What is the expected return to a portfolio of 25% Stock X and 75% Stock Y? r i = 0.0290 + 0.908Forex + 1.292Default + 0.9180Size 28 Stock a i b i,m b i,s b i,v X 0.05 1.1 1.4 .6 Y 0.02 0.8 1.2 1.0
ARBITRAGE PRICING THEORY The APT, as it is known, describes the expected return to an asset as a linear function of the risk of the asset with respect to a set of factors. • The APT is an equilibrium model where the s represent factor sensitivities and the s represent risk premiums. • The APT relies on three assumptions: 1. A factor model describes asset returns. 2. There are many assets, so investors can form well-diversified portfolios that eliminate asset-specific risk. 3. No arbitrage opportunities exist among well-diversified portfolios. • In contrast to multifactor models, the APT models the expected return in equilibrium (the first term of the equation), in essence restricting the first term in the general multifactor expression to the APT value for that term. • 29
ARBITRAGE PRICING THEORY Focus On: Calculations • You are considering purchasing shares in Cleveland Corp., and you believe the APT with three priced risk factors is an accurate description of the expected return to Cleveland Corp. The first risk factor, Macro, has a risk premium of 3% and Cleveland Corp. has a for this risk factor of 1.1. The second risk factor, Term, has a risk premium of 2% and Cleveland has a of 0.74. Finally, the last risk factor, Inflation, has a risk premium of 1.3% and Cleveland has a of 0.27. • If the current risk-free rate is 3.5%, what is the APT three-factor expected return to Cleveland Corp. shares? • 30
THE APT AND ARBITRAGE Focus On: Calculations • Consider the following stock returns and factor sensitivities for a single factor APT. • Can we combine X and Y to achieve an arbitrage possibility with Z? - What weights create a portfolio with equal sensitivities so that the sensitivity of the portfolio = the sensitivity of Z? - Is the expected return to this portfolio the same as the expected return to Z? 31 Stock Expected Return Sensitivity X 0.10 1.625 Y 0.14 2.625 Z 0.11 2.375
THE APT AND ARBITRAGE Focus On: Calculations - What weights create a portfolio with equal sensitivities so that the sensitivity of the portfolio = the sensitivity of Z? - Is the expected return to this portfolio the same as the expected return to Z? - No. Therefore, if we go short Z, we can use the proceeds to go long X and Y in weights 25% and 75%, respectively. We will generate a risk-free profit of 2% = 13% – 11%. • 32
FUNDAMENTAL FACTOR MODELS In contrast to macroeconomic models, fundamental models use expected returns (instead of surprises) as factors. • Because the expected returns no longer have an expected value of zero, as do the surprises in macroeconomic factor models, the intercept, a i , is no longer an expected return but the intercept term from a regression. • The b i terms are typically factor sensitivities that have been standardized by the sensitivity across all stocks.
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