Sharpe Ratio of a combined portfolio with negative information ratio of the active portfolio.

[vsenthil]

I am bit confused about the Shrpe Ratio of a managed portfolio. Based in the reading 54, Sharpe Ratio(Portfolio)^2 = Sharpe Ratio(Benchmark)^ + IR^2. Now let us say I got two managed portfolios, portfolio A with IR -0.25 and portfolio B with IR 0.05 and the benchmark’s sharpe ratio is 0.47. Now combining benchmark with portfolio A
Sharpe Ratio (Portfolio) ^ 2 = 0.47^2 + (-0.25)^2 = 0.5325
Sharpe Ratio (Portfolio) ^2 = 0.47^2 + 0.05^2 = 0.2234
What I am I missing? How is this possible? Fund A has lower Information ratio than fund B, how can it give an higher Sharpe Ratio for the portfolio. This does not make sense. Anybody can explain?

 

[Gigaloo]

The equation is for the highest optimal Sharpe Ratio. I think this might be because the IR you are using might not be the optimal IR for the repective portfolios.

 

[bfry]

Where did you see that example? Remember the IR that we are slicing and dicing into a thousand different pieces is based on EX-ANTE returns. What manager out there would project he has a negative EX-ANTE IR? If so he is just begging to be fired I guess - or at least setting the bar so low that ex-post he will look good maybe
I think that is the source of your confusion and yes Gigaloo is correct - those formulas are for the sharpe ratio for the optimal portfolio which again would never be negative. This is one of the major flaws with the fundamental law of active management; its all based on ex-ante information. If you go through all the mathematics you will see that that relationship holds. I suggest not doing that though…..