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IFT Notes for Level I CFA® Program

R53 Portfolio Risk and Return Part II

Part 3


 

5. Beyond the Capital Asset Pricing Model

Note: there is no explicit learning outcome associated with this section.

In this reading, we saw one return-generating model, the CAPM. But there are more models to estimate the return of an asset.

CAPM is popular because of its simplicity to estimate the expected return. However, there are several theoretical and practical limitations because of its unrealistic assumptions.

5.1 Limitations of the CAPM

Theoretical limitations of the CAPM are as follows:

  • Single–factor model: Only systematic or market risk is priced. This assumption is restrictive as no other investment characteristic is considered.
  • Single-period model: One of the assumptions of the model is that all investors hold assets for a single period, which is practically not true.

Practical limitations of the CAPM are as follows:

  • Market Portfolio: A market portfolio must comprise all assets, including non-investable assets like Eiffel Tower, human capital, etc.
  • Proxy for a Market Portfolio: When a true market portfolio comprising all assets cannot be created, a proxy such as S&P500 is used. But different analysts may use different proxies for the same asset based in the country. For example, one analyst may use S&P 500 as the proxy for equity while another may use DAX.
  • Estimation of beta risk: Beta is an important input for the CAPM model. If not estimated correctly, the expected return will not be accurate as well. Beta is estimated using a long history of returns, which may vary according to the period used. For example, beta calculated using daily returns will be different than a one-year or five-year beta. Similarly, the risk of a company in the past may not represent its current or future risk.
  • CAPM does not predict returns accurately: Studies have shown that actual returns do not reflect predicted returns.
  • Homogeneity in investor expectations: CAPM assumes that all investors have the same expectations for securities that result in one optimal risky portfolio, the market portfolio. If investors have different views, then it will result in multiple optimal risky portfolios and SMLs.

5.2 Extensions to the CAPM

Other models are considered because of the limitations of CAPM. Of course these models, too, come with their own limitations. The models can be broadly categorized into theoretical and practical models.

Theoretical models

In principle, theoretical models are similar to the CAPM but with additional risk factors. One example is the arbitrage pricing theory (APT) which takes the following form:

E(R_p) = R_F + \lambda_1 \beta_p_{,1} + ... + \lambda_K \beta_{p,k}

where k is the number of risk factors,\lambda_1 is the risk premium and \beta_{p,k} is the sensitivity of the portfolio to factor k.

The drawback of this model is that it is difficult to identify risk factors and estimate sensitivity to each factor.

Practical Models

The Fama-French three-factor model and four-factor model have been found to predict asset returns better than the CAPM, which considers only beta risk. The three factors included in the Fama-French model are relative size, relative book-to-market value, and beta of the asset. The four-factor model adds one more momentum factor to the three-factor model. These models have limitations too. They cannot be applied to all assets and there is no certainty that these will work in the future.

Sharpe Ratio  \frac {Portfolio \ risk \ premium}{Portolio \ total \ risk} = \frac {R_p - R_f}{\sigma_p}
M-Squared M^2 = \frac {(R_p - R_f ) \sigma_m}{\sigma_p} - (R_m - R_f)
Treynor Ratio  \frac {Portfolio \ risk \ premium}{beta \ risk} = \frac {R_p - R_f}{\beta_p}
Jensen’s Alpha Actual portfolio return – expected return = R_p-[R_f+β(R_m-R_f )]

6.1 The Sharpe Ratio

Sharpe ratio is the excess return of the portfolio over the risk-free rate divided by the portfolio risk. It is the excess return per unit of risk. The higher the Sharpe ratio the better, all else equal. Sharpe ratio is the slope of the capital allocation line and represents the reward-to-variability ratio. The ex-ante and ex-post Sharpe ratio are given by:

Ex ante SR = \frac {E(R_p )-R_F} {\sigma_p}

→ evaluate the expected risk-adjusted return of the portfolio

Ex post \widehat{SR}=\frac{{\hat{R}}_p-{\hat{R}}_F}{{\hat{\sigma}}_p}

→ to evaluate historical risk-adjusted returns.

Advantages of Sharpe ratio

  • Sharpe ratio can be easily calculated by using readily available market data.
  • The Sharpe ratio is easy to interpret.
  • The Sharpe ratio is the most widely recognized and used appraisal measure.

Limitations of Sharpe ratio

  • The Sharpe ratio uses total risk; not systematic risk.
  • The ratio itself is not informative. The Sharpe ratio of one portfolio must be compared with another to see which one is better. For instance, if a portfolio has a Sharpe ratio of 0.7, the number does not convey anything. But if there is another portfolio with a Sharpe ratio of 0.9, then we know it is superior to the one with 0.7.

6.2 The Treynor Ratio

Treynor ratio is the excess return of the portfolio over the risk-free rate divided by the systematic risk of the portfolio. The numerator must be positive for meaningful results. It does not work for negative beta assets.

The ex-ante and ex-post Treynor ratios are provided below.

TR=\frac{E\left(R_p\right)-R_f}{\beta_p}
\widehat{TR}=\frac{{\hat{R}}_p-{\hat{R}}_f}{{\hat{\beta}}_p}

Limitations of Treynor ratio

  • The ratio itself is not informative. When two portfolios are compared, we know which one is superior but we do not know if its performance is better than the market portfolio.
  • There is no information about the amount of underperformance or over performance. For instance, assume there are two portfolios A and B with Treynor ratios of 0.6 and 0.7 respectively. Though B is better than A, we have no information as to how much B’s performance is better than A.

6.3 M-Squared

If M-Squared return (also known as risk adjusted performance measure or RAP) is greater than zero, the manager (portfolio) has positive risk-adjusted return. One way to get a positive M-Squared is when the risk is same as the market but RP is greater than RM.  Another way is when the return is same as the market but at a lower risk.  If M-Squared is zero, then the manager (portfolio) has the same risk-adjusted return as the market. If M-Squared return is negative, the manager (portfolio) has a lower risk-adjusted return than the market.

The ex-ante and ex-post M2 are given by:

M^2=\left[E\left(R_p\right)-R_f\right]\frac{\sigma_m}{\sigma_p}+R_f=\mathrm{SR}\mathrm{\times }\sigma_m+R_f

(ex ante)

\widehat{M^2}=\left({\hat{R}}_p-{\hat{R}}_f\right)\frac{{\hat{\sigma}}_m}{{\hat{\sigma}}_p}+R_f=\widehat{\mathrm{SR}}\times {\hat{\sigma}}_m+R_f

(ex post)

where: \sigma_m is the standard deviation of the market portfolio and \sigma_{m}$/$\sigma$${}_{p} is the portfolio-specific leverage ratio.

The difference between the risk-adjusted performance of the portfolio and the performance of the market is called M2 alpha.

Advantage: M-Squared is in units of the percent return which makes it more intuitive for the interpretation by the user.

Limitation: A limitation of this measure is that it uses total risk and not systematic risk.

6.4 Jensen’s Alpha

Jensen’s alpha is the difference between the actual return on a portfolio and the CAPM calculated, expected or required return. In other words, it is the plot of the excess return of the security on the excess return of the market. The intercept is Jensen’s alpha and beta is the slope.  Jensen’s alpha can be calculated on both ex ante and ex post basis.

Like the Treynor ratio, it is based on systematic risk. Alpha is used to rank different managers and their portfolios. Since Jensen’s alpha uses systematic risk, it is theoretically superior to M-Squared.

Interpreting Jensen’s Alpha

  • If Jensen’s alpha > 0, the manager (portfolio) has positive risk-adjusted return.
  • If Jensen’s alpha = 0, then the manager (portfolio) has the same risk-adjusted return as the market.
  • If Jensen’s alpha < 0, the manager (portfolio) has a lower risk-adjusted return than the market.

Instructor’s Note:

Sharpe ratio and M-squared are total risk measures. Use these measures when a portfolio is not fully diversified.

Treynor ratio and Jensen’s alpha are based on beta risk and should be used when a portfolio is well diversified.

7. Applications of the CAPM in Portfolio Construction

7.1 Security Characteristic Line

The SCL is a plot of the excess return of a security over the risk-free rate on the y-axis, against the excess return of the market on the x-axis. We saw earlier that the SML’s intercept on the y-axis is the alpha and the line’s slope is its beta. Similarly, the SCL’s slope is the security’s beta.

The SCL is obtained by regressing excess security return on the excess market return.

SCL: R_i-R_f=α_i+ β_i*(R_m-R_f )

where:

R_i-R_f = excess security return

R_m-R_f= excess market return

α_i= Jensen’s alpha or excess return

7.2 Security Selection

In CAPM, we assumed that investors have homogeneous expectations and assign the same value to all assets. So all the investors arrive at the same optimal risky portfolio: the market portfolio. But, in reality, it does not actually happen.

The SML can also be used for security selection. Investors can plot a security’s expected return and beta against the SML. The security is undervalued if it plots above the SML. The security is overvalued if it plots below the SML. The security is fairly priced if it plots on the SML.

As you can see in the exhibit below, security X is undervalued as it plots above the SML. At a risk level of β=0.5, the return of X is greater than the security that plots on the SML. Similarly, Y must not be bought because the security on SML at a risk level of β=0.7 has a higher return than Y.

7.3 Implications of the CAPM for Portfolio Construction

Theoretically, investors should hold a combination of the risk-free asset and the market portfolio but it is impractical to own the market portfolio as it has a large number of securities. For example, S&P 500 is a good representation of the market as it has 500 stocks. But it can be shown that holding as few as 30 stocks can diversify away the non-systematic risk.

Interpretation of the exhibit below:

  • It shows how variance decreases – nonsystematic risk, in particular – as stocks are added to the portfolio.
  • Much of the non-systematic risk is diversified away with 30 stocks. As more stocks are added, the non-systematic risk progressively decreases approaching the systematic risk for 30 stocks.
  • It is important that these stocks are not correlated with each other and must be randomly selected from different asset classes.


Portfolio Management Portfolio Risk and Return Part II Part 3