Portfolio return?

Question

A person invests 10,000 in January first.
In March 1st he adds 15,000 to his investment.
On September 1st he adds another 20,000.
On December 31st he collects 50,000.
What is his return on investment (%) and what is the formula to calculate it?

Answer 1

Future value = present value * (1 + r)^t
where r is the monthly return and t is the number of months

50000 = 10000 (1+r)^12 + 15000 (1+r)^10 + 20000 (1+r)^4

You cannot directly solve for r. You have to try different values of r until you get an equality. If the right-hand-side is too low, increase your estimate of r and vice versa.

For r=0.0135, the right-hand-side is 50000.4871, which is close enough. The annualized return on investment is 12r or 16.2%.

The reason the annual rate is 12r and not (1+r)^12-1 is that by convention, if s is the annual rate compounded monthly, the formula for FV is (1+s/12)^t. I just used r=s/12 to make things look a little simpler.

Answer 2

Small error at the end of that otherwise good prior answer.

Annualized return isn't 12r.

It's (1+r)^12 - 1.