Many investors mistakenly base the success of their portfolios solely on returns. Few investors consider the risk involved in achieving those returns. Since the 1960s, investors have known how to quantify and measure risk with variability in returns, but no single measure actually looked at both risk and return together. Currently, there are three sets of performance measurement tools to assist with portfolio evaluations.
The Treynor, Sharpe, and Jensen indices combine return on risk and return into a single security, but each is slightly different. Which is the best? Perhaps a combination of all three.
Jack L. Treynor was the first to provide investors with a composite measure of portfolio performance that also included risk. Treynor’s goal was to find a performance measure that could be applied to all investors regardless of their personal risk preferences. Treynor suggested that there are really two components of risk: the risk produced by fluctuations in the stock market and the risk that arises from fluctuations in individual securities.
Treynor introduced the concept of the stock market line, which defines the relationship between portfolio returns and market rates of return, whereby the slope of the line measures the relative volatility between the portfolio and the market (represented by beta). The beta coefficient is the measure of volatility of a portfolio of stocks in the market itself. The steeper the line, the better the trade-off between risk and return.
The Treynor measure, also known as the reward / volatility ratio, is defined as:
Treynor Measure=BPR–RFRwhere:PR=portfolio performanceRFR=risk-free rateB=beta
The numerator identifies the risk premium and the denominator corresponds to the portfolio risk. The resulting value represents the portfolio’s return per unit of risk.
As an illustration, suppose that the 10-year annual return on the S&P 500 (market portfolio) is 10%, while the average annual return on T-bills (a good proxy for the risk-free rate) is 10%. 5%. Then, suppose the evaluation is of three different portfolio managers with the following 10-year results:
|Managers||Average annual profitability||Beta|
The Treynor value for each is as follows:
|T (market)||(0.10-0.05) / 1||0.05|
|T (manager A)||(0.10-0.05) / 0.90||0.056|
|T (manager B)||(0.14-0.05) / 1.03||0.087|
|T (manager C)||(0.15-0.05) / 1.20||0.083|
The higher the Treynor measurement, the better the portfolio. If the portfolio manager (or portfolio) is evaluated solely on performance, manager C appears to have performed the best. However, when considering the risks each manager took to achieve their respective returns, Manager B demonstrated a better result. 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 is more applicable to investors who have diversified portfolios.
How to measure the performance of your portfolio
The Sharpe index is almost identical to the Treynor measure, except that the risk measure is the standard deviation of the portfolio rather than considering only the systematic risk represented by beta. Conceived by Bill Sharpe,This metric 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.
Sharpe’s relationship is defined as:
Sharpe relationship=SDPR–RFRwhere:PR=portfolio performanceRFR=risk-free rateSD=Standard deviation
Using the Treynor example above, and assuming that the S&P 500 had a standard deviation of 18% over a 10-year period, we can determine Sharpe’s ratios for the following portfolio managers:
|Manager||Annual return||Portfolio standard deviation|
|S (market)||(0.10-0.05) / 0.18||0.278|
|S (manager X)||(0.14-0.05) / 0.11||0.818|
|S (manager Y)||(0.17-0.05) / 0.20||0.600|
|S (manager Z)||(0.19-0.05) / 0.27||0.519|
Again, we find that the best portfolio is not necessarily the portfolio with the highest yield. In contrast, a superior portfolio has the superior risk-adjusted return or, in this case, the fund headed by manager X.
Unlike the Treynor measure, the Sharpe index assesses the portfolio manager based on both rate of return and diversification (it considers the total risk of the portfolio as measured by the standard deviation in its denominator). Therefore, the Sharpe index is more appropriate for well-diversified portfolios because it takes into account portfolio risks more accurately.
Similar to the previous performance measures discussed, Jensen’s measure is calculated using the CAPM. Named after its creator, Michael C. Jensen, the Jensen measure calculates the excess return a portfolio generates over its expected return. This performance measure is also known as alpha.
The Jensen index measures how much of the portfolio’s rate of return is attributable to the manager’s ability to generate 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 breaks down as follows:
Jenson’s Alpha=PR–CTOPSUBWAYwhere:PR=portfolio performanceCTOPSUBWAY=risk-free rate+B(market risk-free rate of return return)
Assuming a risk-free rate of 5% and a market return of 10%, what is the alpha of the following funds?
|Manager||Average annual profitability||Beta|
We calculate the expected return of the portfolio:
|ER (D)||0.05 + 0.90 (0.10-0.05)||0.0950 or 9.5% return|
|ER (E)||0.05 + 1.10 (0.10-0.05)||0.1050 or 10.5% return|
|ER (F)||0.05 + 1.20 (0.10-0.05)||0.1100 or 11% return|
We calculate the portfolio alpha by subtracting the expected return on 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 better? Manager E did better because while manager F had the same annual return, manager E was expected to return a lower return because the beta of the portfolio was significantly lower than that of portfolio F.
Both the rate of return and the risk of the securities (or portfolios) will vary by time period. Jensen’s measure requires the use of a different risk-free rate of return for each time interval. Evaluating the performance of a fund manager over a five-year period using annual intervals would also require examining the fund’s annual returns minus the risk-free return for each year and relating it to the market portfolio’s annual return minus the same risk. free fee.
In contrast, the Treynor and Sharpe ratios examine average returns for the total period in consideration for all the variables of the formula (the portfolio, the market and the risk-free asset). However, similar to the Treynor measure, Jensen’s alpha calculates risk premiums in terms of beta (systematic risk, not diversifiable) and therefore assumes that the portfolio is already adequately diversified. As a result, this relationship is best applied to an investment such as a mutual fund.
The bottom line
Portfolio performance measures are a key factor in the investment decision. These tools provide the information necessary for investors to evaluate how effectively their money has been invested (or can be invested). Remember, portfolio returns are only part of the story. Without evaluating risk-adjusted returns, an investor cannot see the full investment picture, which can inadvertently lead to confusing decisions.
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Jack L. Treynor. “Treynor on Institutional Investing, “Chapter 1. John Wiley & Sons, 2011.
William F. Sharpe. “Sharpe’s relationship. “Accessed April 2, 2020.
The Nobel Prize. “William F. Sharpe. “Accessed April 2, 2020.
Michael C. Jensen. “The performance of mutual funds in the period 1945-1964, “Page 1. Accessed April 2, 2020.