# MWRR

Diego invests in a mutual fund with the return of the fund over four years are mentioned underneath:

Year

1

2

3

4

New investment made at the start of the year

5000

3000

2000

0

Return during the year

45%

-20%

35%

-15%

Investor’s withdrawal at year-end

0

-800

-1000

-400

How is the **money-weighted rate of return** (or IRR) computed for this?

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whenever you have to solve any problem involving cashflows, i strongly recommend that you start by drawing a timeline, with the first time being time-0, followed by time-1, time-2, etc. then proceed by drawing arrows with information on the cashflows. this allows you to have a better intuition on the problem at hand.

for this problem, you are given the cashflows and the returns during the 1st, 2nd, 3rd and 4th year. you can use this information to calculate the amount in the mutual fund at time-4, right after the last cash outflow of $400.

i would recommend calculting this in steps.

you start with $5,000 at time-0. the rate of return for the first year is 45%, so at the beginning of time-1 (that is, at the beginning of the 2nd year),

you have $5,000 x 1.45 = $7250. you then deposit $3,000 and bring the account balance to $10,250. the rate of return for the second year is -20%. that means at time-2 (beginning of the third year) is $10,250 x (1 - .20) = $8200. you then deposit $1200 and bring the account balance to $9400. repeating these steps, and assuming i didn’t make a mistake, you should get $9,536.50 at the beginning of time-4, right after the last cash outflow of $400.

you are now equipped with all the info needed to calculate Rd, the money-weighted aka dollar-weighted rate of return.

Rd is the solution to the equation

B0 + {sum from k = 1 to n of Ck / (1+ Rd)^tk } - B1 / (1+ Rd)^m = 0

where B0 is the fund balance at time-0 right after any cashflow that happens that day

B1 is the fund balance at time m, right after the last cashflow that happens that day.

Ck is the cashflow occurring at time-k

for this problem,

B0 = 5,000

B1 = 9,536.50

C1 = 3,000 (the cashflow at time-1)

C2 = 1,200 (net of 2,000 inflow and 800 outflow)

C3 = -1,000

and the equation becomes

5,000 + [3,000 / (1+ Rd)] + [1200 / (1 + Rd)^2]

- [1,000 / (1 + Rd)^3] - [9,536.50 / (1 + Rd)^4] = 0

solving for Rd gives Rd = 4.161832838 = about 4.16%