Arbitrage Pricing Theory

If you find other factors that explain asset returns, then add them to your model. Starting from the CAPM, you have a market risk premium. If that's not enough, then you're moving into the world of __arbitrage pricing theory (APT)__. This arose in the 1970s as an alternate to the CAPM, and it's still in use.

APT relies on three assumptions. First, that asset returns can be described by a multifactor model. That's obviously important, since that's where this is going. The second is that there are a lot of assets around that can be used for diversification in a portfolio. How would that affect risk?

That's right. At least, that's the idea in theory. This assumption isn't too strong, as most asset-specific risk can be diversified away—just not all of it. Studies show that about 1-3% of asset-specific risk is still there in well-diversified portfolios. By the way, just to be clear: asset-specific risk here is the same thing as diversifiable risk, unsystematic risk, firm-specific risk, and unique risk. It has a lot of names depending on what you've read before.

No. The idea of diversification is to reduce this.

No. This isn't a consequence of diversification. It's always the case.

Along with returns being modeled by a multifactor model and portfolios being well-diversified, there should probably be something about arbitrage in APT. In a model where returns are predicted, do you think arbitrage opportunities should exist?

No, actually they shouldn't.

You got it!

If riskless profits can be obtained, then the factors used to predict returns would be predicting different things at the same time. That wouldn't work.

With these three assumptions, expected returns are found with the risk-free rate (_R__{_F_}), the _K_ factors ( λ), and their estimated sensitivities (β). $$\displaystyle E[R_P] = R_F + \lambda_1 \beta_{P,1} + ... + \lambda_K \beta_{P,K} $$ How many risk factors? It's up to you. There's no right number, and APT doesn't number them or name them. It just basically sets up a multiple regression opportunity where you're finding a best-fit line. If you think about it this way, what's your intercept term?

No, look again. It's the only explanatory variable without a sensitivity beta.

Exactly. If all sensitivities are zero, you have the risk-free rate as the expected return. If not, just plug in all of the betas with the factor values, and you have your expected return. For example, a risk-free rate of 2.5%, with factor values of 0.9 and 5.4 and respective sensitivities of 0.11 and -0.003, would give you the following. $$\displaystyle E[R_P] = 0.025 + 0.9(0.11) + 5.4(-0.003) = 0.1078 = 10.78 \%$$ There's a wide selection of factors to choose from. Enjoy.

No. This is the dependent variable on the left side, not the intercept from the right side.

To summarize: [[summary]]

Only systematic risk would remain

Only asset-specific risk would remain

Total risk would be the sum of asset-specific risk and systematic risk

No

Yes

Zero

The risk-free rate

The expected return

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