Discount Factors for Dated Cash Flows

What a Discount Factor Represents

A term-structure table may list one number for each future date. That number is a discount factor: it tells how much a payment at that date is worth at the valuation date. A factor of $0.94$, for example, means that one unit payable on the specified date has present value $0.94$ units at the valuation date. It does not mean that the payment itself is $0.94$, nor is the factor an interest rate.

The date attached to a factor matters. A factor for a payment in one year is not automatically the factor for a payment in two years. Likewise, the time unit matters: a row labelled 18 months is not the same as a row labelled 18 years. Before using a table, identify its valuation date, the dates or maturities shown, and the units in which payments are measured.

Definition: For a fixed valuation date, the discount factor $D(t)$ is the present value at that date of one unit payable at maturity $t$. Thus $D(0)=1$, because one unit payable immediately is already worth one unit. If a payment of $C$ units is due at maturity $t$, its present value is $C D(t)$, provided the payment and factor use the same currency and unit.

The notation $D(t)$ is used here to label the factor by maturity. A table might instead print the maturity in a separate column and give only the numerical factor. Either way, each factor belongs to its stated date and to the valuation date for which the table was prepared. A table with a different valuation date describes a different set of present values.

A discount factor is a price-like quantity, not a percentage return. It can be multiplied by a payment amount to obtain a present value. A spot rate, by contrast, is a rate associated with a maturity. The earlier material on spot rates distinguishes rates by maturity; this tutorial focuses on reading the corresponding factors as values attached to dated unit payments. The relationship between a spot rate and a discount factor is a separate calculation.

Reading a Table by Date

A useful way to read a discount-factor table is to treat each row as answering one question: “What is one unit payable on this date worth at the valuation date?” For example, if a table is valued at time 0 and gives $D(1.5)=0.93$, then one unit payable 1.5 years after time 0 has present value $0.93$ units at time 0. The value is not $0.93$ units at the payment date; at that date the payment is one unit.

When the payment amount is not one unit, scale the factor by that amount. A payment of 600 units at the date associated with $D(t)=0.93$ has present value $600(0.93)=558$ units. This follows from the basic property that doubling a fixed payment doubles its present value under the same valuation basis. The factor does not change merely because the payment amount changes.

For a cash flow that is split into payments at different dates, each payment must be matched with the factor for its own date. If all factors are supplied, the present values of the separate payments can be added. This operation relies on valuing each dated amount using the same valuation date and a consistent currency and unit. A factor from the wrong row changes the value being calculated.

Worked Example: Scaling a Factor for a Single Dated Payment

A table valued at time 0 gives $D(0.5)=0.985$, $D(1)=0.960$, and $D(2)=0.910$, with maturities measured in years. A contract pays \$2,500 exactly two years after time 0. Find the present value indicated by the table.

Method. Match the payment date to the two-year row, interpret that row as the present value of one dollar, and scale by the \$2,500 payment.

The payment is due at $t=2$, so the applicable factor is $D(2)=0.910$. The one-dollar interpretation is that \$1 due in two years has a present value of \$0.910 at time 0.

\text{Present value}=2{,}500(0.910)=2{,}275

The table therefore assigns a present value of \$2,275.00 to the payment. As a check, the result is less than the amount due because the factor used is less than 1. The common mistake is to use $D(1)=0.960$ because it is the middle row, or to treat $0.910$ as a rate and calculate a different amount. The payment date—not the row’s position or a rate interpretation—selects the factor.

Interpreting Dated Factors Carefully

The maturity label must be read relative to the table’s valuation date. Suppose the valuation date is 1 January and a table labels a row “18 months.” That row refers to a payment 18 months after 1 January, not 18 months after the date on which the table happens to be read. If the problem gives calendar payment dates rather than maturities, first determine the time from the stated valuation date using the problem’s date convention.

Also check whether the table reports factors per dollar, per \$100, or in another unit. If a factor is stated per \$100 of payment, multiplying it by a payment expressed as a number of dollars without first reconciling the units produces a scale error. Unless stated otherwise, examples here interpret each factor as the present value of one unit of the same currency as the payment.

A table ordinarily provides factors only at specified maturities. If a payment falls between those maturities, the table alone does not say what factor to use at the intervening date. The problem must supply an additional rule or enough information to determine it. Do not silently select the nearest row, assume linear interpolation, or treat the factor as constant between rows. Such choices require an explicit assumption.

Nor should a reader assume that discount factors must decline from one row to the next. A factor is attached to a maturity and reflects the value of a unit payable at that maturity. Depending on the term structure, a later factor can be higher than an earlier one; negative rates are one possible reason. A table with a factor greater than 1 is therefore not automatically erroneous. Check the table’s assumptions and labels before deciding whether a value is inconsistent.

Key interpretation: A factor below 1 means the unit payment is assigned a present value below one unit at the valuation date; a factor above 1 means it is assigned a value above one unit. Neither number is itself a percentage rate, and a factor’s meaning depends on its maturity and valuation date.

Why Scaling and Adding Factors Works

The basic use of a discount-factor table follows from proportionality and additivity of present values under a fixed valuation basis. For one unit payable at $t$, the definition gives present value $D(t)$. A payment of $C$ units is $C$ copies of that unit payment, so its present value scales to $C D(t)$. For payments at several dates, each dated payment is valued at its own date’s factor; the total is the sum of those separate present values.

This justification depends on using factors for the same valuation date and consistent units. It does not license applying one maturity’s factor to a payment due at another maturity, or inventing a factor for a date absent from the table.

\begin{aligned} \text{PV of }C\text{ payable at }t &= C D(t),\\ \text{PV of payments }C_1,\ldots,C_n\text{ at }t_1,\ldots,t_n &= \sum_{k=1}^{n} C_kD(t_k). \end{aligned}

Worked Example: Matching Payments to Different Maturities

A valuation table has a time-0 valuation date and gives $D(0.5)=0.980$ and $D(2)=0.900$, with times measured in years. A hypothetical agreement pays \$1,200 after six months and \$3,000 after two years. Find the present value indicated by the table.

Method. Value each payment separately using the factor for its own maturity, then add the two present values. This avoids treating the two payments as if they shared a payment date.

The six-month payment uses $D(0.5)=0.980$:

$$ 1{,}200(0.980)=1{,}176 $$

The two-year payment uses $D(2)=0.900$:

$$ 3{,}000(0.900)=2{,}700 $$

Adding the present values gives:

$$ 1{,}176+2{,}700=3{,}876 $$

The table therefore indicates a total present value of \$3,876.00. As a check, both factors are below 1, so each payment has a present value below its stated amount, and the total is below the total payments of \$4,200. The common mistake is to apply the two-year factor to both payments, which would ignore the six-month payment date.

Worked Example: Reading a Maturity and Unit Correctly

A table valued at 1 January lists the following factors, with maturities measured from that valuation date: $D(90\text{ days})=0.992$, $D(1\text{ year})=0.965$, $D(18\text{ months})=0.941$, and $D(3\text{ years})=0.875$. A hypothetical invoice for \$10,000 is due exactly 18 months after 1 January. Find its table-based present value, and explain why the one-year factor is not used.

Method. Express the payment date as a maturity from the valuation date, select the matching row, and multiply by the payment amount. Here the problem directly states that the due date is 18 months after 1 January.

The matching table entry is $D(18\text{ months})=0.941$, interpreted as the present value of one dollar due on that date. Thus:

$$ 10{,}000(0.941)=9{,}410 $$

The indicated present value is \$9,410.00. The one-year factor $0.965$ applies to a payment one year after 1 January, not to this invoice. Using it would give $10{,}000(0.965)=\$9,650$, but that calculation prices a payment at the wrong maturity. The check is the date label: 18 months matches the invoice due date exactly, so no interpolation is needed.

Common Table-Reading Errors

Discount-factor questions are often short, but a small reading error can affect every subsequent calculation. Use a consistent sequence: locate the valuation date, identify the payment date as a maturity from that date, match the factor, check the units, and only then multiply by the payment amount.

Using the wrong maturity. The closest row is not necessarily the correct row. Use the exact date shown, or an explicitly stated rule if the date is not listed.
Reading a factor as a rate. A factor such as $0.941$ is the present value of one unit at a specified maturity. It is not a rate of 94.1% or a discount rate of 5.9% unless a separate relationship and convention are supplied.
Forgetting the valuation date. “One year” is measured from the table’s valuation date, not automatically from today or from another event in the problem.
Mixing units. Confirm whether amounts and factors are stated per dollar, per \$100, or in another scale before multiplying.
Inventing missing values. A table that lists one and two years does not, by itself, determine the factor for 18 months.
Assuming factors always fall with maturity. The entries are maturity-specific values; do not reject a table solely because a later factor is larger.

These checks are useful whenever a problem supplies a term structure as a table rather than as a list of rates. The table should be read as a collection of dated values, not as a single rate applied uniformly across all future payments. Keeping the date attached to each factor is the central discipline.

Check Your Understanding

For each question, use only the stated valuation date and discount factors. Amounts and factors are in dollars and per dollar unless specified otherwise.

A table valued at time 0 gives $D(2)=0.88$. What present value does it assign to \$7,500 due in two years? (Answer: \$6,600.00)
A table gives $D(6\text{ months})=0.99$, $D(1\text{ year})=0.96$, $D(18\text{ months})=0.93$, and $D(2\text{ years})=0.90$. What is the indicated present value of \$4,000 due in 18 months? (Answer: \$3,720.00)
At time 0, a table gives $D(1)=0.97$ and $D(3)=0.86$. A hypothetical portfolio pays \$500 in one year and \$800 in three years. What total present value do the factors indicate? (Answer: \$1,173.00)
A table gives $D(1.5)=0.94$ and $D(2.5)=0.89$. Compare the indicated present values of two separate \$1,000 payments, one due in 1.5 years and one due in 2.5 years. (Answer: \$940.00 at 1.5 years and \$890.00 at 2.5 years; the first is \$50.00 higher.)
A table gives $D(1)=1.02$. What present value does it assign to \$500 due in one year, and does the factor alone establish that the table is erroneous? (Answer: \$510.00; no, a factor above 1 is not by itself evidence of an error.)

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