Scale Invariance Of Dollar-Weighted Return

Why Scale Invariance Matters

A dollar-weighted return depends on the timing and size of an investor’s cash flows relative to the ending account value. If every dollar amount is expressed in a different unit, however, the calculated rate should not change. For example, recording an account in dollars rather than in hundreds of dollars changes the numbers but not the investment’s return.

This property is called scale invariance. It is useful both as a mathematical result and as a calculation check: rescaling all amounts should preserve the entire set of dollar-weighted return solutions. The condition that all amounts be scaled by the same positive constant is essential.

Definition: For investor-perspective cash flows $C_0,C_1,\ldots,C_n$ at times $t_0,t_1,\ldots,t_n$, the dollar-weighted return $i$ is a rate that satisfies the present-value equation $$ \sum_{k=0}^{n} C_k(1+i)^{-t_k}=0. $$ Here, time is measured in years, $i$ is an annual effective rate, contributions by the investor are negative, and amounts received by the investor are positive. The ending account value is included as a positive cash flow at the valuation time if it has not already been included.

This equation assumes that the cash flows and their dates are specified, and that the rate is interpreted consistently as an annual effective rate. For a solution representing a positive accumulation factor, require $i>-1$. The equation may have no admissible solution or more than one; scale invariance does not guarantee existence or uniqueness. It says that scaling preserves the solutions that the original equation has.

The Scale-Invariance Result

Theorem: Suppose the dollar-weighted return equation is $\sum_{k=0}^{n} C_k(1+i)^{-t_k}=0$. If every cash flow, including the ending account value, is replaced by $cC_k$, where $c>0$, then the new equation has exactly the same solutions $i$ as the original equation.

The reason is that the new present-value equation is just the old equation multiplied by $c$. Multiplying both sides of an equation by a nonzero constant does not change which rates make the equation equal to zero.

\sum_{k=0}^{n} (cC_k)(1+i)^{-t_k} =c\sum_{k=0}^{n} C_k(1+i)^{-t_k}.

Since $c>0$, it is nonzero. Therefore, the scaled sum equals zero exactly when the original sum equals zero. This also shows why the result preserves the whole solution set: if there are multiple returns, each remains a solution; if there is no return, scaling does not create one.

Write the original equation.
Include every investor cash flow at its stated time, using the appropriate sign.

Apply one scale factor.
Multiply each cash flow, including the terminal account value, by the same positive $c$.

Factor out the scale.
The scaled present-value equation equals $c$ times the original equation.

Conclude that the return is unchanged.
Because $c\ne 0$, both equations have the same rates at which their present-value sums are zero.

Worked Applications

Worked Example: Scaling a Two-Period Investment

An investor pays $\$600$ at time 0 and contributes a further $\$330$ at the end of year 1. The account is worth $\$1{,}089$ at the end of year 2. Find the dollar-weighted return, then find the return if every amount is recorded in units of $\$2.50$. Use an annual effective rate.

Method. Form the investor-perspective present-value equation, solve for a rate, and verify it by substitution. Then use the theorem to check the rescaled amounts. This avoids treating a change in the size or unit of the account as a change in investment performance.

The cash flows are $-600$ at time 0, $-330$ at time 1, and $+1{,}089$ at time 2. Thus

-600-\frac{330}{1+i}+\frac{1{,}089}{(1+i)^2}=0.

Try $i=0.10$. The present value of the year-1 contribution is $330/1.10=300$, and the present value of the ending account value is $1{,}089/1.21=900$. The equation gives

$$ -600-300+900=0. $$

So $i=10\%$ is a solution. To see that it is the only solution greater than $-1$, let $x=1+i$, so $x>0$. Multiplying the equation by $x^2$ gives

-600x^2-330x+1{,}089=0.

The left side is strictly decreasing for $x>0$, since its derivative is $-1{,}200x-330<0$. It can therefore have at most one positive root. Since $x=1.10$ is a root, the annual effective return is $10\%$.

Recording amounts in units of $\$2.50$ divides each amount by $2.50$. The new flows are $-240$, $-132$, and $+435.60$. At $10\%$, their present values are $-240$, $-120$, and $+360$, which sum to zero. The return remains 10%.

Check: The scaled values are exactly the original values multiplied by $0.4$, a positive constant. The theorem predicts that the return is unchanged.

Worked Example: Scaling With a Fractional-Year Cash Flow

An investor contributes $\$1{,}000$ at time 0 and another $\$300$ at $0.5$ years. The account is worth $\$2{,}160$ at $1.5$ years. Find the annual effective dollar-weighted return, and determine what happens if all amounts are multiplied by $3$.

Method. Use the actual fractional times as exponents in the annual effective return equation. This avoids incorrectly treating a half-year cash flow as if it occurred after a full year.

-1{,}000-300(1+i)^{-0.5}+2{,}160(1+i)^{-1.5}=0.

Test $i=0.44$, so $1+i=1.44$. Since $\sqrt{1.44}=1.2$, the discount factors are

(1.44)^{-0.5}=\frac{1}{1.2}=\frac{5}{6}, \qquad (1.44)^{-1.5}=\frac{1}{1.44(1.2)}=\frac{1}{1.728}.

The present values of the three cash flows are

-1{,}000,\qquad -300\left(\frac{5}{6}\right)=-250,\qquad \frac{2{,}160}{1.728}=1{,}250.

Their sum is $-1{,}000-250+1{,}250=0$. Thus $44\%$ is a solution. To verify uniqueness for $i>-1$, set $x=\sqrt{1+i}$, so $x>0$. Multiplying the equation by $x^3$ gives

-1{,}000x^3-300x^2+2{,}160=0.

The left side has derivative $-3{,}000x^2-600x<0$ for $x>0$, so it is strictly decreasing there. Since $x=1.2$ makes the equation zero, there is exactly one positive root and the return is $44\%$.

Multiplying every amount by $3$ gives $-3{,}000$, $-900$, and $+6{,}480$. Their present values at $44\%$ are $-3{,}000$, $-750$, and $+3{,}750$, again summing to zero. The dollar-weighted return is still 44%.

Check: The fractional-year discount factors were retained, and every cash flow was scaled by the same factor. Neither the return nor the timing changed.

Worked Example: A Withdrawal and a Rescaled Account

An investor contributes $\$1{,}000$ at time 0, withdraws $\$100$ at the end of year 1, and has an ending account value of $\$1{,}100$ at the end of year 2. Find the dollar-weighted return and the return when every amount is multiplied by $0.4$.

Method. Treat the withdrawal as a positive investor cash flow, since it is money received by the investor. Solve and verify the original equation before applying the scale factor. This avoids reversing the sign of a withdrawal.

-1{,}000+\frac{100}{1+i}+\frac{1{,}100}{(1+i)^2}=0.

At $i=0.10$, the year-1 withdrawal has present value $100/1.10=90.909090\ldots$, and the ending value has present value $1{,}100/1.21=909.090909\ldots$. The two positive present values sum to $\$1{,}000$, so the equation is zero and $10\%$ is a solution.

For uniqueness, let $x=1+i>0$ and multiply the equation by $x^2$:

-1{,}000x^2+100x+1{,}100=0.

The two roots are $x=1.1$ and $x=-1$, as can be checked by factoring the quadratic as $-1{,}000(x-1.1)(x+1)=0$. Only $x=1.1$ is positive, so the only admissible rate is $i=10\%$.

After multiplying all amounts by $0.4$, the cash flows are $-400$, $+40$, and $+440$. At $10\%$, the present values are $-400$, $36.363636\ldots$, and $363.636363\ldots$; the two positive amounts sum to $400$. The return remains 10%.

Check: The withdrawal and ending value were both scaled along with the initial contribution. The sign of the withdrawal is positive from the investor’s perspective before and after scaling.

What Must Be Scaled, and What Must Stay Fixed?

The theorem applies to cash-flow amounts, not to dates or the rate convention. Every contribution, withdrawal, distribution, and ending account value in the return equation must be multiplied by the same positive constant. The dates $t_k$ remain the same, as does the interpretation of $i$ as an annual effective rate.

Change to the data	Does scale invariance apply?	Reason
Multiply all cash flows, including the ending value, by the same $c>0$	Yes	The whole equation is multiplied by $c$.
Multiply contributions but leave the ending value unchanged	No	The new equation is not a constant multiple of the original.
Multiply amounts by $c$ and also change their dates	No	The discount factors change as well as the amounts.
Multiply all amounts by a negative constant	The algebraic roots are unchanged, but this is not a positive rescaling of investor cash flows	All signs reverse, changing the cash-flow interpretation.

A common exam trap is to scale only the beginning investment and later contributions, while leaving the ending account value as originally stated. That changes the equation rather than merely changing its units. Another trap is to reason that a return is unchanged because the account is “the same proportion larger” without checking whether every cash-flow amount, including the terminal value, was in fact scaled.

Worked Example: Detecting an Incomplete Rescaling

An investor contributes $\$500$ at time 0 and $\$110$ at time 1, then receives $\$605$ at time 2. The original return is $10\%$, since $-500-110/1.10+605/1.21=-500-100+500=-100$. Correct the calculation and decide whether doubling only the two contributions preserves the return.

Method. Substitute the proposed rate directly into the original equation before asserting it is a solution. This avoids carrying forward an unchecked arithmetic claim.

The stated data do not give a $10\%$ return: the present-value sum is $-100$, not zero. In fact, the proposed example values need to be tested against the rate equation before drawing a scaling conclusion. At $10\%$, the accumulated contributions are $500(1.21)=605$ and $110(1.10)=121$, for a required ending value of $\$726$, not $\$605$.

Use the consistent ending value $\$726$. The original equation is then

-500-\frac{110}{1+i}+\frac{726}{(1+i)^2}=0.

At $10\%$, its present values are $-500$, $-100$, and $+600$, so the sum is zero. If only the two contributions are doubled and the ending value remains $\$726$, the new equation at $10\%$ has present-value sum $-1{,}000-200+600=-600$, not zero. This incomplete change is not a rescaling and does not preserve the return.

If all three amounts are doubled, the flows become $-1{,}000$, $-220$, and $+1{,}452$. At $10\%$, their present values are $-1{,}000$, $-200$, and $+1{,}200$, which sum to zero as required.

Check: The corrected original figures satisfy the equation, and the fully doubled figures give exactly twice each original present value. Doubling only the contributions does not.

Exam Checks and Practical Use

Scale invariance is a short result, but it is a strong consistency check for a dollar-weighted return calculation. When comparing two sets of cash flows, first confirm whether one is truly a constant multiple of the other. If it is, there is no need to solve the return equation again: every solution is already known to be the same.

List all external cash flows and the ending account value with investor-perspective signs.
Keep the same dates and annual effective rate convention in both equations.
Check that the same positive factor applies to every amount.
When a numerical return is claimed, substitute it into the equation and verify that the present-value sum is zero.

The key limitation is that scale invariance is not a shortcut for changing the cash-flow pattern. Changing one amount independently, moving a cash flow to another date, or changing how the terminal value is included creates a different equation. This result is aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Key takeaway: Multiplying every investor cash flow and the ending account value by one positive constant multiplies the entire dollar-weighted return equation by that constant. Its zeroes, and therefore its return solutions, remain unchanged.

Check Your Understanding

For each problem, use investor-perspective signs and verify any proposed rate by substitution.

An investor contributes $\$400$ at time 0 and $\$110$ at time 1, then receives $\$605$ at time 2. Find the annual effective return. What is the return if every amount is multiplied by $4$? (Answer: $10\%$ in both cases.)
An investor contributes $\$500$ at time 0 and $\$100$ at $0.5$ years, then receives $\$786.50$ at $1.5$ years. Find the annual effective return. If every amount is multiplied by $0.2$, what is the new return? (Answer: $21\%$ in both cases.)
An investor contributes $\$1{,}000$ at time 0 and $\$200$ at time 1, with an ending account value of $\$1{,}312.50$ at time 2. Verify the return, then state the return if each amount is multiplied by $1.5$. (Answer: $5\%$ in both cases.)
A set of dated cash flows has two dollar-weighted return solutions. Every amount, including the ending value, is multiplied by $7$. How many return solutions does the scaled equation have? (Answer: The same two solutions.)
An investor’s initial contribution and later contribution are each doubled, but the ending account value is left unchanged. Is scale invariance sufficient to conclude that the return is unchanged? (Answer: No; the amounts are not all multiplied by the same constant.)

Change to the data	Does scale invariance apply?	Reason
Multiply all cash flows, including the ending value, by the same \(c>0\)	Yes	The whole equation is multiplied by \(c\).
Multiply contributions but leave the ending value unchanged	No	The new equation is not a constant multiple of the original.
Multiply amounts by \(c\) and also change their dates	No	The discount factors change as well as the amounts.
Multiply all amounts by a negative constant	The algebraic roots are unchanged, but this is not a positive rescaling of investor cash flows	All signs reverse, changing the cash-flow interpretation.

What You'll Learn