Portfolio Management 3-228-07 Albert Lee Chun. Evaluation of Portfolio Performance. Lecture 11. 2 Dec 2008. Introduction. As portfolio managers, how can we evaluate the performance of our portfolio? We know that there are 2 major requirements of a portfolio manager’s performance:
Evaluation of Portfolio Performance
2 Dec 2008
1. The ability to derive above-average returns conditioned on risk taken, either through superior market timing or superior security selection.
2. The ability to diversify the portfolio and eliminate non-systematic risk, relative to a benchmark portfolio.
Concept de mesures ajustées pour le risque
Mesures de Sharpe, Treynor et Jensen
Mesure des habilités de timing
(.10 + .0566) / 2 = 7.83%
[ (1.1) (1.0566) ]1/2 - 1
The arithmetic average provides unbiased estimates of the expected return of the stock. Use this to forecast returns in the next period.
The fixed rate of return over the sample period that would yield the terminal value is know as the geometric average.
The geometric average is less than the arithmetic average and this difference increases with the volatility of returns.
The geometric average is also called the time-weighted average (as opposed to the dollar weighted average), because it puts equal weights on each return.
0 Purchase 1 share of Eggbert’s Egg Co. at $50
1 Purchase 1 share of Eggbert’s Egg Co. at $53
Eggbert pays a dividend of $2 per share
2 Eggbert pays a dividend of $2 per share
Sell both shares for $108
0 -50 share purchase
1 +2 dividend -53 share purchase
2 +4 dividend + 108 shares sold
Internal Rate of Return:
Dollar Weighted: The stocks performance in the second year,
when we own 2 shares, has a greater influence on the overall return.
[ (1.1) (1.0566) ]1/2 - 1
Time Weighted: Each return has equal weight in the geometric
A risk-adjusted measure of return that divides a portfolio's
excess return by its beta.
The Treynor Measure is given by
The Treynor Measure is defined using the average rate of return for portfolio p and the risk-free asset.
A larger Tp is better for all investors, regardless of their risk preferences.
Because it adjusts returns based on systematic risk, it is the relevant performance measure when evaluating diversified portfolios held in separately or in combination with other portfolios.
Q has higher alpha, but P has steeper T-line.
P is the better portfolio.
It adjusts returns for total portfolio risk, as opposed to only systematic risk as in the Treynor Measure.
Thus, an implicit assumption of the Sharpe ratio is that the portfolio is not fully diversified, nor will it be combined with other diversified portfolios.
It is relevant for performance evaluation when comparing mutually exclusive portfolios.
Sharpe originally called it the "reward-to-variability" ratio, before others startedcalling it the Sharpe Ratio.
Measure can be extended to a multi-factor setting, for example:
where the numerator is Jensen’s alpha and the denominator is the standard error of the regression. Recalling that
Note that the risk here is nonsystematic risk, that could, in theory,
be eliminated by diversification.
Measures excess returns relative to a benchmark portfolio.
Sharpe Ratio is the special case where the benchmark equals the risk-free asset.
Risk is measured as the standard deviation of the excess return (Recall that this is the Tracking Error)
For an actively managed portfolio, we may want to maximize the excess return per unit of nonsystematic risk we are bearing.
Excess Return relative
to benchmark portfolio b
Average Excess Return
Variance in Excess Difference
Information to noise ratio.
Developed by Leah and her grandfather Franco Modigliani.
M2 = rp*- rm
rp* is return of the adjusted portfolio that matches the volatility of the market index rm. It is mixed with a position in T-bills.
If the risk of the portfolio is lower than that of the market, one has to increase the volatility by using leverage.
Because the market index and the adjusted portfolio have the same standard deviation, we may compare their performances by comparing returns.
Managed Portfolio: return = 35% st dev = 42%
Market Portfolio: return = 28% st dev = 30%
T-bill return = 6%
30/42 = .714 in P (1-.714) or .286 in T-bills
Return = (.714) (.35) + (.286) (.06) = 26.7%
Since the return of the portfolio is less than the market, M2 is negative, and the managed portfolio underperformed the market.
Introduced by William Sharpe
1992 study of mutual fund performance
Later studies show that 97% of the variation in return could be explained by the funds’ allocation to set of different asset classes.
Monthly returns on Magellan Fund over five year period.
Regression coefficient only positive for 3.
They explain 97.5% of Magellan’s returns.
2.5 percent attributed to security selection within asset classes.
Fund vs Style and Fund vs SML
Impact of positive alpha on abnormal returns.
Beginning with $1 dollar in 1926, and ending in 2005....
P = P1 + P2 – 1
An forecaster who always guesses correctly will show P1 = P2 = P =1, whereas on who always predicts rain will have P1 = 1, P2 = P = 0.
If an investor holds only the market and the risk free security, and the weights remained constant, the portfolio characteristic line would be a straight line.
Adjusting portfolio weights for up and down movements in market returns, we would have:
Henriksson (1984) showed little evidence of market timing. Evidence of market efficiency.
No Market Timing
Beta Increases with Return
Two Values of Beta
Portfolio Risk Premium
Selectivity measures the distance between the return on portfolio p and the return on a benchmark portfolio with beta equal to the beta of portfolio p.
Portfolio managers add value to their investors by
1) selecting superior securities
2) demonstrating superior market timing skills by allocating funds to different asset classes or market segments.
Attribution analysis attempts to distinguish is the source of the portfolio’s overall performance.
Total value added performance is the sum of selection and allocation effects.
Set up a ‘Benchmark’ or ‘Bogey’ portfolio
Where B is the bogey portfolio and p is the managed portfolio.