Hi all,

I just went through Assignment X6 model solution, part (i) and I was surprised to see how the performance fee was calculated there.

When I worked out the question, I thought that the performance fee is based on excess return that the manager delivered compared to the benchmark. In the question, we started with the fund value £200m, the benchmark return over the first period was 5%, the fund return was 8% and the performance fee was 10%. Thus, thanks to the manager’s ability we are better off £6m, because the end-of-period fund value is £216m and it would have been £210m if the benchmark return was achieved. I therefore thought that the performance fee should be £600k.

However, according to the model solution the approach above is wrong and the correct answer is £648k. It says that normal way how the performance fees are calculated is to take the end-of-period fund value and calculate the outperformance as the excess return applied to end-of-period fund value. In this case this means that performance fee is 10% of (0.08-0.05) × £216m, thus £648k.

I don’t know how these fees are calculated in practice and I don’t want to argue with the claim that this second approach is the most widespread. However, I would like to point out on the inconsistency present in this second approach, which I will firstly present on a simple example.

Imagine for a moment that the fund in question was some crazy hedge fund aiming for the extreme returns using some highly leveraged (almost casino-like) strategies. Let us start with the fund value £1m, the benchmark return over the first period 0%, the fund return 500% and the performance fee of 20%. Using my approach, we would calculate that we are better off £5m, and thus we can calculate the performance fee to be £1m. However, if we use the approach of the model solution, then we apply excess return (which is 500%) to the end-of-period fund value, £6m. In this case this means that performance fee is 20% of (5-0) × £6m, thus £6m. The whole fund’s value is thus used as the fee to the fund manager and investors lose all their money (if there is also a flat fee component, the investors will be left with a negative balance and will owe another money to the fund manager).

Although the example speaks for itself, I would like to also explain in pure mathematical terms why I consider the first approach to be mathematically robust (unlike the second approach). Let the initial fund value be L, the benchmark return over the first period rb, the fund return over the first period rf (rf > rb) and the performance fee p. The end-of-period fund value (before fees) is (1+rf) × L and would be (1+rb)× L if the benchmark return was achieved. Using my approach, the outperformance in absolute terms is (rf - rb) × L and the performance fee (rf - rb)× L × p. The end-of-period fund value (after fees) would thus be

(1+rf) × L - (rf - rb)× L × p = L × [1 + (1 – p) × rf + p × rb].

This is a very nice mathematical expression, because achieved return (after performance fee) is the weighted average of fund return (before performance fee) and the benchmark return with weights used 1 – p and p respectively. This corresponds to my expectation of how the performance fee should affect the net return. However, using the normal way (as per model solution) we get that the performance fee equals p × (rf – rb)× (1+rf)× L. The end-of-period fund value (after fees) would thus be

(1+rf) × L - p × (rf – rb)× (1+rf)× L = L × [1 + (1 – p) × rf + p × rb - p × rf × (rf – rb)].

The achieved return (after performance fee) thus contains a second-order term p × rf × (rf – rb). This term does not fit well with the result for me and can create awkwardness, such as in the example above.