# How to quantify the impact of management cost on return?

Suppose funds X and Y are the same but X has 0.25% higher management cost. Suppose we are analyzing a 2 year interval. The simple models with discrete/continuous interval -assumptions are not really right below. I will get more specific in the end.

1. discrete intervals 0.9975**2
2. continuous intervals $e^{0.0025*2}$
3. $\sum_{0}^{days} a_{i} 0.0025^\frac{t_{i}}{t_{total Days}}$, where $a_{i}$ is amount of capital invested.

This is not really right, as the prices change frequently. Let $P_{1}=P_{0}r_{1}$, $P_{N}=P_{N-1}r_{N}=P_{0} \prod_{0}^{t}r_{i}$ and now to charge the customer we could take charge with as small intervals as possible:

1. $C_{0} = 0$, $C_{1} = P_{1} pp_{1}$ and $C_{N} = P_{N} pp_{N}$,

where $pp_{N}$ is a charging function. It is a function because the economic cost to the investor is not really just day-adjusted cost but the foregone opportunity to invest into the fund. Now the sum of such costs differ with convexity. If we have a rising market, the accounting cost is less than the economic cost according to Jensen (note that the accounting cost is just taken from the end result while the economic cost of each single CF):

$EC = \sum^{n}_{i=1} p_{i} pp(x_{i}) \geq pp( \sum_{i=1}^{n} p_{i} x_{i}) = AC$,

It is opposite for the opposite convexity. Because of changing convexity, I have found it very hard to track the impact of cost on the return. Its impact does change over time as described.

How can I quantify the impact of management cost on return? What kind of approximations do you use and how do they differ with different convexity? It is very hard for me to separate the impact of management cost from other issues such as style-drifting. I need to find some proper ways with proper significance levels to investigate this issue. Any paper about this?

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## 1 Answer

Is the management fee deducted daily or annually? Or perhaps are you trying to quantify the difference between the two? If annually, are fees in this example deducted based on average daily balance, or ending balance?

I think you are also confusing yourself regarding economic vs. accounting cost. Look at how much money is left in the fund after fees under different scenarios versus how much the manager collected in fees.

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@hhh I'm trying to work out your problem but having difficulty with all the notation. Can you clarify the notation a bit? I assume $C_i$ is the cost (management fee). What is the charging function? What is $x_i$, the argument of this function? Why is the 2-year interval relevant? –  Tal Fishman Aug 1 '11 at 20:37
@hhh Also, since you ask for references, do you have any references yourself that outline the problem itself, or is this something you are exploring on your own? –  Tal Fishman Aug 1 '11 at 20:51
@hhh I haven't been able to find any references, so links would be helpful. Working out some of the math on my own, it would seem that if the charging function is affine (which it usually is) and the price process has zero drift (which is close to true for the horizons involved), then the economic and accounting cost are equal. –  Tal Fishman Aug 2 '11 at 16:09
sheegaon: Suppose you have 1USD for 50 years. It would have produced $1*e^{0.14*50}$ in a fund if you had not lost it. The AC is just "1USD" while the EC is $e^{0.14*50}$. AC and EC are rarely the same unless short-term. In short-term, the fund could produce -14% every year so $AC > EC$. Sorry you should read some introduction Economics books if you cannot understand this, EC is the foregone possibility and AC is just the accounting cost. Very different issues! –  hhh Aug 2 '11 at 19:01
...I am pretty sure this issue is some very basic problem in some field, not sure whether in Econometrics (because this issue certainly requires some understanding from Economics and related to even understand it). It would be more useful if you left this answer as a comment. I am trying to quantify this issue and it does contain different pricing schemes such as annual or daily. –  hhh Aug 2 '11 at 19:05