Present Value of a Single Dated Payment

Why the Exact Payment Date Matters

A single future payment is valued by asking what amount at the valuation date is financially equivalent to that payment. The answer depends on the time remaining until payment and the interest information for that maturity. In a term structure, rates can differ by maturity, so a rate for one date is not automatically appropriate for another.

This tutorial considers one certain payment, with no default, contingency, or embedded option. The valuation date is time 0, and the payment is due at time $t>0$. Time $t$ is measured in years unless the rate is explicitly stated per another period. The task is to find the present value using the discount factor or spot rate that applies to this exact payment date.

Definition: The discount factor $v(0,t)$ is the time-0 value of one unit of currency payable with certainty at time $t$, under the stated market terms. Thus, the present value of a certain payment $C$ due at time $t$ is $C v(0,t)$. The spot rate for maturity $t$ is the rate that expresses this same discount factor over the interval from time 0 to time $t$.

A discount factor is specific to both the valuation date and the payment date. It is not necessarily the same as the one-year discount factor $v=1/(1+i)$ from a single-rate model. In the spot-rate setting, use the factor for the payment's exact maturity, not a factor chosen merely because its rate is familiar or easy to use.

Present Value Using a Spot Rate or Discount Factor

If $i_t$ is the annual effective spot rate for maturity $t$ years, it is defined so that one unit invested at that rate accumulates to $(1+i_t)^t$ over those $t$ years. Therefore, the discount factor and the present value of $C$ due at time $t$ are:

v(0,t) = \frac{1}{(1+i_t)^t}, \qquad \operatorname{PV}_0(C) = C v(0,t) = \frac{C}{(1+i_t)^t}.

This formula uses an annual effective spot rate that applies to the exact maturity $t$. The exponent is the number of years to the payment, including a fraction of a year when appropriate. If the given rate is instead effective per period, use the number of those periods to the payment date. If a discount factor for that date is already provided, multiply the payment by that factor directly rather than converting it to a rate unnecessarily.

Rate-unit rule: Match the exponent to the rate's period. An annual effective rate uses a time in years; an effective quarterly rate uses a count of quarters. A continuously compounded spot rate uses a time in years in an exponential discount factor.

For a continuously compounded spot rate $\delta_t$ applying to maturity $t$, accumulation over the interval is $e^{\delta_t t}$. Its corresponding discount factor is $e^{-\delta_t t}$, and the present value of $C$ is $C e^{-\delta_t t}$. Do not use this expression for a rate quoted as an effective annual rate; the compounding convention is part of the rate information.

Why the Discounting Formula Holds

The spot rate is defined by the accumulation relationship for one unit invested from time 0 to time $t$. If the annual effective spot rate is $i_t$, that unit becomes $(1+i_t)^t$. Present value reverses this accumulation: a present amount $P$ that grows to $C$ must satisfy $P(1+i_t)^t=C$. Solving for $P$ gives the spot-rate present-value formula.

P(1+i_t)^t=C \quad\Longrightarrow\quad P=\frac{C}{(1+i_t)^t} =Cv(0,t).

The same reasoning gives the continuous version. If $P$ accumulates at force $\delta_t$ for $t$ years, then $P e^{\delta_t t}=C$, so $P=C e^{-\delta_t t}$. These relationships assume the stated rate convention applies over the full interval and that the payment is certain. A rate quoted for a different maturity, or a rate with a different compounding convention, cannot simply be substituted without adjustment.

Worked Examples

Worked Example: A One-Year Payment with a Quarterly Effective Rate

A certain payment of $5,000 is due exactly one year from today. The effective spot rate is 25% per quarter for this one-year maturity. Find its present value. State the rate period and the number of periods used.

Method. The rate is effective per quarter, not per year. One year contains four quarters, so discount through four quarterly periods:

The quarterly accumulation factor is $1+0.25=1.25$. Over four quarters it is:

$$ (1.25)^4 = (1.5625)^2 = 2.44140625. $$

Therefore:

\operatorname{PV}_0 = \frac{5{,}000}{2.44140625} = 2{,}048.

The present value is $2,048.00. Check: $2,048 accumulated for four quarters at 25% per quarter is $2{,}048(2.44140625)=5{,}000$. The common mistake this method avoids is using one year as one period even though the quoted rate is per quarter.

Worked Example: Continuous Spot Discounting to 18 Months

A certain payment of $10,000 is due 18 months from today. The continuously compounded spot rate for this maturity is 4% per year. Find its present value to the nearest cent.

Method. Convert the payment date to years: 18 months is $18/12=1.5$ years. The rate is a continuous force, so use $e^{-\delta_t t}$, with $\delta_t=0.04$ and $t=1.5$:

\delta_t t = 0.04(1.5)=0.06, \qquad e^{-0.06}\approx 0.9417645336.

Multiply the payment by this maturity-specific discount factor:

\operatorname{PV}_0 =10{,}000e^{-0.06} \approx 10{,}000(0.9417645336) =9{,}417.645336.

Rounded to the nearest cent, the present value is $9,417.65. Check: $9{,}417.645336e^{0.06}\approx 10{,}000$. The common mistake this method avoids is applying an annual effective-rate power formula to a continuously compounded rate.

Worked Example: Using a Discount Factor for an Exact Nonannual Date

A certain payment of $24,000 is due 11 months from today. The discount factor from today to exactly 11 months is 0.985. Find the present value and explain why a one-year rate is not needed.

Method. The given factor is already for the exact payment date. Multiply it by the payment:

\operatorname{PV}_0 =24{,}000(0.985) =24{,}000(1-0.015) =24{,}000-360 =23{,}640.

The present value is $23,640.00. Check: $23{,}640/24{,}000=0.985$, the supplied discount factor. No one-year spot rate is needed: the problem gives the discount factor for 11 months directly. The common mistake this method avoids is replacing that exact-date factor with a one-year factor without a stated basis for doing so.

A Reliable Procedure for One Dated Payment

A short, deliberate sequence helps prevent errors, particularly when a problem gives a nonannual date or a rate in an unfamiliar unit.

Identify the payment date.
Measure the time from the valuation date to the payment. Convert months, quarters, or days to the unit required by the stated rate.

Identify what is supplied.
Determine whether the problem gives an exact-date discount factor, an annual effective spot rate, an effective rate per period, or a continuous spot rate.

Choose the matching formula.
Use $C v(0,t)$ for a supplied discount factor, $C/(1+i_t)^t$ for an annual effective spot rate, a period count for a per-period effective rate, or $C e^{-\delta_t t}$ for continuous discounting.

Calculate and check.
Keep the unrounded factor through the calculation, round the final money amount to cents, and verify that re-accumulating the present value gives the stated payment.

Common Mistakes and Exam Traps

The arithmetic in a one-payment valuation is usually short. Most errors come from interpreting the rate or the date incorrectly.

Using the wrong number of periods. A rate effective per quarter must be applied once for each quarter to the payment date. For a one-year payment, that is four periods, not one.
Using the wrong maturity's spot rate. A spot rate for one year is not automatically the rate for an 11-month or 18-month payment. Use the rate or factor for the actual maturity specified.
Mixing rate conventions. An annual effective spot rate, a rate effective per quarter, and a continuously compounded spot rate imply different accumulation formulas. Do not treat their numerical percentages as interchangeable.
Rounding too early. Keep the discount factor at full calculator precision until multiplying by the payment. Round the final dollar value, not each intermediate quantity.
Reversing accumulation and discounting. A future payment is brought back by dividing by its accumulation factor, or equivalently multiplying by its discount factor. Multiplying by $(1+i_t)^t$ moves a present amount forward instead.

Key takeaway: For one certain payment, present value equals the payment multiplied by the discount factor for its exact maturity. When using a spot rate, the rate's compounding convention and the payment-date units determine the correct factor.

Check Your Understanding

For each problem, identify the relevant maturity and rate convention before calculating. Round monetary answers to cents where needed.

A certain $6,300 payment is due in two years. The annual effective spot rate for this maturity is 5%. Find its present value. (Answer: $5,714.29)
A certain $1,331 payment is due in nine months. The effective spot rate is 10% per quarter. Find its present value. (Answer: $1,000.00)
A certain $10,000 payment is due in two years, and the continuously compounded spot rate for that maturity is 3% per year. Find its present value to the nearest cent. (Answer: $9,417.65)
A certain $6,250 payment is due in 14 months. Its discount factor from today to exactly 14 months is 0.972. Find its present value. (Answer: $6,075.00)
A certain $10,000 payment is due in 18 months. The effective spot rate is 2% per quarter. Find its present value to the nearest cent. (Answer: $8,879.71)

What You'll Learn