Applying the CAPM to Performance Measurement

8 important questions on Applying the CAPM to Performance Measurement

What does the Treynor measure do?

Adjustment of Sharpe ratio, by replacing full portfolio/stock volatility by the beta of the portfolio. It thereby only takes into account systematic risk; excess return/ beta.
Deriving a bit, you get the security market line (SML) as the beta should be 1 for the market portfolio.

Treynor helpt to see whether a portfolio is sufficiently rewarded, in comparison to systematic risk.THerefore particular useful for well-diversified portfolios.

Critics: Roll criticizes that a market portfolio is needed for the beta, but that such a market portfolio does not exist.

What is the relation between Treynor and Jensen?

Only exact relation of the three. Take Jensen's formula: excess return = alpha + beta * expected excess return. Divide both sides by beta. Left side will be Treynor and the right side will be the Treynor-Black adjustment of Jensen's alpha + expected excess return.

What is the relation between Sharpe and Jensen?

Approximation. Note that beta = correlation*sd(p)*sd(m)/variance(m). If the portfolio is well-diversified then the correlation coefficient is very close to 1. Replacing that in Jensen's formula and correspondingly dividing both sides by sd(p) gives the Sharpe ratio on the left side.
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What is the relation between Treynor and Sharpe?

Approximation. Still assume well-diversified, therefore correlation close to 1. Easy to solve then as one uses beta and other standard deviation.

What extensions have been made to Jensen's measure?

Elton and Gruber, have adjusted Jensen's formula to a measure of total (sigma, volatility) risk. A shift from the SML to the CML. Due to the total risk measure this allows to evaluate stock picking skills.

Other generalisations have been made to get manager's value-added evaluation and market riming strategy. THe convenience of Jensen's Alpha is that it doesn't rely on the model chosen whether it is four-factor or CAPM, it is still about the alpha that remains.

What is the Sortino ratio?

There is no measuer on whether the differentials of differing from the market are produced above or below the mean. That's why we can use a sem-variance (dispersion below the mean). This measure is specifically suitable for assymetrical return distributions. Same principle of Sharpe ratio, but the risk-free rate is replaced by the minimum acceptable return (MAR). SD of MAR are measured on the returns that fall below the MAR, threshold.

Sortino = (Expected return - MAR) / wortel(1/t * sum of (Rp - MAR)^2.
For which Rp should be below MAR.

What is the actuarial approach?

Aversion to risk is measured by W, the gain-shortfall equilibrium. This is the relation between the expected gain desired to make up for a fixed shortfall risk.

RAR = R - (W-1)S. Average individual has W-2. This makes R-S. This W implies that the investor agrees to invest of the expected amount of gain is double the shortfall.

How about taking into account the management style?

Due to mandate constraints, the investment style among managers differs. Lobosco (1999) introduced SRAP, style risk adjusted performance. It is calculated by sd(market)/sd(portfolio) * ( Rp - Rf) + Rf = Risk adjusted performance. It is all about adjusting the benchmark to reflect the constraints of the portfolio.

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