Dollar-Weighted Rate of Return

Section 5.2 Dollar-Weighted Rate of Return

In general, for a sequence of investments starting with an initial investment of $V_0$ at time 0 and subsequent investments (i.e. cash flows) of value $C_1, C_2, ... , C_n$ at times $t_1, t_2, ... , t_n$ then the future value of these at time t is given by

\begin{equation*} A = V_0(1+r)^t + \sum_{k=1}^n C_k(1+r)^{t-t_k}\text{.} \end{equation*}

One can approximate this equation with an easier one using "linearization" where one replaces $(1+r)^t$ with it's first-order Taylor approximation $1+rt\text{.}$ This gives

\begin{equation*} A = V_0(1+rt) + \sum_{k=1}^n C_k(1+r(t-t_k)) \end{equation*}

which is the future value of the investments if only simple interest were used. Solving for the interest rate yields the "dollar weighted rate of return"

\begin{equation*} r = \frac{A - V_0 - \sum_{k=1}^n C_k}{V_0 t + \sum_{k=1}^n (t-t_k)C_k}\text{.} \end{equation*}

For these, $V_0$ can be thought of as the initial valance in the account and $C_k$ as deposits made at times $t_k$

Example 5.2.1. Dollar Weighted Rate of Return.

On January 1, 2000, the balance in an account is $\$ 25200\text{.}$ On April 1, 2000, $\$500$ is deposited in this account and on July 1, 2001, a withdrawl of $\$ 1000$ is made. The balance in the account on October 1, 2001 is $\$ 25900\text{.}$ Determine the annual rate of ionterest in this account according to the dollar-weighted method.

So, the cashflow has the following:

\begin{equation*} \begin{matrix} \text{Investment} & 25200 & 500 & -1000 \\ \hline \\ \text{Time} & 0 & 3/12 & 18/12 \end{matrix} \end{equation*}

with a future value of $A = \$ 25900\text{.}$ Therefore, the annual dollar-weighted rate of interest is

\begin{equation*} r = \frac{25900 - 25200 - 500 + 1000}{25200 (21/12) + 500 (18/12) - 1000 (3/12)} \approx 0.026906\text{.} \end{equation*}