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.
- discrete intervals
0.9975**2 - continuous intervals $e^{0.0025*2}$
- $\sum_{0}^{days} a_{i} 0.0025^\frac{t_{i}}{t_{total Days}}$, 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:
- $C_{0} = 0$, $C_{1} = P_{1} pp_{1}$ 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?
