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# P2.T9.20.2. Factor theory: stochastic discount factors

#### Nicole Seaman

##### Director of FRM Operations
Staff member
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Learning objectives: Describe multifactor models and compare and contrast multifactor models to the CAPM. Explain how stochastic discount factors are created and apply them in the valuation of assets. Describe efficient market theory and explain how markets can be inefficient.

Questions:

20.2.1. In the capital asset pricing model (CAPM), the expected excess return of an asset is a product of the quantity and price of risk, E(r_i) - Rf = β(i,M)×[E(r_M) - Rf]. Similarly, in factor theory, the expected excess return of an asset is given by E(r_i) - Rf = β(i,m)×λ(m). Here β(i,m) = cov(r_i, m)/var(m) and it is the beta of the asset with respect to the stochastic discount factor (SDF; aka, pricing kernel). Importantly, β(i,m) and λ(m) are both negative. Each of the following statements is true EXCEPT which statement is false?

a. λ(m) = -var(m)/E(m) and it is the price of bad times risk
b. An increase in β(i,m), for example from -0.95 to -0.33, implies a decrease in the asset's expected return; i.e., higher beta implies lower risk premium
c. Because the value of the bad times index (m) is negative, an asset with a non-zero beta with respect to the SDF, β(i,m), has a negative expected excess return
d. An increase in the asset's covariance with the bad times index, cov(r_i, m), for example from -0.0980 to -0.0210, implies a decrease in the asset's expected return

20.2.2. In factor theory, an asset's expected excess return is given by E(r_i) - Rf = β(i,m)×λ(m) where the asset's beta is with respect to the stochastic discount factor (SDF; aka, pricing kernel) and (m) in an index of bad time. Let's make the following assumptions about an asset:
• The riskfree rate is 3.0% such that E(m) = 1/1.03 = 0.9709
• The variance(m) equals 30.0%^2 = 0.090
• The covariance(r_i, m) equals -0.10680
Which is nearest to the asset's expected excess return, E(r_i) - Rf?

a. -5.0%
b. +4.0%
c. +8.0%
d. +11.0%

20.2.3. Although the classical efficient market hypothesis (EMH) has three forms (i.e., weak, semi-strong, and strong), Richard Thaler helpfully reminds us that its assertion has two components: "the price is right", and "there is no free lunch" (see https://review.chicagobooth.edu/behavioral-science/2018/article/behavioral-economics-nuts-nudges).
• To believe "the price is right" is to believe that asset prices in the efficient market already equal their intrinsic or true value.
• To believe that "there is no free lunch" is to believe that active management does not add sustainable value because expected excess returns can only be achieved by adding more risk.
Ang's Factor Theory has something to say about the efficient market hypothesis. According to Ang, which of the following is TRUE?

a. Factor theory says that markets are efficient; that is, there are no deviations from efficiency and no enduring risk premiums
b. The rational explanation for market inefficiency is that high returns compensate investors for their exposure to idiosyncratic risk vectors
c. The behavioral explanation for market inefficiency is that high returns result from the inefficient updating of beliefs; and from the over- or under-reaction to news
d. Risk premiums should not persist because rational inefficiencies will be arbitraged away and behavioral inefficiencies by definition cannot persist beyond a few days