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Section 5.3 Time-weighted annual rate of return

Suppose now that we make investments into a fund over time and know the outstanding balance befoer each deposit or withdrawl occurs. Notationally, let \(B_0\) be the initial balance in the fund and \(B_k\) the balance in the fund immediately before time \(t_k\) with \(W_k\) the amount of each deposit (if negative, then withdrawal). On a time line:

\begin{equation*} \begin{matrix} \text{Time} & 0 & t_1 & t_2 & \ldots & t_{n-1} & t_n \\ \hline \\ \text{Balance Before} & 0 & B_1 & B_2 & ... & B_{n-1} & B_n \\ \hline \\ \text{Deposit/Withdrawal} & 0 & W_1 & W_2 & ... & W_{n-1} & W_n \\ \hline \\ \text{Balance After} & B_0 & B_1 + W_1 & B_2 + W_2 & ... & B_{n-1} + W_{n-1} & B_n + W_n \\ \end{matrix} \end{equation*}

From the beginning of time, the growth over the first time period is given by \(B_1 = B_0(1+r_1)^{t_1}\) or

\begin{equation*} \frac{B_1}{B_0} = (1+r_1)^{t_1}\text{.} \end{equation*}

For subsequent time periods, the growth is given by \(B_{k+1} = B_k(1+r_2)^{t_{k+1}-t_k}\) or

\begin{equation*} \frac{B_{k+1}}{B_k} = (1+r_2)^{t_{k+1}-t_k}\text{.} \end{equation*}

Multiplying these together yields

\begin{align*} \frac{B_1}{B_0} \cdot \frac{B_2}{B_1} ... \cdot \frac{B_n}{B_{n-1}}\\ & = (1+r_1)^{t_1} \cdot (1+r_2)^{t_2-t_1} ... \cdot (1+r_n)^{t_n-t_{n-1}} \\ & = (1+r)^{t_n} \end{align*}

noting that \(r_k = r\) for all k would allow all of the powers to cancel except for the last. Define the time-weighted annual rate of return to be the value of r that makes this happen.

Example 5.3.2.

Consider the sequence of investments described in the table below:

\begin{equation*} \begin{matrix} \text{Time} & \text{11/1/16} & \text{3/1/17} & \text{8/1/17} & \text{2/1/18} & \text{4/1/18} \\ \hline \\ \text{Balance Before} & \$ 14516 & \$ 14547 & \$ 18351 & \$ 16969 & \$ 18542 \\ \hline \\ \text{Deposit/Withdrawal} & 0 & \$ 3000 & -\$ 2000 & \$ 2500 & 0 \\ \end{matrix} \end{equation*}

Then the time-weighted annual effective yield rate is given by

\begin{equation*} (1+r)^{17/12} = \frac{14547}{14516} \cdot \frac{18351}{14547+3000} \cdot \frac{16969}{18351-2000} \cdot \frac{18524}{16969+2500} \approx 1.035877 \end{equation*}

and by solving yields \(r \approx 0.025193371\)