Why Reinvestment-Rate Sensitivity Matters
A cash receipt earns interest only after it arrives. If the reinvestment rate changes, a receipt received early in an investment horizon has more time to accumulate at the changed rate than a receipt received later. The effect on terminal wealth therefore depends on both the size of each receipt and its remaining reinvestment period.
This tutorial quantifies that effect. The method is to calculate terminal wealth under each rate scenario, then decompose the difference into the contribution from each cash receipt. A derivative gives a useful local measure of sensitivity, while a direct scenario comparison gives the exact change between two specified rates. These calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus.
The model assumes the stated annual effective rate applies for every year after each receipt arrives, compounding annually. Cash-flow amounts and dates are fixed across the scenarios; only the reinvestment rate changes. It also assumes no taxes, fees, withdrawals, or uncertainty in the rates. If rates vary by year, the accumulation factor must instead reflect the rate for each year; the constant-rate formula above does not apply unchanged.
Exact Change Between Two Rate Scenarios
Let \(j_0\) be a baseline rate and \(j_1\) an alternative rate. Subtracting the two terminal-value calculations gives the exact change:
Each term in the sum is the contribution of one receipt to the total difference. This decomposition is useful even when a single total difference is all that is requested: it reveals which receipts are driving the result. If all receipts are nonnegative and \(j_1>j_0>-1\), every receipt with at least one year to earn interest contributes a nonnegative amount. A receipt at the horizon contributes zero because its exponent is zero under both scenarios.
Record each receipt amount and the number of years from its receipt date to the terminal date.
For each receipt, multiply its amount by the appropriate accumulation factor for each scenario.
Subtract the baseline accumulation from the alternative accumulation, then add the contributions.
With nonnegative receipts and a higher rate, the total should not decrease. Identify which receipt amounts and reinvestment periods explain the largest contributions.
Worked Example: Decomposing a Rate Increase Across Receipts
A fund receives $250 at time 0 and $600 at time 2. Both amounts are accumulated to time 5. Compare terminal wealth at annual effective reinvestment rates of 6% and 6.5%, and identify each receipt’s contribution to the change.
Method. Accumulate each receipt for its own remaining number of years, then subtract the 6% result from the 6.5% result. The time-0 receipt earns interest for 5 years; the time-2 receipt earns interest for 3 years.
For the time-0 receipt, the relevant powers are \(1.065^5=1.370086663415625\) and \(1.06^5=1.3382255776\). Its contribution to the change is
For the time-2 receipt, the powers are \(1.065^3=1.207949625\) and \(1.06^3=1.191016\). Its contribution is
The baseline terminal wealth is \(250(1.3382255776)+600(1.191016)=\$1{,}049.1659944\). The alternative terminal wealth is \(250(1.370086663415625)+600(1.207949625)=\$1{,}067.29144085390625\). The difference is \(\$1{,}067.29144085390625-\$1{,}049.1659944=\$18.12544645390625\), which agrees with the sum of the two receipt-level contributions. Rounded to the nearest cent, the increase is $18.13.
Check and common mistake avoided. The higher-rate terminal value is larger, as expected for positive receipts. A tempting error is to use \(1.370086397166\) for \(1.065^5\); the correct power is \(1.370086663415625\). Although that earlier value happens not to change the answer rounded to cents, it gives an inaccurate receipt contribution and total difference.
Marginal Sensitivity to a Common Rate
An exact scenario comparison answers how much terminal wealth changes between two specified rates. A derivative answers a different question: how rapidly would terminal wealth change for a very small rate change near a specified rate? Differentiating the terminal-value formula term by term gives
To justify the result, hold each receipt amount and date fixed. For a receipt at time \(t\), its terminal value is \(C_t(1+j)^n\), where \(n=T-t\). For \(n\geq 1\), the power rule gives a derivative of \(C_tn(1+j)^{n-1}\). When \(n=0\), the receipt’s terminal value is simply \(C_T\), a constant, and its derivative is zero. Adding the derivatives of all receipts yields the displayed formula.
The derivative measures dollars of terminal wealth per unit change in the rate, where a unit change means a change of 1.00, or 100 percentage points. For a small change \(\Delta j\), the first-order estimate is \(W(j+\Delta j)-W(j)\approx W'(j)\Delta j\). For a material change, calculate both scenario values exactly rather than treating this approximation as exact.
Worked Example: Using the Derivative and Checking an Exact Change
A fund receives $500 at time 1 and $300 at time 3, with terminal value measured at time 4. The annual effective reinvestment rate is initially 4%. Find terminal wealth and its local sensitivity at 4%, then compare the derivative estimate for a one-percentage-point increase with the exact change to 5%.
Method. The first receipt is reinvested for 3 years and the second for 1 year. Thus \(W(0.04)=500(1.04)^3+300(1.04)\). Since \(1.04^3=1.124864\), this is \(500(1.124864)+300(1.04)=\$562.432+\$312=\$874.432\).
The local sensitivity is
per unit change in the rate. A one-percentage-point increase is \(\Delta j=0.01\), so the derivative estimates an increase of \(\$1{,}922.40(0.01)=\$19.224\), or about $19.22.
For the exact change, calculate the terminal-value difference directly:
The exact increase is $19.38 to the nearest cent, slightly greater than the local estimate of $19.22. The exact result is the appropriate answer for the full one-percentage-point change; the derivative is a local sensitivity measure, not an exact finite-change formula.
Check and common mistake avoided. The one-percentage-point change was entered as 0.01, not 1. If one incorrectly multiplied the derivative by 1, the estimate would be $1,922.40 rather than about $19.22.
Which Receipts Drive the Change?
The exact contribution formula shows two sources of sensitivity: the size of a receipt and the number of years it has to earn the rate difference. For a small rate change, the derivative contribution from a receipt is \(C_t(T-t)(1+j)^{T-t-1}\). This depends on the receipt amount, its remaining term, and the rate level. The remaining term matters directly through \(T-t\), as well as through the accumulation factor.
For positive receipts, a higher common rate produces a larger terminal value. For a receipt with at least two years to accumulate, its terminal value is also convex in the rate: its second derivative is \(C_t(T-t)(T-t-1)(1+j)^{T-t-2}\geq 0\) for \(j>-1\). A one-year receipt has a terminal value linear in the rate, and a receipt at the horizon is unaffected. These observations help explain why a rate change generally has a larger dollar effect on earlier receipts, but amounts matter too: a sufficiently large late receipt can contribute more than a small early one.
Worked Example: Comparing Early and Late Receipts
A fund receives $500 at time 0 and another $500 at time 3. Both receipts are accumulated to time 5. Compare the change in terminal wealth when the annual effective reinvestment rate rises from 5% to 6%. Which receipt contributes more?
Method. The time-0 receipt has 5 years to accumulate, while the time-3 receipt has 2 years. Their respective contributions to the increase are
and
The total increase is \(\$30.97200755+\$10.55=\$41.52200755\), or $41.52 to the nearest cent. The early receipt contributes more because the same $500 is exposed to the higher reinvestment rate for five years rather than two.
As a check, terminal wealth at 5% is \(500(1.2762815625)+500(1.1025)=\$1{,}189.39078125\). At 6% it is \(500(1.3382255776)+500(1.1236)=\$1{,}230.9127888\). Their difference, $41.52200755, matches the receipt-by-receipt calculation.
Common mistake avoided. Applying five years of accumulation to both receipts would overstate the late receipt’s contribution. The exponent must be determined from the receipt date, not from the length of the entire investment horizon.
Interpretation and Limits
Receipt-level attribution is valuable when a terminal-value gap needs to be explained rather than merely reported. It can show whether the difference is mainly due to a large receipt, an early receipt, or several smaller effects combined. The derivative can summarize local exposure at a benchmark rate, while the exact difference formula is preferable when comparing specified scenarios.
The monotonicity conclusion requires care. It applies when the relevant cash flows are nonnegative. If a cash flow is an outflow, its signed contribution can move terminal wealth in the opposite direction; with both inflows and outflows, the total sensitivity need not be positive. The rate change also needs to be interpreted consistently: a quoted percentage-point change must be converted to a decimal rate change before multiplying by a derivative. Finally, changing the receipt dates or amounts between scenarios is not a pure reinvestment-rate sensitivity comparison, because more than one feature has changed.
Check Your Understanding
For each question, use annual compounding and the stated annual effective reinvestment rates. Round dollar amounts to the nearest cent where needed.
- A fund receives $300 at time 0 and $200 at time 2. Accumulate both to time 4. Find the increase in terminal wealth when the rate rises from 3% to 4%. (Answer: $17.44)
- A $800 receipt arrives at time 1 and is accumulated to time 5. Find its terminal-value increase when the rate changes from 5% to 5.5%. (Answer: $18.65)
- A fund receives $250 at time 0 and $400 at time 2, with terminal value at time 4. At a 4% rate, find the derivative of terminal wealth with respect to the rate, and use it to estimate the increase for a 0.5-percentage-point rise. (Answer: \(W'(0.04)=\$1{,}956.864\) per unit rate; estimated increase is $9.78)
- Two separate $600 receipts arrive at times 0 and 3, and both are accumulated to time 5. At a rate increase from 5% to 6%, calculate each receipt’s contribution and state which contributes more. (Answer: time-0 receipt, $37.17; time-3 receipt, $12.66; the time-0 receipt contributes more)
- Explain why the conclusion that a higher reinvestment rate increases terminal wealth may fail if the cash-flow schedule includes outflows as well as receipts. (Answer: outflows have negative signed amounts, so their accumulated values can decrease terminal wealth as the rate rises; the net sensitivity can therefore be negative or positive.)