfolio which includes the ABC stock.
i. E(rNP) = wOP E(rOP ) + wABC E(rABC ) = (0.9  0.0067) + (0.1  0.0125) = 0.728%
ii.
Cov = ρ  OP  ABC = 0.40  0.0237  0.0295 = 0.00028
iii. NP = [wOP2 OP2 + wABC2 ABC2 + 2 wOP wABC (CovOP , ABC)]1/2
= [(0.9 2  0.02372) + (0.12  0.02952) + (2  0.9  0.1  0.00028)]1/2
= 2.27%
b. If Grace sells the ABC stock, she will invest the proceeds in risk-free government
securities yielding 0.42% monthly. Assuming Grace sells the ABC stock and replaces it
with the government securities, calculate the
i. Expected return of her new portfolio, which includes the government securities.
ii. Covariance of the government security returns with the original portfolio returns.
iii. Standard deviation of her new portfolio, which includes the government securities.
Subscript OP refers to the original portfolio, GS to government securities, and NP to
the new portfolio.
i.
E(rNP) = wOP E(rOP ) + wGS E(rGS ) = (0.9  0.0067) + (0.1  0.0042) = 0.645%
ii.
Cov = ρ  OP  GS = 0  2.37%  0 = 0
iii. NP = [wOP2 OP2 + wGS2 GS2 + 2 wOP wGS (CovOP , GS)]1/2
= [(0.92  0.02372) + (0.12  0) + (2  0.9  0.1  0)]1/2
= 2.13%
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
SELF-STUDY PROBLEM SET 1
The following are estimates for two stocks.
Stock Expected Return
Beta
Firm-Specific Standard Deviation
A
13%
0.8
30%
B
18
1.2
40
The market index has a standard deviation of 22% and the risk-free rate is 8%.
a. What are the standard deviations of stocks A and B?
σ2i = β2i σ2M + σ2ei
σA = √0.82 × 0.222 + 0.302 = 34.78%
σB = √1.22 × 0.222 + 0.402 = 47.93%
b. Suppose that we were to construct a portfolio with proportions:
Stock A: 0.30
Stock B: 0.45
T-bills: 0.25
Compute the expected return, beta, nonsystematic standard deviation, and standard
deviation of the portfolio.
E(rP ) = wA × E(rA ) + wB × E(rB ) + wf × rf
E(rP ) = 0.30 × 13% +(0.45 × 18% + 0.25 × 8%
= 14%
βP = wA × βA + wB × βB + wf × βf
βP = 0.30 × 0.8 + 0.45 × 1.2 + 0.25 × 0.0
= 0.78
σ2ep = wA2 × σ2eA + wB2 × σ2eB + wf2 × σ2ef
= 0.302 × 0.302 + 0.452 × 0.402 + 0.252 × 0
= 0.0405
σep = √0.0405 = 𝟎. 𝟐𝟎𝟏𝟐
σ2p = β2p σ2M + σ2ep
= 0.782 0.222 + 0.20122
= 0.069928
σp = √0.069928 = 𝟎. 𝟐𝟔𝟒𝟒
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
SELF-STUDY PROBLEM SET 2
Consider the following two regression lines for stocks A and B in the following figure with
the same standard deviation.
a. Which stock has higher firm-specific risk?
The two figures depict the stocks’ security characteristic lines (SCL). Stock A has
higher firm-specific risk because the deviations of the observations from the SCL are
larger for Stock A than for Stock B. Deviations are measured by the vertical distance of
each observation from the SCL.
b. Which stock has greater systematic (market) risk?
Beta is the slope of the SCL, which is the measure of systematic risk. The SCL for Stock
B is steeper; hence Stock B’s systematic risk is greater.
c. Which stock has higher R2 ?
The R2 (or squared correlation coefficient) of the SCL is the ratio of the explained
variance of the stock’s return to total variance, and the total variance is the sum of the
explained variance plus the unexplained variance (the stock’s residual variance):
R2 =
β2i σ2M
Explained Variance
=
2 2
2
Total Variance
βi σM + σei
Since the explained variance for Stock B is greater than for Stock A (the explained
variance isβ2p σ2M , which is greater since its beta is higher), and its residual variance σ2eB is
smaller, Stock B’s R2 is higher than Stock A’s.
d. Which stock has higher alpha?
Alpha is the intercept of the SCL with the excess return axis. Stock A has a small positive
alpha whereas Stock B has a negative alpha; hence, Stock A’s alpha is larger.
e. Which stock has higher correlation with the market?
The correlation coefficient is simply the square root of R2, so Stock B’s correlation with
the market is higher.
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
SELF-STUDY PROBLEM SET 3
Consider the two (excess return) index model regression results for A and B:
a. Which stock has more firm-specific risk?
Firm-specific risk is measured by the residual standard deviation. Thus, stock A has
more firm-specific risk: 10.3% > 9.1%
b. Which has greater market risk?
Market risk is measured by beta, the slope coefficient of the regression. A has a larger
beta coefficient: 1.2 > 0.8
c. For which stock does market movement explain a greater fraction of return variability?
R2 measures the fraction of total variance of return explained by the market return.
A’s R2 is larger than B’s: 0.576 > 0.436
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
SELF-STUDY PROBLEM SET 4
Use the following data for sections a through f. Suppose that the index model for
stocks A and B is estimated from excess returns with the following results:
a.
What is the standard deviation of each stock?
β2i σ2M
R2 = 2 2
βi σM + σ2ei
β2A σ2M
R2
0.72 × 0.22
2
σA =
0.20
σ2A =
σA = √0.0980 = 𝟎. 𝟑𝟏𝟑𝟏
β2B σ2M
R2
1.22 × 0.22
2
σB =
0.12
σ2B =
σB = √0.48 = 𝟎. 𝟔𝟗𝟐𝟖
b.
Break down the variance of each stock to the systematic and firm-specific components.
β2A σ2M = 0.72 × 0.22 = 𝟎. 𝟎𝟏𝟗𝟔
σ2A = β2A σ2M + σ2eA
0.0980 = 0.0196 + σ2eA
σ2eA = 𝟎. 𝟎𝟕𝟖𝟒
β2B σ2M = 1.22 × 0.22 = 𝟎. 𝟎𝟓𝟕𝟔
σ2B = β2B σ2M + σ2eB
0.48 = 0.0576 + σ2eB
σ2eB = 𝟎. 𝟒𝟐𝟐𝟒
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
c.
What are the covariance and correlation coefficient between the two stocks?
Cov(rA , rB ) = βA βB σ2M
= 0.70 × 1.2 × 0.202
= 𝟎. 𝟎𝟑𝟑𝟔
βA βB σ2M
σA σB
0.0336
=
0.3131 × 0.6928
= 𝟎. 𝟏𝟓𝟓
ρA,B =
d.
What is the covariance between each stock and the market index?
Cov(rA , rM ) = βA βM σ2M
= 0.70 × 1 × 0.202
= 𝟎. 𝟎𝟐𝟖
Cov(rB , rM ) = βB βM σ2M
= 1.2 × 1 × 0.202
= 𝟎. 𝟎𝟒𝟖
e.
For portfolio P with investment proportions of 60% in A and 40% in B, what is the
variance related to the systematic and unsystematic components, the standard deviation
of P and the covariance of the portfolio P and the market index?
σ2p = β2p σ2M + σ2ep
β P = w A × β A + wB × β B
βP = (0.6 × 0.7) + (0.4 × 1.2)
= 0.90
σ2ep = wA2 × σ2eA + wB2 × σ2eB
= (0.62 × 0.0784) + (0.42 × 0.4224)
= 0.09581
β2p σ2M = 0.902 × 0.202 = 𝟎. 𝟎𝟑𝟐𝟒
σ2p = 0.09581 + 0.0324
σ2p = 0.1282
σp = √0.1282 = 𝟎. 𝟑𝟓𝟖𝟏
Cov(rP , rM ) = βP βM σ2M
= 0.90 × 1 × 0.202 = 𝟎. 𝟎𝟑𝟔
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
Tutorial 5 - Unit III/Part III Portfolio Theory II/II
f.
Rework section e for portfolio Q with investment proportions of 50% in P, 30% in the
market index, and 20% in T-bills.
σ2Q = β2Q σ2M + σ2eQ
βQ = wP × βP + wM × βM + wf × βf
βQ = (0.5 × 0.9) + (0.3 × 1) + 0
= 0.75
2
σ2eQ = wP2 × σ2eP + wM
× σ2eM + wF2 × σ2eF
= (0.52 × 0.09581) + (0.32 × 0) + 0
= 0.02395
β2Q σ2M = 0.752 × 0.202 = 𝟎. 𝟎𝟐𝟐𝟓
σ2p = 0.0225 + 0.02395
σ2p = 0.04645
σp = √0.4645 = 𝟎. 𝟐𝟏𝟓𝟓
Cov(rQ , rM ) = βQ βM σ2M
= 0.75 × 1 × 0.202 = 𝟎. 𝟎𝟑𝟎
Tutorial 5 - Unit III/Part III Portfolio Theory II/II