Dollar-Weighted Rate of Return

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Section 5.2 Dollar-Weighted Rate of Return

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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

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

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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

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

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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"

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

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For these,

$V_0$ can be thought of as the initial valance in the account and

$C_k$ as deposits made at times

$t_k$

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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

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

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pretty_print('Dollar Weighted Annual Rate of Return')
c=1
@interact
def _(n = slider(1,10,1,3,label='Number of Investment Changes')):
    @interact(layout=dict(top=[['M','FV'],['endtime']]))
    def __(M=('Investment / Time ', input_grid(2, n, default = [[25200,500,-1000,0,0,0,0,0,0,0,0],[0,3/12,18/12,2,3,4,5,6,7,8]])),
           FV = input_box(25900,label='Final Value',width=15),
           endtime = input_box(21/12,label='Final time (yrs)',width=8)):
        pretty_print(html('State of Investments'))
        
        Inv = M[0][0:n]
        times = M[1][0:n]
        A = matrix(QQ,[Inv,times])
        show(A)
        top = FV - Inv[0]
        bot = Inv[0]*endtime
        for k in range(1,n):
            top = top - Inv[k]
            bot = bot + Inv[k]*(endtime-times[k])
 
        r = (top/bot).n()
        pretty_print(html('The Dollar-Weighted Rate of Return = %1.6f'%r))