Why Reconcile a Dollar-Weighted Return?
A dollar-weighted return is tied to the amounts invested and the dates those amounts enter or leave an account. Once a candidate return has been calculated, a direct check is available: grow the opening balance to the ending date, grow each contribution or withdrawal from its own date to that same ending date, and add the signed amounts. The result must equal the observed ending account value under the assumed return model.
This check is useful because a return equation can be set up with the wrong cash-flow sign, date, or time unit and still look algebraically plausible. A forward account roll makes the meaning of every term visible. The methods here are aligned with the SOA Exam FM (Financial Mathematics) syllabus.
The equation assumes the same effective annual return applies throughout the measurement period, with compound accumulation over fractional years. It also assumes each dated external flow is applied to the account at its stated time and earns the return only after that time. The equation does not describe a situation with a different return in each interval unless those interval-specific accumulation factors are used instead.
Why Forward Accumulation Gives the Reconciliation Equation
The equation follows from rolling the account forward one date at a time. Over an interval of \(s\) years, a balance \(B\) grows to \(B(1+i)^s\). When an external cash flow \(C\) occurs, the account balance changes by \(C\): a contribution increases the balance, while a withdrawal decreases it. Repeating these two operations carries the opening balance and every flow to the ending date.
From one event date to the next, multiply the current balance by \((1+i)^s\), where \(s\) is the interval in years.
Add the signed cash flow at its date. A contribution is positive; a withdrawal is negative.
After the final flow, accumulate the resulting balance to \(T\). By compound accumulation, this produces the opening balance and each flow grown separately to \(T\), as in the forward-balance equation.
For example, a flow \(C_j\) at time \(t_j\) is followed by \(T-t_j\) years of accumulation, so its contribution to the ending value is \(C_j(1+i)^{T-t_j}\). This term is positive or negative according to the flow’s sign; accumulation does not change the sign.
Worked Reconciliations
Worked Example: Checking a Two-Year Return
An account starts with $1,000. The investor contributes $250 at the end of year 1. The ending account value at the end of year 2 is $1,436.40. Check whether an annual dollar-weighted return of 8% reconciles with that ending value. Assume a constant annual effective return and that the contribution is made after the first year’s investment growth.
Method. Accumulate each amount to time 2. The opening balance is invested for 2 years, while the contribution is invested for 1 year:
Calculate each term:
The accumulated value is $1,436.40, exactly the stated ending value, so the 8% return reconciles. As a chronological check, the balance after year 1 is \(1000(1.08)=1080\); after the $250 contribution it is $1,330. It then grows to \(1330(1.08)=1436.40\) at the end of year 2.
Common mistake avoided. The $250 contribution does not earn two years of interest: it enters at time 1 and earns for one year. Using \(250(1.08)^2\) would incorrectly treat it as if it were in the account from time 0.
Worked Example: Reconciling Fractional-Year Cash Flows
An account begins with $1,000. A contribution of $500 is made at time \(t=0.5\), and a withdrawal of $200 is made at \(t=1.5\). The ending value at \(T=2\) is $1,909.60. Check a candidate annual effective return of 21%. Assume the return is constant and each flow is applied at its stated time.
Method. The dates are in years, so the accumulation periods are \(2\), \(1.5\), and \(0.5\) years. For half a year, the accumulation factor is
Thus, for 1.5 years, the factor is \(1.1^3=1.331\), and for 2 years, it is \(1.1^4=1.4641\). Accumulate the three signed amounts:
The computed value matches the observed ending value, so the candidate return reconciles. The same result follows by rolling the account chronologically: at \(t=0.5\), the initial balance is \(1000(1.1)=1100\), and the contribution raises it to $1,600. One year later, the balance is \(1600(1.21)=1936\); after the $200 withdrawal, it is $1,736. The final half-year gives \(1736(1.1)=1909.60\).
Common mistake avoided. The annual rate is 21%, but the half-year accumulation factor is not \(1+0.21/2\). Under an annual effective rate, it is \((1.21)^{0.5}=1.1\). The half-year and 1.5-year factors must follow compound accumulation.
Worked Example: Keeping a Withdrawal’s Sign Correct
An account starts with $2,000. The investor withdraws $500 at the end of year 1 and contributes $300 at the end of year 2. The observed ending value at the end of year 3 is $2,079. Check a candidate annual effective return of 5%, assuming each flow is applied after that year’s growth.
Method. At time 3, the opening balance has accumulated for 3 years, the withdrawal for 2 years, and the contribution for 1 year. The withdrawal is negative from the account’s perspective:
The three terms are
Adding them gives \(2315.25-551.25+315.00=2079.00\). The candidate return therefore reconciles with the observed ending value.
The chronological rollforward confirms the calculation. After year 1, the balance is \(2000(1.05)=2100\); subtracting the withdrawal leaves $1,600. After year 2, the balance is \(1600(1.05)=1680\); adding the contribution gives $1,980. After year 3, it is \(1980(1.05)=2079\).
Common mistake avoided. Treating the withdrawal as a positive $500 would add its accumulated value rather than subtract it, producing \(2315.25+551.25+315.00=3181.50\). That amount does not represent the account balance because a withdrawal removes money from the account.
Using the Check Reliably
A reconciliation is a verification of a proposed return, not a replacement for solving the return equation. When a candidate \(i\) is supplied or has been calculated, substitute it into the forward-balance equation and compare the result with \(V_T\). If the values differ beyond any stated rounding tolerance, inspect the setup before concluding that the return is wrong.
| Check | What to verify |
|---|---|
| Cash-flow sign | Contributions are positive and withdrawals are negative when cash flows are written from the account’s perspective. |
| Cash-flow date | Use the time the flow enters or leaves the account, not the start or end of a different interval. |
| Accumulation period | A flow at \(t_j\) accumulates for \(T-t_j\) years. A flow at \(T\) has no remaining accumulation period. |
| Rate convention | For an annual effective rate and fractional years, use \((1+i)^s\) for an interval of \(s\) years. |
| Rounding | Keep unrounded intermediate values when possible; round the final money amount only if requested. |
An ending-balance difference can be a useful diagnostic. A discrepancy equal to a cash flow grown for the wrong interval may indicate a timing error. A discrepancy with the same sign as a wrongly included withdrawal may indicate a sign error. These clues do not prove the cause; recheck every date, sign, rate convention, and interval.
Check Your Understanding
For each question, assume a constant annual effective return and apply each cash flow at its stated time. Reconcile by accumulating all amounts to the ending date.
- An account has an initial value of $800, receives a $100 contribution at \(t=1\), and ends at \(t=2\). At 6% per year, what ending value should the forward-balance check produce? (Answer: $1,004.88)
- An account starts at $1,500, receives $300 at \(t=0.5\), and has a $250 withdrawal at \(t=1.5\). What ending value at \(T=2\) reconciles with a 21% annual effective return? (Answer: $2,320.45)
- An account starts at $4,000, has a $700 withdrawal at \(t=1\), and receives a $500 contribution at \(t=2\). What value at \(T=3\) reconciles with a 4% annual effective return? (Answer: $4,262.34)
- An account starts at $2,500, receives $400 at \(t=1\), and has a $300 withdrawal at \(t=2\). Find the ending value at \(T=3\) that reconciles with a 10% annual effective return. (Answer: $3,481.50)
- An account starts at $6,000, receives $500 at \(t=1\), and has a $200 withdrawal at \(t=2\). What ending value at \(T=3\) is consistent with a 5% annual effective return? (Answer: $7,287.00)