Question

Course: Theory of Interest (Actuarial Science)
Chapter: Yield Rates

Problem:

On 1/1/2006, Anthony deposits $90 into an investment account. On 4/1/2006 when the amount in the account is $x, a withdrawal of $W is made. The dollar-weighted rate of return is 20%. The time-weighted rate of return is 16%. Find W and x.
Answer: W = 24, x = $104.4

Answer 1

Invested amount on 1/1/2006: $90

Principal + Interest on 4/1/2006: $X

Here we have only one period of 4 months from 1st January to 1st April.

Time-Weighted return is the holding period return for 4 months. From the question we know that Time Weighted return is 16%

Hence,

($X - $W- $90)/ ($90) = 16%

Solving this equation, we get $X - $W = $104.40

For Dollar Weighted Return, we need to compute IRR as it takes timing of cash flows into account.

Cash Flows with dates

1/1/2006: -$90 (Negative sign denotes that the money was invested)

4/1/2006: $X (As this is the amount which is in the account prior to withdrawal of $W)

So, the calculation the value of $X using the IRR of 20%, we get $X as $108

(I used Solver to find the solution by setting the objective of getting IRR = 20% and by changing the value of cash flow on 4/1/2006)

So, we can see that $X = $108 and putting this value in the equation, we can find the value of $W = $3.60