Project Valuation Under a Nonflat Term Structure

Why Project Values Depend on Payment Dates

Two projects can have the same total receipts and still have different present values if those receipts arrive at different times. Under a nonflat term structure, the discount factor for a payment in one year need not match the discount factor for a payment in three years. A sound comparison therefore values each cash flow at its own maturity before comparing the resulting net present values.

The calculations in this tutorial assume that the cash flows are deterministic and that the stated term structure is the appropriate one for valuing them. In an exam problem, use the curve specified in the question. In an actual investment analysis, risky expected cash flows may require a risk adjustment; a risk-free spot curve alone does not automatically account for project risk.

Definition: Let $D_t$ be the discount factor from time $t$ to time 0. A cash flow of $C_t$ at time $t$ has present value $C_tD_t$. For a project with cash flows $C_0,C_1,\ldots,C_n$, where receipts are positive and outlays are negative, its net present value is $$ \operatorname{NPV}=\sum_{t=0}^{n} C_tD_t,\qquad D_0=1. $$ If the annual effective spot rate for maturity $t$ is $i_t$, then $D_t=(1+i_t)^{-t}$. The rate $i_t$ is maturity-specific; it is not necessarily the same for each $t$.

The expression $C_tD_t$ places each cash flow at time 0. The initial cash flow is already at time 0, so its discount factor is 1. A positive NPV means that the present value of receipts exceeds the present value of outlays, under the stated valuation assumptions. When two comparable, mutually exclusive projects are being judged by NPV, the project with the higher NPV is preferred, provided the decision-maker’s objective is to maximize value and no other constraint changes the choice.

Valuing a Stream with Maturity-Specific Factors

A useful way to organize the calculation is to attach a time and a discount factor to every cash flow. The factor is selected by the cash flow’s maturity, not by the order in which the cash flows are listed or by the project’s average payment date.

Set the valuation date.
Use the stated focal date. Here, project NPVs are measured at time 0.

Record signed cash flows by date.
Enter outlays as negative amounts and receipts as positive amounts. Do not omit an initial investment.

Match each date to its factor.
Use $D_t$ for a cash flow at time $t$, even when the curve is not flat.

Sum present values and compare.
Calculate each project’s NPV separately, then rank the results using the decision criterion stated in the problem.

The formula follows directly from the meaning of a discount factor. By definition, one unit payable at time $t$ is worth $D_t$ at time 0, so $C_t$ units payable then are worth $C_tD_t$. Present value is additive across cash flows: valuing each component and adding the results gives the value of the whole stream. Including the time-0 cash flow gives the stated NPV sum, since $C_0D_0=C_0$.

\operatorname{NPV} =C_0+C_1D_1+C_2D_2+\cdots+C_nD_n

For example, if a curve supplies annual effective spot rates rather than discount factors, first convert each rate at its own maturity: $D_t=(1+i_t)^{-t}$. Do not apply the one-year rate repeatedly unless the curve actually specifies that rate for every period. If the problem gives discount factors directly, use them directly; converting them to spot rates and back is unnecessary and can introduce rounding error.

Worked Example: Finding a Break-Even Receipt

The applicable annual spot discount factors are $D_1=0.96$, $D_2=0.90$, and $D_3=0.80$. Project G costs $90 at time 0 and pays $40 at year 1 and $X$ dollars at year 3. Project H costs $90 at time 0 and pays $80 at year 2. Find $X$ that makes the two NPVs equal. Then state which project has the higher NPV if $X=$50$. The cash flows are assumed deterministic and the given curve applies to both projects.

Method. Calculate each NPV using the factor for each payment date, equate the expressions to find the break-even amount, and then compare at the specified value of $X$.

For G, the time-0 cost is $-$90$, the year-1 receipt has present value $40(0.96)=$38.40$, and the year-3 receipt has present value $0.80X$. Thus,

\operatorname{NPV}_G=-90+40(0.96)+0.80X =-90+38.40+0.80X =-51.60+0.80X.

For H, the year-2 receipt has present value $80(0.90)=$72.00$, so

\operatorname{NPV}_H=-90+80(0.90)=-90+72=-18.00.

At break-even, $-51.60+0.80X=-18.00$. Therefore $0.80X=33.60$, and $X=33.60/0.80=$42.00$. Substitution checks the result: G’s NPV is $-51.60+0.80(42)=-51.60+33.60=-$18.00$, equal to H’s NPV.

If $X=$50$, then G’s NPV is $-51.60+0.80(50)=-51.60+40=-$11.60$, which exceeds H’s $-$18.00$ by $6.40. G ranks higher at that receipt amount. This method avoids treating the year-3 payment as though it had the year-2 discount factor.

Comparing Projects with Different Signs and Dates

The same NPV calculation applies when a project has an additional outlay after the initial investment. Such an outlay must be entered as a negative cash flow at its actual date. Omitting it, or discounting it as a receipt, can reverse the ranking.

Worked Example: Ranking Projects with a Later Outlay

Use $D_1=0.96$, $D_2=0.90$, and $D_3=0.80$. Project J has cash flows of $-$120$ at time 0, $+$100$ at year 2, and $-$15$ at year 3. Project K has cash flows of $-$120$ at time 0, $+$50$ at year 1, and $+$55$ at year 3. Find each NPV and identify the higher-NPV project. Assume the same applicable curve for both projects.

Method. Multiply each signed cash flow by the factor for its date and sum. The year-3 cash flow for J is an outlay, so its present value is negative.

For J, the year-2 receipt has present value $100(0.90)=$90.00$. The year-3 outlay has present value $-15(0.80)=-$12.00$. Therefore,

\operatorname{NPV}_J=-120+90-12=-\$42.00.

For K, the year-1 receipt has present value $50(0.96)=$48.00$, and the year-3 receipt has present value $55(0.80)=$44.00$. Therefore,

\operatorname{NPV}_K=-120+48+44=-\$28.00.

Since $-28>-42$, K has the higher NPV, by $(-28)-(-42)=$14.00$. A direct check of the arithmetic gives J: $-120+90-12=-42$, and K: $-120+48+44=-28$. Both NPVs are negative, so both projects have present-value costs exceeding present-value receipts under these assumptions; K is the better of the two by the stated NPV criterion.

This calculation avoids two frequent errors: treating J’s year-3 outlay as positive and reporting $-$52$ instead of the correctly summed $-$42$. The signed cash-flow timeline should be checked before interpreting the ranking.

Payment Timing Can Change the Ranking

A nonflat curve makes the payment dates especially important. Even when two projects have the same initial cost and the same total receipts, their present values may differ because their receipts are discounted by different factors. The following example isolates this timing effect: the receipt amounts are equal across projects in total, but each receipt arrives at different dates.

Worked Example: Equal Total Receipts, Different Timing

The discount factors are $D_1=0.96$, $D_2=0.90$, $D_3=0.80$, and $D_4=0.72$. Project P costs $100 at time 0 and receives $60 at year 1 and $60 at year 3. Project Q costs $100 at time 0 and receives $60 at year 2 and $60 at year 4. Compare their NPVs. Assume deterministic cash flows and that the stated factors apply.

Method. Discount the two receipts in each project separately. Each project has total nominal receipts of $60+60=$120$, but the dates differ.

For P, the year-1 receipt is worth $60(0.96)=$57.60$, and the year-3 receipt is worth $60(0.80)=$48.00$. Thus,

\operatorname{NPV}_P=-100+57.60+48.00=\$5.60.

For Q, the year-2 receipt is worth $60(0.90)=$54.00$, and the year-4 receipt is worth $60(0.72)=$43.20$. Thus,

\operatorname{NPV}_Q=-100+54.00+43.20=-\$2.80.

P has the higher NPV by $5.60-(-2.80)=$8.40$. The check is that both projects receive $120 in total, yet P’s receipts have the greater present value: $57.60+48.00=$105.60$, compared with $54.00+43.20=$97.20$. The difference is $8.40, matching the NPV difference because their initial costs are equal.

P’s receipts are both earlier than Q’s corresponding receipts, and the supplied factors decrease with maturity. That is why P’s present value is higher here. Do not assume that moving a receipt earlier always has a particular dollar effect without checking the relevant factors; the factors and amounts determine the result.

Common Mistakes and Decision Checks

A nonflat term structure does not change the basic idea of present value, but it removes the convenience of applying one common factor to every maturity. Before finalizing a comparison, check that the date, sign, and factor of each cash flow agree.

Using one rate for every maturity. If the curve supplies different spot rates or factors by maturity, apply the maturity-specific value to each cash flow. A single-rate calculation is appropriate only if the problem specifies a flat rate for the relevant maturities.
Discounting by the wrong date. A payment at year 3 uses $D_3$, not $D_2$ because it is the third item listed or because another payment is at year 2.
Dropping the initial investment. The time-0 outlay is included in NPV with factor $D_0=1$; it is not discounted again.
Ignoring a later outlay or its sign. A payment made by the project is negative even when it occurs after positive receipts.
Ranking by total receipts alone. Nominal totals do not reflect the timing of the cash flows. Compare present values when the decision criterion is NPV.
Using an unsuitable curve without comment. A supplied curve is usable for the project comparison under the question’s valuation assumptions. For risky cash flows outside that setup, risk must be treated consistently rather than assumed away.

These calculations are aligned with the SOA Exam FM (Financial Mathematics) syllabus. On an exam-style problem, a compact timeline is often the safest first step: write each signed amount beside its payment date, then attach its corresponding factor and sum.

Check Your Understanding

Use the stated discount factors for each question, assume deterministic cash flows, and measure NPV at time 0.

With $D_1=0.97$, $D_2=0.91$, and $D_3=0.82$, Project A costs $200 and pays $75 at year 1 and $150 at year 3. Project B costs $200 and pays $110 at year 2 and $100 at year 3. Calculate both NPVs and identify the higher one. (Answer: A has NPV $-$4.25$, B has NPV $-$17.90$; A is higher by $13.65.)
A project costs $75 now and pays $40 at year 1 and $50 at year 2. If $D_1=0.95$ and $D_2=0.88$, calculate its NPV. (Answer: $7.00.)
Using $D_1=0.96$, $D_2=0.89$, and $D_3=0.78$, Project A costs $250 and pays $100 at year 1 and $100 at year 3. Project B costs $250 and pays $200 at year 2. Compare their NPVs. (Answer: A has NPV $-$76.00$, B has NPV $-$72.00$; B is higher by $4.00.)
With $D_1=0.96$, $D_2=0.90$, and $D_3=0.80$, Project U costs $100 and pays $25 at year 1 and $X$ at year 3. Project V costs $100 and pays $80 at year 2. Find the value of $X$ that makes their NPVs equal. (Answer: $60.00.)
With $D_1=0.97$, $D_2=0.92$, and $D_3=0.84$, Project R pays $30 at year 1, $80 at year 2, and $100 at year 3. Project S costs $180 now and pays $100 at year 1 and $100 at year 3. If R’s initial cost is $175, calculate both NPVs and rank them. (Answer: R has NPV $11.70, S has NPV $1.00; R is higher by $10.70.)

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