Time-weighted versus money-weighted returns

Key takeaways

• TWR links the returns of sub-periods separated by each external cash flow, so the size and timing of contributions and withdrawals do not affect it.
• MWR is the IRR of the investor's cash flows plus ending value, so it rises when more money is invested before good periods and falls when more is invested before bad ones.
• The same portfolio can show a positive TWR and a negative MWR: in this topic's example, 3.9% a year time-weighted and -5.2% a year money-weighted.
• Public and hedge fund managers report TWR because investors control the cash flows; private funds report IRR because the GP controls them.
• Placing a public manager's TWR next to a private fund's IRR is an apples-to-oranges comparison; a public market equivalent (PME) method puts both on the same cash flows.
• The GIPS standards set out when each type of return is appropriate for a manager's performance presentations.

Two questions, two returns

Imagine a portfolio manager who runs a strategy for a client. The client adds money and withdraws money on the client's own schedule. If the manager's return is measured in a way that depends on those decisions, the manager will look better or worse because of choices it did not make. A fair measure of the manager must strip them out. That is the time-weighted return.

Now look from the client's side. The client does not care about an abstract return on a hypothetical dollar held throughout. It cares about the money it actually had at risk. If it doubled its investment just before a crash, it lost more money than the manager's time-weighted return suggests. A measure of the client's experience must include the effect of those decisions. That is the money-weighted return.

The distinction is not about which asset class is involved. It is about who controlled the external cash flows. The measure that evaluates a decision-maker should include the decisions that party made and exclude the ones it did not.

Time-weighted return: linking sub-periods

To compute a TWR, split the measurement period into sub-periods at every external cash flow (a contribution or a withdrawal). Within each sub-period there are no external flows, so its return is simply the ending value divided by the starting value, minus one. The starting value of each sub-period includes the cash flow that just arrived. Then multiply one plus each sub-period return together (geometric linking) and subtract one.

Because each sub-period return is a ratio, it is unaffected by how many dollars were in the portfolio during that sub-period. A 20 percent gain on $100 and a 20 percent gain on $500 are both 20 percent. That is exactly what removes the influence of cash flow size and timing.

The requirement is data: the portfolio must be valued on the date of every external cash flow. For a daily-priced public equity portfolio this is routine. For an illiquid portfolio valued quarterly, the sub-periods can only be approximated, which is one practical reason private funds do not report TWR.

Money-weighted return: the IRR of the investor

The money-weighted return treats the investor's contributions as outflows and withdrawals as inflows, adds the ending value as a final inflow (and the beginning value as an initial outflow, if measurement starts partway through), and solves for the IRR. It is exactly the since-inception IRR described earlier in this section, applied to any portfolio.

Each dollar's return counts in proportion to how long and how heavily it was invested. In Portfolio P, $100m experienced the good year and $500m experienced the bad year. The bad year therefore dominates the result.

Why they diverge: the mirror case

The gap in Portfolio P came entirely from timing: the investor added most of its money just before the weak year. Reverse the order of the returns and keep the same kind of contribution, and the gap reverses too. In Portfolio Q, the first year loses 10 percent and the second gains 20 percent, and the investor adds $410m after the loss. The TWR is identical, 8.0 percent over two years, because the product 0.90 x 1.20 equals 1.20 x 0.90. But now most of the money experiences the good year, and the MWR is strongly positive.

The pattern generalizes. MWR is above TWR when contributions precede strong periods (or withdrawals precede weak ones), and below TWR when contributions precede weak periods (or withdrawals precede strong ones). The gap is larger when the cash flows are large relative to the portfolio and when returns vary a lot between sub-periods. If there are no external cash flows at all, the two measures are identical.

A shortcut: the Modified Dietz approximation

Before daily valuation was routine, managers approximated money-weighted returns within a period with the Modified Dietz method. It divides the investment gain by the average capital invested, where each cash flow is weighted by the fraction of the period it was in the portfolio. It avoids iteration and is still used to estimate returns for sub-periods when a valuation on the exact date of a cash flow is not available.

Modified Dietz is a money-weighted measure and is close to the IRR over the same period when flows and returns are moderate. It is not a time-weighted return on its own, though linking Modified Dietz returns for short sub-periods (such as months) is a common way to approximate a TWR.

Why public managers use TWR and private funds use IRR

In a mutual fund, a separately managed account or a hedge fund, the investor decides when to subscribe and when to redeem (subject to notice periods, lockups and gates in hedge funds). The manager has to invest whatever arrives. Holding the manager responsible for the timing of those flows would be unfair, so these managers report TWR, and hedge funds typically publish monthly returns computed from NAV per share, which is time-weighted by construction.

In a private fund, the GP decides when to call capital and when to distribute it. Timing is part of the manager's skill: calling capital only when an attractive deal is ready, and returning it promptly after an exit, are decisions the GP makes and should be judged on. The IRR includes exactly those decisions. That, together with the practical fact that private assets are valued only periodically, is why private fund performance is reported as IRR.

The GIPS standards, the global investment performance standards maintained by CFA Institute, reflect this logic. In broad terms, they call for time-weighted returns in most cases and permit money-weighted returns where the manager controls the external cash flows, as in a closed-end private fund. A manager's GIPS report should state which method was used.

The comparison trap

A typical board report lists each asset class with its return over the same horizon: public equity 10-year return, fixed income 10-year return, private equity 10-year return. The public figures are almost always TWRs from the managers or the custodian. The private equity figure is almost always a pooled IRR. They look alike and sit in the same column. They are not comparable.

Portfolios P and Q show why. Suppose Portfolio P is an index fund and the investor is a pension plan contributing on its own schedule. The index fund reports 3.9 percent a year. Now suppose a private fund happened to call capital on the same dates and produced exactly the index's returns. It would report an IRR of -5.2 percent a year. A board reading the two figures side by side would conclude that private equity underperformed public equity by 9.1 points a year, when their underlying performance was identical.

The fix is to evaluate the public alternative on the private fund's own cash flows. That is what public market equivalent (PME) methods do. The Kaplan-Schoar PME, for example, discounts the fund's contributions and distributions by the index's cumulative return and divides the present value of distributions (including ending NAV) by the present value of contributions. A result above 1.0 means the fund beat the index on the same cash flows; below 1.0, it lagged. Other methods, such as Long-Nickels PME and direct alpha, answer the same question in IRR terms.