SOA sample question 48

[Samad3]

ds1/s1 = .06dt + .02dZ
ds2/s2 = .03dt + k * dZ

s1(0) = 100
s2(0) = 50

Stocks are non-dividend and r=.04.

Want to contruct zero-investment, risk-free portfolio with stocks and risk-free bonds. Have 1 share of Stock 1 in portfolio, how many shares of Stock 2 do we need to buy?


My approach was to solve this by using the abitrage strategy for sharpe ratios. (If share ratio 1>sharpe ratio 2, buy 1/S(1)*sigma1, sell 1/S(2)sigma2 and lend ...) I got the question wrong because I missed that selling -4 shares of stock 2 equals buying 4 shares.

The part I don't understand is I thought two perfectly correlated assets had the same sharpe ratio. (both have dZ so that means they are perfectly correlated?). It seems contradictory to solve this problem as an Arbitrage problem after we have already set the sharpe ratios equal to eachother to solve for k. How do we decide Stock 1 has the higher sharpe ratio, I thought they were equal?

[Abraham Weishaus]

I'm not sure what your problem is. The Sharpe ratios ARE equal, arbitrage is impossible, and therefore you immediately know that the risk-free portfolio you construct must earn 4%, which gives you the answer.

[Samad3]

I thought I was solving it correctly by doing the following:

Buy 1/(100*02) Shares of S1
=.5
But told we have 1 share of Stock 1 so scale this up
=.5 x (2) = 1

Need to sell 1/(50*-.01) Shares of S2
=-2
Scale up by same factor
=-2*2 = -4.

Sell -4 should mean buy 4 shares of S2. (There would also be an amount to lend)

I think this is the correct answer but I think my reasoning in using this approach is off. Doesn't SRatio #1 > SRatio #2 need to be valid to use this approach?

[Abraham Weishaus]

No, you're just saying the same thing. You've constructed a risk-free portfolio, and since the Sharpe ratios are equal, it will only earn the risk-free rate. The proof that the Sharpe ratios are equal is if they weren't, the portfolio you constructed would earn less than the risk-free rate, so you could sell it, invest the money risk-free, and have a guaranteed profit at no cost.