Indigo Fund expected annual return: 10.5%, STD: 25.0%, SR: 0.30, active return: 1.2%, active risk.
The formula is STD (RA) = IR/SRB * STD (RB) = 0,15/0,33 * 18% = 8,11% optimal active risk. The weight on the active portfolio (Indigo) would be 8.11%/8.0% = 1.014 and the weight on the benchmark portfolio would be 1 – 1.014 = –0.014.
My question is how the STD for the portfolio can be 19,7% when the portfolio is 1,014 Indigo and -0,014 S&P 500, and taken into consideration that Indigo’s STD is as high as 25%? How come the portfolio’s STD is as low as 19,7% when more than 100% is invested in Indigo which has a STD of 25%?
Simple answer is because you are short the benchmark standard deviation of 18%…
Remember your optimal active weight was 101.35% in the active portfolio and short 1.35% of the benchmark portfolio. The math behind this works out as well if you calculate the SR for the optimal portfolio as SR^2= [SR^2bm + IR^2]^.5. Now that you know the sharpe ratio and you have the expected return of the optimal portfolio (given by 101.35% of the active return of 1.2% plus the benchmark return of 9%) you can backsolve for the Rf rate by taking the Sharpe ratio of the benchmark and the E® and St. Dev of the benchmark. It works out to 3%. Now taking the Sharpe ratio of the optimal portfolio which is .3655 and knowing the expected return of the optimal portfolio and the risk free rate solve for the St. Dev of the optimal portfolio - whalah it equals 19.743%
I hope you didn’t lose too much sleep over this one.
Thanks. I understand the logic and formulas behind it. Perhaps it is just the constructed example and the numbers in it that makes it a bit weird.The case is that if we invest 100% in only Indigo fund our STD will be 25% (given in the text). However, when we invest 1,014 in Indigo and -0,014 in the S&P 500 the STD changes to 19,7% which is a very big change. How come the new STD is not much closer to 25%?
Keep in mind you are not investing in the fund per se. You are investing in “The Fund 2.0.” Its a completely new portfolio offering a higher expected return and a reduced amount of risk - hence the name the new optimal portfolio. If you go through the math you will see that this relationship holds.
Also remember back to MPT and the standard deviation of the portfolio. It’s not a simple weighted average. There is some risk reduction going on here behind the scenes - I mean think about it if you are short the benchmark and long the active portfolio that is comparable to the benchmark one will be zigging and one will be zagging.