# Jensen's measure, Sharpe's measure, A comparison of...

## Jensen's Measure

Jensen approached the study of the results of investment activity, estimating the excess of the portfolio return r i, t over the risk-free yield at each moment of the period under consideration (in our case - for the last three years). The Jensen measure, denoted by , is nothing more than a free term in the regression equation describing the linear excess regression () of the portfolio in question excess () of the market portfolio:

Jensen's measure can be calculated in two ways:

a) if the average values ​​are defined , , and , then

b) in another case, we can take the observed values ​​, for every month of the previous three years, calculate the excesses , substitute them in the regression formula and calculate as a free regression member. The possibility of direct calculation of Jensen's measure as a free regression member is one of the main advantages of Jensen's method. Then the measure is compared with the measure of the market portfolio that is zero.

The ability of the manager to exceed the performance of the market portfolio will be expressed by the fact that the measure becomes greater than zero, and if the manager fails, the measure becomes negative.

If on the graph, the values ​​of the excesses are plotted on the vertical axis, and the portfolio beta βi is horizontal, then the line SML (according to the realized data) in these coordinates will pass through the origin (Figure 5.2).

Fig. 5.2. Graphic interpretation of the Jensen measure

On the vertical axis in this graph, the excess values ​​ are plotted. The line corresponding to the fish -

night portfolio, passes through the origin. The portfolio E has a positive value , so its risk/yield ratio is higher than the market, but the portfolio F since his

In this case, each time the estimated portfolio has and the corresponding line passes above the market, cutting off the positive segment on the vertical axis (as for the conditional portfolio Е ), the activity of the manager of such a portfolio should be assessed positively. With (as for the conditional portfolio F ), the manager's activity is considered to be unsuccessful.

## Sharpe's Measure

Sharpe proposed to evaluate the portfolio using a measure , showing the ratio of the excess of the average values ​​ over , i.e. , and of the summary (rather than a systematic) portfolio risk, expressed by the standard deviation :

Hence, we can assume that the Sharpe measure shows how the excess of changes when the total portfolio risk per unit is changed.

For the market portfolio M the magnitude of the Sharpe measure

It is with this value that we need to compare Sharpe's measure for each portfolio: the higher and the more significant it is , the more successful portfolio results.

We will illustrate the use of the Sharp measure on the conditional example of three portfolios R, S, T, corresponding data on which are given in Table. 5.5.

Sharpe's measures for each portfolio are calculated as follows:

As follows from Table. 5.5, portfolios R and S dominate the market portfolio, as and

Table 55

Calculating measure S i for three conditional portfolios

 Options Portfolios R 5 T Market 0.123 0.131 0.118 0.125 0.090 0.120 0.135 0.138

## Comparison of the measures of Traynor, Jensen and Sharpe

Although all three measures are derived using the CAPM theory, they can give inconsistent results in evaluating portfolios, since the results of applying each measure depend on the characteristics of the portfolios that are subject to analysis.

So, the Traynor and Jensen measures give absolutely identical results if we evaluate a portfolio in relation to the market portfolio: if the Jensen measure J K for K - of the first portfolio is positive, hence the Traynor measure T K is greater than Traynor's market measure T M. Accordingly, in the graph (Figure 5.3), the line of possible portfolios passing through the point K, lies above the line SML (built on actual data). At the same time, Jensen's measure J k & gt; 0, then both measures show that the portfolio K dominates the market portfolio.

However, the Traynor and Jensen measures can give opposite results if they use their help to rank portfolios, i. try to determine which of the portfolios K or L (both superior to market) is dominant. This is due to the fact that these measures take different account of the systematic risk: in the Traynor model, the coefficient "beta" is derived by regressing the real data r i, t by r m, t, and in the Jensen model - by regressing the excess ( r i, t - r f, t) to the excess ( r M, t - r f, t). In this regard, the measure of Jensen should not be used to rank portfolios, especially if the risk of portfolios is very different.

Fig. 5.3. Comparison of the measures of Traynor and Jensen

For example, let's compare two portfolios K and L (see Figure 5.3). Which one is better? If you follow the measure of Jensen, then the portfolio L, is better, as ero J L & gt; J K. However, this is not true, and the lines of possible portfolios disprove this conclusion, since composite portfolios composed of a portfolio K plus a risk-free narrowing, or a loan, always dominate similar portfolios compiled on the portfolio L.

Even greater differences can be obtained using the Sharpe measure. For example, you can pick up such portfolios that for them, the Traynor measure will indicate superiority over the market portfolio, and Sharpe's measure - on the contrary, about the superiority of the market portfolio. All this is explained by the fact that the Sharp model takes into account summary , and not just systematic risk. If the portfolio is sufficiently diversified and the total risk approaches market risk, then all three measures will yield the same results. However, if a manager tries to acquire undervalued funds and sell overvalued ones in an attempt to exceed the market portfolio, then diversification may decrease and diversified risk will affect portfolio estimates. As a result, the measures of Jensen and Traynor can show that the portfolio dominates the market, and Sharpe's measure does not.

What is the guideline in this case? Strange as it may seem, but everything will be determined by what share in the overall state of the investor is occupied by investment costs. If they make up almost all of his condition, then one must be guided by Sharpe's measure. When a small part of the state is invested - a measure of Traynor or Jensen. It is believed that the activity of new managers is better evaluated as Traynor progresses. The measure of Jensen is better when comparing portfolios with slightly different levels of risk.

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