© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Primer I: Top-Down Portfolio Management
• Capital
vs
. Asset allocation
• Markowitz security selection model
• Primer II: Asset Pricing Models
• CAPM (Theory & Practice)
• Index & Multi-Factor Models
• Primer III:
• Active Portfolio Management
• Performance Measurement: Benchmarks & Rewards
Part VI:
Active Portfolio
Management
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Two examples so far
•
stocks
-- Markowtiz security selection model (need inputs!)
•
bonds
-- fixed-income portfolio management
• Relevant questions
• why?
• what?
– market timing
– security analysis
» index model
» multi-factor model
• how?
– key worry = control for risk
Active Portfolio Management -- Why?
• APM -- a contradiction in terms?
• Nope!
– market efficiency requires many investors to manage actively
• intuition
mis -priced securities
– > deviations from passive strategies pay off
– > price pressures eliminate mis-pricing
– > active management does not pay off
– > securities become mis -priced again
– > …
• Theory
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Evidence
– other managers may beat the market
» small but statistically significant
» noise in security returns
−>
hard to disclaim
– some portfolio managers are really good
» hard to argue
– anomalies
» January effect, …
» disappearance?
What is Active Portfolio Management?
• Isn’t every strategy active?
• 1. Security selection -- clearly
– identify mis -priced securities
• 2. Asset allocation -- yep
– different asset categories
» require different forecasts
– example
» long-term bond return determinants
»
≠
equity return determinants
– international assets
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Isn’t every strategy active?
• 3. capital allocation -- even that!
– proportion invested in market portfolio
– requires to forecast
and
– might also lead to
market timing
» market conditions change over time
σ
2
*
005
.
0
2
]
[
M
f
m
x
x
x
A
r
r
E
w
=
−
]
[
r
m
E
σ
2
M
What is Active Portfolio Management? 3
• Approach
• definition
– purely passive strategy
» invest only in index funds
» one fund per asset category (equity, bonds, bills)
» proportions unchanged regardless of market conditions
• example
» 60% equity + 30% bonds + 10% bills
» fixed for 5 years = entire investment horizon
• active management
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Objectives
• concentrate on portfolio construction
– CAPM
−>
we can separate
– construction of efficient portfolio
– and allocation of funds
» between risky asset and bills
• two components
– security analysis
» maximize Sharpe ratio (CAPM)
– market timing
» shift assets in and out of risky portfolio
Market Timing
• Idea
• market timers shift money
» from money market (MM) to risky portfolio
• based on their forecasts of market return
• potential profits
• huge
• example
($1,000 reinvested from 1927 till 1978)
» 30-day T-bills:
$3,600
(
r
= 2.49%)
» NYSE:
$ 67,500
(
r
= 8.44%)
» perfect market timing:
$5,360,000,000
(
r
= 34.71%)
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Reasons for differences
• compounding
» for all assets
» importance for pension funds
• risk
– mainly for equities (
Fig. p. 985 6th edition
)
» T-bills are mostly risk-free
– irrelevant for market timing
» exception: T-bill rate varies a little
• information
» for market timing
Market Timing 3
• Market timing as an option
– much less risky than equities
• standard deviation is misleading
– perfect market timing
– yields dominant payoffs in each state of the world
(Fig. 27.1)
» gives minimum return guarantee
» + a non-negative random number
– market timing fees
• market timers will charge for service
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• In practice
• clients
– want managers to pick efficient portfolios
» maximize Sharpe ratio
– still need to pick the optimal proportion
» to invest in the risk-free asset
• managers
– need to update customers continuously
» relative attractiveness of risky portolio changes
– costly
• solution
– let managers shift money in and out of MM funds
» = solution used by most funds
Market Timing 5
• Evaluating market timers
• Basic idea
– Risk-return trade-off (Q1c, Assignment 2)
– fund performance should improve with the market
• Intuition
– as market improves,
a good market timer
shifts more money to market
»
caveat
: true if short sales are ruled out
• Formally
– Non-linear regressions (Fig. 24.5, BKM6)
» Regress portfolio excess returns (ER)
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Map
• A. idea
• B. portfolio construction
• C. numerical example
• D. multi- factor models
• E. use in practice
» industry use
» advantages
vs
. dangers
Security Selection 2
• A. Idea
(Treynor-Black)
• 1. consider the entire set of securities
» assume the entire set is there
» index model (passive portfolio = market porfolio)
• 2. focus on a small subset
» as many as analysts can reasonably handle
• 3. analyze
– use index (single - or multi-) factor model
» to estimate alpha, beta(s) and residual risk
» of securities in subset
– identify securities with positive expected “alpha”
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• A.
(continued)
• 4. mix non-zero alpha securities
» with passive portfolio (= market porfolio)
– why?
» want to maximize return
» but need to control for risk
» small subset
−>
too much risk if invest only in subset
– how?
» use the beta, alpha and residual risk estimated
• 5. optimal risky portfolio
» = mix of active and passive portfolio
Security Selection 4
• B. Portfolio construction (NOT Exam Mat’l)
• 1. assumptions
(index model)
– market portfolio M = efficient portfolio
–
and
have been estimated
» use them for passive portfolio
» no need for market timing
– beta relationship
]
[
r
m
E
σ
2
M
i
f
M
i
i
f
i
r
r
r
e
r
−
=
α
+
β
(
−
)
+
0
)
,
cov(
)
,
cov(
e
i
e
j
=
e
i
r
M
=
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• B.
(continued)
• 2. research or estimate
• 3. active portfolio
»
−>
done (i.e., keep security in passive portf.)
»
−>
go long
»
−>
go short
» optimal weights:
k
k
f
M
k
f
k
r
r
r
e
r
=
+
β
(
−
)
+
+
α
0
=
k
α
0
>
k
α
0
<
k
α
∑
=
=
n
j
j
j
k
k
k
e
e
w
1
2
2
)
(
/
)
(
/
σ
α
σ
α
1
1
=
∑
=
n
k
k
w
Security Selection 6
• B.
(continued)
• active portfolio
» “A” comprises all the assets with non-zero alpha
∑
=
=
n
k
k
k
A
w
1
α
α
∑
=
=
n
k
k
k
A
w
1
β
β
)
(
2
2
2
2
A
M
A
A
β
σ
σ
e
σ
=
+
∑
=
+
=
n
k
k
k
M
A
w
e
1
2
2
2
2
)
(
σ
σ
β
cov(
e
i
,
e
j
)
=
0
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• B.
(continued)
• 4. mixing active (A) & passive (M) portfolios
– portfolio A may lie above CML
» interpretation
(given analysis, M is not efficient after all)
» no need to know the “original” efficient frontier
– new frontier
(BKM6 Fig. 27.2)
» combine A and M
» A and M not perfectly correlated
– optimal risky portfolio
» tangency point, given risk-free asset
Security Selection 8
• B.
(continued)
• 5. formal construction
– intuition
» optimal combo of 2 risky assets & T-bills
(Lecture 11)
1
s.t.
]
[
=
+
−
M
A
P
f
P
w
w
w
r
r
E
Max
A
σ
σ
σ
σ
σ
M
A
A
A
A
ρ
A
M
A
M
A
A
A
M
A
w
w
w
w
R
E
w
R
E
w
Max
w
A
(
1
)
2
2
2
2
2
(
1
)
,
]
[
]
[
)
1
(
−
+
+
−
+
−
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• B.
(continued)
• 6. Optimal risky allocation
σ
σ
2
(
[
]
)
2
)
]
[
(
E
r
A
r
f
M
E
r
M
r
f
A
Den
=
−
+
−
Den
Num
w
A
=
)
,
cov(
)
]
[
(
)
]
[
(
E
r
A
r
f
M
2
E
r
M
r
f
r
A
r
M
Num
=
−
σ
−
−
)
,
cov(
)
2
]
[
]
[
(
E
r
M
+
E
r
A
−
r
f
r
A
r
M
−
Security Selection 10
•
Optimal risky allocation
– intuition for wo
» wo = ratio of reward-to-risk ratios for A and M
» we mix for risk diversification reasons
» the higher the reward for the extra risk taken
» the more we invest in the (very) risky portfolio A
σ
σ
α
/
2
2
)
]
[
(
)
(
/
M
f
M
A
A
o
r
r
E
e
w
−
=
o
A
o
A
w
w
w
)
1
(
1
*
β
−
+
=
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
•
Optimal risky allocation
– intuition for w*
» w* = adjustment for beta
» the weight for A depends on diversif. opportunities
» if
then more can be gained by diversifying
w
A
M
w
=
1
−
*
o
A
o
A
w
w
w
)
1
(
1
*
β
−
+
=
1
<
A
β
o
A
w
w
<
⇒
*
Security Selection 12
• 7. Optimal security weights:
– intuition
» to “max” the composite portfolio’s Sharpe ratio
» given the Sharpe ratio of the market (M) is fixed
» we must maximize the “appraisal” ratio of A:
∑
=
=
n
j
j
j
k
k
k
e
e
w
1
2
2
)
(
/
)
(
/
σ
α
σ
α
2
2
2
2
2
2
)
(
]
[
)
(
+
−
=
+
=
A
A
M
f
M
A
A
M
P
e
r
r
E
e
S
S
σ
σ
α
σ
α
)
(
A
A
e
σ
α
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
•
Optimal security weights
» given
» and
» we must have:
∑
=
=
n
j
j
j
k
k
k
e
e
w
1
2
2
)
(
/
)
(
/
σ
α
σ
α
∑
=
=
n
k
k
k
A
w
1
α
α
∑
=
+
=
n
k
k
k
M
A
A
w
e
1
2
2
2
2
2
β
σ
σ
(
)
σ
Security Selection 14
• B.
(continued)
• 8. Individual security contributions
» the appraisal ratio of each security
» measure its contribution
» to the performance of the active portfolio
∑
=
=
n
k
k
k
A
A
e
e
1
2
2
)
(
)
(
σ
σ
α
α
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• C. Numerical example
• data
(BKM6 pp. 992-995 and interpretation)
• Sharpe ratio
%
20
%;
7
%;
15
]
[
r
M
=
r
f
=
M
=
E
σ
2
/
1
1
2
2
2
2
2
)
(
]
[
)
(
+
−
=
+
=
∑
=
n
k
k
k
M
f
M
A
A
M
P
e
r
r
E
e
S
S
σ
σ
α
σ
α
S
S
P
M
2
2
/
1
2
2
2
.
16
20
8
222
.
0
%
20
%
8
1154
.
1563
.
1556
.
=
>
=
=
+
+
+
=
Security Selection 16
• optimal weights
(see table)
• portfolio characteristics
» large alpha, but large idiosyncratic risk
%
56
.
20
03
.
0
4735
.
1
)
05
.
0
(
)
6212
.
1
(
07
.
0
1477
.
1
+
−
−
+
=
=
x
x
x
∑
=
=
n
k
k
k
A
w
1
α
α
9519
.
0
1
=
=
∑
=
n
k
k
k
A
w
β
β
%
62
.
82
)
(
e
A
=
σ
σ
2
(
e
A
)
=
0
.
6828
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• optimal risky portfolio
– despite high alpha, small proportion in active portfolio
» large risk needs to be balanced out
– small adjustment for beta
» beta is close to 1
%
95
.
14
1495
.
0
)
1
(
1
*
=
=
−
+
=
o
A
o
A
w
w
w
β
1506
.
0
)
]
[
(
)
(
/
/
2
2
=
−
=
σ
σ
α
M
f
M
A
A
o
r
r
E
e
w
%
05
.
85
1
−
*
=
=
w
A
M
w
Security Selection 18
• performance gain
– Sharpe ratio
– M2 measure = 1.42%
(large number, given only 3 securities)
» match risk (i.e., std-dev) of portfolio M
» by mixing optimal risky portfolio and T-bills
» in proportions
and
respectively
S
S
P
=
=
>
=
0
.
4
=
M
%
20
%
8
4711
.
0
2219
.
0
σ
σ
2
2
M
P
σ
σ
2
2
1
M
P
−
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• D. Multi-factor models (NOT Exam Mat’l)
• so far: index model
• now: 2- factor illustration of multi- factor extension
• extension
from index model
is straightforward
» the entire analysis is based on residual analysis
» computations required, then proceed as before
f
f
f
k
f
k
f
k
r
E
r
r
E
r
r
e
r
E
[
]
−
=
β
1
(
[
1
]
−
)
+
β
2
(
[
2
]
−
)
+
+
α
)
(
)
,
cov(
2
2
2
1
2
1
2
2
2
2
2
1
2
1
2
k
k
k
k
k
k
β
σ
β
σ
β
β
r
r
σ
e
σ
=
+
+
+
)
,
cov(
)
)
,
cov(
r
i
r
j
=
β
i
1
β
j
1
σ
1
2
+
β
i
2
β
j
2
σ
2
2
+
(
β
i
1
β
j
2
+
β
i
2
β
j
1
r
1
r
2
Security Selection 20
D a t a :
A p o r t f o l i o m a n a g e m e n t h o u s e a p p r o x i m a t e s t h e r e t u r n g e n e r a t i n g p r o c e s s b y a t w o
-f a c t o r m o d e l a n d u s e s t w o - -f a c t o r p o r t -f o l i o s t o c o n s t r u c t i t s p a s s i v e p o r t -f o l i o . T h e i n p u t
t a b l e t h a t i s c o n s i d e r e d b y t h e h o u s e a n a l y s t s l o o k s a s f o l l o w s :
M i c r o F o r e c a s t s ---A s s e t E x p e c t e d R e t u r n ( % ) B e t a o n M B e t a o n H R e s i d u a l S D ( % ) ---S t o c k A 20 1 . 2 1.8 5 8 S t o c k B 18 1 . 4 1.1 7 1 S t o c k C 17 0 . 5 1.5 6 0 S t o c k D 12 1 . 0 0.2 5 5 ---M a c r o F o r e c a s t s ---A s s e t E x p e c t e d R e t u r n ( % ) S t a n d a r d D e v i a t i o n ( % ) ---T-bills 8 0 F a c t o r M p o r t f o l i o 1 6 23 F a c t o r H p o r t f o l i o 1 0 18---T h e c o r r e l a t i o n c o e f f i c i e n t b e t w e e n t h e t w o - f a c t o r p o r t f o l i o i s 0 . 6 .
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
Questions:
(a)
What is the optimal passive portfolio?
(b)
By how much is the optimal passive portfolio superior to the single-factor passive
portfolio, M, in terms of Sharpe’s measure?
(c)
What is the Sharpe measure of the optimal risky portfolio and what is the contribution
of the active portfolio to that measure?
Security Selection 22
• optimal combo of 2 risky assets & T-bills
(Lecture 9)
1
s.t.
]
[
=
+
−
M
H
P
f
P
w
w
w
r
r
E
Max
M
σ
σ
σ
σ
σ
H
M
M
M
M
ρ
M
H
M
H
M
M
M
H
M
w
w
w
w
R
E
w
R
E
w
Max
w
M
(
1
)
2
2
2
2
2
(
1
)
,
]
[
]
[
)
1
(
−
+
+
−
+
−
Den
Num
w
M
=
)
,
cov(
)
]
[
(
)
]
[
(
E
r
M
r
f
2
H
E
r
H
r
f
r
M
r
H
Num
=
−
σ
−
−
σ
σ
2
(
[
]
)
2
)
]
[
(
E
r
M
r
f
H
E
r
H
r
f
M
Den
=
−
+
−
)
,
cov(
)
2
]
[
]
[
(
E
r
H
+
E
r
M
−
r
f
r
M
r
H
−
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
σ
σ
2
(
[
]
)
2
)
]
[
(
E
r
M
r
f
H
E
r
H
r
f
M
Den
=
−
+
−
Den
Num
w
M
=
)
,
cov(
)
]
[
(
)
]
[
(
E
r
M
r
f
2
H
E
r
H
r
f
r
M
r
H
Num
=
−
σ
−
−
)
,
cov(
)
2
]
[
]
[
(
E
r
H
+
E
r
M
−
r
f
r
M
r
H
−
Security Selection 24
Answers:
(a)
The optimal passive portfolio is obtained from equation (7.8) in Chapter 7 on Optimal
Risky Portfolios – see Lecture 9.
wM = [E(RM)
σ
H 2– E(RH)Cov(rH, rM)/{E(R
M)σ
H 2+ E(RM)
σ
M 2– [E(R
H)+E(RM)]Cov(rH, rM)}where R
M= 8%, R
H= 2% and Cov(r
H, r
M) =
ρσ
Mσ
H= 0.6 x 23 x 18 = 248.4.
Thus,
w
M= 8 x 18
2
– (2 x 248.4)/[8 x 18
2+ (2 x 23
2) – (8 + 2) 248.4] = 1.797,
and
w
H= -0.797.
Because the weight on H is negative, if short sales are not allowed, portfolio H would
have to be left out of the passive portfolio.
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
Answers:
(b)With short sales allowed,
E(Rpassive) = 1.797 x 8 + (-0.797) x 2 = 12.78%
σ
2passive = (1.797 x 23)2
+ [(-0.797) x 18]
2+ 2 x 1.797 x (-0.797) x 248.4 = 1202.54
σ
passive = 34.68%.Sharpe’s measure in this case is given by:
S
passive= 12.78/34.68 = 0.3685,
and compared with the (simple) market’s Sharpe measure of
SM = 8/23 = 0.3478.
Security Selection 26
• We now must
• research or estimate
• find the active portfolio
»
−>
done (i.e., keep security in passive portf.)
»
−>
go long
»
−>
go short
» optimal weights:
k
k
f
M
k
f
k
r
r
r
e
r
=
+
β
(
−
)
+
+
α
0
=
k
α
0
>
k
α
0
<
k
α
∑
=
=
n
j
j
j
k
k
k
e
e
w
1
2
2
)
(
/
)
(
/
σ
α
σ
α
1
1
=
∑
=
n
k
k
w
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
Answers:
(c) The first step is to find the beta of the stocks relative to the optimized passive
portfolio. For any stock i, the covariance with a portfolio is the sum of the
covariances with the portfolio components, accounting for the weights of the
components. Thus,
β
i= Cov(r
i, r
passive)/
σ
2passive= (
β
iMw
Mσ
M2+
β
iHw
Hσ
H2)/
σ
2passive.
Therefore,
β
A= [1.2 x 1.797 x 23
2+ 1.8 x (-0.797) x 18
2]/1202.54 = 0.5621
β
B= [1.4 x 1.797 x 23
2+ 1.1 x (-0.797) x 18
2]/1202.54 = 0.8705
β
C= [0.5 x 1.797 x 23
2+ 1.5 x (-0.797) x 18
2]/1202.54 = 0.0731
β
D= [ 1 . 0 x 1 . 7 9 7 x 2 3
2+ 0.2 x (-0.797) x 18
2]/1202.54 = 0.7476
Security Selection 28
Now the alphas relative to the optimized portfolio can be computed:
α
i = E(ri) – rf -β
i, passive x E(rpassive)so that
α
A = 20 – 8 – (0.5621 x 12.78) = 4.82%α
B = 18 – 8 – (0.8705 x 12.78) = -1.12%α
C = 17 – 8 – (0.0731 x 12.78) = 8.07%© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
And the residual variances are now obtained from:
σ
e 2(i:passive) =
σ
i 2– (
β
2i:passivex
σ
2 passive),
where
σ
i 2=
β
M 2σ
M 2+
σ
e 2(i).
σ
e 2(A) = (1.3 x 23)
2+ 58
2– (0.5621 x 34.68)
2= 3878.01
σ
e 2(B) = (1.8 x 23)
2+ 71
2– (0.8705 x 34.68)
2= 5843.59
σ
e 2(C) = (0.7 x 23)
2+ 60
2– (0.0731 x 34.68)
2= 3852.78
σ
e2(D) = (1.0 x 23)
2+ 55
2– (0.7476 x 34.68)
2= 2881.80
Security Selection 30
From this point, the procedure is identical to that of the index model:
Stock
α
/
σ
e 2(
α
/
σ
e 2)/(
Σα
/
σ
e 2)
A
0.001243
1.0189
B
-0.000192
-0.1574
C
0.002095
1.7172
D
-0.001926
-1.5787
Total
0.001220
1.0000
The active portfolio parameters are:
α
= 1.0189 x 4.82 + (-0.1574) (–1.12) + (1.7172 x 8.07) + (-1.5787)(–5.55) = 27.7%
β
= 1.0189 x 0.5621 + (-0.1574)(0.8705) + 1.7172 x 0.0731 + (-1.5787)(0.7476) = -0.619.
σ
e 2= 1.0189
2x 3878.01 + (-0.1574)
2x 5843.59 + 1.7172
2x 3852.78
+ (-1.5787)
2x 2881.80 = 22,714.03
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
•
Optimal risky allocation (index model)
– intuition for wo
» wo = ratio of reward-to-risk ratios for A and P
» we mix for risk diversification reasons
» the higher the reward for the extra risk taken
» the more we invest in the (very) risky portfolio A
σ
σ
α
/
2
2
)
]
[
(
)
(
/
M
f
M
A
A
o
r
r
E
e
w
−
=
o
A
o
A
w
w
w
)
1
(
1
*
β
−
+
=
Security Selection 31
The proportions in the overall risky portfolio can now be determined:
w0 = (
α
/
σ
e2
)/[E(Rpassive)/
σ
2passive] = (27.71/22,714.03)/(12.78/1202.54) = 0.1148.w* = 0.1148/[1 + (1 + 0.6190) x 0.1148] = 0.0968.
Sharpe’s measure for the optimal risky portfolio is:
S
2= S
2passive + (α
/
σ
e)2= 0.3685
2+ [27.71
2/22,714.03] = 0.1696
S = 0.4118,
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• E. Potential benefits
(Treynor-Black)
• in practice
– not yet used often widely
» hard to estimate alphas
(bias correction needed)
» correction requires constant monitoring & appraisal
» shows alphas imprecise, second-guesses analysts
» do you think analysts like that?
• yet, significant benefits
» easy to implement
» allows for decentralized decisions
» can add significant return
» amenable to multi-factor analysis
Part VI.B:
Active Portfolio
Management Evaluation
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Returns
• return measurement over several periods
• Performance measures
• market timing
• security analysis
» Treynor, Sharpe, Jensen, appraisal ratio, M2
» practical cases
• Performance attribution
• bogey; asset allocation; sector and security decisions
Portfolio Return Measurement
• One period
vs
. multiple periods
• easy
vs
. unclear
» depends on number of periods
» affected by intermediate investments/withdrawals
• Time-weighted
vs
. dollar-weighted
• average return
vs
. IRR
• examples
(Table 24.1)
• why use a time average?
» performance measurement assigns responsibilities
» cash ins and outs?
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Geometric
vs
. arithmetic averages
• arithmetic
– unbiased forecast of expected future performance
» oriented towards the future
• geometric
– constant rate
» compounded, would yield same total return over period
» downward bias relative to arithmetic
» oriented towards the past
2
2
σ
−
≈
A
G
r
r
Portfolio Return Measurement 3
Question:
XYZ stock price and dividend histories are as follows:
---Year
Beginning of Year Price
Dividend Paid at Year-End
---1991
$100
$ 4
1992
$110
$ 4
1993
$ 9 0
$ 4
1994
$ 9 5
$ 4
---An investor buys three shares of XYZ at the beginning of 1991, buys another two shares
at the beginning of 1992, sells one share at the beginning of 1993, and sells all four
remaining shares at the beginning of 1994.
(a)
What are the arithmetic and geometric average time-weighted rates of return for the
investor?
(b)
What is the dollar-weighted rate of return?
(Hint: Carefully prepare a chart of cash flows for the four dates corresponding to the
turns of the year for January 1, 1991 to January 1, 1994. Calculate the internal
rate of return).
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
Answer:
(a)
Time-weighted average returns are based on year-by-year rates of return.
Year
Return [(capital gains + dividend)/price)]
---1991-1992
[(110-100) + 4]/100 = 14%
1992-1993
[(90 – 110) + 4]/110 = -14.55%
1993-1994
[(95 – 90) + 4 ]/90 = 10%
---Arithmetic mean = 3.15%
Geometric mean = 2.33%
Portfolio Return Measurement 5
(b)
T i m e
Cash Flow
Explanation
---0
-300
Purchase of 3 shares at $100 each.
1
-208
Purchase of 2 shares at $110 less dividend income on 3 share s held
2
110
Dividends on 5 shares plus sale of one share at price of $90 each.
3
396
Dividends on 4 shares plus sale of 4 shares at price of $95 each.
---$110
$396
______________________________________|____________|_____
Date
1/1/91
1/1/92
1/1/93
1/1/94
|
|
($300)
($208)
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Idea
• market timers shift money (MM
<−>
risky portfolio)
» based on their forecasts of market return
• return from market timing
» depends on # of times the timer is correct
• two scenarios: bull
vs
. bear
– must be correct in each scenario
» example 1: always predict snow in Winter in Montreal
right 95% of the time
but always wrong when no snow
» example 2: forward hedges
• overall quality
vs
. risk adjustment
Market Timing Evaluation 2
• Overall quality
• measure 1
– measure = P1 + P2 - 1
» P1
= proportion of correct bull predictions
= 1 if 100% correct
» P2
= proportion of correct bull predictions
= 1 if 100% correct
– example: return maximization
» correct 100% of bulls, 0% of bears
−>
measure = 0
» correct 50% of bulls, 50% of bears
->
measure = 0
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Overall quality
• measure 2
– unclear
» how large (P1 + P2 - 1) must be
» to ensure that we have seen “good performance”
– solution
» statistical significance
» application: hedging decisions (
as time allows
)
Market Timing Evaluation 4
• Risk adjustment
• problems
» neither measure so far accounts for risk
» market timers constantly change portfolio risk profile
• solutions
» time-varying dummy D (=1 for bull, 0 for bear)
» Squared term
P
f
M
f
M
f
P
r
a
b
r
r
c
r
r
D
e
r
−
=
+
(
−
)
+
(
−
)
+
P
f
M
f
M
f
P
r
a
b
r
r
c
r
r
D
e
r
−
=
+
(
−
)
+
(
−
)
+
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Bottom line -- evaluating market timers
• Basic idea
– Risk-return trade-off (Q1c, Assignment 2)
– fund performance should improve with the market
• Intuition
– as market improves,
a good market timer
shifts more money to market
»
caveat
: true if short sales are ruled out
• Formally
– Non-linear regressions (Fig. 24.5, BKM6)
»
regress
portfolio excess returns (ER)
on
market ER and ER^2
or on
ER and ER*timing dummy
Evaluating Security Selection
• Basic
• idea and problems
• Traditional
• response to problems
» Sharpe, Treynor, Jensen, appraisal ratio
• time-changing beta and market timing
• In practice
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Basic
• idea
– compare returns with those of “similar” portfolios
– depict percentiles
(BKM 4-5-6, Fig. 24.1)
» ex.: manager outperforms 90 out of 100 fund managers
is in the 90th percentile
» 5th, 95th percentiles; median, 25th and 75th_
• problems
– equities: allocations differ within groups
– fixed income: durations vary
Evaluating Security Selection 3
• Traditional
• idea
– account for risk taken by manager
– assume index model holds (and past performance matters)
» and compute risk-adjusted excess returns
• problems
– does extra performance cover fees
» difficulty to beat S&P 500
– estimation in practice
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Traditional measures
(continued)
• Sharpe
» appropriate for entire risky investment
• Treynor
» appropriate for one of many portfolios
(Fig. 24.3)
P
f
P
r
r
σ
−
P
f
P
r
r
β
−
Evaluating Security Selection 5
• Traditional measures
(continued)
• Jensen
• appraisal ratio
» benefit-to-cost ratio
» appropriate for active portfolio
(active PM)
)
(
P
P
e
σ
α
[
f
P
(
M
f
)
]
P
P
=
r
−
r
+
β
r
−
r
α
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
Question
Consider the two (excess return) index-model regression results for Stocks A and B. The
risk-free rate over the period was 6%, and the market’s average return was 14%.
i.
r
A- r
f= 1% + 1.2(r
M- r
f)
R-square = 0.576; residual std deviation ,
σ
(e
A) =10.3%;
standard deviation of (r
A-r
f) = 26.1%.
ii. r
B- r
f= 2% + 0.8(r
M- r
f)
R-square = 0.436; residual std deviation ,
σ
(e
B) =19.1%;
standard deviation of (r
B-r
f) = 24.9%.
(a)
Calculate the following statistics for each stock:
i.
Alpha.
ii.
Appraisal ratio.
iii.
Sharpe measure.
iv.
Treynor measure.
Evaluating Security Selection 7
Answer:
(a)
To compute the Sharpe measure, note that for each portfolio, (r
p– r
f) can be computed
from the right-hand side of the regression equation using the assumed parameters r
M=
14% and r
f= 6%.
The standard deviation of each stock’s returns is given in the problem.
The beta to use for the Treynor measure is the slope coefficient of the regression equation
presented in the problem.
Portfolio A
Portfolio B
(i)
α
is the intercept of the regression
1%
2%
(ii) Appraisal ratio =
α
/
σ
(e)
0.097
0.1047
(iii) Sharpe measure = (r
p– r
f)/
σ
0.4061
0.3373
(iv) Treynor measure = (r
p– r
f)/
β
8.833
10.5
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
(b)
Which stock is the best choice under the following circumstances?
i.
This is the only risky asset to be held by the investor.
ii.
This stock will be mixed with the rest of the investor’s portfolio, currently
composed solely of holdings in the market index fund.
iii.
This is one of many stocks that the investor is analyzing to form an actively
managed stock portfolio.
Evaluating Security Selection 9
Answer:
(a)
(i) If this is the only risky asset, then Sharpe’s measure is the one to use.
A’s is higher, so it is preferred.
(ii) If the portfolio is mixed with the index fund, the contribution to the overall Sharpe
measure is determined by the appraisal ratio.
Therefore, B is preferred.
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Problems
• does extra performance cover fees?
» difficulty to beat S&P 500
• estimation in practice?
» statistical significance?
» time-varying beta?
(Fig. 24.4)
» solution: add a quadratic term in regression
(Fig. 24.5)
• bottom line
» still used
» but not so much any more
Performance Attribution
• Idea
• hard to evaluate managers on risk-adjusted basis
• important to allocate bonuses
• Split
• excess returns
• between contributions
» broad asset allocation
» industry choices within each market
» security choices within each sector
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
• Bogey
(BKM6 Table 24.5)
• base-line passive portfolio
• assumed fixed for investment horizon
• Splits
(BKM6 Tables 24.6 to 24.8)
• broad asset
» compare to bogey
• industry
» given weights, compare with market weights
• security
Performance Attribution 3
Question 9 (20 points)
Consider the following information regarding the performance of a money manager in a
recent month. The table represents the actual return of each sector of the manager’s
portfolio in Column 1, the fraction of the portfolio allocated to each sector in Column 2,
the benchmark or neutral sector allocations in Column 3, and the returns of sector indices
in Column 4.
Actual Return
Actual Weight
Benchmark Weight
Index Return
---Equity
2%
0.70
0.60
2.5% (S&P 500)
Bonds
1%
0.20
0.30
1.2% (SB Index)*
Cash
0.5%
0.10
0.10
0.5%
---* S&B Index = Salomon Brothers Index.
(a)
What was the manager’s return in the month? What was his or her overperformance
or underperformance?
(b)
What was the contribution of security selection to relative performance?
(c)
What was the contribution of asset allocation to relative performance? Confirm that
the sum of selection and allocation contributions equals his or her total “excess”
return relative to the bogey.
© Michel A. Robe. No part of this primer may be reproduced by or transferred to anyone, in whole or in part, without written prior permission.
A n s w e r :
(a) Bogey:
0.60 x 2.5% + 0.30 x 1.2% + 0.10 x 0.5% = 1.91%
Actual:
0.70 x 2.0% + 0.20 x 1.0% + 0.10 x 0.5% = 1.65%
Underperformance:
0.26%
( a )
Security Selection:
Market
Differential Return
Manager’s Portfolio
Contributio n
Within Market
Weight
to
Performance
-Equity
-0.5%
0 . 7 0
-0.35%
B o n d s
-0.2%
0 . 2 0
-0.04%
C a s h
0
0 . 1 0
0%
-Contribution of security selection
-0.39%
