TWR vs MWR

2024-10-04

Our two best friends, Alice and Bob, decided to invest money last year. Alice invested 200$ at the beginning of the year, whereas Bob invested 100$ at the beginning of the year and another 100$ halfway through the year. Both ended up with 400$ at the end of the year. Whose portfolio grew more? (i.e. who got luckier?)

This seems like a stupid question: they both invested 200$ and earned another 200$. When you think about it though, Bob invested his second 100$ for a shorter time period than Alice. If we assume that invested money on average grows at a constant rate, then Bob should have earned less than Alice, because his second 100$ were only invested for six months, compared to a whole year in Alice’s case. Since Bob’s returns were equal to Alice’s, his portfolio must have grown more.

In other words, if Bob invested for a shorter amount of time and earned the same as Alice, we could say that he got luckier (or invested in better assets). In finance, there are 2 ways of measuring this.

The Time-Weighted Return (TWR) for Alice and Bob is exactly the same:

TWR=B−AATWR = \frac{B - A}{A}

The Money-Weighted Return is where things start to differ. Here’s the modified Dietz formula for MWR:

MWR=B−A−FA+∑iWi⋅FiMWR = \frac{B - A - F}{A + \sum_i W_i \cdot F_i}

This time, A is the initial investment amount, which is 200$ for Alice and 100$ for Bob. B is the final amount, as before. F=∑iFiF = \sum_i F_i is the net of all cash flows (everything that was added minus everything that was taken out). WiW_i are weights representing what fraction of the time period FiF_i was invested for. In Alice’s case, F = 0, but in Bob’s case F1=100, W1=0.5F_1 = 100, \ W_1 = 0.5 , because he invested 100$ halfway through the year. So for Bob F = 100$ and ∑iFi⋅Wi=100⋅0.5=50$\sum_i F_i \cdot W_i = 100 \cdot 0.5 = 50\$ , whereas for Alice F=∑iFi⋅Wi=0F = \sum_i F_i \cdot W_i = 0 (both values are zero).

This means that for Alice, the TWR and MWR values are the same. For Bob, however, the MWR value is higher:

400−100−100100+3652⋅100≈1.33\frac{400 - 100 - 100}{100 + \frac{365}{2} \cdot 100} \approx 1.33

According to the MWR measurement, Bob got a 133% return on his investment. With this growth rate, if Bob had invested at the beginning of the year like Alice, by the end of the year he would have earned 200 x (1 + 1.33) = 466$.