External Performance Analysis

Risk-Adjusted Performance Measures

Sharpe Ratio
The Sharpe Ratio (also known as Reward-to-Volatility-Ratio) indicates the excess return per unit of risk associated with the excess return. The higher the Sharpe Ratio, the better the performance.

[1]

Graphically, the Sharpe Ratio is the slope of a line between the riskfree rate and the portfolio in the mean/volatility space. Actually, the task of finding the efficient portfolio in the Markowitz' mean-variance framework with a riskfree asset is equal to maximizing the Sharpe Ratio of the portfolio.

The Sharpe Ration is closely related to the t-statistic for measuring the statistical significance of the mean excess return. The t-statistic will equal the Sharpe Ratio times the square root of the number of returns used for the calculation.

The Sharpe Ratio does not refer to the market portfolio or any other benchmark. Actually, the implicit benchmark is the riskfree rate of return. The excess return can be interpreted as a zero-investment strategy: It can be obtained by taking a long position in the portfolio and a short position in the riskfree rate, with the funds from the latter used to finance the purchase of the former.

Because the total risk of a portfolio - its standard deviation - is used in the Sharpe Ratio, diversification does not play any role in performance analysis. The Sharpe Ratio is a useful measure for an investor which puts all his money in one fund; in this situation, only total risk matters

Usually, the Sharpe Ratio is measured and used without any tests about statistical significance. But one can test whether the difference between two Sharpe Ratios is zero. H0: S(1) - S(2) = 0. This hypothesis is rejected when the test statistic z(12) is larger than the critical value z(c), which is standard normally distributed.

[2]

[3]

[4]

But this test is only powerful for a large number of observations T.

Application: US stocks and bonds in historic perspecive.

Treynor Ratio
Like the Sharpe Ratio, the Treynor Ratio (sometimes called Reward-to-Variability-Ratio) also relates excess return to risk; but systematic risk instead of total risk is used. The higher the Treynor Ratio, the better the performance under analysis.

[1]

In mean/beta-space, the Treynor Ratio is graphically represented by the line between the riskfree rate and the portfolio.

Like the Sharpe Ratio, T does not quantify the value added of active portfolio management. It is a ranking criterion only. But it can be expected that portfolio managers which possess private information will have a higher T than the T of the uninformed market strategy. A ranking of portfolios based on the T measure is only useful if the funds under consideration are sub-funds of a broader, fully diversified portfolio. If this is not the case, portfolios with identical systematic risk, but different total risk, will be rated the same. But the portfolio with a higher total risk is less diversified and therefore has a higher unsystematic risk which is not priced in the market.

Alpha
The Sharpe and Treynor Ratios discussed above can only be used in relative performance comparison between portfolios and between a portfolio and a benchmark. Alpha (also called Jensen's Alpha) measures the value added by selection activities. Alpha is defined as the difference between the average realized return of a portfolio manager with private information and the expected return of the passive strategy based upon public information only with equal systematic risk.

[1]

In the context of the CAPM, a portfolio manager with private information will choose securities which will have a positive return surprise:

[2]

Alpha can be estimated together with Beta by introducing a constant in a linear regression between portfolio and benchmark excess returns:

[3]

Graphically, a Alpha shifts the Security Market line up or down in mean/beta-space.

The statistical significance of Alpha can be measured with the usual t-tests for the parameters of a linear regression. More powerful statistcal tests to identify significant superior performance have been developed.

A direct comparison of Alphas between different portfolios is only valid when they have equal systematic risk (equal beta).

The implementation of private selection information means overweighting securities which have positive Alphas. This means taking more unsystematic risk compared to the passive strategy and results in a higher total risk of the actively managed portfolio. Alpha does not capture this increase in risk.

Information Ratio
The Information Ratio (also known as Appraisal Ratio) is basically an risk-adjustement of Alpha. It measures the Alpha per unit of active risk, i.e. tracking error.

[1]

There exists a close connection between the IR and the statistical significance of excess returns. The hypothesis that the set of relative returns is positive and statistically significant on average can be tested with the t-statistic:

[2] t-statistic =

The excess return of a fund with a beta equal to one can be described with the following expression:

[3]

The relative return of this fund is:

[4]

Therefore, the expected relative return is:

[5]

Substituting [5] in [2] leads to:.

[6] t-statistic =

If a fund's beta is close to one, its information ratio times the square root of the number of observations is about equal to the t-statistic for testing the significance of positive relative returns. A statistical test for overperformance is therefore also a test for a significant information ratio.

Linear Relationship between Sharpe, Treynor Ratios and Alpha
There exist simple linear relationships between these risk-adjusted performance measures.

[1] Relationship Alpha - Sharpe Ratio:

[2] Relationship Alpha - Treynor Ratio:

Because of these relationships, rankings of funds using Sharpe Ratios, Treynor Ratios or Alphas will be similar, but not identical. Therefore, the three measures are not substitutes and each measure should be assessed separately in performance analysis.

Positive Period Weighting Measure
Grinblatt/Titmann

Measuring Market Timing

Squared Regression

[1]

[2]

Note that we have assumed that variations in beta are non-stochastic and caused by market timing activities only.

Inserting equation [2] into equation [1] yields the following expression:

[3]

Equation [3] can be estimated with a simple linear regression. As before, alpha will measure selectivity capabilities. The variable gamma will measure timing capabilities: A positive gamma will indicate that timing activities have added value to portfolio performance. Comparing the gammas of different funds will indicate the relative importance of timing activities in their investment policies.

Henrikson/Merton

Decomposition of Risk-Adjusted Performance

Style Analysis

Technically speaking, style analysis is about finding the set of style index weights which minimizes the tracking error between the resulting benchmark and the portfolio. Moreover, this set of style index weights is required to contain only positive or zero weights (no-leverage assumption).

subject to

with

Using all returns available to calculate the style index weights as described above yields an Average Style Model: The result reflects the average style profile of the portfolio over the observation period. Conclusions drawn from this model make the implicit assumption of a constant style profile.

To account for changes in the style profile over time, one can work with the Out-Of-Sample Benchmark Style Model.

Peer Group Analysis

Quartiles

Chart

Persistence

Rolling Analytics