Portfolio Evaluation

1
Portfolio Evaluation

• Outline
• Investment return measurement
• conventional measurement theory
• Evaluation with changing portfolio composition
• Evaluation with market timing
• Performance attribution procedures and evaluation

2
Measuring Returns

• Dollar-weighted return is the internal rate of
  return. It is a return equal across a
  multiperiod.
• Time-weighted return is the arithmetic average of
  each one- period return
• Time-weighted return is important for money
  managers. Because they cannot control cash inflow
  and outflow for each period, return per period
  measure is more relevant.

3
Arithmetic Average is simply the average of
returns over several periods.Geometric return
average is the return over several periods is
computed as(1rG)(1r1)(1r2)...(1rn)1/n Fo
r past returns performance evaluation, the
geometric return is a better measure than
arithmetic average. For estimating the expected
future return, using historic average,
arithmetric average is a better as it is an
unbiased estimator.
4
Conventional Approaches to Performance Evaluation

• Sharpe measure (rp-rf)/sp is the excess return
  per unit risk of standard deviation
• Treynor measure (rp-rf)/bp is the excess return
  per unit systematic risk.
• Jensen measure abnormal returnap rp -
  rfbp(rm-rf)
• Appraisal ratio ap/s(ep), which is the alpha
  (abnormal return) divided by the nonsystematic
  risk.

5
Evaluations among Different Measures
Excess Return
Treynor lines
. Q
. P
SML
Market
1.0
Beta
6
Treynor measure assumes (1) the portfolio is
well-diversified and (2) accurate
estimates. Illustration according to security
characteristic line (SCL), a0.2,
b1.2,s(e)2. The standard error for the a
is roughly equal to s(a)s(e)/N1/2 which means
for 5 significance, we have the following t
1.96 (a-0)/s(a) 0.2N0.5/2 N 384 months
(too long to be reliable!)
7
In practice, the portfolio management industry
uses a benchment for performance measurement. In
academics, other measurements include stochastic
dominance method.
Frequency
g(y) f(x)
Return
G(y)
F(x)
1
8
Changing Portfolio Composition
excess return
27
3
Quarter
-1
-9
Mean return (first 4 quarters) (-13-13)/41 sd
(4...4)/40.52
9
Mean of the last 4 quarters (-927-927)/49 S
d (18x18.../40.518 The two years have a
Sharpe Measure of 0.5 but the distribution of the
return is different. Combination of the two
years would yield a mean excess return is 5 and
its sd is (6)2...(22)2/80.513.42 The
Sharpe index 5/13.420.37(inferior to 0.4
which is the passive strategy and 0.5 individual
year) Portfolio mean shift will bias the
evaluation performance
10
Market Timing and slope shift of beta

• If the proportion between risky asset and
  riskfree asset is constant, the beta of the
  entire portfolio remains the same over time as
  shown below

rp-rf
slope0.6
rm-rf
11
If the portfolio manager shifts funds from the
riskfree assets to the risky asset in
anticipation of the rise in market return, then
we will observe
rp-rf
rm-rf
Slope of the beta rises
12
That is, there is a regime shift in the
regression analysis. To capture the regime
shift, we can formulate the several regression
models as (1) rp-rfab(rm-rf)c(rm-rf)2ep Hypo
thesis cgt0 (2) rp-rfab(rm-rf)c(rm-rf)Dep wh
ere D is a (0,1) dummy - 1 when rmgt rf 0
elsewhere. Empirical results show no market
timing evidence, i.e., we cannot reject c0 in
both regressions
13
Performance Attribution

• Portfolio managers constantly make broad-brush
  asset market allocation and sector and security
  allocation within markets
• Performance is measured in terms of managed
  portfolio performance and the benchmark portfolio

14
Benchmark Performance and Excess Return

• Component Benchmark Return
  WeightSP500 0.6 5.81Bond Index
  0.3 1.45Money Mkt 0.1
  0.48
• Benchmark return0.6x5.810.3x1.450.1x0.483
  .97
• Managed portfolio excess returnactual return -
  benchmark5.34-3.971.37

15
Asset Allocation Decisions The performance of the
managed fund is due to different proportion of
funds allocated as shown MKT
Equity Fixed Inc. TB
Actual wt 0.7 0.07 0.23 Benchmark 0.6
0.30 0.10 Excess wt. 0.1
-0.23 0.13 (a) Mkt
excess return 1.84 -2.52
-3.49 (b) (5.81-3.97)
(1.45-3.97) (0.48-3.97) Contribution 0.184
0.5796 -0.4537 (a x b) Total
contribution 0.18400.5796-0.45370.3099
16
Sector and Security Selection This analysis
captures the super results of the portfolio due
to their greater performance Mkt
Equity Fixed Income Return 7.28 1.89 Index
5.81 1.45 Excess ret 1.47 0.44 (a) Port.
wt. 0.7 0.07
(b) Contribution 1.03 0.03 (a x
b) Total contribution1.030.031.06
17
Portfolio Attribution Summary Asset allocation
0.31 Sector/security selection
1.06 Total excess return 1.37