Time-weighted vs money-weighted rate of return

I have recently attended two different investor days where the respective asset managers were explaining the past performance and giving an outlook on the deal pipeline.

Both asset managers had nothing to do with each other, and yet they both independently referred to the IRR (“internal rate of return”) as a “time-weighted” performance measure. Given that their respective IRRs were seemingly quite impressive, I saw no necessity to engage in an academic conversation about their statements. I did see the need for this blog post though.

Time-weighted rate of return

If you are investing money yourself, it is very likely that you will intuitively turn to a time-weighted rate of return (“TWRR”) in order to measure (your) performance. I’m super-bad with mathematical notation so I’ll spare you the formula and jump straight to an example:

Let’s take four simple transactions:

· Buy 100 shares for EUR 10 each on 01/01/2020 (paying EUR 1,000) — value EUR 1,000

· Buy 100 shares for EUR 8 each on 01/07/2020 (paying EUR 800) — value EUR 1,600 (because you now own 200 shares in total but they are valued at EUR 8)

· Sell 50 shares for EUR 4 each on 01/04/2021 (receiving EUR 200) — value EUR 600 (because you now own 150 shares … you get the story)

· Sell 150 shares for EUR 10,5 each on 01/10/2021 (receiving EUR 1,575) — value EUR nil

OK we might still need some formulae. You’ll have to take the ending value (e.g. EUR 1,600), deduct any cash deposits&add-back any withdrawals (e.g. EUR 800), divide by the starting value (e.g. EUR 1,000) and deduct 1: (1,600–800)/1,000–1 = -20%. Alternatively you could spare the “-1” if you put the starting value in the denominator as well [(1,600–800–1,000)/1,000 = -20%] but that’s a matter of style.

So if you run this for each sub-period (which is measured from whenever-there-is-a-cash-flow to the-next-cash-flow), you can write down a holding period return next to each sub-period: -20% (example above), -50%, 163%. Note that right now we don’t care whether the sub-period is 182 or 274 days long.

Still with me? Great. Now the fun part: we link these returns geometrically, not arithmetically! Arithmetic is for the hoipolloi — they take the average over the returns: (-20%-50%+163%)/3 = 31%. Yikes … that’s just … irrelevant, mkay? Just don’t ever do that.

Geez, next time just upload a spreadsheet, will you?

Geometric is the charm, and it works like that: each return period is expressed by a factor by adding back 1 (Woooot? We just deducted “1” now we add “1”? — Answer: yes, please). So that’s 0.8, 0.5 and 2.63. Now you link it by multiplying, not by adding: 0.8 x 0.5 x 2.63 = 1.05. What does that mean? Well, if you deduct 1 (I know, I know …), it tells you that your overall performance is +5%.

I can hear you saying “HANG ON NIC” — wait for it — “I could have just looked at the share price at the end of the total period (10.5) and the beginning of the period (10), and you see immediately that the entire holding period return is +5%.”

Haha you’re right. And that’s the whole point ;-) Share price performance, market indices … all simply expressed in easy-to-calculate percentages are nothing else but TWRR real life. What is illustrated above is that TWRR specifically takes out any cash flows in and out, because that is “your own thing” and has — in principal — nothing to do with the performance of the investment itself.

The End? Almost — note that the return in the example above spans from 01/01/2020 to 01/10/2021. People like to refer to an annual performance though, so let’s apply some math to get it straight:

Total number of days in the entire period is 639, and we want to base it on 365 — easy: you raise the performance expressed as a factor (1.05) to the power of the reciprocal of number of years in the period and deduct 1 :o) number of years is: 639days/365 days in a year = 1.751 years, reciprocal is 1/1.751 = 0.571. That gives you 1.05⁰.574 = 1.028–1 = 2.83%. All we are saying here is that the 5% over the ~ 21 months period is equivalent to an annual performance of 2.83%.

Is that the right number now?

Can we check that? Sure, if you don’t trust you own numbers you could do the following:

Scroll all the way up to our different sub-periods (*rrrk rrrk rrrk*) and calculate the holding period in days for each sub-period: that should give you 182, 274 and 183 days. Now for something fancy, you use the concept of linking returns geometrically that we have introduced above (*rrrk rrrk*) in order to calculate factors that give you the overall holding return when you multiply them. No worries, it’s easier than you think: for period 1 you take your annual performance factor (1+ 2.83% = 1.0283) and raise it to the power of

a) The number of years in the period;

b) 1, or 1–1, or whatever leave me alone thanks bye; or

c) 42?

Let’s try number of years: that would be 182days/365 days in a year = 0.5. 1.0283⁰.5=1.014. Same for periods 2 and 3: 1.021 and 1.014. Now we link these numbers: 1.014x1.021x1.014 = *drumroll* 1.05 — your 5% performance. Next time you open a scary financial model and see the acronym “CAGR” (compound annual growth rate) which is exactly the same thing: the (artificial) number that you could apply to each sub-period in order to link the ending value to the beginning value with a nice straight line.

Just realized that I wrote a lot of paragraphs for the simpler of the two concepts :o) In a nutshell, TWRR is what everyone uses, kind of, without calling it TWRR most of the time.

Money-weighted rate of return

If it is so simple (leaving aside my desperate attempt above to make it appear much more complex than it is), why do we need anything else then? That’s a fair question after all. Let’s turn once more to the example above (*rrrk rrrk*). Now that we have figured out that the performance is 5%, let’s look at the absolute amounts (in EUR and Cents) that we have earned: -1,000–800+200+1,575 = -25.

-25? Like, the performance is +5% but my result is — 25 EUR??

Welcome to the world of money-weighted rate of return (“MWRR”) (Hint: nobody on this planet ever called it MWRR.)

Never heard of MWRR

The money-weighted rate of return is nothing else but your IRR. It follows a slightly different logic: instead of stripping out intermediate cash flows (deposits/withdrawals) and concentrate on the ending values of each sub-period, in the IRR you concentrate solely on the cash flows, without giving consideration to the ending values, or even bother breaking the periods into sub-periods at all along the way.

The IRR is the discount rate that sets the net present value of the cash flows to 0. Even if that statement leaves you staring at your screen with a blank face, please memorize the following: Timing! Matters! Timing Matters A LOT!

No ending-value-at-each-sub-period, no geometric voodoo, just figuring out which discount rate you need to plug in: mathematically it’s sort of a trial-and-error thing (smart people call it “iteration”): need to find r, e.g. period 1: -800 / (1+r)^(182/365) => in our example that’s -0.95% (if you discount each of the cash flows outlined above with -0.95%, the present value of the inflows will equal the present value of the outflows, have a try). No need to pull out pencil and paper though, Excel’s “XIRR” function is your friend.

Note that you do not need the ending value at each cash-flow point in order to calculate IRR, and in our example the IRR of -0.95% seems to be much closer to the “intuitive” return (-EUR 25) than the (annualized) TWRR of 2.83%.

Which one is the good one? Which one to avoid?

Let’s compare and contrast the two different methods now:

Assume we swap the transactions 2 and 3 in our example (*rrrrrrrrrrrrrrrrk*): instead of buying 100 and 8 and selling 50 at 4, we first sell 50 at 8 and buy 100 at 4 — note that the market did not change at all, just the investment/divestment decisions did. I spare you the pain of going through the steps of calculating TWRR: it’s unchanged, lads and ladies! The IRR though goes through the roof and only stops at 35%, which seems reasonable given that you have a net gain of EUR 575 through these transactions.

In principle, the TWRR is probably better if you want to evaluate the performance of an investment — after all, is it the investment’s (or the investment manager’s) fault that you made stupid buy/sell decisions along the way which are, like, your own thing?

Imagine our example was a closed-ended Venture Capital fund. And you, super-excited about all these concepts and equipped with your latest Texas Instruments BA II Plus™ Professional, want to calculate the TWRR (because, cash flow don’t matter haha). In order to do just that, you know that you need to know the value at the end of each cash flow once you adjusted for the cash flow.

There is just a slight issue with all of this: there is no way you can easily retrieve that number. Part of the problem is the fact that valuations are only done on a quarterly basis, and not everytime there is a cash flow (lest to speak as frequently as weekly or daily). You may be able to rebuild something close to the underlying value movements 3–6 months after the fact taking a lot of assumptions and digging deep into the GP reports. Alternatively, you could surrender and acknowledge that TWRR is simply rarely used in private funds.

How can I memorize that?

I try to do the following: the “time-weighted rate of return” is not the rate with the word “money” in it (😊), so it is the one that does eliminate the cash in’s/out’s in the portfolio and looks at the various sub-periods (=time) linked geometrically.

The “money-weighted rate of return” (IRR) is the one which uses the concept of discounting cash flows (=money) in order to arrive at a performance metric.

The absolute sum of money moved around does not matter in the time-weighted return, and it matters a lot in the money-weighted return.

Is IRR so much better then? Not necessarily, but that’s for another article …