Time-weighted annual rate of return

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:

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

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Theorem 5.3.1. Time Weighted Annual Rate of Return Formula.

The "time-weighted annual rate of return" r is the solution of

$\begin{equation*} (1+r)^{t_n} = \frac{B_1}{B_0} \cdot \frac{B_2}{B_1+W_1} \cdot \ldots \cdot \frac{B_n}{B_{n-1}+W_{n-1}} \end{equation*}$

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Proof.

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.

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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 $r \approx 0.025193371$

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pretty_print('Time Weighted Annual Rate of Return')
​
@interact
def _(n = slider(2,10,1,5,label='Number of Investment Changes')):
    head = 'Times  BalanceBefore  Dep/Withdraw'
    @interact
    def __(M=(head, input_grid(3, n, default = [[0,4/12,9/12,15/12,17/12,4,5,6,7,8,9],[14516,14547,18351,16969,18542,1,1,1,1,1,1],[0,3000,-2000,2500,0,0,0,0,0,0,0]]))):
        pretty_print('State of Investments')
        times = M[0][0:n]
        BalBefore = M[1][0:n]
        Inv = M[2][0:n]
        A = matrix(QQ,[times,BalBefore,Inv])
        show(A)
        s=1
        for k in range(n-1):
            s=s*BalBefore[k+1]/(Inv[k]+BalBefore[k]) 
​
        r = (e^(ln(s)/times[n-1])-1).n()
        pretty_print(html('The Dollar-Weighted Rate of Return = %1.6f'%r))