Idea Transcript
CHAPTER 8: INDEX MODELS ,
PROBLEM SETS
'I.
The advantage of the index model, compared to the Markowitz procedure, is the vastly reduced number of estimates required. In addition, the large number of estimates required for the Markowitz procedure can result in large aggregate estimation errors when implementing the procedure. The disadvantage of the index model arises from the model's assumption that return residuals are uncorrelated. This assumption will be incorrect if the index used omits a significant risk factor.' "
2.
The trade-off entailed in departing from pure indexing in favor of aq. actively managed portfolio is between the probability (or possibility) of superior performance against the certainty of additional manag~ment fees.
3.
The answer to this question can be seen from the formul~s for WO and w*. Other things held equal, w is smaller the greater the re,sidual variance" of a canCtiaate asset for inClusion in the portfolio. Furtner, we see that regardless of beta, when WO decreases, so does w*. Therefore, other things equal, the greater the residual 'variance of an asset; the smaller its position in the optimal risky portfolio. That is, ,increased firm-specific risk reduces the extent to which an active investor wi~l be wil~ing to depart from an indexed portfolio.
4.
The total risk premium equals: ex + (~ x market risk premium). We call alpha a "nonmarket" return premium because it is the portion 'of the return premium that'is independent of market performance.
°
.
,
The Sharpe ratio indicates that a higher alpha makes a security more desirable. Alpha, the numerator of the Sharpe ratio, 'is a fixed number that is not affected by the standard deviation of returns, the denominator of the Sharpe ratio. Hence, an increase in alpha increases the Sharpe ratio. Since the portfolio alpha is the portfolio-weighted average of the securities' alphas, then, holding all other parameters fixed, an increase in a security's alpha results in an increase in the portfolio Sharpe ratio. .
8-1
5.
a.
To optimize this portfolio one would need: n
= 60 estimates
of means
n
= 60 estimates
of variances
2
n -n 2
= 1,770 estimates
Therefore, In total: n 2 + 3n
.
b.
2
of covariances
= 1,890 estimates
In a single index model: rj - rr
= 0:8' '. .
c.
R2 measures th~ fraction of total variance of.return explained by the'market return. A's R2 is larger than B"s: 0.576 > 0.436 '-
d.
Rewriting the SCL equation in terms of total return (r) rather than excess return (R): . . rA - rf = a + 13(rM-:-rr) ~ rA = a + rf (l - 13)+ 13rM The intercept is now equal to:
a + rf(l - ~) = 1 + rf(1- 1.2) 'Since rf = 6%, the intercept would be: I
8-4
---=
1.2
= -0.2%
A has a
9.
The standard deviation of each stock can be derived from the followi!lg equation forR2:
_~;(j~ ...;. Explained
R2
i -
(j2
variance Total variance.
-
'" Therefore: =
(j2
~i
2
(j~.
0.7
~
Ri
A
2
X
20 = 980
0.20
=31.30%
(jA
For stock B: 2 B
= 69.28%
(jB
10.
2
= 1.2 X 20 = 4 800 0.12 ,.
(j2
The systematic risk for A is:
~i
(j~
2
= 0.70
X
•
2
20 = 1~6
The firm-specific risk of A (the residual variance) is the difference between A's total risk and"its systematic risk: . . 980 - 196 = 784 The systematic risk for B is: ~~(j~
= 1.202x 202 = 576
B's firm-specific risk (residual variance) is: 4800 - 576 = 4224
8-5
I
12.
Note that the c0l!elation is the square root of R2:p=.JR2, Cov(rA,fM) = PcrAcrM= 0.201/2 X 31.30 x 20 = 280 Cov(rB,!M) = PcrBcrM~ 0.12112X 69.28 x 20 = 480
13. .for portfolio P we can compute: , crp= [(0.62 x 980) + (0042 X 4-800) + (2 x 004 x 0.6 x 336]112= [1282.08]112
= 35.81 %
~p = (0.6 x 0.7)+ (004 x 1.2) = 0.90 cr2 (ep) = cr~- ~~cr~ = 1282.08 - (0.902 x 400) = 958.08 Cov(rp,rM) = ~pcr~ =0.90 x 400=360 This same result can also be attained using the covariances of the individual stocks with the market: Cov(rp,rM) = Cov(0.6r~ + OArB,rM) = 0.6Cov(rA, rM) + OACov(rB,rM) • . = (0.6 x 280) + (004 x 480) = 360
14. ,Note that the variance of T -bills is zero, and the c~)Variance of T-bills with any ass'etis zero. Therefore, for portfolio Q:
= [(0.52 xl,282.08)+(0.32 ~Q
= wp~p +
W
x400)+(2xO.5xO.3x360)
M~M= (0.5xO.90)+
r/
2
= 21.55%
(O.3xl) +0 = 0.75
cr2 (eQ) = cr~ -.:~~cr~ = 464.52 ~ (0.752 x400) == 239.52 Cov(rQ,rM) ='~Qcr~ = 0.75x400
15.
a.
= 300
Merrill Lynch adjusts beta by taking the sample estimate of beta and averaging it with 1.0, using.the weights of 2/3 and 1/3, as follows: adjusted beta = [(2/3) x i,24] + [(1/3) x 1.0]
b.
:= 1.16,
If you use your current estimate of beta to be ~t-l :::;1.24, then ~t
= 0.3 + (0.7 x 1.24) = 1.16~
8-6
16.
For Stock A: . aA ;,.rA .:...[rr + ~A(rM-rr)] = 11- [6 +0.8(12 - 6)]
= 0.2%
For stock B: aB = 14 - [6 + 1.5(12 -6)]= -1% Stock A would be a good addition to,a well-diversified portfolio. A short position in Stock . B may be desirable.
17.
a. ai
a
A
= fj
Alpha (a), - [rr + ~i(rM- rr)]
Expected excess return E(n) - rf
= 20% - [8% +• 1'.3(16% - 8%)] = 1.6%
20% - 8% = 12%
aB =18% - [8% + 1.8(16% - 8%)] = - 4,4%
18% - 8% = 10%
ac = 17% - [8% + 0.7(16% - 8%)] = 3,4%
17% - 8% = 9%
aD = 12% - [8% +1.0(16% - 8%)] = - 4.0%
12% - 8% = 4%
Stocks A and C have positive alphas, whereas stocks Band D have negative alphas, The residual variances are: cr2(eA) = 582 = 3,364 cr2(eB) = ,712 = 5,041
cr2( ec) = 602 = 3~600 cr2(eD) = 552 = 3,025
b.
To construct the optimal risky portfolio, we first determine the optimal active po~folio. ~sing the Treynor-Black techniq~e, we construct the active portfolio:
0.1 ~(e)
a cr\e) A
...0,000476
B
.............................................
C
D:1. / ~(e) .......
-0.000873
0.000944
-----.-.- .................•....................
........ :::0,6144.............. 1.1265
...-------- ....... _----_ ....... __ ...............
-1.2181
................................................
D
-0.001322
1.7058
Total
-0.000775
1.0000
Do not be concerned that the positive alpha stocks hav~ negative weights andyice versa. We will see that the entire position in the active portfolio will be negative; returning everything to good order. . .
8-7
With.the.se weights, the forecast for the active portfolio is:' u = [-0.6142 x 1.6] +"[1.1265 x (- 4.4)]-
[1.2181 x 3.4] + [1.7058 x (~4.0)]
=-16.90% ~ = [-0.6142 x 1.3] + [1.1265 x 1.8] -,-[1.2181 x 0.70] + [1.7058 x 1] = 2.08 The high beta (higher than any individual beta) results from the short positions in the relatively low beta stocks and the long positipns in the relatively high beta stocks. a2(e) = [(-0.6142)2 x 3364]'+ [1.12652 x 5041] + [(-1.2181)2 x 3600] + [1.70582 x 3025]
=: 21,809.6 a(e) = 147.68%
.
Here, again, the levered position in stock B [with high a2(e)] overcomes the diversification effect, and results in a high residual standard deviation. The optimal risky portfolio has a proportion w* in the active portfolio, computed as follows: 2
=u/a (e) o [E(rM)-rf]/'a~
w
. = -16.90/21,809.6=_0.05124 8/232 .
The negative position is justif~ed for the reason stated earlier. The adjustment for beta is: . w* =
. w0 = - 0.05124 1+ (1- ~).w 0 . 1 + (1- 2.08)(-0.05124)
= -0.0486
. Since w* is negative, the result is a positive position in stocks with positive alphas and a negative position in stocks with negative alphas. The position in the index portfolio is: ' '. 1 - (-0.0486) = 1.0486
. "
8-8
Compare this to the market's Sh':lrpe measure: SM = 8/23 = 0.3478 The difference is: 0.0184 Note that the only-moderate improvement in performance results from the fact that only a small position is taken in the active portfolio A because of its large residual variance. d.
To calculate the exact makeup of the complete portfolio, we first compute the mean excess return of the optimal risky portfolio and its variance. The risky portfolio beta is given by: . ~p
=
WM
+ (WA
X ~A)
= 1.0486 + [(-O.0~86) x 2.08] = 0.95
E(Rp) = Up
+ ~pE(RM) = [(-0.0486) x (-16.90%)] + (0.95 x 8%) = 8.42%
cr; = ~;(J~
+ (J2(ep)
= (0.95
X
23)2 +((-0.04862)x
21,809.6) = 528.94
(Jp = 23.00% . Since A =2.8, the optimal position in this portfolio is: y= 8.42 = 0.5685 . 0.0Ix2 ..8x528.94 In contrast, with a passive strategy:
8
y =-----= 0.0Ix2.8x232
0.5401
This is a,difference of: 0.0284 The final positions of the complete portfolio are: Bills M ,A B
C D
0.5685 0.5685 0.5685 0.5685
43.l5% 59.61% 1.70% -3.11% 3.37% -4.71% 100.00% [sum is subject to rounding error]
1- 0.5685'= 0.5685 x 1.0486 = x (-0.0486) x (-0.6142) = x (-0.0486) x 1.1265 = x (-0.0486) x (-1.2181) = x (-0.0486) x 1.7058 =
Note that M may include positive proportions of stocks A through D .
.8-9
18. a.
If a rrianager is not allowed to sell short he will not include stocks with negative alphas in his portfolio, so he will consider only A and C:
A
1.6 3.4,
C
a
a\e)
a
2
a ( e)
3,364 3,600
0.000476 0.000944 0.001420.
ala2(e) "La / a\e) 0.3352 0.6648' 1.0000
The forecast for. the active portfolio is: a
= (0.3352
x 1.6) + (0.6648 x 3.4)
= 2.80%
f3= (0.3352 x 1.3) + (0.664.8 x 0.7) = 0.90 a2(e)
= (0.33522
x 3,364)+: ~ (0.66482 x 3,6(0) = 1,969.03
. ,f":
i
.-
, •
a(e) = 44.37%
The weight in the active portf?lio is: =
w o
2
ala (e) E(R' M ) I aM2
= 2.80/1,969.03 .. 8 I 232
.
= 0.0940 0,
Adjusting for beta: w* = __ w_o__ = 0.094 . = 0.0931 1+ (1- f3)w° 1 + [(1- 0.90) x 0.094] The information ratio of the active portfolio is: A = a la(e) =2.80/44.37 == 0.0631 Hence, the square of Sharpe's measure is: • S2
= (8/23)2
+ 0.06312 = 0.1250
Therefore: S =0.3535 The market's Sharpe measure 'is: SM = 0.3478 When sh'ort sales are allowed (Problem 18), the manager's Sharpe measure is higher (0.3662). The reducti0!1 in the Sharpe measure ,is the cost of the short sale restriction.
8-10
The characteristics' of the optimal risky portfolio are:
+ WA X ~A
= ('1- 0.0931) + (0.0931 x 0.9) = 0.99 . .' E(Rp) = f3= 0.75
4.
d.
5.
b.
8-13 il ~
I.