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P1.T1.64. Arbitrage pricing model (APT) versus CAPM

David Harper CFA FRM

David Harper CFA FRM
Staff member
AIMs: Explain the relationship between the CAPM and the APT. Describe how the APT can be used in both active and passive portfolio management.


63.1. An arbitrage pricing model (APT) characterizes excess security returns as a linear function of two indexes, I(1) and I(2). In this way, a security's excess return in percentage terms, ER(i), is given by ER(i) = R(i) - Rf = a + b(1)*I(1) + b(2)*I(2), where b(i) is the factor sensitivity to the index, I(i). We observe three securities that fit the APT model, as follows: Security #1 ER(1) = a + 2.0*I(1) + 3.0*I(2) = 8.0; Security #2 ER(2) = a + 4.0*I(1) + 2.5*I(2) = 3.5; Security #3 ER(3) = a + 1.0*I(1) - 2.0*I(2) = -5.5. Which is the specification of the model?

a. ER(i) = 1.0 + b(1)*2.5 + b(2)*3.0
b. ER(i) = 2.0 - b(1)*1.5 + b(2)*3.0
c. ER(i) = 3.0 + b(1)*0.5 + b(2)*0.5
d. ER(i) = 4.0 - b(1)*3.0 - b(2)*1.0

63.2. Which of the following is TRUE about the relationship between the CAPM and the arbitrage pricing model (APT)?

a. CAPM assumes that the market is the only source of covariance between returns
b. If we employ a procedure (e.g., Roll and Ross) and identify more than one common factor, we can logically reject the CAPM
c. Similar to the CAPM, in order to test the APT we need to identify a "Market Portfolio" of all risky assets
d. The APT solution with multiple factors appropriately priced is fully consistent with the Sharpe-Lintner-Mossin form of the CAPM

63.3. In regard to multi-factor models, including the arbitrage pricing model (APT), each of the following is true EXCEPT:

a. A disadvantage of APT models, in general, is the curse of dimensionality; i.e., k factors requires the identification of k*(k+1)/2 values
b. A multi-factor risk model is likely to employ fewer factors than a multi-factor alpha (i.e., expected return) model
c. APT has advantages in flexibility over CAPM: APT is more flexible; does not require that returns are normally distributed; and merely assumes investors are risk-averse
d. The factor sensitivities (betas) in APT are equal to Covariance(security's return, factor risk premium)/Variance(factor risk premium)

63.4. Your colleague Janet Smith just earned her FRM and argues that multi-factor models like APT offer the following benefits:

I. A multi-index model can assist in PASSIVE portfolio management by closely, if not perfectly, tracking an index without holding all of the index's stocks in the same proportion​
II. A multi-index model can assist in ACTIVE portfolio management by determining under- or over-valued stocks​
III. A multi-index model can assist in ACTIVE portfolio management by allowing the manager to make factor bets; i.e., skill-based beta​
IV. A multi-index model can assist in performance measurement and attribution by re-classifying single-factor alpha as common factor beta​

According to Elton and Gruber, which of her assertion are true?

a. I. only
b. I. and III. only
c. II. and IV. only
d. All of the above

(Source: Edwin J. Elton, Martin J. Gruber, Stephen J. Brown, William N. Goetzmann, Modern Portfolio Theory and Investment Analysis (John Wiley & Sons, Nov 16, 2009))

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