Dollar-Weighted vs. Time-Weighted Returns

Dollar-Weighted Rate of Return

The dollar-weighted rate of return (DWRR) is the internal rate of return (IRR) of a fund, accounting for the timing and amounts of deposits and withdrawals. It solves the equation: B_0*(1+i)^T + sum of C_t*(1+i)^(T-t) = B_T, where B_0 is the initial balance, B_T is the ending balance, and C_t are net contributions (positive for deposits, negative for withdrawals) at time t.

This equation sets the accumulated value of all cash flows equal to the ending balance. Alternatively, you can set it up as a present value equation where all cash flows are discounted to time 0: B_0 + sum of C_t * v^t = B_T * v^T. Both forms give the same DWRR.

Example: A fund starts the year with $100,000. At mid-year (t = 0.5), the investor deposits $20,000. The fund ends the year at $130,000. The DWRR solves: 100,000*(1+i) + 20,000*(1+i)^0.5 = 130,000. Testing i = 0.08: 100,000*1.08 + 20,000*1.0392 = 108,000 + 20,784 = 128,784 (too low). Testing i = 0.10: 100,000*1.10 + 20,000*1.0488 = 110,000 + 20,976 = 130,976 (too high). By interpolation, i is approximately 9.4%.

The Simple Interest Approximation

Because solving for the exact DWRR often requires iteration, Exam FM frequently uses the simple interest approximation: i is approximately (I) / (B_0 + sum of C_t * (1 - t/T)), where I = B_T - B_0 - sum of C_t is the total interest earned (ending balance minus starting balance minus net contributions).

For the example above: I = 130,000 - 100,000 - 20,000 = $10,000. The denominator is 100,000 + 20,000*(1 - 0.5) = 100,000 + 10,000 = 110,000. So i is approximately 10,000/110,000 = 9.09%. This approximation is close to the exact answer and is much faster to compute.

The denominator in the approximation represents the "weighted" exposure: each dollar is weighted by the fraction of the year it is invested. The initial balance is invested for the full year (weight 1), and the mid-year deposit is invested for half the year (weight 0.5).

Time-Weighted Rate of Return

The time-weighted rate of return (TWRR) eliminates the effect of cash flow timing by computing the geometric mean of sub-period returns. The fund's time horizon is divided into sub-periods at each cash flow date. For each sub-period, the return is computed as (ending value) / (beginning value including the cash flow). The TWRR is then:

(1+i)^T = (V_1 / B_0) * (V_2 / (V_1 + C_1)) * (V_3 / (V_2 + C_2)) * ..., where V_t is the fund value just before the t-th cash flow, and C_t is the net contribution at time t.

Example: Using the same fund. The fund starts at $100,000. Just before the mid-year deposit, the fund value is $105,000. After the $20,000 deposit, the fund value is $125,000. At year-end, the fund is worth $130,000. Sub-period 1 return: 105,000/100,000 = 1.05. Sub-period 2 return: 130,000/125,000 = 1.04. TWRR: (1+i) = 1.05 * 1.04 = 1.092, so i = 9.2%.

Why the Two Returns Differ

The DWRR and TWRR measure different things. The DWRR measures the investor's actual return, accounting for when money was added or removed. The TWRR measures the fund's investment performance, independent of the investor's cash flow decisions.

In the example above, the DWRR (approximately 9.4%) exceeds the TWRR (9.2%) because the investor contributed $20,000 just before a period of positive returns (4% in the second half). The additional money benefited from the good performance, boosting the investor's return relative to the fund's pure performance.

The general rules are: when contributions precede strong performance, DWRR > TWRR. When contributions precede poor performance, DWRR < TWRR. When there are no external cash flows, DWRR = TWRR.

When to Use Each Measure

DWRR is appropriate for measuring an individual investor's return because it reflects the actual dollars earned relative to the dollars invested and their timing. A pension fund member, for instance, cares about DWRR because it tells them how their specific contribution pattern performed.

TWRR is appropriate for evaluating a fund manager because it isolates investment performance from cash flow decisions that the manager cannot control. If an investor makes a large deposit right before a market downturn, the DWRR will be poor, but this is not the manager's fault. The TWRR removes this effect.

Industry standards (such as the CFA Institute's Global Investment Performance Standards) require TWRR for reporting fund performance precisely because it eliminates the distortion from investor-driven cash flows.

Linked Internal Rate of Return

In practice, computing the TWRR requires knowing the fund value at every cash flow date, which may not always be available. The linked IRR is an approximation that divides the year into sub-periods (such as months or quarters) and computes the IRR within each sub-period, then links them geometrically. This provides a reasonable approximation to the TWRR when daily valuations are unavailable.

On Exam FM, most TWRR problems provide the fund values at each cash flow date, so the exact TWRR can be computed directly. If the problem does not provide intermediate values, you may need to use the DWRR instead.

Study Tips and Common Pitfalls

The most common mistake is confusing which method to use. Remember: DWRR for the investor's perspective, TWRR for the manager's perspective. Another frequent error is setting up the DWRR equation incorrectly. Make sure deposits are positive and withdrawals are negative (or be consistent with your own sign convention). Also verify that the time exponents match the fraction of the period each cash flow is invested.

For the simple interest approximation, be careful with the weighting in the denominator. Each cash flow is weighted by the remaining fraction of the investment period. A deposit at time t in a period of length T has weight (T - t)/T = 1 - t/T. Practice at least 3 DWRR problems and 3 TWRR problems, including at least one that asks you to compare the two results and explain which is higher and why.