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Chapter 13 Alternative Models of Systematic RiskPowerPoint Presentation

Chapter 13 Alternative Models of Systematic Risk

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Chapter 13 Alternative Models of Systematic Risk. Chapter Outline. 13.1 The Efficiency of the Market Portfolio 13.2 Implication of Positive Alphas 13.3 Multifactor Models of Risk 13.4 Characteristic Variable Models of Expected Return 13.5 Methods Used in Practice.

Chapter 13 Alternative Models of Systematic Risk

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Chapter 13 Alternative Models of Systematic Risk

13.1 The Efficiency of the Market Portfolio

13.2 Implication of Positive Alphas

13.3 Multifactor Models of Risk

13.4 Characteristic Variable Models of Expected Return

13.5 Methods Used in Practice

Describe the empirical findings about firm size, book-to-market, and momentum strategies that imply that the CAPM does not accurately model expected returns.

Discuss two conditions that might cause investors to care about characteristics other than expected return and volatility of their portfolios.

Use multi-factor models, such as the Fama-French-Carhart model, to calculate expected returns.

Illustrate how multifactor models can be written as the expected return on a self-financing portfolio.

Discuss the use of characteristic models in calculating expected returns and betas.

If the market portfolio is efficient, securities should not have alphas that are significantly different from zero.

For most stocks the standard errors of the alpha estimates are large, so it is impossible to conclude that the alphas are statistically different from zero.

However, it is not difficult to find individual stocks that, in the past, have not plotted on the SML.

Researchers have studied whether portfolios of stocks plot on this line and have searched for portfolios that would be most likely to have nonzero alphas.

Researchers have identified a number of characteristics that can be used to pick portfolios that produce high average returns.

Size Effect

Stocks with lower market capitalizations have been found to have higher average returns.

Portfolios based on size were formed.

Portfolios consisting of small stocks had higher average excess returns than those consisting of large stocks.

Book-to-Market Ratio

The ratio of the book value of equity to the market value of equity

Portfolios based on the book-to-market ratio were formed.

Portfolios consisting of high book-to-market stocks had higher average excess returns than those consisting of low book-to-market stocks.

When the market portfolio is not efficient, theory predicts that stocks with low market capitalizations or high book-to-market ratios will have positive alphas.

This is evidence against the efficiency of the market portfolio.

Data Snooping Bias

The idea that given enough characteristics, it will always be possible to find some characteristic that by pure chance happens to be correlated with the estimation error of a regression

Problem

Suppose two firms, ABC and XYZ, are both expected to pay a dividend stream $2.2 million per year in perpetuity.

ABC’s cost of capital is 12% per year and XYZ’s cost of capital is 16%.

Which firm has the higher market value?

Which firm has the higher expected return?

Solution

ABC has an expected return of 12%.

XYZ has an expected return of 16%.

Problem

Now assume both stocks have the same estimated beta, either because of estimation error or because the market portfolio is not efficient.

Based on this beta, the CAPM would assign an expected return of 15% to both stocks.

Which firm has the higher alpha?

How do the market values of the firms relate to their alphas?

Solution

αABC = 12% - 15% = -3%

αXYZ = 16% - 15% = 1%

The firm with the lower market value has the higher alpha.

Momentum Strategy

Buying stocks that have had past high returns (and shorting stocks that have had past low returns)

When the market portfolio is efficient, past returns should not predict alphas.

However, researchers found that the best performing stocks had positive alphas over the next six months.

This is evidence against the efficiency of the market portfolio.

If the CAPM correctly computes the risk premium, an investment opportunity with a positive alpha is a positive-NPV investment opportunity, and investors should flock to invest in such strategies.

If small stock or high book-to-market portfolios do have positive alphas, one can draw one of two conclusions:

Investors are systematically ignoring positive-NPV investment opportunities.

If the CAPM correctly computes risk premiums, but investors are ignoring opportunities to earn extra returns without bearing any extra risk, it is because

They are unaware of them or,

The costs to implement the strategies are larger than the NPV of undertaking them.

If small stock or high book-to-market portfolios do have positive alphas, one can draw one of two conclusions:

The positive-alpha trading strategies contain risk that investors are unwilling to bear but the CAPM does not capture. This would suggest that the market portfolio is not efficient.

The true market portfolio is more than just stocks—it includes bonds, real estate, art, precious metals, and any other investment vehicles available.

However, researchers use a proxy portfolio like the S&P 500 and assume that it will be highly correlated to the true market portfolio.

If the true market portfolio is efficient but the proxy portfolio is not highly correlated with the true market, then the proxy will not be efficient and stocks will have nonzero alphas.

The most important example of a non-tradeable wealth is human capital.

If investors have a significant amount of non-tradeable wealth, this wealth will be an important part of their portfolios, but will not be part of the market portfolio of tradeable securities.

Given non-tradeable wealth, the market portfolio of tradeable securities will likely not be efficient.

The expected return of any marketable security is:

When the market portfolio is not efficient, we have to find a method to identify an efficient portfolio before we can use the above equation. However, it is not actually necessary to identify the efficient portfolio itself.

All that is required is to identify a collection of portfolios from which the efficient portfolio can be constructed.

Assume that there are two portfolios that can be combined to form an efficient portfolio.

These are called factor portfolios and their returns are denoted as RF1 and RF2. The efficient portfolio consists of some (unknown) combination of these two factor portfolios, represented by portfolio weights x1 and x2:

To see if these factor portfolios measure risk, regress the excess returns of some stock s on the excess returns of both factors:

This statistical technique is known as a multiple regression.

A portfolio, P, consisting of the two factor portfolios has a return of:

which simplifies to:

Since εi is uncorrelated with each factor, it must be uncorrelated with the efficient portfolio:

Recall that risk that is uncorrelated with the efficient portfolio is diversifiable risk that does not command a risk premium. Therefore, the expected return of portfolio P is rf , which means αs must equal zero.

Setting αs equal to zero and taking expectations of both sides, the result is the following two-factor model of expected returns:

Factor Beta

The sensitivity of the stock’s excess returns to the excess return of a factor portfolio.

Single-Factor versus Multi-Factor Model

A singe-factor model uses one portfolio while a multi-factor model uses more than one portfolio in the model.

The CAPM is an example of a single-factor model while the Arbitrage Pricing Theory (APT) is an example of a multifactor model.

Given N factor portfolios with returns RF1, . . . , RFN, the expected return of asset s is defined as:

β1…. βN are the factor betas.

A self-financing portfolio can be constructed by going long in some stocks and going short in other stocks with equal market value.

In general, a self-financing portfolio is any portfolio with portfolio weights that sum to zero rather than one.

If all factor portfolios are self-financing then:

A trading strategy that each year buys a portfolio of small stocks and finances this position by short selling a portfolio of big stocks has historically produced positive risk-adjusted returns.

This self-financing portfolio is widely known as the small-minus-big (SMB) portfolio.

A trading strategy that each year buys an equally-weighted portfolio of stocks with a book-to-market ratio less than the 30th percentile of NYSE firms and finances this position by short selling an equally-weighted portfolio of stocks with a book-to-market ratio greater than the 70th percentile of NYSE stocks has historically produced positive risk-adjusted returns.

This self-financing portfolio is widely known as the high-minus-low (HML) portfolio.

Each year, after ranking stocks by their return over the last one year, a trading strategy that buys the top 30% of stocks and finances this position by short selling bottom 30% of stocks has historically produced positive risk-adjusted returns.

This self-financing portfolio is widely known as the prior one-year momentum (PR1YR) portfolio.

This trading strategy requires holding the portfolio for a year and the process is repeated annually.

Currently the most popular choice for the multifactor model uses the excess return of the market, SMB, HML, and PR1YR portfolios.

Fama-French-Carhart (FFC) Factor Specifications

Problem

You are considering making an investment in a project in the semiconductor industry.

The project has the same level of non-diversifiable risk as investing in Intel stock.

Problem (continued)

Assume you have calculated the following factor betas for Intel stock:

Determine the cost of capital by using the FFC factor specification if the monthly risk-free rate is 0.5%.

Solution

The annual cost of capital is .0099691 × 12 = 11.96%

Although it is widely used in research to measure risk, there is much debate about whether the FFC factor specification is really a significant improvement over the CAPM.

One area where researchers have found that the FFC factor specification does appear to do better than the CAPM is measuring the risk of actively managed mutual funds.

Researchers have found that funds with high returns in the past have positive alphas under the CAPM. When the same tests were repeated using the FFC factor specification to compute alphas, no evidence was found that mutual funds with high past returns had future positive alphas.

Calculating the cost of capital using the CAPM or multifactor model relies on accurate estimates of risk premiums and betas.

Accurately estimating these quantities is difficult as both risk premiums and betas may not remain stable over time.

Characteristic Variable Model

An approach to measuring risk that views firms as a portfolio of different measurable characteristics that together determine the firm’s risk and return.

There is an important difference between this and the multifactor models considered earlier.

In the multifactor models, the returns of the factor portfolios are observed, and the sensitivity of each stock to the different factors is estimated.

In the characteristic variable model, the weight of each stock on each characteristic is observed, and then we estimate the return Rcn associated with each characteristic.

One way to estimate relation between the characteristic variables and returns is to use the relation to estimate each stock’s expected return.

If you view a stock as portfolio of characteristic variables, then the stock’s expected return is the sum over all the variables of the amount of each characteristic variable the stock contains times the expected return of that variable.

Researchers have evaluated the usefulness of the characteristic variable approach by ranking stocks based on their characteristics model.

They put stocks into 10 ranked portfolios based on their characteristics model’s prediction of expected return. They then measured the return of each portfolio over the following month. They found that the top-ranked portfolios had the highest returns.

Another approach is to use the estimated returns of the characteristic variables to estimate the covariance between pairs of stocks, or between a stock and the market index.

By viewing each stock as a portfolio of characteristics, one can calculate the covariance between two different stocks i and j as:

The beta of a stock is equal to the weighted-average of the characteristic variable betas where the weights are the amounts of each characteristic variable the stock contains.

As the firm evolves in time, its beta will change accordingly to reflect its new level of risk.

There is no clear answer to the question of which technique is used to measure risk in practice—it very much depends on the organization and the sector.

There is little consensus in practice in which technique to use because all the techniques covered are imprecise.