Question

A client invested €100 at the start of the month. Assume that the manager tracks an assigned benchmark index. After 20 days, the portfolio gained 10% (i.e., value=€110), as did the index, and the client added an extra €50 (total portfolio value=€160). From day 20 to day 30, the portfolio, and the index, lost 9.09%--the final portfolio value is €160×(1-0.0909)= €145.46. Calculate the money-weighted return and time-weighted return. Which rate of should you use to evaluate the performance of the manager relative to its benchmark?

Answer 1

Start of the Month -
Investment =

Return in 20 days = 10%
Value of investment in 20 days = 100*(1 + 10%) =  

Additional Investment at the end of 20th day =
Portfolio Value at the end of 20th day =

Return in next 10 days = -9.09%
Value of investment at the end of 30th day = 160*(1 - 9.09%) =

Money-weighted Return:
Money-weighted rate of return is that discount rate which makes Present Value of all future cashflows to equal to the initial investment. In other words, money-weighted return is nothing but the IRR of the cashflows.

Let us assume the money-weighted return = r% annualized

Cash flow at t = 0
Cash flow at t = 20
Cashflow at t = 30

Let us put this cashflow in excel and use the "IRR" formula of excel to calculate money weighted return, as given below -

Note that, putting cash of on a daily basis will give you a "Daily Return".
Convert it to annual return as follows -

Values, in excel will look like below -

Time-weighted Return:

Time-weighted return is a measure of compound rate of growth in portfolio. Time-weighted return ignores the timings if different cashflows, and tries to show the return had there been just one initial investment.

Time-Weighted-Return = (1 + return in period1)*(1 + Return in period2)*.....*(1 + return in period_n) - 1

In Our case,
Return in period 1 = +10%
Return in period 2 = -9.09%

Time-weighted Return = (1+10%)*(1 - 9.09%) - 1 = 1.100*0.909 - 1 = 0.001%

[Almost 0, but in case calculation is required to best decimal value possible]

But, the above is return for 30 day period.
Daily Return =
1-day return = = 0.000033333%
Annual Return = = 0.0122%

Which rate of should you use to evaluate the performance of the manager relative to its benchmark?

To measure performance of the manager, Time-weighted return is always utilized as it removes the effect of intermediate cashflows. Timing of cashflows are more investor centric and the manager has no control over it, hence to ingonre imperfect timings of the cashflow, time-weighted -return is utilized.

Even in this case, money-weighted return was negative, due to bad timing of cashflow at the 20th day, when index had already gained 10% in last 20 days. Higher money at the end of 20th day resulted into more dollar lossed in next 10 days. But overall, index was still at the same place as it was 30-days ago, so manager's performance matched that with index.