Time-Weighted and Money-Weighted Return Example

Question:

You are investing for your brother. You purchased 1k Google share ($400 each) for him in 2006. One year later, it became $600. You thought the subprime mortgage was going to drag the economy but your brother insisted to keep the stocks and in addition to that, he wanted to buy another 1k share. In 2008, it became $300. What is the time-weighted and money weighted-rate of returns?

Answer:

Again, let’s draw the time line with cash flows first. In 2006, you bought 1000 share at $400 each. Therefore you invested $400k (-ve, cash outflow). In 2007, you bought another 1000 share at $600 each. So, there is another cash outflow of $600k in 2007. In 2008, although you might not sell the stocks, it is the time you calculate the return, so we can assume there is cash inflow of $600k as if you have sold the 2000 shares at $300 each.

Let’s start with money-weighted return. By definition, it is the IRR (Internal Rate of Return) in the whole period:

400k = -600k/(1+r) + 600k/(1+r)^2

You may find r by iteration or using the calculator. It is -31.4%.

Now, let’s do time-weighted rate of return now. This is to calculate the geometric average of return in each sub-period. The period is defined by any major cash flow. The first subperiod is from 2006 to 2007 before another 1000 shares were bought:

r1 = (600-400)/400= 50%

The 2^nd period is between 2007 and 2008. This period begins with $1200k worth of stocks and ends with only $600k worth of stock.

r2 = (600-1200)/1200 = -50%

So the time-weighted rate of return is just the GM of r1 and r2:

r = ((1+0.5) (1-0.5))^0.5 -1 = -13.4%

So which method is more appropriate?

Time-weighted return is more appropriate in this case because you have NO control over the cash flow (it’s your brother who made wrong judgment.) So your performance is -13.4%

But if it were you persuaded your brother to buy 1k additional share in 2007, it was your wrong judgment. You have control on the cash flow and your performance is only -31.4%