Spot Valuation with Irregular Payment Dates

Why Exact Payment Dates Matter

A cash-flow schedule does not always line up with annual or semiannual dates. A payment may be due in 5 months, 15 months, or on another irregular date. Its value still follows the same principle as any other dated payment: discount it from its exact maturity using the discount factor assigned to that maturity. Using a factor for a nearby date changes the valuation.

This matters whenever a schedule combines dates that do not share a regular payment interval. The calculation is manageable once the dates and units are clear. The main work is matching each amount to the correct point on the spot curve, then summing the present values.

Definition: Let $t$ be a cash flow's maturity, measured in years from the valuation date, and let $v_t$ be the spot discount factor for that exact maturity. A cash flow of $C_t$ dollars due at time $t$ has present value $C_t v_t$. If the annual effective spot rate for maturity $t$ is $i_t$, then $v_t=(1+i_t)^{-t}$, provided the rate is quoted as an annual effective rate and $t$ is measured in years.

The spot rate is maturity-specific: in general, $i_{0.5}$, $i_{1.5}$, and $i_{2.5}$ need not be equal. Likewise, a discount factor quoted at one maturity is not automatically the discount factor for another. If a payment date falls between quoted maturities, a valuation requires a factor for that date or a specified rule for obtaining one. Do not assume an interpolation or extrapolation method unless the problem states it.

A Reliable Valuation Procedure

Before calculating, identify the valuation date and express each payment's time from that date in the stated year units. A month count, for example, is not itself a number of years: 15 months is $15/12=1.25$ years. If a day-count convention is specified, use it; if not, do not invent one. Then match every cash flow to the spot factor or spot rate at its exact maturity.

Set the time origin.
Record the valuation date as time zero and express every payment date as a maturity from that date.

Match maturity to curve information.
Use the discount factor given for the exact payment maturity. If a rate is given instead, confirm its quotation basis before converting it.

Discount each amount.
Multiply each signed cash flow by its matching spot discount factor.

Combine the present values.
Add the discounted amounts, retaining negative signs for outflows.

Why the Present Values Add

Suppose a deterministic schedule has payments $C_{t_1},C_{t_2},\ldots,C_{t_n}$ at times $t_1,t_2,\ldots,t_n$. Under the stated spot-rate model, one dollar invested at time zero to maturity $t_j$ accumulates to $(1+i_{t_j})^{t_j}$. Therefore, the amount that grows to $C_{t_j}$ is $C_{t_j}/(1+i_{t_j})^{t_j}$. This is exactly $C_{t_j}v_{t_j}$.

The total schedule is valued by adding the time-zero values of its separate signed payments. This follows from the additivity of money values at the same date: replacing several present-value amounts with their sum does not change their combined value.

\operatorname{PV}=\sum_{j=1}^{n} C_{t_j}v_{t_j} =\sum_{j=1}^{n}\frac{C_{t_j}}{(1+i_{t_j})^{t_j}}

This relation assumes deterministic cash flows and a spot discount factor available for every payment maturity. It does not make a payment's value independent of its date. Each term uses its own maturity-specific factor.

Worked Example: Three Payments on Irregular Dates

A hypothetical contract pays $1,200 in 6 months, $2,000 in 15 months, and $3,000 in 30 months. At the valuation date, the spot discount factors for exactly 0.5, 1.25, and 2.5 years are 0.9600, 0.9100, and 0.8400, respectively. Find the present value. The payments are certain; there is no mortality or probability assumption.

Method. Convert each payment date to years, match it to its stated discount factor, and add the discounted payments. This avoids treating unequal intervals as if they shared one rate.

Payment in 6 months. Its maturity is $6/12=0.5$ years, so its present value is $1{,}200(0.9600)=\$1{,}152.00$.

Payment in 15 months. Its maturity is $15/12=1.25$ years, so its present value is $2{,}000(0.9100)=\$1{,}820.00$.

Payment in 30 months. Its maturity is $30/12=2.5$ years, so its present value is $3{,}000(0.8400)=\$2{,}520.00$.

The total is $1{,}152.00+1{,}820.00+2{,}520.00=\$5{,}492.00$. As a check, all three present values are below their positive future payments because each stated discount factor is below 1.

When Payments Share a Date

Different items in a schedule can occur at the same maturity. For example, a receipt and a fee may both be due 16 months from now. They can be combined into one signed net payment before discounting, because both use the same factor. Keeping their signs visible helps prevent an outflow from being mistakenly added as if it were an inflow.

Worked Example: Netting Inflows and Outflows at a Shared Maturity

A hypothetical project has a $6,000 receipt due in 9 months, a $3,400 receipt due in 16 months, a $900 expense due on that same 16-month date, and a $5,000 receipt due in 24 months. The exact-maturity discount factors are 0.9700 at 0.75 years, 0.9300 at $16/12$ years, and 0.8800 at 2 years. Find the net present value of these signed cash flows. All amounts are certain; no mortality or probability model is involved.

Method. First net the two payments at 16 months, then apply the factor for $16/12$ years. This avoids discounting either amount using a factor from a different date.

The net cash flow at 16 months is $3{,}400-900=\$2{,}500$. The three dated amounts and their present values are:

At 9 months, $6{,}000(0.9700)=\$5{,}820.00$.

At 16 months, $2{,}500(0.9300)=\$2{,}325.00$.

At 24 months, $5{,}000(0.8800)=\$4{,}400.00$.

Adding gives $5{,}820.00+2{,}325.00+4{,}400.00=\$12{,}545.00$. The calculation also checks against discounting the receipt and expense separately at 16 months: $3{,}400(0.9300)-900(0.9300)=3{,}162.00-837.00=\$2{,}325.00$.

Using Annual Effective Spot Rates at Fractional Maturities

Sometimes a problem gives a spot rate rather than a discount factor. If the rate for maturity $t$ is an annual effective rate $i_t$, the corresponding discount factor is $(1+i_t)^{-t}$. The exponent is the maturity in years, not the number of whole years rounded down or the number of payment dates in the schedule.

Worked Example: Converting Fractional-Year Spot Rates

A hypothetical payment schedule has a $2,100 receipt in 6 months and a $6,000 receipt in 18 months. The annual effective spot rate is 10.25% at each of those exact maturities. Find the present value to the nearest cent. The payments are certain; there is no mortality or probability assumption.

Method. Convert each annual effective spot rate to a discount factor using that payment's exact term in years. The common error to avoid is using one year as the exponent for both payments.

For the 6-month payment, $t=6/12=0.5$. Its accumulation factor is $1.1025^{0.5}=\sqrt{1.1025}=1.05$, so its discount factor is $1/1.05$. The present value is $2{,}100/1.05=\$2{,}000.00$.

For the 18-month payment, $t=18/12=1.5$. Since $1.1025=1.05^2$, its accumulation factor is $1.1025^{1.5}=(1.05^2)^{1.5}=1.05^3=1.157625$. Its present value is $6{,}000/1.157625=\$5{,}183.03$, rounded to the nearest cent.

The total present value is $2{,}000.00+5{,}183.03=\$7{,}183.03$. As a check, each positive payment is discounted by a factor below 1, and the 18-month payment receives more discounting than the 6-month payment at the same positive rate.

Common Traps and Scope of the Calculation

The most important exam trap is matching a date to a convenient but incorrect factor. A 15-month payment is due at 1.25 years; it is not a 1-year payment simply because the curve has a 1-year quote. If the problem supplies a factor only at 1 year and 2 years, that information alone does not specify the factor at 1.25 years. A valuation at the unquoted date needs an additional factor or an explicitly prescribed interpolation rule.

Another frequent error is applying a stated annual effective spot rate with an integer exponent to a fractional-year maturity. For a payment at 18 months, the exponent is $1.5$, not 1 or 2. Also keep valuation date, time units, and signs consistent: an outflow is negative when computing a net present value, and every factor must be associated with a maturity measured from the same valuation date.

Key takeaway: For each payment, establish its exact maturity first. Use only the discount factor for that maturity, or convert the spot rate for that exact maturity using the stated quotation convention. If the necessary curve information is missing, do not silently substitute a nearby rate or invent an interpolation method.

Exact-date spot valuation is a basic tool for comparing dated receipts and payments on a common time-zero basis. The method is aligned with the SOA Exam FM (Financial Mathematics) syllabus: date conversion, maturity-specific discounting, and careful treatment of signed cash flows are essential whenever the payment schedule is irregular.

Check Your Understanding

For each question, use the factor or rate stated for the exact maturity. Treat all cash flows as certain, with no mortality or probability assumptions.

A fund will receive $800 in 6 months and $1,500 in 15 months. The exact-maturity discount factors are 0.9400 at 0.5 years and 0.9000 at 1.25 years. Find the present value. (Answer: $2,102.00)
A contract pays $2,400 in 0.8 years. At 1.6 years, it pays $4,000 and charges a $600 fee. The discount factors at 0.8 and 1.6 years are 0.9650 and 0.9250. Find the net present value. (Answer: $5,461.00)
A schedule pays $2,100 in 6 months and $6,000 in 18 months. The annual effective spot rate is 10.25% at both exact maturities. Find the present value to the nearest cent. (Answer: $7,183.03)
A payment of $2,500 is due in 1 year. A problem provides discount factors only at 0.5 years and 1.5 years, and gives no interpolation rule. Can its present value be determined from the supplied information? (Answer: No; the 1-year discount factor or a specified rule to obtain it is needed.)
A schedule pays $1,000 in 0.4 years, $2,500 in 1.1 years, and $5,000 in 2.3 years. The exact-maturity discount factors are 0.9900, 0.9500, and 0.8600, respectively. Find the present value. (Answer: $7,665.00)

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