# Understanding Time-Weighted Rate of Return

In general I consider myself a relatively maths-literate person, but I'm really struggling to understand TWRR (Time-Weighted Rate of Return); at least as my investment provider is presenting it to me.

I've been investing in an investment account for two years now. It's a fully managed portfolio, I have no direct input into where the money is invested. Each month I invest a fixed amount of money (£100) on a fixed date/schedule, following the pound cost averaging principle. I have never withdrawn from this account (the only outflows of money are the management fees).

On my account dashboard, the provider shows the total amount of money invested as £2400, which makes sense. They also show the total amount currently in the plan as £2300, which makes sense (these are investments so they can lose money as well as gain, and the market isn't doing too well at the moment), and they show the overall gain ("all time") as -£100, which makes sense too.

However, the TWRR figure (for "all time") is shown as a positive percentage, around 3%.

How can this make any sense? I think I get that the point of the TWRR is to adjust for inflows and outflows of money to provide a more representative overall 'interest rate'. However, in this case there have been no outflows and the overall portfolio has fallen - so shouldn't the overall TWRR resolve to a negative figure, since my investments have fallen? Have they calculated it correctly?

Isn't this very misleading? This looks like a positive picture, but of course it isn't, because I've lost money (which is fine, again, it's the stock market, I'm just looking to understand the situation).

(In case it's important - I doubt it is - this is a UK Stocks and Shares ISA account)

(Some of the numbers above simplified for privacy and clarity)

TTWR only considers the compounded return, not the invested amount, and you are steadily increasing the invested amount. So if you had good returns when the invested amount was small and bad returns when it was large you could have a positive return yet have less than the total invested. E.g. good return one year:`600/400 - 1 = 50%`, and in another year `2100/2400 - 1 = -12.5%` loss. So the investment lost `200 - 300 = -100` but the `TTWR = (600/400)*(2100/2400) - 1 = 31.25%`.

``````Annualised TTWR = ((600/400)*(2100/2400))^(1/2) - 1 = 14.56% pa
``````

The TTWR is a good measure of the investment instrument. It doesn't take into account how much anyone has invested at a particular time.

An alternative measure that would take into account the loss would be the money-weighted return. E.g. (figures matching the TTWR example), return after two annual cash flows:

``````NPV = 400/(1 + r)^0 + 1800/(1 + r)^1 - 2100/(1 + r)^2 = 0

∴ r = -3.87% per annum
``````

Taking figures from the OP's example, money-weighted return:

``````NPV = 100/(1 + r)^0 +
100/(1 + r)^1 +
100/(1 + r)^2 +
...
100/(1 + r)^22 +
100/(1 + r)^23 -
2300/(1 + r)^24 = 0

∴ NPV = (100 (1 + r - (1 + r)^-23))/r -
2300/(1 + r)^24 = 0

∴ r = -0.3421% per month

MWR = (1 + r)^12 - 1 = -4.03% per annum
``````
• That helps, thanks. "So if you had good returns when the invested amount was small and bad returns when it was large"... and that is broadly true, the returns a few years ago were good and recently have been poor (see world situation and stock markets generally!), but recently I've had more money in it. So in effect what TWRR is measuring (loosely) is how good that investor is at giving me a return in general (and by extension perhaps one can conclude whether to leave money with them), not whether I'm actually making any money today. Fair? Jun 24, 2022 at 11:31
• Is the TWRR rate in effect an average of all the percentage growths? (over however many subdivisions of time are arbitrarily chosen, presumably daily). Jun 24, 2022 at 11:39
• Yes, TTWR is a measure of "how good that investor is at giving me a return in general". TTWR simply compounds returns achieved over given time periods. Annualisation is a further step. The money-weighted return just looks at cash flows (not the intermediate performance) and gives an overall return. Jun 24, 2022 at 11:59