Geometrically Linking Time-Weighted Returns

Why Time-Weighted Returns Are Linked

A sequence of subperiod returns describes performance over several consecutive intervals. To find the return over the full span, the returns must be combined to reflect how value accumulates from one interval to the next. The operation is geometric linking: convert each subperiod return into an accumulation factor, multiply the factors, and subtract one.

This is not generally the same as adding the subperiod returns. For example, a gain followed by a loss applies the loss to the value after the gain, not to the original value. Multiplication captures that change in the base on which each return is earned.

Definition: Suppose an investment has consecutive subperiod returns $r_1,r_2,\ldots,r_n$, each expressed as a decimal. The accumulation factor for subperiod $j$ is $1+r_j$. The cumulative time-weighted return over all $n$ subperiods is $$ R_{\mathrm{TW}}=\prod_{j=1}^{n}(1+r_j)-1. $$ The result is the cumulative return over the entire sequence, not the sum of the subperiod returns.

For instance, a return of $6\%$ is entered as $0.06$, so its accumulation factor is $1.06$. A return of $-4\%$ is entered as $-0.04$, so its accumulation factor is $0.96$. Each factor represents the multiplier applied to the value over that subperiod.

The inputs $r_j$ are assumed to be valid returns for consecutive subperiods, calculated on a consistent basis. The method applies when the accumulation from one subperiod carries into the next and each factor is defined. It does not calculate the subperiod returns themselves; use the appropriate return calculation for each interval first. If an investment loses all its value, its accumulation factor is zero, and ordinary linking cannot describe subsequent returns on that same invested amount.

Why the Factors Multiply

The formula follows directly from the definition of a return. Let $V_0$ be the value at the start of the first subperiod, and let $V_j$ be the value after subperiod $j$. If the return during subperiod $j$ is $r_j$, then the value at its end equals the starting value multiplied by $1+r_j$.

Apply the first return.
After the first subperiod, $V_1=V_0(1+r_1)$.

Carry the resulting value forward.
After the second subperiod, $V_2=V_1(1+r_2)=V_0(1+r_1)(1+r_2)$.

Continue through all subperiods.
Repeating the same relation gives $V_n=V_0\prod_{j=1}^{n}(1+r_j)$.

Convert the total factor to a return.
By definition, the full-period return is $(V_n-V_0)/V_0=V_n/V_0-1$, which equals $\prod_{j=1}^{n}(1+r_j)-1$.

This derivation also shows why the factors, rather than the percentage returns, are the quantities to multiply: the ending value of one interval becomes the starting value for the next. The order of multiplication does not change the product, but each return must still belong to the correct subperiod when calculating or checking the inputs.

Calculating a Cumulative Return

Worked Example: A Gain, a Loss, and Another Gain

A hypothetical investment earns returns of $8\%$, $-5\%$, and $10\%$ in three consecutive subperiods. Find the cumulative time-weighted return. The stated returns are already calculated for their respective subperiods.

Method. Convert each percentage to a decimal, form the accumulation factors, and multiply them.

The factors are $1+0.08=1.08$, $1-0.05=0.95$, and $1+0.10=1.10$. Thus,

$$ (1.08)(0.95)(1.10) =(1.026)(1.10) =1.1286. $$

The product $1.1286$ is the accumulation factor over all three subperiods. Subtract one to obtain the cumulative return:

R_{\mathrm{TW}}=1.1286-1=0.1286=12.86\%.

Check. Starting with $100$, the successive values are $100(1.08)=108$, $108(0.95)=102.60$, and $102.60(1.10)=112.86$. The increase from $100$ to $112.86$ is $12.86\%$, confirming the linked result. Rounding is not needed for this example.

Common mistake avoided. Adding $8\%-5\%+10\%=13\%$ is incorrect: the second and third returns apply to the changing value, not to the initial $100$.

The product gives the cumulative change over the whole sequence. It does not imply that each subperiod had the same return. In particular, two different sequences can produce the same product, while sequences with the same arithmetic sum can produce different products.

Worked Example: Linking Four Subperiods with a Small Net Change

A hypothetical fund has consecutive subperiod returns of $-4\%$, $6\%$, $2\%$, and $-3\%$. Find its cumulative return. The four subperiods are the full period of interest.

Method. Keep the negative returns as negative decimals when forming their factors. The accumulation factors are $0.96$, $1.06$, $1.02$, and $0.97$.

$$ (0.96)(1.06)=1.0176, $$

$$ (1.0176)(1.02)=1.037952, $$

$$ (1.037952)(0.97)=1.00681344. $$

Therefore,

R_{\mathrm{TW}}=1.00681344-1=0.00681344.

As a percentage, the cumulative return is $0.681344\%$, or $0.681\%$ to three decimal places.

Check. The total accumulation factor exceeds $1$, so the overall return must be positive. The calculated return, about $0.681\%$, has the correct sign. The arithmetic sum of the four returns is $1\%$, which is not the linked result.

Common mistake avoided. A small positive sum does not mean that the linked result must equal that sum. Apply each return to the value reached after the preceding subperiod.

Comparing Sequences and Solving for a Missing Return

Geometric linking is also useful when the task asks for more than a direct cumulative return. You can compare sequences by calculating each product, or solve for an unknown return by isolating its accumulation factor. When solving, divide by the product of the known factors before subtracting one.

Worked Example: Comparing Two Return Sequences

Two hypothetical investment strategies are measured over the same three consecutive subperiods. Strategy A earns $5\%$, $-2\%$, and $4\%$; Strategy B earns $2\%$, $3\%$, and $1\%$. Find each cumulative return and determine which is larger.

Method. Link the factors separately for each strategy. For Strategy A:

(1.05)(0.98)=1.029, \qquad (1.029)(1.04)=1.07016.

So Strategy A’s cumulative return is $1.07016-1=0.07016$, or $7.016\%$.

For Strategy B:

(1.02)(1.03)=1.0506, \qquad (1.0506)(1.01)=1.061106.

So Strategy B’s cumulative return is $1.061106-1=0.061106$, or $6.1106\%$. The difference is $7.016\%-6.1106\%=0.9054$ percentage points. Strategy A has the higher cumulative return.

Check. Each product exceeds $1$, consistent with both sequences having positive cumulative returns. The comparison is based on total accumulation factors, not on the largest individual subperiod return.

Common mistake avoided. Comparing only the sums of each sequence ignores compounding. The correct comparison is between the linked products.

Worked Example: Finding a Return Needed to Reach a Target

A hypothetical investment earns $4\%$, $-2\%$, and $5\%$ in its first three consecutive subperiods. What return $r$ must it earn in a fourth subperiod to produce a cumulative return of $12\%$ over all four subperiods?

Method. A cumulative return of $12\%$ means the required total accumulation factor is $1.12$. The known factors multiply to

(1.04)(0.98)=1.0192, \qquad (1.0192)(1.05)=1.07016.

The fourth factor must satisfy

$$ (1.07016)(1+r)=1.12. $$

Divide by $1.07016$, then subtract one:

1+r=\frac{1.12}{1.07016}, \qquad r=\frac{1.12}{1.07016}-1 =\frac{0.04984}{1.07016} \approx 0.0465725.

The required fourth-subperiod return is therefore $4.657\%$ to three decimal places.

Check. The known factor is $1.07016$, and the required final factor is about $1.0465725$. Their product is $1.12$, up to rounding, as required. The unknown return is positive because the target factor $1.12$ exceeds the known factor $1.07016$.

Common mistake avoided. Subtracting the first three percentage returns from $12\%$ does not solve for the fourth return. The target is a product of factors, so isolate the unknown factor by division.

Common Checks and Exam Traps

A reliable calculation has three stages: convert each return to a decimal, multiply the accumulation factors, and subtract one only after forming the full product. Keep the product as an accumulation factor until the final subtraction; this makes it easier to spot whether the result is plausible.

Adding returns instead of linking factors. The sum is not generally the cumulative return because later returns apply to a changed value.
Using $r_j$ instead of $1+r_j$ in the product. The factor $1+r_j$ includes both the original value and the gain or loss.
Dropping the minus sign on a loss. A return of $-5\%$ gives a factor of $0.95$, not $1.05$.
Subtracting one too early. Multiply all factors first, then subtract one to convert the full factor into a return.
Reporting the factor as the return. A product of $1.12$ means a $12\%$ cumulative return, not a $112\%$ return.

As a quick reasonableness check, a sequence with only nonnegative subperiod returns cannot have a negative linked return, and a sequence with only nonpositive returns cannot have a positive linked return, provided its factors are defined. For mixed gains and losses, use the product rather than relying on the signs or sum alone.

Key takeaway: For consecutive subperiod returns $r_1,\ldots,r_n$, the cumulative time-weighted return is $\prod_{j=1}^{n}(1+r_j)-1$. The factors multiply because each subperiod’s ending value becomes the next subperiod’s starting value.

Check Your Understanding

For each problem, calculate the cumulative return by linking the subperiod accumulation factors. Give the final percentage to three decimal places when rounding is needed.

A hypothetical account earns $3\%$ in one subperiod and $-2\%$ in the next. Find its cumulative return. (Answer: $0.940\%$)
A hypothetical portfolio earns $-6\%$, $4\%$, and $5\%$ in three consecutive subperiods. Find its cumulative return. (Answer: $2.648\%$)
A hypothetical investment earns $2.5\%$, $-1.5\%$, and $3\%$ over three consecutive subperiods. Find its cumulative return. (Answer: $3.991\%$)
A hypothetical account earns $3\%$ and then $4\%$. What return must it earn in a third subperiod for its cumulative return to be $10\%$? (Answer: $2.689\%$)
A hypothetical fund earns $3\%$, $2\%$, $-1\%$, and $4\%$ in four consecutive subperiods. Find its cumulative return. (Answer: $8.170\%$)

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