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CML and CAL
0
2
4
6
8
10
12
14
16
18
0
10
20
Standard Deviation
Ex
30
pected Retrun
CAL: Slope = 0.3571
CML: Slope = 0.20
6-6

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21.
a.
With 70% of his money invested in my fund’s portfolio, the client’s expected
return is 15% per year and standard deviation is 19.6% per year.
If he shifts
that money to the passive portfolio (which has an expected return of 13% and
standard deviation of 25%), his overall expected return becomes:
E(r
C
) = r
f
+ 0.7[E(r
M
)
−
r
f
] = 8 + [0.7
×
(13 – 8)] = 11.5%
The standard deviation of the complete portfolio using the passive portfolio
would be:
σ
C
= 0.7
×
σ
M
= 0.7
×
25% = 17.5%
Therefore, the shift entails a decrease in mean from 14% to 11.5% and a
decrease in standard deviation from 19.6% to 17.5%.
Since both mean return
and
standard deviation decrease, it is not yet clear whether the move is
beneficial.
The disadvantage of the shift is that, if the client is willing to
accept a mean return on his total portfolio of 11.5%, he can achieve it with a
lower standard deviation using my fund rather than the passive portfolio.
To achieve a target mean of 11.5%, we first write the mean of the complete
portfolio as a function of the proportion invested in my fund (
y
):
E(r
C
) = 8 + y(18
−
8) = 8 + 10y
Our target is: E(r
C
) = 11.5%.
Therefore, the proportion that must be invested
in my fund is determined as follows:
11.5 = 8 + 10y
⇒
35
.
0
10
8
5
.
11
y
=
−
=
The standard deviation of this portfolio would be:
σ
C
= y
×
28% = 0.35
×
28% = 9.8%
Thus, by using my portfolio, the same 11.5% expected return can be achieved
with a standard deviation of only 9.8% as opposed to the standard deviation of
17.5% using the passive portfolio.
b.
The fee would reduce the reward-to-variability ratio, i.e., the slope of the
CAL.
The client will be indifferent between my fund and the passive
portfolio if the slope of the after-fee CAL and the CML are equal.
Let f
denote the fee:
Slope of CAL with fee
28
f
10
28
f
8
18
−
=
−
−
=
Slope of CML (which requires no fee)
20
.
0
25
8
13
=
−
=
Setting these slopes equal we have:
20
.
0
28
f
10
=
−
⇒
10
−
f = 28
×
0.20 = 5.6
⇒
f = 10
−
5.6 = 4.4% per year
6-7