Simple portfolio optimization

dirtdog · August 30th, 2013, 3:11 pm

I know this is a simple question, but I am unable to come up with an answer - can someone push me in the right direction?What is the maximum return of the following portfolio?Stock 1 return 15% volatility 15%Stock 2 return 30% volatility 30%I am guessing if you allocate all your funds to stock 2 the max is then 30%?Also, what is the minimum volatility you can achieve?If you don't allocate any funds to any of the stocks, the volatility is 0 -but that probably not the answer.Thanks,dirtdog

acastaldo · August 30th, 2013, 6:11 pm

Let w be the fraction invested in the 1st stock, 1-w the fraction invested in the second stockwhere 0<= w <=1Then the expected return on the portfolio is 0.15*w + 0.30*(1-w) = 0.30-0.15*wif you draw this you get a falling straight line segment from (0,0.3) to (1,0.15)Clearly your guess is correct: the best return is for w=0, i.e. everything invested in the second stockNow write the equation for the variance of the portfolio...

ppauper · August 30th, 2013, 7:52 pm

you need the covariance [$]\rho_{12}[$],portfolio variance[$]\sigma_p^2=w_1^2\sigma_1^2+w_2^2\sigma_2^2+2w_1 w_2\sigma_1\sigma_2\rho_{12}[$][$]w_1=w[$], [$]w_2=1-w[$], and minimize [$]\sigma_p^2[$] with respect to [$]w[$]are you restricted to [$]0\le w\le 1 [$] or are you allowed to short one of the stocks ?

dirtdog · August 31st, 2013, 5:59 am

Thanks for your help guys. I arrived at the same conclusions, but I was fearing that I was missing something obvious when the exercise didn't state a cross correlation.