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Ec2723, Fall 2013: Q & A Notes 1. Is the market portfolio in the APT (in a single-factor model) the same as the market portfolio in the CAPM? Answer: In theory, No. This actually speaks to a more fundamental difference between the CAPM and the APT: the former is an equilibrium model but the latter is not. Under the CAPM, the market portfolio is the tangent portfolio, which is also mean-variance efficient. Instead, in the APT with a single factor, the market can be any broadly diversified portfolio that produces uncorrelated residual risks. In practice, however, people usually use some typical market indexes to proxy the market portfolio in both models. More fundamentally, the APT only requires no arbitrage and the fact that errors across stocks are uncorrelated. It does not impose that investors always choose their portfolios optimally, as required by the CAPM. In fact, the APT does not even require everybody to recognize the arbitrage opportunity. Only a few deep-pocketed arbitrageur who are able to take advantage of the arbitrage opportunities will ensure that no arbitrage holds. 2. Does the beta anomaly in stock returns (that is, the fact that the expected return of high-beta stocks is too low relative to the CAPM prediction) provide evidence against the efficient market hypothesis? Answer: This is uncertain. With knowledge of the beta anomaly, investors can beat the market by shorting high-beta stocks while longing low-beta stocks. Nevertheless, there are two responses to this observation. On the one hand, it could be the case that the market is inefficient. On the other hand, it could also be the case that our market model is incorrect, and we should have used the Black CAPM instead. This is the joint hypothesis problem, which implies that we can only test market efficiency and a given market model jointly. Hence, we are not sure whether such evidence is really against the efficient market hypothesis. 3. In Chapter 5, page 134, how can we derive condition (5.56) from condition (5.55)? Answer: We use three approximations: 1) for small y, we have exp(y) 1+ y, 2) for small y, we have y 3 0, and 3) unexpected log stock returns are approximately equal to unexpected changes in log stock prices. Here it goes: E t (1 + R t+1 ) = D t+1 P t + exp(E t g t+1 ) exp Var t (p t+1 - p t ) 2 ! D t+1 P t + exp(E t g t+1 ) " 1+ Var t (p t+1 - p t ) 2 # = D t+1 P t + exp(E t g t+1 ) + exp(E t g t+1 ) Var t (p t+1 - p t ) 2 D t+1 P t + exp(E t g t+1 ) + (1 + E t g t+1 ) Var t (p t+1 - p t ) 2 1

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  • Ec2723, Fall 2013: Q & A Notes

    1. Is the market portfolio in the APT (in a single-factor model) the same as the marketportfolio in the CAPM?

    Answer: In theory, No. This actually speaks to a more fundamental difference betweenthe CAPM and the APT: the former is an equilibrium model but the latter is not. Under theCAPM, the market portfolio is the tangent portfolio, which is also mean-variance efficient.Instead, in the APT with a single factor, the market can be any broadly diversified portfoliothat produces uncorrelated residual risks. In practice, however, people usually use sometypical market indexes to proxy the market portfolio in both models.

    More fundamentally, the APT only requires no arbitrage and the fact that errors acrossstocks are uncorrelated. It does not impose that investors always choose their portfoliosoptimally, as required by the CAPM. In fact, the APT does not even require everybody torecognize the arbitrage opportunity. Only a few deep-pocketed arbitrageur who are able totake advantage of the arbitrage opportunities will ensure that no arbitrage holds.

    2. Does the beta anomaly in stock returns (that is, the fact that the expected return ofhigh-beta stocks is too low relative to the CAPM prediction) provide evidence against theefficient market hypothesis?

    Answer: This is uncertain. With knowledge of the beta anomaly, investors can beatthe market by shorting high-beta stocks while longing low-beta stocks. Nevertheless, thereare two responses to this observation. On the one hand, it could be the case that themarket is inefficient. On the other hand, it could also be the case that our market model isincorrect, and we should have used the Black CAPM instead. This is the joint hypothesisproblem, which implies that we can only test market efficiency and a given market modeljointly. Hence, we are not sure whether such evidence is really against the efficient markethypothesis.

    3. In Chapter 5, page 134, how can we derive condition (5.56) from condition (5.55)?

    Answer: We use three approximations: 1) for small y, we have exp(y) 1 + y, 2) forsmall y, we have y3 0, and 3) unexpected log stock returns are approximately equal tounexpected changes in log stock prices. Here it goes:

    Et(1 +Rt+1) =Dt+1Pt

    + exp(Etgt+1) exp

    (Vart(pt+1 pt)

    2

    )

    Dt+1Pt

    + exp(Etgt+1)

    [1 +

    Vart(pt+1 pt)2

    ]

    =Dt+1Pt

    + exp(Etgt+1) + exp(Etgt+1)Vart(pt+1 pt)

    2

    Dt+1Pt

    + exp(Etgt+1) + (1 + Etgt+1)Vart(pt+1 pt)

    2

    1

  • =Dt+1Pt

    + exp(Etgt+1) + (1 + Etgt+1)Vart(pt+1 pt)

    2

    =Dt+1Pt

    + exp(Etgt+1) +Vart(pt+1 pt)

    2+

    Etgt+1Vart(pt+1 pt)2 0

    Dt+1Pt

    + exp(Etgt+1) +Vart(rt+1)

    2.

    4. In Chapter 6, page 160, between conditions (6.32) and (6.33) it says The dividend on thewealth portfolio is aggregate consumption, and the expected return on the wealth portfolio isa constant, plus expected consumption growth times (1/). How can we understand this?

    Answer: This actually tells you how to derive (6.33) from (6.32). First, it says that

    dw,t = ct ,

    which helps you substitute dw,t+1+j out in (6.32). The second step is more tricky. Recallcondition (6.30) (on page 159), which says

    rf,t+1 =1

    Et[ct+1] + const .

    This comes from the Euler equation (6.29). Actually, for any risky asset, including the wealthportfolio, this relation still holds, but with a different constant:

    rw,t+1 =1

    Et[ct+1] + constw .

    This will further help you substitute rw,t+1+j out in (6.32). Note that the constant will becancelled out when taking the expectation innovation between two time periods, as it is nottime-varying. This leads to (6.33) eventually.

    5. We know that the coefficient relative risk aversion () is roughly in the range of 5 to 15.How does the elasticity of intertemporal substitution (EIS, ) look like?

    Answer: In the long-run risk model we know that we need > 1. However, in empirics,we dont have a good consensus on this. In Lettau, Ludvigson and Wachter (2006) theestimate of EIS () is larger than 1. But in an earlier paper by Vissing-Jorgensen (2002),she found that the EIS for stockholders is in the range of 0.3 to 0.4, 0.8 to 1 for bondholders,and larger for households with larger asset holdings within these two groups.

    6. In the lecture notes for bonds, page 3, the first expression (as below) looks confusing.What does it mean?

    1 + Y1t = Et[1 +R2,t+1] = (1 + Y2t)2Et

    [1

    1 + Y1,t+1

    ]

    Answer: This follows the definition of the yield to maturity and the holding-periodreturn. The first term is the yield to maturity of a one-period bond, i.e., the return of

    2

  • buying a one-period bond at t = 0 and holding it until maturity (t = 1). The second term isthe holding-period return of a two-period bond, i.e., the return of buying a two-period bondat t = 0 and selling it at t = 1. The third term is the total return of buying a two-periodbond at t = 0 and holding it until maturity (t = 2), through issuing a one-period bond att = 1 to finance the position between t = 1 and t = 2. When the expectations hypothesisholds, these three returns should be the same.

    7. How can we test if risksharing is perfect?

    Answer: This follows Cochrane (1991). Under full insurance, ct+1 should be cross-sectionally independent of idiosyncratic variables. So we could run cross-sectional regressionsof ct+1 on a variety of exogenous variables and test if the coefficients are zero.

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