Many investors mistakenly base the success of their portfolios on returns alone. Few consider the risk that they took to achieve those returns. Since the 1960s, investors have known how to quantify and measure risk with the variability of returns, but no single measure actually looked at both risk and return together. Today, we have three sets of performance measurement tools to assist us with our portfolio evaluations. The Treynor, Sharpe and Jensen ratios combine risk and return performance into a single value, but each is slightly different. Which one is best for you? Why should you care? Let's find out.

## Treynor Measure

Jack L. Treynor was the first to provide investors with a composite measure of portfolio performance that also included risk. Treynor's objective was to find a performance measure that could apply to all investors, regardless of their personal risk preferences. He suggested that there were really two components of risk: the risk produced by fluctuations in the market and the risk arising from the fluctuations of individual securities.

Treynor introduced the concept of the security market line, which defines the relationship between portfolio returns and market rates of returns, whereby the slope of the line measures the relative volatility between the portfolio and the market (as represented by beta). The beta coefficient is simply the volatility measure of a stock portfolio to the market itself. The greater the line's slope, the better the risk-return tradeoff.

The Treynor measure, also known as the reward-to-volatility ratio, can be easily defined as:

## (Portfolio Return â€“ Risk-Free Rate) / Beta

The numerator identifies the risk premium and the denominator corresponds with the risk of the portfolio. The resulting value represents the portfolio's return per unit risk.

To better understand how this works, suppose that the 10-year annual return for the S&P 500 (market portfolio) is 10%, while the average annual return on Treasury bills (a good proxy for the risk-free rate) is 5%. Then assume you are evaluating three distinct portfolio managers with the following 10-year results:

 Managers Average Annual Return Beta Manager A 10% 0.90 Manager B 14% 1.03 Manager C 15% 1.20

Now, you can compute the Treynor value for each:

T(market) = (.10-.05)/1 = .05

T(manager A) = (.10-.05)/0.90 = .056

T(manager B) = (.14-.05)/1.03 = .087

T(manager C) = (.15-.05)/1.20 = .083

The higher the Treynor measure, the better the portfolio. If you had been evaluating the portfolio manager (or portfolio) on performance alone, you may have inadvertently identified manager C as having yielded the best results. However, when considering the risks that each manager took to attain their respective returns, Manager B demonstrated the better outcome. In this case, all three managers performed better than the aggregate market.

Because this measure only uses systematic risk, it assumes that the investor already has an adequately diversified portfolio and, therefore, unsystematic risk (also known as diversifiable risk) is not considered. As a result, this performance measure should really only be used by investors who hold diversified portfolios.

## Sharpe Ratio

The Sharpe ratio is almost identical to the Treynor measure, except that the risk measure is the standard deviation of the portfolio instead of considering only the systematic risk, as represented by beta. Conceived by Bill Sharpe, this measure closely follows his work on the capital asset pricing model (CAPM) and by extension uses total risk to compare portfolios to the capital market line.

The Sharpe ratio can be easily defined as:

## (Portfolio Return â€“ Risk-Free Rate) / Standard Deviation

Using the Treynor example from above, and assuming that the S&P 500 had a standard deviation of 18% over a 10-year period, let's determine the Sharpe ratios for the following portfolio managers:

 Manager Annual Return Portfolio Standard Deviation Manager X 14% 0.11 Manager Y 17% 0.20 Manager Z 19% 0.27

S(market) = (.10-.05)/.18 = .278

S(manager X) = (.14-.05)/.11 = .818

S(manager Y) = (.17-.05)/.20 = .600

S(manager Z) = (.19-.05)/.27 = .519

Once again, we find that the best portfolio is not necessarily the one with the highest return. Instead, it's the one with the most superior risk-adjusted return, or in this case the fund headed by manager X.

Unlike the Treynor measure, the Sharpe ratio evaluates the portfolio manager on the basis of both rate of return and diversification (as it considers total portfolio risk as measured by standard deviation in its denominator). Therefore, the Sharpe ratio is more appropriate for well diversified portfolios, because it more accurately takes into account the risks of the portfolio.

## Jensen Measure

Like the previous performance measures discussed, the Jensen measure is also based on CAPM. Named after its creator, Michael C. Jensen, the Jensen measure calculates the excess return that a portfolio generates over its expected return. This measure of return is also known as alpha.

The Jensen ratio measures how much of the portfolio's rate of return is attributable to the manager's ability to deliver above-average returns, adjusted for market risk. The higher the ratio, the better the risk-adjusted returns. A portfolio with a consistently positive excess return will have a positive alpha, while a portfolio with a consistently negative excess return will have a negative alpha

The formula is broken down as follows:

 Jensen\'s Alpha = Portfolio Return â€“ Benchmark Portfolio Return Where: Benchmark Return (CAPM) = Risk-Free Rate of Return + Beta (Return of Market â€“ Risk-Free Rate of Return)

So, if we once again assume a risk-free rate of 5% and a market return of 10%, what is the alpha for the following funds?

 Manager Average Annual Return Beta Manager D 11% 0.90 Manager E 15% 1.10 Manager F 15% 1.20

First, we calculate the portfolio's expected return:

ER(D)= .05 + 0.90 (.10-.05) = .0950 or 9.5% return

ER(E)= .05 + 1.10 (.10-.05) = .1050 or 10.50% return

ER(F)= .05 + 1.20 (.10-.05) = .1100 or 11% return

Then, we calculate the portfolio's alpha by subtracting the expected return of the portfolio from the actual return:

Alpha D = 11%- 9.5% = 1.5%

Alpha E = 15%- 10.5% = 4.5%

Alpha F = 15%- 11% = 4.0%

Which manager did best? Manager E did best because, although manager F had the same annual return, it was expected that manager E would yield a lower return because the portfolio's beta was significantly lower than that of portfolio F.

Of course, both rate of return and risk for securities (or portfolios) will vary by time period. The Jensen measure requires the use of a different risk-free rate of return for each time interval considered. So, let's say you wanted to evaluate the performance of a fund manager for a five-year period using annual intervals; you would have to also examine the fund's annual returns minus the risk-free return for each year and relate it to the annual return on the market portfolio, minus the same risk-free rate. Conversely, the Treynor and Sharpe ratios examine average returns for the total period under consideration for all variables in the formula (the portfolio, market and risk-free asset). Like the Treynor measure, however, Jensen's alpha calculates risk premiums in terms of beta (systematic, undiversifiable risk) and therefore assumes the portfolio is already adequately diversified. As a result, this ratio is best applied with diversified portfolios, like mutual funds.

## The Bottom Line

Portfolio performance measures should be a key aspect of the investment decision process. These tools provide the necessary information for investors to assess how effectively their money has been invested (or may be invested). Remember, portfolio returns are only part of the story. Without evaluating risk-adjusted returns, an investor cannot possibly see the whole investment picture, which may inadvertently lead to clouded investment decisions.

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