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Time-weighted return

Remove external cash-flow jumps from investment performance. Link 10% and 20% subperiod returns into a 32% two-year return and reconcile it with ending account wealth.

Time-weighted return (TWR) measures compounded investment performance after removing the direct effect of external contributions and withdrawals. It splits the measurement horizon at those flows, calculates a return on the capital actually invested in each subperiod, then links the growth factors. This lets a quant compare portfolio performance without treating a client's deposit as a trading gain.
TWR answers how the investment performed through time. Money-weighted return instead reflects the size and timing of the investor's cash flows. Neither number alone establishes manager skill: risk, the benchmark and costs still matter.

The deposit is not a return

Consider this synthetic USD account over two exact one-year periods. All investment income remains inside the account, values include that income, and there are no fees, taxes or other external flows. At year 1, the contribution occurs immediately after the first valuation; no investment return accrues between the before-flow and after-flow snapshots.
TimeAccount valueWhat happened
Year 0USD 100Starting investment
Year 1, immediately before contributionUSD 110First year's investment result
Year 1, immediately after contributionUSD 160Investor adds USD 50
Year 2USD 192Second year's investment result
The first holding-period return is measured on USD 100. The second is measured on USD 160, because the new capital participates in the second year's investment performance:
r1=110100−1=10%,r2=192160−1=20%.r_1=\frac{110}{100}-1=10\%, \qquad r_2=\frac{192}{160}-1=20\%.

Link the factors rather than adding the percentages:

RTWR=(1+r1)(1+r2)−1=1.10×1.20−1=32%.R_{\mathrm{TWR}}=(1+r_1)(1+r_2)-1 =1.10\times1.20-1=32\%.
Account value rises from USD 100 to 110, jumps to 160 after a USD 50 contribution at year 1, then reaches 192 at year 2. A flow-adjusted index follows 100, 110, 110 and 132, giving a two-year time-weighted return of 32%.
Account value rises from USD 100 to 110, jumps to 160 after a USD 50 contribution at year 1, then reaches 192 at year 2. A flow-adjusted index follows 100, 110, 110 and 132, giving a two-year time-weighted return of 32%.
Original synthetic USD account over two exact one-year subperiods. The index keeps the investment growth factors and removes the contribution's jump; no fees, taxes or other external flows.
The index ends at 132, while the account ends at USD 192. There is no contradiction. The original USD 100 grows to USD 132; the additional USD 50, invested only for the second year, grows to USD 60. Those two capital cohorts reconcile the ending account value.
Dividing USD 192 by USD 100 and subtracting one produces 92%, mixing the contribution with investment earnings. Subtracting the contribution first produces a USD 42 investment profit, but dividing that by the original USD 100 gives 42%, which still assigns the contributed capital's earnings to the initial capital base. Adding 10% and 20% gives 30% and misses compounding.

Split at the flow, then link

For an exact flow-by-flow calculation, let Vi−1+>0V_{i-1}^{+}>0 be the portfolio value just after the previous external flow and Vi−V_i^{-} its value just before the next one. The initial and terminal valuations close the horizon. With no external flows inside subperiod ii,

ri=Vi−Vi−1+−1,RTWR=∏i=1n(1+ri)−1.r_i=\frac{V_i^{-}}{V_{i-1}^{+}}-1, \qquad R_{\mathrm{TWR}}=\prod_{i=1}^{n}(1+r_i)-1.

If the signed external flow at the boundary is CiC_i, positive for a contribution and negative for a withdrawal, then Vi+=Vi−+CiV_i^{+}=V_i^{-}+C_i. Use the before-flow value to close one subperiod and the after-flow value to open the next. The same timestamp can therefore have two valid account values.

“Time-weighted” does not mean an arithmetic average weighted by days. Subperiods may have unequal lengths; linking their actual growth factors produces the return over their combined horizon.

Buying a security with cash already inside the portfolio is an internal transaction. A dividend retained inside that same portfolio is investment income, not a fresh client contribution. Keep the account boundary and total-return convention consistent.

What the percentage leaves open

32% is the total return over two years, not an annual rate. Its constant annualised growth equivalent is 1.321/2−1≈14.89%1.32^{1/2}-1\approx14.89\%. Use elapsed years for annualisation, rather than the number of cash-flow-defined subperiods.
TWR neutralises the direct accounting effect of flows, conditional on the observed subperiod returns. It cannot remove their economic consequences: a withdrawal may force sales, and a contribution may sit in cash before investment. Those effects remain in the portfolio's returns. Compare portfolios over the same dates, currency, income treatment and fee basis, with an appropriate benchmark.
Valuations matter. Without a reliable valuation at each external flow, the exact calculation above is unavailable; methods such as Modified Dietz can approximate it. The GIPS Standards Handbook for Firms, provision 2.A.24 discusses cash-flow treatment, geometric linking and permitted approximations in its applicable reporting framework. Zero or negative capital also needs separate treatment: a zero denominator does not define an ordinary percentage return, and an emptied account cannot simply be chained through as though it remained invested.
For the next step from flow-adjusted performance to risk-aware comparisons, Risk-Adjusted Performance Measures begins with TWR and then develops Sharpe, Treynor and alpha measures. A public preview is available; the full lesson requires membership.

Sources

  • Edwin J. Elton, Martin J. Gruber, Stephen J. Brown and William N. Goetzmann, Modern Portfolio Theory and Investment Analysis, 9th edition (Wiley, 2014), Chapter 26, “Measures of Return,” pp. 661–662: external cash flows, compounded subperiod returns and the equivalent unitised view.
  • CFA Institute, GIPS Standards Handbook for Firms, provision 2.A.24 and its discussion: portfolio TWR methodology, valuation frequency, external flows and approximate methods.
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  • time weighted return
  • investment performance
  • cash flows
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