CAPM can be extended in a number of areas and provide additional applications beyond the estimation of security returns. A key area is in the evaluation of performance where a number of commonly used metrics are used.

**Performance Evaluation**

There are various performance metrics that can be computed that are extensions of CAPM and allow for the assessment of portfolio performance and evaluation. Active managers are expected to perform better than passive managers or to at least cover the costs of active management. There are four ratios commonly used in performance evaluation. All measures assume that the benchmark market portfolio is the correct portfolio and if not, may cause the results to be misleading. The benchmark should be appropriate for the portfolio being measured and should exhibit similar characteristics.

**Sharpe Ratio**

The Sharpe Ratio is defined as the portfolio risk premium divided by the portfolio risk:

- Sharpe ratio = (R
_{p}– R_{f}) / σ_{p}

The Sharpe ratio, or reward-to-variability ratio, is the slope of the capital allocation line (CAL). The greater the slope (higher number) the better the asset. Note that the risk being used is the total risk of the portfolio, not its systematic risk which is a limitation of the measure. The portfolio with the highest Sharpe ratio has the best performance but the Sharpe ratio by itself is not informative. In order to rank portfolios, the Sharpe ratio for each portfolio must be computed. A further limitation occurs when the numerators are negative. In this instance, the Sharpe ratio will be less negative for a riskier portfolio resulting in incorrect rankings.

**Treynor Ratio**

The Treynor ratio is an extension of the Sharpe ratio that instead of using total risk uses beta or systematic risk in the denominator.

- Treynor ratio = (R
_{p}– R_{f}) / β_{p}

As with the Sharpe ratio, the Treynor ratio requires positive numerators to give meaningful comparative results and, the Treynor ratio does not work for negative beta assets. Also, while both the Sharpe and the Treynor ratio can rank portfolios, they do not provide information on whether the portfolios are better than the market portfolio or information about the degree of superiority of a higher ratio portfolio over a lower ratio portfolio.

**M-Squared (M²) Ratio**

The concept behind the M² ratio is to create a portfolio P’ that mimics the risk of the market portfolio by altering the weights of the actual portfolio P and the risk-free asset until portfolio P’ has the same total risk as the market. The return on the mimicking portfolio P’ is determined and compared with the market return. The weight in portfolio P (wp) which sets the portfolio risk equal to the market risk can be written as:

- w
_{p}= σ_{m}/ σ_{p}with the balance (1 – w_{p}) invested in the risk-free asset

The return for the mimicking portfolio P’ is as follows:

- R
_{p’}= w_{p}R_{p}+ (1 – w_{p})R_{f}which we can reformulate as R_{p’ }= (σ_{m}/ σ_{p }) x R_{p }+ (1 – {(σ_{m}/ σ_{p })} x R_{f} - R
_{p’}= R_{f }+ σ_{m}([R_{p }– R_{f}]/ σ_{p })

The difference in return between the mimicking portfolio and the market return is M² which is expressed as:

- M² = (R
_{p }– R_{f }) x (σ_{m}/ σ_{p }) – (R_{m }– R_{f })

A portfolio that matches the return of the market will have a M² value equal to zero while a portfolio that outperforms will have a positive value. By using the M² measure, it is possible to rank portfolios and also to determine which portfolios beat the market on a risk-adjusted basis.

**Jensen’s Alpha**

Jensen’s alpha is based on systematic risk. The daily returns of the portfolio are regressed against the daily returns of the market in order to compute a measure of this systematic risk in the same manner as the CAPM. The difference between the actual return of the portfolio and the calculated or modeled risk-adjusted return is a measure of performance relative to the market.

- Jensen’s alpha = α
_{p}= R_{p }– [R_{f}+ β_{p}(R_{m }– R_{f })]

If α_{p} is positive, the portfolio has outperformed the market whereas as negative value indicates underperformance. The values of alpha can also be used to rank portfolios or the managers of those portfolios with the alpha being a representation of the maximum an investor should pay for the active management of that portfolio.

**Security Selection**

The CAPM assumes investors have homogeneous expectations, are rational and risk-averse and thus assign the same value to all assets to create the same risky market portfolio. If investors are heterogeneous, their different beliefs could result in a valuation or price for a security different from the CAPM-calculated price. The CAPM-calculated price is the current market price because it reflects the beliefs of all other investors in the market. If the investor estimated price is higher than the current market price, this could then provide an indication to buy the asset as it is considered undervalued by the market.

A Jensen’s alpha for individual securities can also be computed with positive values indicating the security is likely to outperform the market on a risk-adjusted basis.

Similar information can be represented graphically by a Security Market Line (SML). The expected return and beta for a security can be assessed against the SML with those securities that are undervalued relative to market consensus appearing above the SML line. Securities overvalued relative to market consensus will appear below the SML line.

**Portfolio Construction**

CAPM suggests that investors should hold the market portfolio and a risk-free asset. The true market portfolio consists of a large number of securities and it may not be practical for an investor to own them all. Much of the non-systematic risk can be diversified away by holding 30 or more individual securities, however, these securities should be randomly selected from multiple asset classes. An index may serve as the best method of creating diversification.

Securities not included within the index can be evaluated relative to the index to determine their suitability for portfolio inclusion. The alpha and beta of the security can be estimated relative to the index and those with a positive alpha should be included. The same exercise can be conducted for securities within the index – those with negative alphas relative to the index should be excluded, or sold short.

To determine the weight of each security within the portfolio, those securities with higher alpha should be given more weight with this weight proportional to the alpha divided by the non-systematic variance (risk) of the security:

- w
_{i}∝ α_{i}/ σ²_{ei }where α_{i}/ σ²_{ei }is the information ratio

The larger the information ratio, the more valuable the security.

**Limitations of CAPM**

The CAPM is subject to theoretical and practical implications. From a theoretical perspective, it is both a single factor and single period model. There may be other factors over multi-time periods that would be more appropriate in modeling expected returns. Practically, the following are the limitations:

- Market portfolio: the true market portfolio includes all assets – financial and non-financial which may not be investable or tradeable
- Proxy for market portfolio: typically, a proxy for the market portfolio is used, but different analysts tend to use different proxies
- Estimation of beta risk: a long history is required to estimate beta, however, the history may not be an accurate representation of the future beta and different historical time periods (3 years versus 5 years) and different data frequency (daily versus monthly) are likely to produce different betas
- Poor predictor of returns: the empirical support for CAPM is weak – the model is not good at predicting future returns which in turn indicates that asset returns cannot be determined solely by systematic risk
- Homogeneity in investor expectations: in reality, investors are unlikely to have homogeneous expectations – there will be many optimal risky portfolios and numerous SMLs

QuestionGiven the following data, which option correctly gives the Sharpe ratio, Treynor ratio, and M² ratio?

- R
_{p}= 12%- R
_{m}= 10%- R
_{f}= 2%- σ
_{m}= 18%- σ
_{p}= 22%- ρ
_{p,m}= 0.8A. Sharpe = 0.45 ; Treynor = 10.2 ; M² = 0.2

B. Sharpe = 0.54 ; Treynor = 9.8 ; M² = 1.2

C. Sharpe = 1.22 ; Treynor = 6.4 ; M² = 0.3

SolutionThe correct answer is A.

Sharpe ratio = (R

_{p}– R_{f}) / σ_{p }= (12% – 2%) / 22% = 0.45Treynor ratio = (R

_{p}– R_{f}) / β_{p }= (12% – 2%) / β_{p }where β_{p }= ρ_{p,m}σ_{p}/ σ_{m }= (0.8)(22%)/18% = 0.98 therefore Treynor ratio = (12% – 2%) / 0.98 = 10.2M² = (R

_{p }– R_{f }) x (σ_{m}/ σ_{p }) – (R_{m }– R_{f }) = (12% – 2%) x (18% / 22%) – (10% – 2%) = 0.18 ≈ 0.2

*Reading 42 LOS 42h:*

*Describe and demonstrate applications of the CAPM and the SML*