What is the difference between TWR and IRR?

The time-weighted return (TWR) and the internal rate of return (IRR, also called the money-weighted return) both describe the performance of the same portfolio, and they often disagree. That is not an error. They answer two different questions:

TWR: how well did the investments perform? Every period counts equally, however much money was invested during it.
IRR: how well did the invested money do? Every euro counts for as long as it was invested, so periods with more money in them weigh more.

The two formulas side by side

The TWR chains the returns $r_i$ of the sub-periods between cash flows:

\text{TWR} = \prod_{i=1}^{n} (1 + r_i) - 1

The IRR is the rate $r$ at which all cash flows $C_k$ , dated $t_k$ , balance out:

\sum_{k} \frac{C_k}{(1 + r)^{(t_k - t_0)/365}} = 0

The TWR uses only the return of each sub-period: the size of the portfolio drops out. The IRR uses only the cash flows and the final value: the individual sub-period returns never appear.

Why they differ

The gap between the two comes entirely from the timing of deposits and withdrawals:

Situation	Result
No deposits or withdrawals	TWR and IRR are equal (once both are annualized)
Money added before a strong period	IRR above TWR
Money added before a weak period	IRR below TWR
Money withdrawn before a fall	IRR above TWR

A common case is a monthly savings plan in a market that falls and then recovers. Each purchase during the fall buys cheaply, so the IRR ends up above the TWR even if the investments merely returned to where they started.

A simple illustration

€10,000 is invested in January and grows 10% by the end of June. Another €10,000 is added, and the portfolio then loses 5% by December.

	TWR	IRR
Return over the year	+4.5%	−0.33%
Question answered	How did the holdings do?	How did the invested money do?

The investments did well while the portfolio was small and badly once it was large. The TWR gives the good half-year the same weight as the bad one; the IRR gives the bad one twice the weight, because it held twice the money.

Which one to use

Comparing with an index, an ETF or a fund: the TWR. They report time-weighted returns, and a fair comparison has to remove the effect of the investor's own deposits.
Judging a savings plan, or what the money earned: the IRR. It is the rate a savings account would have had to pay to produce the same result.
Judging the timing of deposits and withdrawals: the gap between the two. An IRR persistently below the TWR suggests that money tends to arrive at unfavourable moments.

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