24 Dollar-Weighted and Time-Weighted Rate of Return
24.1 Dollar-Weighted Rate of Return
The dollar-weighted rate of return is the internal rate of return for the fund. But usually most people refer the concept to the simple interest approximation of the internal rate of return.
Example: A pension fund receives contributions and pays benefits from time to time. The fund began the year 2018 with a balance of \(\$1,000,000\). There were contributions ot the fund of \(\$200,000\) at the end of February and again at the end of August. There was a benefit of \(\$500,000\) paid out of the fund at the end of October. The balance remaining in the fund at the start of the year 2016 was \(\$1,100,000\). Find the dollar-weighted return on the fund, assuming each month is \(1/12\) of a year.
Solution: The equation of value for the dollar-weighted return is:
\[\begin{align}1,000,000(1+i_D) &+ 200,000 \bigg(1+\frac{10}{12}i_D\bigg) \\&+ 200,000 \bigg(1+\frac{4}{12}i_D\bigg) - 500,000 \bigg(1+\frac{2}{12}i_D\bigg) =1,100,000\end{align}\] Solving for \(i_D\), we get:
\[\begin{align} i_D &= \frac{1,100,000 - 1,000,000 - 200,000 - 200,000 + 500,000}{1,000,000 + 200,000 \times \frac{10}{12}+ 200,000 \times \frac{4}{12} - 500,000 \times \frac{2}{12}}\\ &= \frac{200,000}{1,150,000} = .1739 = 17.39\%\end{align}\]
24.1.1 Formula
The dollar-weighted rate of return during a year is determined by:
\[\begin{align} i_{D} &= \frac{\text{Net Interest}}{\text{Fund Exposure}} =\frac{\text{Withdrawals - Deposits}}{ \sum \text{(Net deposit)(Time deposit is in the fund)}} \\&=\frac{B - (A + \sum_{k=1}^n C_k)}{A + \sum_{k=1}^n C_k(1-t_k)}\end{align}\] where:
- \(C_k\) is the net deposit at time \(t_k\)
- \(A\) is the balance in the fund at the start of the year
- \(B\) is the balance in the fund at the end of the year.
24.2 Time-Weighted Rate of Return
The time-weighted rate of return for a one year period is found by compounding the returns over successive parts of the year.
24.2.1 Formula
The time-weighted rate of return during a year is determined by: \[1+i_T = \frac{F_1}{A}\times \frac{F_2}{F_1 + C_1}\times\frac{F_3}{F_2 + C_2} ...\times \frac{B}{F_{n-1} + C_n}\] where
- \(C_k\) is the net deposit at time \(t_k\)
- \(F_k\) is the value of the fund just before the net deposit at time \(t_k\)
- \(A\) is the balance in the fund at the start of the year
- \(B\) is the balance in the fund at the end of the year
Example: A pension fund receives contributions and pays benefits from time to time. The fund value is reported after every transaction and at year end. The details during the year 2018 are as folows:
| Date | Deposit Amount | Fund Value after Deposit |
|---|---|---|
| 01 Jan 2018 | 1,000,000 | |
| 28 Feb 2018 | 200,000 | 1,240,000 |
| 31 Aug 2018 | 200,000 | 1,600,000 |
| 31 Oct 2018 | -500,000 | 1,080,000 |
| 31 Dec 2018 | 200,000 | 900,000 |
Calculate the time-weighted rate of return.
Solution: The time-weighted rate of return for 2018 is: \[\begin{align}i_T &= \bigg(\frac{1,040,000}{1,000,000}\bigg)\bigg(\frac{1,580,000}{1,240,000}\bigg)\bigg(\frac{1,130,000}{1,600,000}\bigg)\bigg(\frac{1,100,000}{1,080,000}\bigg) - 1 \\ &= .1809 = 18.09\%\end{align}\]