Putting Project Value, Reinvestment, and Rate Sensitivity Together
A project can look attractive at the valuation date yet produce a different terminal result if interim receipts are reinvested at rates that differ from the rates used to discount them. A further question is how much the project’s value changes when the spot curve moves. These are related questions, but they use different calculations: NPV discounts dated cash flows, forward rates accumulate interim receipts, and duration estimates a value change under a specified rate shift.
The calculations below assume annual cash-flow dates and annual effective rates unless stated otherwise. All cash flows are measured in dollars, and a positive receipt increases project value. The spot and forward rates are hypothetical inputs for the examples; no actual reinvestment outcome is guaranteed by an implied forward rate.
A spot rate is the annual effective rate for investing from time 0 to one specified maturity. Thus, different maturities can have different rates. Applying one rate to every cash flow is appropriate only when the valuation assumption actually specifies a flat curve. With a nonflat curve, replacing the maturity-specific rates by an average rate generally changes the NPV.
Worked Example: NPV on a Nonflat Spot Curve
Worked Example: NPV on a Nonflat Spot Curve
A hypothetical project requires an outlay of $250 now. It pays $100 at the end of year 1 and $180 at the end of year 2. The annual effective spot rate is 4% for one year and 5% for two years. Find the project’s NPV. State the method and one common mistake to avoid.
Method. Discount each receipt at the spot rate matching its payment date, then subtract the time-0 outlay.
The year-1 receipt has present value $$ \frac{100}{1.04}=96.153846\ldots $$ The year-2 receipt has present value $$ \frac{180}{(1.05)^2} =\frac{180}{1.1025} =163.265306\ldots $$ Therefore, the present value of receipts and the project NPV are $$ V=96.153846\ldots+163.265306\ldots=259.419152\ldots $$ $$ \operatorname{NPV}=259.419152\ldots-250=9.419152\ldots $$ The project’s NPV is $9.42, rounded to the nearest cent.
Check. The receipts are worth about $259.42 at time 0, which exceeds the $250 outlay by about $9.42. The positive NPV agrees with that comparison.
Common mistake avoided: discounting both receipts at 4%, or both at 5%, would ignore the stated maturity-specific spot rates. The year-1 payment uses 4%; the year-2 payment uses 5%.
Accumulating Interim Receipts with Forward Rates
A forward rate applies to a future period beginning at a specified date. Write \(f_{t-1,t}\) for the annual effective rate applying from time \(t-1\) to time \(t\). For a spot curve with annual effective spot rates \(s_t\), the implied one-year forward rate for that period is
where the time-0 accumulation factor is 1. The formula follows because investing from time 0 to time \(t\) must have the same accumulated value as investing to time \(t-1\), then investing for one more year at the implied forward rate. This is a no-arbitrage relationship under the applicable market assumptions; it does not say that the forward rate will be the rate actually available for future reinvestment.
To express receipts at a common horizon \(n\), accumulate a receipt at time \(t\) through each later period:
A receipt at time \(n\) has no remaining periods to accumulate, so its product is 1. The initial outlay, if included in terminal project wealth, must also be accumulated from time 0 through all \(n\) periods. This gives a common-date comparison; it is not the same as reporting NPV at time 0.
Worked Example: Accumulating Receipts to a Common Horizon
A hypothetical project costs $120 at time 0 and pays $50 at the end of year 1 and $80 at the end of year 2. For this scenario, the forward rates are 3% from year 0 to 1, 4% from year 1 to 2, and 5% from year 2 to 3. Find the terminal value of the receipts at year 3 and the project’s net terminal wealth at that date.
Method. Accumulate each receipt only over the periods after it is received. Accumulate the time-0 cost over all three periods.
The year-1 receipt is reinvested for the year-1-to-2 and year-2-to-3 periods: $$ 50(1.04)(1.05)=54.60. $$ The year-2 receipt is reinvested only for the year-2-to-3 period: $$ 80(1.05)=84.00. $$ Thus, the year-3 terminal value of receipts is $$ 54.60+84.00=138.60. $$ The initial cost accumulated to year 3 is $$ 120(1.03)(1.04)(1.05) =120(1.12476) =134.9712. $$ Net terminal wealth is therefore $$ 138.60-134.9712=3.6288. $$ The project has net terminal wealth of $3.63 at year 3, rounded to the nearest cent.
Check. The year-1 receipt earns two forward periods, while the year-2 receipt earns one. The initial outlay earns all three in the comparison. The receipt total exceeds the accumulated outlay by $3.6288.
Common mistake avoided: multiplying the year-1 receipt by the year-0-to-1 forward factor would incorrectly give it an extra period of interest. That receipt already exists at year 1.
Why Forward Accumulation and NPV Agree on One Curve
If the forward rates are implied by the same spot curve used for valuation, accumulating a time-0 value to a later date and discounting a dated cash flow back to time 0 are consistent operations. In particular, the product of forward accumulation factors from time 0 to \(t\) equals \((1+s_t)^t\). Therefore,
This identity is useful as a check on dates and rate periods. It does not require all forward rates to be equal. It does require the forward rates and spot rates to belong to the same curve and use compatible annual effective conventions. If assumed realized reinvestment rates differ from the implied forwards, the resulting terminal wealth can differ from the curve-consistent value.
Curve-Based Duration and a Parallel Spot-Rate Shift
Duration is a local measure of value sensitivity. For this tutorial, consider the present value \(V\) of positive project receipts. A parallel spot-rate shift of \(\Delta y\) means that every spot rate \(s_t\) changes to \(s_t+\Delta y\), with the cash flows and their dates held fixed. The initial time-0 cost is not discounted and is unchanged by this shift.
The denominator in \(D_{\parallel}\) is the present value of receipts, not the project’s NPV. NPV can be small or negative because it nets receipts against an initial cost; it is not the value of the receipt stream whose sensitivity is being measured.
Why the duration formula holds. Under a parallel shift, the present value of receipt \(C_t\) becomes \(C_t(1+s_t+\Delta y)^{-t}\). Differentiating with respect to the common shift and evaluating at \(\Delta y=0\) gives
Adding these derivatives over all receipt dates gives \(dV/d(\Delta y)\). Dividing the negative derivative by \(V\) yields \(D_{\parallel}\), so the first-order change is \(\Delta V\approx -V D_{\parallel}\Delta y\). This is a local, first-order estimate: it is most appropriate for a small shift and does not include curvature.
When the curve is flat at rate \(i\), the Macaulay duration is the present-value-weighted average payment time, \(\sum tC_t/(1+i)^t \div V\). The parallel modified duration above is Macaulay duration divided by \(1+i\). With a nonflat curve, the maturity-specific formula for \(D_{\parallel}\) is the direct sensitivity measure for a parallel additive shift in spot rates.
Worked Example: Estimate NPV Sensitivity from a Nonflat Curve
A hypothetical project costs $240 now and pays $130 at the end of year 1 and $130 at the end of year 2. The annual effective spot rates are 4% for year 1 and 5% for year 2. Estimate the new NPV if both spot rates rise in parallel by 0.50 percentage points, or 0.005.
Method. First calculate the receipt present value and the old NPV. Then calculate the parallel modified duration of the receipts and use the first-order sensitivity formula. Keep unrounded values until the final answer.
The present values of the receipts are $$ V_1=\frac{130}{1.04}=125, \qquad V_2=\frac{130}{(1.05)^2} =\frac{130}{1.1025} =117.9138321995\ldots $$ Therefore, $$ V=125+117.9138321995\ldots=242.9138321995\ldots $$ and $$ \operatorname{NPV}_{\mathrm{old}} =242.9138321995\ldots-240 =2.9138321995\ldots $$ The numerator of parallel modified duration is $$ \frac{1(130)}{(1.04)^2} + \frac{2(130)}{(1.05)^3} = 120.1923076923\ldots+224.597776\ldots = 344.7900837\ldots $$ Thus, $$ D_{\parallel} =\frac{344.7900837\ldots}{242.9138321995\ldots} =1.4193925\ldots $$ The estimated change in receipt value is $$ \Delta V\approx -(242.9138321995\ldots)(1.4193925\ldots)(0.005) =-1.7239504\ldots $$ The estimated new NPV is $$ 2.9138321995\ldots-1.7239504\ldots =1.1898818\ldots $$ The first-order estimated new NPV is $1.19, rounded to the nearest cent.
Check. A rate increase lowers the present value of positive future receipts, so NPV should fall. The estimate falls from about $2.91 to $1.19, as expected. It remains positive under this first-order estimate.
Common mistake avoided: multiplying duration by the project NPV of about $2.91 would use the wrong base. The sensitivity formula uses the receipts’ present value, about $242.91; the fixed time-0 cost is then included when updating NPV.
Practical Checks and Exam Traps
Integrated project questions often combine dates, signs, and rates. A reliable solution keeps the three calculations separate until each has been checked.
Record every outflow and receipt at its actual date. Treat the initial outlay as a negative time-0 cash flow in NPV.
Discount each cash flow with the spot rate for its maturity. Do not use a one-year forward rate as if it were a spot rate from time 0.
For each receipt, apply the forward rates for periods after the receipt date, not before it.
Use the parallel duration of discounted receipts for a parallel spot shift. Leave a fixed time-0 outlay unchanged.
A frequent rate-convention error is entering a percentage-point change as a whole percentage number: a 0.50 percentage-point increase is \(0.005\), not \(0.50\). Another is interpreting an implied forward rate as a promise about future realized reinvestment. For valuation consistency, the forward curve is derived from the spot curve; for a realized terminal result, the rates actually earned on reinvestment matter.
These linked calculations are useful when comparing staged projects, testing how a rate scenario affects a project’s value, or distinguishing initial value from terminal wealth. They are aligned with the SOA Exam FM (Financial Mathematics) syllabus. The key discipline is to state the valuation date, the cash-flow date, and the rate period before calculating.
Check Your Understanding
For each problem, use annual effective rates and retain unrounded values until the final answer.
- A project costs $150 at time 0 and pays $60 at year 1 and $110 at year 2. The one-year and two-year spot rates are 4% and 5%, respectively. Find the curve-based NPV. (Answer: $7.47)
- A project costs $90 now, pays $30 at year 1 and $70 at year 2, and is evaluated at year 3. The forward rates for years 0–1, 1–2, and 2–3 are 2%, 3%, and 4%. Find net terminal wealth at year 3. (Answer: $6.60)
- A project costs $190 and pays $103 at year 1 and $108.16 at year 2. The spot rates are 3% and 4%. Find the receipts’ parallel modified duration and estimate the new NPV after a parallel increase of 0.50 percentage points. (Answer: duration \(1.4470\) years; estimated NPV $8.55)
- The annual effective spot rates are 3% for one year and 4% for two years. Find the implied forward rate for the period from year 1 to year 2. (Answer: 5.0097%)
- A project’s receipt value is $500 and its parallel modified duration is 2.4. Its initial outlay is fixed at $470. Estimate the new NPV after a parallel decrease of 0.25 percentage points, given an original NPV of $30. (Answer: estimated new NPV $33.00)