Identify Arbitrage strategy from list of portfolio

CFA Level II Portfolio Management
#1

The Schweser Notes, reading 53, SS 18 has a question in concept checker which gave the details of three portfolio and asked to identify arbitrage strategy

Portfolio Expected Return Factor Sensitivity

A 10% 1.2

B 20% 2.0

C 13% 1.76

Question they have asked is which portfolio will involve short position in executing a arbitrage strategy

My doubts are

  1. Unless the one factor model eqution is not given, does it make any sense of this question? Shouldn’t it be given in question?

  2. Although using any of the two portfolio we can calculate risk free rate and premium but when it is given that one of the portfolio is not diversified so will it make any sense for us first calculate one factor equation?

  3. Assuming that equation is given, is there any logical technique to find the arbitrage portfolio in one step? I used to do some trial and error and then find the best combination. I appreciate if someone can share the sure shot method that works in one step.

#2

Here’s the approach:

  • Portfolio C’s factor sensitivity is 70% of the way from Portfolio A’s to Portfolio B’s ((1.76 − 1.2) / (2.0 − 1.2) = 0.7).
  • You can duplicate Portfolio C’s factor sensitivity with 30% of Portfolio A and 70% of Portfolio B (0.3(1.2) + 0.7(2.0) = 1.76).
  • If you duplicate Portfolio C’s factor sensitivity with 30% of Portfolio A and 70% of Portfolio B, the return on this new portfolio will be 0.3(10%) + 0.7(20%) = 17%.
  • Because 17% > 13% (the return on Portfolio C), you want to be long 30% A and 70% B, and short 100% C.

In general, take the middle factor sensitivity (here, 1.2 < 1.76 < 2.0, so choose 1.76) and duplicate it with the other two factor sensitivities. Then follow the steps above. Note that if the return on the combined portfolio were less than that of Portfolio C, you would be long C and short both A and B.

#3

Thanks S2000. So you mean to say

  1. We should not try to calculate one factor equation when ask is to determine the arbitrage portfolio?

  2. And does that also mean that if we are asked to find risk free and risk premium then assumption is that in given portfolios the arbitrage strategy won’t exist?

#4

You’re welcome.

I wouldn’t. I’d go through the process I outlined above.

I believe so, but I’d have to think on that one a little while.

#5

Alright. I will wait for your inputs on this.

#6

When asked to calculate the weights for the long and short positions in a portfolio to generate arbitrage profit, is there a simple way of working this out? I know it some form of simple algebra but i cant get my head around it…

#7

The four step approach mentioned by S2000 is good way to determine weights. Try and let us know if these steps doesn’t help in coming up answer. I tried on two questions and it worked

#8

I tried it and it works - the explanation in the cfa curriculum is very different to S2000magician’s…

#9

Mathematicians are the laziest people on earth.

We’d rather solve a general problem once and be done with it (e.g., the quadratic formula) than risk having to solve two or more similar problems.

The solution I gave is general: it always works.