T HE E CONOMICS OF F INANCIAL M ARKETS
R. E. BAILEY
Solution Guide to Exercises for
Chapter 5 Portfolio selection: the mean-variance model
1. An investor uses the mean-variance criterion for selecting a portfolio of two risky assets. Asset 1 has an expected return of 20% and a variance of 4. Asset 2 has an expected return of 60%
and a variance of 36. There is no risk-free asset available.
(a) Explain how to construct the efficient portfolio frontier for the cases in which the correlation coefficient between the returns, ρ12 , is equal to +1 and also when it is equal to
−1.
Answer:
The expected rate of return on the portfolio is given by:
1
3
µP = a + (1 − a),
5
5
where a is the proportion of the portfolio invested in asset 1.
The variance of the rate of return on the portfolio is:
σP2 = 4a2 + 36(1 − a)2 + 2a(1 − a) × 2 × 6 × ρ12 ,
where ρ12 is the correlation coefficient between the rates of return on the two assets.
Case: ρ12 = +1:
σP2
= 4a2 + 36(1 − a)2 + 24a(1 − a)
2
σP
(1)
= (2a + 6(1 − a))
(2)
= 2a + 6(1 − a)
(3)
Case: ρ12 = −1:
σP2
= 4a2 + 36(1 − a)2 − 24a(1 − a)
2
= (2a − 6(1 − a))
σP
(4)
(5)
= ±(2a − 6(1 − a))
(6)
σP
= +(2a − 6(1 − a))
for
σP
= −(2a − 6(1 − a))
for
3
a≥
4
3
a<
4
(7)
(8)
Case: ρ12 = +1:
σP = 6 − 4a
Hence:
a=
µP
=
=
6 − σP
4
a=
−2 + σP
4
1 6 − σP
3 −2 + σP
+
5
4
5
4
6 − σP − 6 + 3σP
2σP
σP
=
=
.
20
20
10
1
(9)
(10)
σP 6
6 a
a
a
a
a
a
a
l
a
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a
ρ12 = +1
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ρ12 = −1
l
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H
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ρ
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12 = −1
,
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l
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l,
a
3
1
4
Figure 1: Standard deviation and a with ρ12 = ±1.
Case: ρ12 = −1 and a ≥ 3/4:
σP = 2a − 6(1 − a) = −6 + 8a
Hence:
a=
µP
=
=
6 + σP
8
a=
2 − σP
8
1 6 + σP
3 2 − σP
+
5
8
5
8
6 + σP + 6 − 3σP
12 − 2σP
3
σP
=
=
−
.
40
40
10
20
(11)
(12)
Case: ρ12 = −1 and a < 3/4:
σP = −2a + 6(1 − a) = 6 − 8a
Hence:
a=
µP
=
=
6 − σP
8
a=
2 + σP
8
1 6 − σP
3 2 + σP
+
5
8
5
8
6 − σP + 6 + 3σP
12 + 2σP
3
σP
=
=
+
.
40
40
10
20
(13)
(14)
(b) Describe, in general terms, how to construct the portfolio frontier when −1 < ρ < +1.
Answer:
When −1 < ρ < +1, the portfolio proportions a and 1 − a are chosen to minimize σP
(or σP2 ) for each level of µP . For each level of µP , the solution provides one point on the
portfolio frontier. As µP is chosen at different levels, so the frontier is traced out. The
multiple asset case is similar, except that now there are n portfolio proportions to choose:
a1 , a2 , . . . , an (such that the proportions sum to 1).
For −1 < ρ < +1, the frontier is a hyperbola in the space of (µP , σP ):
2
µP 6
3
10
ρ12 = −1
r Asset 2
%
%
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b
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%
b
b
%
Q
b
b
%
br
ρ12 = +1
Asset 1
σP
Figure 2: Efficiency Frontier with Two Assets and ρ12 = ±1.
µP 6
3
10
ρ12 = −1
r Asset 2
%
%%
%
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b
%
b
Q
b
b
b%
r%
ρ12 = +1
Asset 1
σP
Figure 3: Efficiency Frontier with Two Assets and ρ12 = ±1.
2. You are given the following information about three assets:
µj
8%
16%
11%
Asset:
1
2
3
σj
8%
50%
10%
The risk-free interest rate, r0 , equals 6%.
(a) Calculate and interpret the Sharpe ratio for each of the three assets.
Answer:
Asset µj
σj
µj − r0
sj
1
0.08
0.08
0.02
0.25
2
0.16
0.50
0.10
0.20
3
0.11 0.10
0.05
0.50
The Sharpe ratio for asset j is defined by sj = (µj − r0 )/σj . In words, it is the excess
of the expected rate of return above the risk-free rate divided by the standard deviation
of return for each asset. The Sharpe can be interpreted for each asset as its rate of return
“normalised” for variability, that is, the hypothetical expected return (in excess of the
3
risk-free rate) per unit of return variability. It is as if the excess expected return is scaled
by the return variability (standard deviation) to make all assets comparable with a “unit
standard deviation”.
(b) Suppose that the risk (standard deviation) of a benchmark portfolio is given by 20%.
What, if anything, does this imply about the comparative performance of the assets?
Answer:
The Risk-Adjusted Performance, RAP , provides a natural way of measuring asset returns at a benchmark risk (standard deviation):
σB
RAPj = r0 +
(µj − r0 ),
σj
or RAPj = r0 + σB sj , substituting j’s Sharpe ratio. In words, RAPj is the expected
rate of return on asset j if its risk equalled that of the benchmark but leaving its Sharpe
ratio unchanged. For the information provided here:
RAPj
= r0 + σB sj
(15)
RAP1 = 0.06 + 0.2 × 0.25 = 0.11
(16)
RAP2 = 0.06 + 0.2 × 0.20 = 0.10
(17)
RAP3 = 0.06 + 0.2 × 0.50 = 0.16
(18)
That is:
Asset RAPj
1
0.11
2
0.10
3
0.16
If all assets had the same standard deviation (risk, as measured by σB = 0.20) but retain
their Sharpe ratios, then it is as if asset 1 yields 11%, asset 2 yields 10%, and asset 3
yields 16%. Hence, asset 2 is the riskiest and asset 3 the least risky.
Of course, the RAP provides the same information as the Sharpe ratio (the ranking is
the same) but gives the information in a slightly different form.
(c) You are now told that a mean-variance efficient portfolio, ‘E’, is available with µE =
12% and σE = 15%. What inferences can you draw from this information?
Answer:
The Sharpe ratio for the efficient portfolio is:
sE =
0.12 − 0.06
0.06
=
= 0.4.
0.15
0.15
But note that sE = 0.4 < 0.5 = s3 .
There is a contradiction, because sE ≥ sj for every asset or portfolio of assets. Hence,
either “E” is not mean-variance efficient or asset 3 does not have µ3 = 0.11 and σ3 =
0.10.
3. Consider a world with several risky assets and in which an investor can borrow at a given
interest rate which is different (higher) than the rate at which the investor can lend.
(a) Construct the efficient portfolio frontier.
Answer:
First, construct the frontier for a portfolios consisting of risky assets only. This is a
hyperbola, depicted as F F in the following figures.
4
Next construct the set of efficient portfolios when lending occurs (i.e. when there is a
non-negative proportion of the portfolio invested in the risk-free asset). To do this, with
a lending rate of r0L , start from the point r0L on the vertical axis: this corresponds to a
portfolio invested entirely in the risk-free asset. Now draw a straight line (ray) from this
point to any point on the F F frontier. Any point on the line is a feasible portfolio. Try
drawing a sequence of rays, each with higher µP for a given σP (i.e. steeper straight
lines). The steepest ray that connects with F F will be a ray that is just tangential to the
frontier F F . This ray depicts the efficient portfolios for the given lending rate. In the
figure it is depicted by the line r0L Y .
Note that, because the investor can lend but not borrow at r0L , only the segment of the
line between r0L and Y is relevant. Points to the right of Y are not feasible, because the
investor would be borrowing.
The efficient portfolios with borrowing at r0B are constructed in a similar way to provide
the set depicted by the straight line r0B Z. For this set only points to the right of Z are
feasible: the investor can borrow but not lend at the rate r0B .
Hence the overall set of efficient portfolios is depicted by the line r0L Y ZE. In the segment r0L Y the portfolio has a positive amount of the risk-free asset (lending). In the segment Y Z the portfolio is invested entirely in risky assets (neither borrowing nor lending).
In the segment ZE the portfolio has a negative amount of the risk-free asset (borrowing).
µP
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6
r0B
r0L
F
σP
Figure 4: The Efficient Portfolios with different lending and borrowing rates.
(b) Depict an equilibrium for an investor who chooses to borrow.
Answer:
Begin with the previous diagram and imagine that there is a set of indifference curves in
(µP , σP ) space, representing preferences. Suppose that preferences are such that there
is an indifference curve II 0 tangential to the line ZE (see figure 5 on page 6).
The point of tangency represents an optimal portfolio for an investor that borrows. (No
higher indifference curve is attainable.)
5
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6
r0B
r0L
F
σP
Figure 5: An optimal portfolio depicted by tangency of an indifference curve with the efficient
portfolio constraint.
(c) Suppose that the interest rate at which the investor can borrow increases. Examine the
implications for the investor’s optimal investment decisions.
Answer:
0
Suppose that the borrowing rate increases from r0B to r0B (see figure 6 on page 7). Now
the set of efficient portfolios is depicted by the line: r0L Y Z 0 E 0 . Every investor is certainly
no better off than before the rate increase. An investor who is a borrower at the initial
rate is worse off (will be on a lower indifference curve).
If the investor continues as a borrower at the higher interest rate, there will be a point
of tangency between a different (lower) indifference curve and the segment of the line
Z 0E0.
The investor will be worse off as a consequence of the increase in the borrowing rate but
may end up borrowing more (taking more risk) or less depending on “substitution” and
“scale” effects. The substitution effect (a movement along a given indifference curve)
of an increase in the borrowing rate will be to reduce borrowing. The scale (“income”)
effect could result in either an increase or decrease in borrowing.
Note that it is possible that borrowing is reduced to zero as a consequence of the increase
in the borrowing rate.
*****
6
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r0B
0
r0B
r0L
F
Figure 6: The Efficient Portfolios after an increase in the borrowing rate.
7
σP