Time-Weighted Yield¶

The time-weighted yield is a measure of how well a fund was managed. Mathematically, it is defined as:

$j_tw = \left[\prod_{k=1}^{r+1} (1 + j_k) \right] - 1,$

effective over the investment interval, where

$1 + j_k = \begin{cases} \frac{B_{t_1}}{B_0} & k = 1\\ \frac{B_{t_k}}{B_{t_{k-1}} + C_{t_{k-1}}} & k = 2, 3, \cdots, r+1 \end{cases}.$

The Bs represent the balances at points in time and the Cs represent the contributions. If we desire an annualized yield:

$i_{tw} = (1 + j_{tw})^{\frac{1}{T}} -1 = \left[ \prod_{k=1}^{r+1} (1 + j_k)\right]^{\frac{1}{T}} - 1$

Examples¶

Suppose we deposit 100,000 in a bank account. It grows to 105,000 at time 1, and we immediately deposit an additional 5,000. It then grows to 115,000 at time 2. what is the time-weighted yield?

We can solve this problem by passing the balance amounts and times to the time_weighted_yield method of the Payments() class:

In [1]: from tmval import Payments

In [2]: pmts = Payments(
   ...:    amounts=[100000, 5000],
   ...:    times=[0, 1]
   ...: )
   ...: 

In [3]: i = pmts.time_weighted_yield(
   ...:    balance_times=[0, 1, 2],
   ...:    balance_amounts=[100000, 105000, 115000],
   ...:    annual=True
   ...: )
   ...: 

In [4]: print(i)
Pattern: Effective Interest
Rate: 0.0477248077273309
Unit of time: 1 year