Arbitrage with portfolios

CFA Level II
#1

Someone posted the below on a portfolio question on factors on CFAI, and I have no idea the answer (both in the real world or for the exam).

Is the factor (ROE) assumed to be the same across the world? I understand that if we were comparing portfolios in the same country and using GDP as a factor then it would influence them all the same. It seems unlikely that ROE will be consistent across all of those areas. It doesn’t give you an option to choose this but should we just always assume the factor will be the same for all portfolios?

Test question: is there arbitrage possibilities?

Portfolio** Expected Return **Factor Sensitivity Eurozone 11.9% 0.3 North America 10.7% 0.8 Pacific Rim 13.7% 0.5

#2

Looking at factor sensitivities, write the middle number as a linear combination of the other two (i.e., interpolate):

0.5 = w(0.3) + (1 − w)(0.8)

0.5 = 0.3_w_ + 0.8 − 0.8_w_

−0.3 = −0.5_w_

w = 0.3 / 0.5 = 0.6

1 − w = 0.4

Now figure out the return of the weighted portfolio:

0.6(11.9%) + 0.4(10.7%) = 11.42%

Because 13.7% > 11.42%, you want to be long Pacific Rim and short Eurozone and North America, the latter in the same proportions as your weights:

  • Long Pacific Rim, 100%
  • Short Eurozone, 60%
  • Short North America, 40%

Your net factor sensitivity is zero and you have a positive expected return.

#3

Thanks magician, that’s very useful.

Is my understanding correct: using the portfolio you created, ie with a net factor sensitivity of zero, you expect positive return regardless of what the actual value of ROE is?

#4

That’s correct: we have, in essence, removed the effect of ROE.

#5

Much thanks (and for all your other posts as well)

#6

You’re quite welcome.