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Having read a chapter in Bodie, Kane and Marcus' Investment, I came across a formula I do not quite understand. It states that the percentage fee in excess of what an index fund would charge on active management of an optimal portfolio is given by

$$f = \frac{1}{2A}\sum_{i=1}^{n} \left[\frac{\alpha_{i}}{\sigma(e_{i})}\right]^2.$$

When applying the formula to the following question, I can not seem to get the correct answer:

One mutual fund has a team of analysts that performs security analysis. They are able to produce forecasts of annual alphas of 1%, on average, in a universe of 100 stocks with an accuracy of 5% (measured in terms of r-square). The standard deviation of the residuals is 6%. The assets under management of the fund are $50,000,000. What is the amount of fees the fund can charge from a mean-variant investor with a risk aversion of 3 (in excess of a passive index fund)?

A few things I can immediately deduce is the value of $A$. It then becomes quite ambiguous. I am assuming that the $i$'s of the alpha's and standard deviation of residuals correspond to stocks in the active portfolio. However, what am I exactly summing here? I only have one value for alpha, and it seems that it is representative of the whole universe of stocks. Any help would be greatly appreciated. It seems that the answer is "$57,870,37". As you can see, there is a typo in the answer which makes matters worse.

Thank you to all in advanced for you help.

Gus.

EDIT

After a tiring amount of trial and error I have come to a solution I do not understand.

$$f=\frac{1}{2\cdot3}\cdot 100 \cdot \left[\frac{1\%\cdot \frac{5}{10}}{6\%}\right]^{2}.$$

Why multiply the alpha by $\frac{5}{10}$?.

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It is supposed to be multiplied by 5/100 (5%). You should then be able to get $57,870.37 if you multiply it by the fund value.

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  • $\begingroup$ Yes, that's correct. Alpha's have to be multiplied by the $R^{2}$. If you could elaborate on your solution (for future viewers) I will gladly give you the tick $\endgroup$ – Gustavo Louis G. Montańo Jun 12 '15 at 11:35

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