answer the queston
7. Capital asset pricing and arbitrage pricing theory
Instructor: Seongcheol Paeng
7/25/2020
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Assignments
Problem Sets (Paraphrase with your own words.)
Explain Implications of the CAPM.
1. All investors will choose to hold the market portfolio (M), which includes all assets of the security universe. For simplicity, we shall refer to all assets as stocks. The proportion of each stock in the market portfolio equals the market value of the stock (price per share times the number of shares outstanding) divided by the total market value of all stocks.
2. The market portfolio will be on the efficient frontier. Moreover, it will be the optimal risky portfolio, the tangency point of the capital allocation line (CAL) to the efficient frontier. As a result, the capital market line (CML), the line from the risk-free rate through the market portfolio, M, is also the best attainable capital allocation line. All investors hold M as their optimal risky portfolio, differing only in the amount invested in it versus in the risk-free asset.
3. The risk premium on the market portfolio will be proportional to the variance of the market portfolio and investors’ typical degree of risk aversion. Mathematically, (7.1) where is the standard deviation of the return on the market portfolio and represents the degree of risk aversion of the average investor.
4. The risk premium on individual assets will be proportional to the risk premium on the market portfolio (M) and to the beta coefficient of the security on the market portfolio. Beta measures the extent to which returns respond to the market portfolio. Formally, beta is the regression (slope) coefficient of the security return on the market return, representing sensitivity to fluctuations in the overall security market.
7/25/2020
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Assignments
Problem Sets (Paraphrase with your own words.)
2. Explain the Security Market Line (SML).
Security Market Line (SML): Graphical representation of the expected return-beta relationship of the CAPM.
The CML graphs the risk premiums of efficient complete portfolios (made up of the market portfolio and the risk-free asset) as a function of portfolio standard deviation.
This is appropriate because standard deviation is a valid measure of risk for portfolios that are candidates for an investor’s complete portfolio.
The SML, in contrast, graphs individual-asset risk premiums as a function of asset risk (which we measure by beta).
The relevant measure of risk for an individual asset (which is held as part of a well-diversified portfolio) is not the asset standard deviation but rather the asset beta. The SML is valid both for individual assets and portfolios. Because the SML is the graphical representation of the mean–beta relationship, “fairly priced” assets plot exactly on the SML. The expected returns of such assets are commensurate with their risk.
Whenever the CAPM holds, all securities must lie on the SML. Underpriced stocks plot above the SML: Given beta, their expected returns are greater than is indicated by the CAPM.
Overpriced stocks plot below the SML. The difference between fair and actual expected rates of return on a stock is the alpha, denoted α. The expected return on a mispriced security is given by
7/25/2020
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Assignments
Problem Sets (Paraphrase with your own words.)
Explain what we learn from Table 7.2.
Despite these qualifications, we can safely say that Google is a cyclical stock.
Its returns vary equally with or more than the overall market, as its beta is higher than the average value of 1, albeit not significantly so.
Thus, we would expect Google’s excess return to respond, on average, more than one-for-one with the market index.
Suppose the current T-bill rate is 2.75%, and our forecast for the market excess return is 5.5%. Then the required rate of return for an investment with the same risk as Google’s equity would be
7/25/2020
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Assignments
Problem Sets (Paraphrase with your own words.)
4. Explain the Two-Factor model by Merton.
Such a multifactor CAPM was first presented by Merton (1973). In the two-factor economy of Equation 7.5, the expected rate of return on a security would be the sum of three terms:
1. The risk-free rate of return.
2. The sensitivity to the market index (i.e., the market beta, ) times the risk premium of the index, .
3. The sensitivity to interest rate risk (i.e., the T-bond beta, ) times the risk premium of the T-bond portfolio, .
This assertion is expressed mathematically as a two-factor security market line for security i: (7.6)
The multifactor model clearly gives us a richer way to think about risk exposures and compensation for those exposures than the single-index model or the CAPM.
7/25/2020
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Assignments
Problem Sets (Paraphrase with your own words.)
5. Explain the Fama-French Three-Factor Model.
Fama and French proposed a three-factor model that has become a standard tool for empirical studies of asset returns.
They add to the market-index portfolios formed on the basis of firm size and book-to-market ratio to explain average returns.
These additional factors are motivated by the observations that average returns on stocks of small firms and firms with high ratios of book value of equity to market value of equity have been higher than predicted by the CAPM.
This observation suggests that size and book-to-market (B/M) ratio may be proxies for exposures to sources of systematic risk not captured by beta and thus result in return premiums.
7/25/2020
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Assignments
Problem Sets (Paraphrase with your own words.)
6. Explain what we learn from Table 7.4.
While the FF three-factor model offers a richer and more accurate description of asset returns, applying this model requires two more forecasts of future returns, namely, for the SMB and HML portfolios.
We have so far in this section been using a T-bill rate of 2.75% and a market risk premium of 5.5%. If we add to these values a forecast of 2.5% for the SMB premium and 4% for HML, the required rate for an investment with the same risk as Google’s equity would be
=2.75 + (1.51 × 5.5) + (−.20 × 2.5) + (−1.33 × 4) = 5.24%
which is considerably lower than the rate (9.35%) derived from cyclical considerations alone (i.e., single-beta models).
7/25/2020
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Assignments
Problem Sets (Paraphrase with your own words.)
7. Compare the APT with the CAPM.
A violation of the APT’s pricing relationships will cause extremely strong pressure to restore them even if only a limited number of investors become aware of the disequilibrium. Moreover, the APT provides an expected return–beta relationship using a well-diversified portfolio that can be constructed from a large number of securities.
In spite of these apparent advantages, the APT does not fully dominate the CAPM. The CAPM provides an unequivocal statement on the expected return–beta relationship for all securities, whereas the APT implies that this relationship holds for all but perhaps a small number of securities.
Moreover, while the APT is built on the foundation of well-diversified portfolios, we’ve seen, for example in Table 10.1, that even large portfolios may have non-negligible residual risk.
Despite these shortcomings, the APT is valuable.
First, recall that the CAPM requires that almost all investors be mean-variance optimizers. The APT frees us of this assumption. It is sufficient that a small number of sophisticated arbitrageurs scour the market for arbitrage opportunities.
Moreover, when we replace the unobserved market portfolio of the CAPM with an observed, broad index portfolio that may not be efficient, we no longer can be sure that the CAPM predicts risk premiums of all assets with no bias. Therefore, neither model is free of limitations.
In the end, however, it is noteworthy and comforting that despite the very different paths they take to get there, both models arrive at the same security market line.
Most important, they both highlight the distinction between firm-specific and systematic risk, which is at the heart of all modern models of risk and return.
7/25/2020
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