Question

On January 1, a fund is worth 100,000. On May 1, the value has increased to 120,000 and then 30,000 of new principal is deposited. On July 1, the value has increased to 200,000 and then 30,000 of new principal is deposited. On November 1, the value has declined to 130,000 and then 50,000 is withdrawn. On January 1 of the following year, the fund is again worth 100,000. Calculate the dollar-weighted rate of return for the year.

Answer 1

Let the dollar weighted return be r% per months i.e. r*12% monthly compounded rate p.a.

Then the fund is equivalent to a project where

$100000 is invested on January 1 (t=0)

$30000 is invested on May 1 (t=4)

$30000 is invested on July 1 (t=6)

$50000 is withdrawn on November 1 (t=10)

$100000 is withdrawn on next year Janaury 1 (t=12)

So, r is given as

-100000/(1+r)^0 -30000/(1+r)^4-30000/(1+r)^6+50000/(1+r)^10+100000/(1+r)^12 = 0

=> 50000/(1+r)^10+100000/(1+r)^12 - (100000 +30000/(1+r)^4+ 30000/(1+r)^6) = 0

Using hit and trial method

let r = -0.01, Left hand side of equation = 5008.85

let r = -0.007, Left hand side of equation = 287.27

let r = -0.0065, Left hand side of equation = -480.86

So, r lies between -0.007 and -0.065

Using linear approximation

r = -0.007 +((287.27-0)/(287.27-(-480.86))*(-0.0065-(-0.007)) = -0.006813

which is the correct value of r

So, Annual Dollar weighted return = -0.006813*12 = -0.0817 or -8.18% compounded monthly