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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

- Dollar-weighted returnis 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.

- 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 - [rf+bp(rm-rf)]
- Appraisal ratio: ap/s(ep), which is the alpha (abnormal return) divided by the nonsystematic risk.

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

G(y)

F(x)

1

% 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

- 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

- 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

- Component Benchmark Return WeightS&P500 0.6 5.81%Bond Index 0.3 1.45Money 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 EquityFixed Inc.TB

Actual wt 0.7 0.070.23

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 =0.1840+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 EquityFixed Income

Return 7.28%1.89%

Index 5.811.45

Excess ret 1.470.44 (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