WebCab Portfolio Web Services for .NET v5.0 Demo

PerformanceEvaluation.TreynorsMeasure Method 

We calculate Treynor's performance measure which takes into account the systematic risk (or beta) and the average return when assessing the overall risk adjusted performance of a portfolio.

public double TreynorsMeasure(
   double averageReturn,
   double beta,
   double riskFree
);

Parameters

averageReturn
The average return from the portfolio over the period considered expressed in decimal format (i.e. 1 percent = 0.01).
beta
The beta of the portfolio with respect to a given market variable (see remarks for further details).
riskFree
The (constant) risk free interest rate expressed in decimal format (i.e. 1 percent = 0.01). The risk free interest rate of a given maturity within a given currency is usual taken to be the yield to maturity of a Government Bond of the same maturity. Here we should taken the yield to maturity of the Government bond at the beginning of the period under consideration which matures at the end of the period over which Sharpe's ration is evaluated. Note that if no such interest rate security exists then you could use the yield to maturity of the closest instrument or better still use such instruments and the boot-strapping technique.

Return Value

A double equal to Treynor's measure of the risk-adjusted performance of a portfolio.

Remarks

The Treynor measure differs from the SharpesRatio measure in that the return per unit measure of `risk' in standardized by uses the beta of the portfolio rather than the standard deviation of the price.

The Beta of a Portfolio

The beta of a portfolio measures the sensitivity of a portfolios return to some underlying market index (for example, the S&P500, FT100 etc). That is, if the underlying index increase by x percent then the Portfolio increase by ax percent then the beta of the portfolio against this index is said to be a. Care should be taken that the time period used in measuring the return of the portfolio and market index are the same. Moreover, there are good grounds for using the same the time period of the same length as the time period over which the average return and risk free interest rate are quoted.

Remark: An effective means by which to estimate the beta is by applying the least squares regression line from regression analysis.

Selection of a suitable Market Index to measure the Portfolio's Beta against

The market index used in order to measure the portfolios performance with respect to Treynor's measure such reflect the range of asset from which the portfolio can be constructed or is benchmarked against. For example, for a fund which can select investments form the S&P500 and is benchmarked against the S&P500, then S&P500 should be the market index used in the evaluation of the beta of the portfolio. If on the other hand the fund invests in Eastern European Stocks then a more natural benchmark could be Barrings European European Benchmark.

Applying Treynor's Measure

Suppose an US stock investment gives a return over a given year of 10 percent and the rate available from a risk free Treasury Bond over the same year is 6 percent. Then the reward for taking on the higher level of `risk' (as measured by the beta of the portfolio) of the stock investment over the risk free Treasury bond investment is 4 percent. Say the risk (i.e. standard deviation) of the stock over the year is 2 percent. Then the excess return of the stock over the Treasury per unit of risk (i.e. per 1 percent of standard deviation) is 2 percent. That is, the Sharpe's ratio here is 2 percent.

Application of Treynors Measure

The Treynor measure allows you to compare the performance of portfolios which invest within similar market sectors. It also help in differentiation of skill of a fund manager from the performance of the sector in which he often is obliged to invest it. Below we describe two instances when this can be particularly helpful:

  1. Select Fund from Category - In order to select a mutual fund from within a category (i.e. global general, bond fund etc) you could select a corresponding index for the given category under consideration and then evaluate the corresponding Treynors measure for all the mutual funds within that category. Now accordance to the Treynors measure the fund which offers the best return per unit `beta' is the portfolio with the highest (Treynor) value.
  2. Compare Funds in Different Categories - In a similar fashion the Treynor measure can be used in order to measure risk adjusted return of funds which invests within different sectors. Though the measure is far from idea it does allow the skill of the fund manager to (approximately) be factored out from the performance of the underlying sector. In order to compare two fund which invest in different sectors of the market you will need to evaluate there Treynors measure using the appropriate `beta' (and market index) in each case. Then according to Treynors measure the fund which offers the best risk (and sector) adjusted return is the fund with the highest (Treynor) value.

See Also

PerformanceEvaluation Class | Portfolio Namespace