 # Multifactor models question

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Based on the answer, it is long 30% A, long 70% B and short 100% C. While I can understand this, I am puzzled as to how these numbers came about. Would it be just purely trial and error? If so, wouldn’t this take up a huge amount of time? Any shortcuts here?

Thank you.

All they’re doing is creating the middle factor sensitivity from a weighted average of the high factor sensitivity and the low factor sensitivity. So:

wA(1.20) + wB(2.00) = 1.76

wA + wB = 1

Then,

wB = 1 − wA

wA(1.20) + (1 − wA)(2.00) = 1.76

1.20wA − 2.00 wA + 2.00 = 1.76

−0.80wA = 1.76 – 2.00 = −0.24

wA = −0.24 / −0.80 = 0.30

wB = 1 − wA = wB = 1 – 0.30 = 0.70

Then it’s just a matter of determining whether you want to be long 30% A and 70% B and short 100% C or the other way round. You check your expected return if you’re long A, long B, and short C:

0.30(10%) + 0.70(20%) – 1.00(13%) = 4%.

Therefore, you want to be long 30% A, long 30% B, and short 100% C.

(Note, by the way, that if you know you’re going to be short only one stock, then it has to be the one with the middle sensitivity, as here. If you know that you’re going to be short two stocks, then it has to be the other two. You really don’t have to do the calculations.)

Thanks a lot S2000magician!

My pleasure.

Can someone explain me the logic behind these types of questions.

I get that with APT we are trying to determine arbitrage opportunities. However, I do not get how we can determine if there is an arbitrage when we are given information to three portfolios with their corresponding betas and expected returns. Why do we create a forth portfolio and why do we choose the middle value of beta to create it?

hey onda, i have the same question as you did. did you solve the problem yet?

hi Magician, can you please explain the logic behind it? how can i tell which portfolio to short or long at the first place?

Did you go through my calculations, above?

You create a fourth portfolio so that you have two (the new one and the existing, middle one) with the same factor exposure, so that you can be long one portfolio and short the other and have zero factor exposure.

You determine whether or not there’s an opportunity by comparing the returns on those two portfolios: if they’re equal then there’s no arbitrage opportunity, and if they’re not, then there is. Buy the portfolio with the higher return, sell the portfolio with the lower return.