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T1 CH6 question on APT

ktrathen

Member
Subscriber
Hi,

This is my first post on this forum. I have a question about the following definition:

The basic idea of APT is that investors can create a zero-beta portfolio with zero net investment.

What is meant by zero net investment in this context?

Thanks!
 

David Harper CFA FRM

David Harper CFA FRM
Staff member
Subscriber
Hi @ktrathen Welcome! To me, that is merely a premise of the APT. I don't know about you, but I need examples. Using the CAPM to illustrate (as a special case of one factor) where E(r) = Rf + (Rm - Rf)*β, assume I told you that we can select among the following securities:
  • Security A has β(A,M) = 1.0 and expected return, Er(A) = 9.0%
  • Security B has β(B,M) = 1.25 and expected return, Er(A) = 10.50%
  • Security C has β(C,M) = 2.0 and expected return, Er(A) = 15.0%
If you buy $200.00 of A, then your portfolio beta is 1.0. How can you create a zero-beta portfolio? Since C has a beta of 2.0, you can short $100.0 of Security C. Now have you a zero-beta portfolio as given by (200% * +1.0) + (-100% * -2.0) = 0. The expected return is ($200 * 1.09) - ($100 * 1.15) = $18.00 - $15.00 = $3.00. That is a $3.00 / $100.0 = +3.00% return on your zero-beta portfolio. As a zero-beta portfolio, it has no risk, so its 3.0% should be the risk-free rate. And in fact, 3.0% must be the risk-free rate, if we solve for two unknowns:

C: Rf + 2.0*ERP = 15.0%
A: Rf + 1.0*ERP = 9.0%

... subtracting the second from the first, we see that ERP = 6.0% and Rf = 3.0%. By design, my three securities all lie on the SML. There exists alternative zero-beta portfolios. My portfolio is not a net-zero investment (it requires $100.0). But I can reduce the net investment (eg) if instead buy $200.00 of Portfolio A and short $200/1.25 = $160 of Portfolio B. This also gives me a zero-beta portfolio but the net investment is now only $200 - 160 = $40, and the expected return is still the risk-free rate (we expect +$1.20 on net $40.0 investment, or +3.0%). Any zero-beta portfolio should return the risk-free rate. I can keep going and get closer to a zero-beta portfolio that requires zero net investment.

My portfolios are all on the SML but if Security C instead has an expected return of 14.0%, then my first zero-beta portfolio would have an expected return of +4.00%, or +1.0% excess return. This would allow your or my to exploit with an arbitrage, in theory forcing the price of Security C to "get back on the SML line" (it begs be shorted more until its price is reduced and its expected return increases to 15% to match its beta) .

The zero-beta portfolio replaces the risk-free rate (and liberates APT relative to CAPM) because, if the factors fully explain excess returns, the zero-beta portfolio should return risk-free rate. So, to me, the more important assumption is that a zero-beta portfolio enforces prices. If you can construct a zero-beta portfolio by investing $100 (as in my first example), you can conduct an arbitrage. But that still requires funds. If we add the assumption that you can construct a zero-beta portfolio with zero net investment, then suddenly anybody/everybody can do that: nothing is required to exploit an arbitrage made possible wherever securities do not lie "on the line;" i.e., where they are too cheap or expensive for their beta. Both assumptions enable a premise that lets us conclude the APT should be valid given our assumptions (i.e., factors and their betas capture the risk/reward; all the idiosyncratic risk is diversified away; and arbitrage eliminates any discrepancies ... precisely because all participants can conduct arbitrage with zero-beta no-investment-required portfolios). I hope that's helpful!
 
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