Estimating Active Management Fees from Forecast Skill
Summary
The document asks how to apply a mean-variance framework for estimating the fee an active fund can charge above a passive index fund. The stated formula aggregates squared alpha-to-residual-risk ratios across securities and adjusts for investor risk aversion. The example supplies a universe of stocks, average forecast alpha, forecast accuracy, residual standard deviation, assets under management, and risk aversion, but the questioner is unsure how accuracy enters the calculation and how to interpret the sum.
The reply identifies a percentage conversion error: an accuracy of five percent should be represented as 5/100, rather than 5/10, when adjusting the alpha in the proposed calculation. The resulting fee rate is then multiplied by fund assets to obtain a dollar amount. This is a narrowly focused correction to the example, not a full derivation of the fee formula or a discussion of how forecast accuracy maps to realized alpha. The result depends on the assumptions and interpretation of the supplied inputs.
Key ideas
- The stated fee formula depends on alpha relative to residual risk and on investor risk aversion.
- Forecast accuracy expressed as five percent must be converted to 0.05 in the example calculation.
- The percentage fee rate is converted to a dollar fee by multiplying by assets under management.
- The reply corrects an arithmetic input but does not derive or validate the broader formula.
Tags
Full text
# Calculating Fees (Kane, Marcus, and Trippi)
# Calculating Fees (Kane, Marcus, and Trippi)
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 $\frac{5}{10}$?.
## Answer by Hellowoodman (score 1)
https://quant.stackexchange.com/a/18240
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.