Portfolio Evaluation. Outline Investment return measurement conventional measurement theory Evaluation with changing portfolio composition Evaluation with market timing Performance attribution procedures and evaluation. Measuring Returns.
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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.
Treynor measure assumes (1) the portfolio is well-diversified and
(2) accurate estimates.
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
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.
% excess return
Mean return (first 4 quarters)
sd =[ (4%+...+4%)/4]0.5=2%
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:
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
from the riskfree assets to the risky asset
in anticipation of the rise in market
return, then we will observe:
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:
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
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)
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 =0.1840+0.5796-0.4537=0.3099
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)
Asset allocation 0.31%
Sector/security selection 1.06
Total excess return 1.37