FP513: Module 7.2 Performance Measures

formula to calculate ALPHA

= r_p - [ r_f + (r_m - r_f) B ] wherein: B = beta of portfolio r_p = average rate of return of portfolio r_f = risk-free rate of return r_m = market's rate of return

4 Performance Measurement tools

Alpha Beta Treynor ratio Sharpe ratio

Why is the Sharpe ratio the preferred performance measure when evaluating performance between undiversified portfolios compared to Treynor Ratio or Jensen's alpha?

Because beta is meaningful in respect to diversified portfolios only, an undiversified portfolio or an individual stock's risk-adjusted return may appear superior when measured with either Treynor or Jensen and inferior when measured with Sharpe that uses standard deviation instead.

formula to calculate the information ratio

IR = (R_p - R_b) / Standard deviation_A wherein: R_p = average rate of return for portfolio for a given period R_b = average rate of return of benchmark portfolio; CAPM's required rate of return may be used standard deviation_A = standard deviation of the excess return during the given period

Formula to calculate the sharpe ratio

S_p = (r_p - r_f) / standard deviation of portfolio

Formula to calculate the Treynor ratio

T_p = (r_p - r_f) / B_p wherein: r_p = average rate of return of portfolio r_f = risk free rate of return B_p = beta for portfolio

Shortfall of using Sharpe ratio as performance indicator

This ratio can be confusing to investors. For example, if the large-cap mutual fund sector has been the "hot" performance over the recent past and the emerging markets sector has been the "dog" or laggard, then investors should expect that many large-cap funds will have high Sharpe ratios and many emerging markets funds will have low sharpe ratios. In other words, high sharpe ratios will appear in asset classes that have performed well in the recent past. Such ratios do not predict that those same asset classes will also perform well in the future. Comparing funds with each other using the Sharpe ratio is best confined into comparisons of funds within a distinct asset class. An investor may want to construct a diversified portfolio consisting of all major asset classes, including emerging market assets. If a decision were made solely on Sharpe ratios across all asset classes, then no emerging markets funds would be likely to be chosen. If, however, emerging market funds were compared with each other, then an emerging markets fund with a relatively high Sharpe ratio could be selected to add to the existing portfolio

How do active managers utilize the information ratio?

active manager takes active risk by deviating from the benchmark's holdings, making bets different from the benchmark bets. In doing so, the manager's risk level will change, and the manager's return will change. If risk level is higher than the benchmark risk level, then it is hoped that the manager's different-from-benchmark bets will have a higher return than the benchmark. If that is the case, the information ratio is likely positive. If the manager's risk level rose, but the bets did not pay off with higher-than-benchmark returns, then the information ratio is likely to be negative.

Jensen's alpha

aka Jensen's performance index used to compare the actual return to the required return; measure of the risk-adjusted value added by a portfolio manager it is measured as the portfolio's actual or realized return in excess of the expected return calculated by the capital asset pricing model (CAPM) dependent on beta and standard deviation; therefore may not be a reliable indicator of portfolio performance if beta is not reliable; because beta is used in calculating this measure. it is only appropriate for evaluating diversified portfolios and stocks that constitute diversified portfolio

Similarities between Sharpe and Treynor ratio

both are used to compare stock or mutual fund's excess return with a benchmark index or with another similar stock or mutual fund

Treynor ratio

focused on the manager's excess return over the benchmark return relative to the tracking error (standard deviation of the difference between the returns on the portfolio and the returns of the benchmark) dependent on beta; therefore may not be a reliable indicator of portfolio performance if beta is not reliable; since beta is used, this ratio can only be used to compare the performance of diversified portfolios or stocks that constitute diversified portfolios. relative performance measure; only when it is used to evaluate an asset's performance relative to another potential investment or a benchmark does it convey useful information the higher this ratio, the better the risk-adjusted performance of the asset. Therefore, when selecting between two portfolios, we want to choose the one with the higher ratio.

Risk-adjusted return

goal is ultimately to determine how much return was actually achieved given the amount of risk expected in order to achieve that return when determining using beta, divide the stock or mutual fund's nominal rate of return by its beta coefficient

Information Ratio

measures the portfolio's average rate of return in excess of a comparison (benchmark) portfolio divided by the standard deviation of the excess return therefore this ratio measures active return (numerator) over active risk (denominator) This ratio should be used only when alpha is positive (ex. when portfolio manager has added value) and beta is meaningful Managers taking on higher levels of risk are expected to achieve higher rates of return. This is what this ratio measures. The higher this ratio, the more likely it is the portfolio manager was able to capitalize on opportunities in the marketplace. A ratio of 0.50 or higher would mean that the fund manager has an excess return (or alpha) that is about half of the volatility of the alpha

What does a positive alpha mean?

portfolio manager has added value on an absolute basis, compared with the required return based on the level of investment risk undertaken

What does a negative alpha mean?

portfolio plots below the SML

Sharpe ratio

useful for comparing one security's risk-adjusted return with that of another security or with a benchmark return relative performance measure - the higher the ratio, the better the risk-adjusted performance uses standard deviation and therefore, is a catchall formula that may be used to compare the performance of all portfolios (can be used whether portfolio/investment is diversified or not) any security can be directly compared with another security using this ratio.