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FRM Part I · FRM Exam Part I · The Arbitrage Pricing Theory and Multifactor Models of Risk and Return

Under a one-factor APT, portfolio A has beta 0.8 and expected return 7%, and portfolio B has beta 1.6 and expected return 11%. Both are well diversified. The risk-free rate is 4%. An investor wants to exploit any arbitrage using a combination of A, B and the risk-free asset. Which statement is correct?

Arbitrage exists. A's excess return per unit of beta is 3% divided by 0.8, or 3.75%, while B's is 7% divided by 1.6, or 4.375%. Unequal ratios violate the APT linear relation, so buying B and shorting a beta-matched mix of A and cash earns a riskless profit.

  1. ANo arbitrage exists, since both portfolios offer the same risk premium per unit of beta
  2. BArbitrage exists, because B's premium per unit of beta exceeds A's
  3. CArbitrage exists, because A's premium per unit of beta exceeds B'sCorrect
  4. DArbitrage cannot exist because the portfolios have different betas

Explanation

A's premium per unit of beta is (7%-4%)/0.8 = 3.75%. B's is (11%-4%)/1.6 = 4.375%. These differ, so B offers more per unit of beta, which means arbitrage exists. The correct choice is therefore that B's ratio exceeds A's, and the option stating that is the second one.

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