Question

Explains how different measures of risk - adjusted performance measures (e.g. Sharpe and Treynor ratios) define the risk s faced in the portfolio and how each measure adjust ed the portfolio’s return perfo rmance for the level of that risk.

Answer 1

The academic and the professional investment literatures have developed several returns-based measures to assess the value of active management. The assumption that risk could be captured by a single factor, such as the portfolio’s standard deviation or its beta. For these measures, the selection of which are appropriate for a client’s specific needs involves evaluating which aspect of risk is most important given the role of the investment in the client’s total portfolio, the plausibility of the assumptions a measure makes about the probability distribution of possible returns, the assumed theoretical pricing model, and possibly other considerations.

The Sharpe ratio measures the additional return for bearing risk above the risk-free rate, stated per unit of return volatility. In performance appraisal, this additional return is often referred to as excess return. This use contrasts with how “excess return” is used in return performance attribution—that is, as a return in excess of a benchmark’s return. Nobel laureate William F. Sharpe introduced the measure in 1966 and used the term reward-to-variability ratio to describe that measure. The measure quickly became known as the Sharpe ratio (and sometimes as the Sharpe measure).

The Sharpe ratio (SR) stems from the mean–variance portfolio theory developed by Markowitz (1952) in which risk-averse investors make portfolio decisions based only on mean return (expected return) and the variance or standard deviation of returns. Such an investor (a so-called “mean–variance investor”) prefers a higher Sharpe ratio and will seek to maximize the Sharpe ratio of his or her total portfolio. According to the CAPM developed by Sharpe (1964), the market portfolio is the portfolio with the highest Sharpe ratio and it is mean–variance efficient in the sense of offering the highest mean return per unit of return volatility.

We calculate the Sharp Ration by deducting the Risk free rate from Portfolio return whole divided by Standard Deviation of the portfolio.

All else being equal, a higher Sharpe ratio is preferable to a lower Sharpe ratio—1 is preferable to 0.5—because it implies greater reward per unit of total risk. In short, the higher the Sharpe ratio, the better a portfolio’s expected risk-adjusted performance. What constitutes a “good” Sharpe ratio, however, needs to be judged in relation to the portfolio’s benchmark or some other accepted point of comparison.

M Square: Risk-Adjusted Performance (RAP): M2 provides a measure of portfolio return that is adjusted for the total risk of the portfolio relative to that of some benchmark. In 1997, Nobel Prize winner Franco Modigliani and his granddaughter, Leah Modigliani, developed what they called a risk-adjusted performance measure, or RAP. The RAP measure has since become more commonly known as M Square reflecting the Modigliani names. It is related to the Sharpe ratio and ranks portfolios identically, but it has the useful advantage of being denominated in familiar terms of percentage return advantage assuming the same level of total risk as the benchmark.

M Square borrows from capital market theory by assuming a portfolio is leveraged or de-leveraged until its volatility (as measured by standard deviation) matches that of its benchmark. This adjustment produces a portfolio-specific leverage ratio that equates the portfolio’s risk to that of its benchmark. The portfolio’s excess return times the leverage ratio plus the risk-free rate is then compared with the benchmark’s actual return to determine whether the portfolio has outperformed or underperformed the benchmark on a risk-adjusted basis.

We calculate the M Square by = Sharp Ration X Standard Deviation of Benchmark + Risk Free return

The Treynor ratio (Treynor 1965) measures the excess return per unit of systematic risk. With the Treynor ratio (TR), and the systematic-risk-based appraisal measures to follow, we must carefully choose an efficient market benchmark against which to measure the systematic risk of the manager’s fund. By contrast, the Sharpe ratio can be compared among different funds without the explicit choice of a market benchmark. Similarly, the M2 measure can, in principle, be computed for a fund with respect to a benchmark of the user’s choice.

We calculate The Treynor ratio by deducting the Risk free rate from Portfolio return whole divided by Beta of the portfolio.

Jensen’s Alpha measures the return of a portfolio in excess of the theoretical required return given the equilibrium model for asset returns known as the CAPM. Also called Jensen measure. Jensen’s alpha measures the return of a portfolio in excess of the theoretical required return given the equilibrium model for asset returns known as the CAPM.

Another advantage of Jensen’s alpha is it is easy to interpret because it is measured as an excess return rather than being a ratio, as is the case for the Treynor ratio. For example, the implications of a Jensen’s alpha of 2.0% versus a Jensen’s alpha of ?0.5% are much easier to understand than are Treynor indexes of 0.90 and 0.72. The market model approach also provides an easy way to test the statistical significance of the Jensen’s alpha estimate.

We calculate Jensen’s Alpha by = Porfolio Retuns – CAPM Return, whereas; CAPM = Expected Return = Risk Free Return + ? of the portfolio (Market Return – Risk Free Return )