Portfolio Evaluation
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Outline
Investment return measurement
conventional measurement theory
Evaluation with changing portfolio
composition
• Evaluation with market timing
• Performance attribution procedures and
evaluation
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.
Arithmetic Average is simply the average of returns
over several periods.
Geometric return average is the return over several
periods is computed as:
(1+rG)=[(1+r1)(1+r2)...(1+rn)]1/n
For 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.
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 return
ap =rp - [rf+bp(rm-rf)]
• Appraisal ratio: ap/s(ep), which is the alpha
(abnormal return) divided by the nonsystematic
risk.
Evaluations among Different
Measures
Excess Return
Treynor lines
. Q
.P
SML
Market
1.0
Beta
Treynor measure assumes
(1) the portfolio is well-diversified and
(2) accurate estimates.
Illustration:
according to security characteristic line
(SCL), a=0.2%, b=1.2,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!)
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
1
G(y)
F(x)
Changing Portfolio Composition
% excess return
27
3
-1
Quarter
-9
Mean return (first 4 quarters)
=(-1+3-1+3)/4=1%
sd =[ (4%+...+4%)/4]0.5=2%
Mean of the last 4 quarters:
= (-9+27-9+27)/4=9%
Sd =[(18%x18%+...]/4]0.5=18%
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%)2/8]0.5=13.42%
The Sharpe index = 5%/13.42%=0.37
(inferior to 0.4 which is the passive strategy and 0.5
individual year)
Portfolio mean shift will bias the evaluation
performance
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
slope=0.6
rm-rf
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
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-rf=a+b(rm-rf)+c(rm-rf)2+ep
Hypothesis: c>0
(2) rp-rf=a+b(rm-rf)+c(rm-rf)D+ep
where D is a (0,1) dummy - 1 when
rm> rf 0 elsewhere.
Empirical results show no market
timing evidence, i.e., we cannot reject
c=0 in both regressions
Performance Attribution
• Portfolio managers constantly make broadbrush asset market allocation and sector and
security allocation within markets
• Performance is measured in terms of
managed portfolio performance and the
benchmark portfolio
Benchmark Performance and Excess Return
• Component Benchmark Return
Weight
S&P500
0.6
5.81%
Bond Index 0.3
1.45
Money Mkt 0.1
0.48
• Benchmark return
=0.6x5.81%+0.3x1.45%+0.1x0.48%
=3.97%
• Managed portfolio excess return
=actual return - benchmark
=5.34%-3.97%
=1.37%
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
(a x b=)
-0.4537
Total contribution =0.1840+0.5796-0.4537=0.3099
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 contribution=1.03+0.03=1.06
Portfolio Attribution Summary:
Asset allocation
0.31%
Sector/security selection
1.06
Total excess return
1.37