Binati, I think that there are a couple of issues with your solution.
Firstly, you seem to assume that the progression of management fee and outperformance fee grows annually by a factor of 1.182141. This was the accumulation factor applicable to fund value without fees, but fees charged scales this down. The accumulation factor for fund value net of fees is
1.18214 – 0.02 – 0.2*(0.18214 – 0.02) = 1.129712
Since both of management fee and outperformance fee are proportional to the fund value they will also grow annually by this factor.
Secondly, you seem to assume that the following equality holds:
Fund value (no fees assumed) = Fund value (annual fees of ‘2 and 20’ assumed) + sum of management fees + sum of outperformance fees.
This however does not hold because we would ignore time value of money. Arguably a small fee charged in the first years would grow to much larger value if it was not subtracted and could grow at fund’s rate of return. Going ad absurdum, if the only fee was $1000 at term 0, then the terminal value of the fund would not be $4,300,000 - $1,000 = $4,299,000, but it would obviously be $0 instead. In this case, we should accumulate or discount all values to the same term and the correct rate we should use to calculate accumulated values at term 50 is fund’s gross annual growth rate, 0.18214. Therefore, if we wanted to fix the equality above, we should use
Fund value (no fees assumed) at term 50 = Fund value (annual fees of ‘2 and 20’ assumed) at term 50+ sum of management fees accumulated to term 50 + sum of outperformance fees accumulated to term 50.
Thirdly, the small difference between your solution and examiner’s report solution seems to be that examiner’s report assumes that the fee is applied as a percentage of the start-of-period fund value, whereas when you are calculating management fee for the first year you are assuming that the fee is applied as a percentage of the end-of-period fund value. However, in the calculation of sum of fees you seem to also assume that the fee is applied as a percentage of the start-of-period fund value, so I am bit unsure about your assumption here. If we adjusted examiner’s report solution so that the fee is applied to the end-of-period value, then the accumulated fund would become:
Accumulated fund = 1000*[ 1.18214 * (1 – 0.02 – 0.2*(0.18214 – 0.02)) ]^50 = $291,110
Now if we follow your approach adjusted for issues/differences described above, we get:
Management fee is charged annually, as 2% of fund value at the beginning of the year.
Therefore, first year = 2% x $1000 = $20
Accumulated value of management fee paid in first year is $20*1.18214^49 = $72,746.5986
Second year = 2% x $1000*1.129712 = $22.59
Accumulated value of management fee paid in first year is $20*1.129712*1.18214^48= $69,519
And so on till the 50th year.
To calculate the accumulated value of management fee paid, we use the formula:
sum of a geometric progression (Sn) = a((r^n-1)/(r-1))
Where a = first term of GP
r = (1 + net annual return) / (1+gross annual return)
using the formula,
Accumulated value of Management fee = $72,746.5986 x ((1.129712/1.182141)^50-1)/( 1.129712/1.182141-1))
= $72,746.5986 x 20.2138943957
= $1,470,492.06
Outperformance = 18.2141% - 2% = 16.2141%
To calculate the accumulated value of outperformance fee, using similar formula
a = 20%*1000*0.162141*1.18214^49
r = (1 + net annual return) / (1+gross annual return) =1.129712/1.182141
accumulated value of outperformance fee = 20%*1000*0.162141*1.18214^49 x ((1.129712/1.182141)^50-1)/( 1.129712/1.182141-1))
= $2,384,237.76
So, accumulated value of fee = accumulated value of management fee + accumulated value of outperformance fee= $3,854,729.82
Accumulate value of fund net fees = $4,300,000 - $3,854,729.82
= $445,270.18
This is consistent with examiner’s report up to the small difference in hundreds of dollars due to rounding error in growth rate. More specifically, we had minor inconsistency between the following three used results - that the Fund value (no fees assumed) at term 50 is $4,300,000, that the fund value at term 0 is $1,000 and that the fund’s gross annual growth rate 0.18214. In fact, if we accumulate $1,000 to term 50 at rate 0.18214 we get $1,000 * 1.18214^50 = $4,299,744 which differs from $4,300,000 assumed by a few hundreds (unsurprisingly, the same order of difference as we observed above).
Note that even though I managed to ‘fix’ your solution to obtain the same result as is presented in examiner’s report, I would not recommend this approach simply because it is unnecessarily much more laborious than the approach used in the examiner’s report.
Hope this helps.
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