Present Value of Multiple Dated Payments

Why Each Payment Needs Its Own Discount Factor

A stream of payments is not a single payment made at one date. Each amount is received or paid at its own maturity, so its present value depends on the discount factor for that date. The present value of the entire stream is found by discounting every dated amount to the same valuation date and adding the results.

This method is useful whenever a contract, investment, or liability specifies payments at several future dates. It makes the timing of each amount explicit and avoids treating a multi-date stream as though all of its cash flows had the same maturity.

Definition: A cash-flow schedule at valuation time 0 consists of amounts $C_{t_1}, C_{t_2}, \ldots, C_{t_n}$, payable at future times $t_1, t_2, \ldots, t_n$. Let $v_t$ be the discount factor from time $t$ to time 0, based on the spot rate for maturity $t$. The present value of the schedule is $$ \operatorname{PV}_0=\sum_{k=1}^{n} C_{t_k}v_{t_k}. $$

The amounts $C_{t_k}$ are signed: a receipt to the investor is positive, while a payment made by the investor is negative. In the examples below, the schedule consists of positive payments being valued as receipts. The valuation date is time 0, amounts are deterministic, and the applicable discount factors are known. No assumption about reinvesting intermediate receipts is needed to calculate present value.

If the spot rate for maturity $t$ is an annual effective rate $i_t$, and $t$ is measured in years, then $v_t=(1+i_t)^{-t}$. Alternatively, the maturity-specific discount factors may be supplied directly. In either case, use the factor corresponding to the cash flow's maturity. For fractional dates, the rate convention and time units must agree; a supplied discount factor already incorporates that convention.

Why the Present Values Add

The sum formula follows from the value of each individual dated payment. Discounting $C_{t_k}$ from time $t_k$ to time 0 gives $C_{t_k}v_{t_k}$. All these discounted amounts are now expressed at the same date. Under the usual additive valuation of deterministic cash flows, the value of the whole schedule is the sum of those amounts.

\begin{aligned} \operatorname{PV}_0 &=\operatorname{PV}_0(C_{t_1})+\operatorname{PV}_0(C_{t_2})+\cdots+\operatorname{PV}_0(C_{t_n})\\ &=C_{t_1}v_{t_1}+C_{t_2}v_{t_2}+\cdots+C_{t_n}v_{t_n}\\ &=\sum_{k=1}^{n} C_{t_k}v_{t_k}. \end{aligned}

This argument depends on putting every cash flow at the same valuation date before adding. Adding the undiscounted amounts first would produce a total at different future dates, not a present value. The formula also does not imply that one discount factor applies to every term: each $v_{t_k}$ belongs to its own maturity.

Calculating a Stream with Annual Maturities

A practical calculation can be organized as a table: list each payment date and amount, identify the applicable discount factor, and multiply row by row. Only after every payment has been discounted should the present values be added.

Worked Example: Three Payments with Different Annual Spot Rates

A hypothetical investment will pay $500 at the end of year 1, $800 at the end of year 2, and $1,500 at the end of year 3. The annual effective spot rates are 4.167% for year 1, 5.409% for year 2, and 7.722% for year 3. Find the present value at time 0. Use the corresponding discount factors $0.96$, $0.90$, and $0.80$, respectively.

Method. Discount each payment using the factor for its own maturity, then add. This avoids the common mistake of applying one rate to all three dates.

At year 1, the present value is $$ 500(0.96)=480. $$ At year 2, it is $$ 800(0.90)=720. $$ At year 3, it is $$ 1{,}500(0.80)=1{,}200. $$

The present value of the complete stream is therefore $$ 480+720+1{,}200=2{,}400. $$

Check. Each row uses a factor for the matching maturity, and the three discounted amounts sum to $2,400. The answer is in dollars at time 0.

The rate figures in that example are maturity-specific annual effective spot rates. For instance, the year-2 factor is $v_2=(1+i_2)^{-2}$, not a second application of the year-1 factor. When a problem gives discount factors directly, using those factors avoids unnecessary rate conversions.

Fractional Payment Dates

The same sum applies when a payment date falls between whole years. The maturity may be measured in fractions of a year, but the time unit and rate convention must be consistent. If the appropriate factor is provided for each date, multiply the payment by that factor just as for an integer maturity.

Worked Example: Payments at Half-Year and Fractional-Year Maturities

A hypothetical contract pays $2,000 at time 0.5, $3,000 at time 1.5, and $5,000 at time 2.5, where time is measured in years. The corresponding annual effective spot rates are approximately 4.123%, 4.211%, and 5.246%. The maturity-specific discount factors are $0.98$, $0.94$, and $0.88$. Find the present value at time 0.

Method. Use the factor supplied for each fractional maturity. The fact that the times are not all whole numbers does not change the rule: a payment is paired with the factor for its own date.

For the payment at time 0.5: $$ 2{,}000(0.98)=1{,}960. $$ For the payment at time 1.5: $$ 3{,}000(0.94)=2{,}820. $$ For the payment at time 2.5: $$ 5{,}000(0.88)=4{,}400. $$

Adding these values gives $$ 1{,}960+2{,}820+4{,}400=9{,}180. $$

Check. The discount factors are all less than 1, so each positive future payment has a present value below its undiscounted amount. The total, $9,180, is below the sum of the payments, $10,000.

A Stream Can Be Valued One Payment at a Time

A useful way to keep a longer schedule clear is to treat each payment as its own single-dated cash flow. Find that payment's present value, record it, and move to the next date. This approach reduces the chance of losing track of which rate, factor, or time unit belongs to which amount.

Set the valuation date and list the schedule.
Record every amount with its payment date, and identify whether each amount is a receipt or a payment from the perspective being valued.

Match each amount to a maturity.
For each payment, identify the spot rate or discount factor for that specific date. Check that the time unit is consistent with the rate convention.

Discount each amount separately.
Multiply each signed cash flow by its own discount factor to express it at the valuation date.

Add the discounted amounts and check.
Confirm that every scheduled cash flow appears once, with the correct sign and maturity, before reporting the total.

Worked Example: Valuing a Four-Date Payment Schedule

A hypothetical supplier agreement provides receipts of $600 at time 1, $1,400 at time 2, and $2,500 at time 3. The maturity-specific annual effective spot rates are approximately 3.093%, 4.813%, and 5.988%, giving discount factors $0.97$, $0.91$, and $0.84$. A separate $700 receipt is due at time 4, with a discount factor of $0.78$. Find the present value at time 0.

Method. Make one row for every dated receipt and apply its own factor. The year-4 amount must not be omitted merely because its rate is specified separately.

The present values are: $$ \begin{aligned} 600(0.97)&=582,\\ 1{,}400(0.91)&=1{,}274,\\ 2{,}500(0.84)&=2{,}100,\\ 700(0.78)&=546. \end{aligned} $$

Thus the present value is $$ 582+1{,}274+2{,}100+546=4{,}502. $$

Check. There are four receipts in the schedule and four discounted amounts in the calculation. Their undiscounted total is $5,200, which exceeds the $4,502 present value because all four discount factors are below 1.

Common Mistake: Reusing One Factor at Every Date

A frequent error is to discount every payment using the same rate or factor, even when the problem supplies different spot rates by maturity. That calculation changes the assumptions: it values the stream using one rate rather than the stated maturity-specific factors.

For the first example, incorrectly applying a 5% annual effective rate to every payment would give $$ \frac{500}{1.05}+\frac{800}{(1.05)^2}+\frac{1{,}500}{(1.05)^3} \approx 476.19+725.62+1{,}295.76=2{,}497.57. $$

The stated factors instead give $2,400. The error is not in adding the discounted amounts; it is in using discount factors that do not correspond to the given maturities. Before calculating, label the factor beside its maturity, then check every multiplication against that pairing.

Key takeaway: The present value of a multi-date schedule is the sum of its cash flows, each multiplied by the discount factor for its own maturity. Keep dates, factors, signs, and time units aligned.

Check Your Understanding

For each schedule, use the discount factors given and report the present value at time 0.

A schedule pays $300 at time 1, $700 at time 2, and $1,200 at time 3. The corresponding annual effective spot rates are approximately 2.041%, 3.709%, and 5.572%, giving factors $0.98$, $0.93$, and $0.85$. (Answer: $1,965)
A schedule pays $1,500 at time 1, $2,500 at time 2, and $4,000 at time 3. The maturity-specific factors are $0.96$, $0.89$, and $0.78$. (Answer: $6,785)
A contract pays $2,000 at time 0.5, $1,000 at time 1.5, and $3,000 at time 2.5. The annual effective spot rates for those maturities are approximately 3.074%, 5.643%, and 6.226%, giving factors $0.97$, $0.92$, and $0.86$. (Answer: $5,440)
A schedule pays $900 at time 1, $1,600 at time 2, and $2,100 at time 4. The corresponding annual effective spot rates are approximately 5.263%, 7.220%, and 8.703%, giving factors $0.95$, $0.87$, and $0.72$. (Answer: $3,759)
A hypothetical agreement pays $500 at time 1, $1,000 at time 2, and $2,000 at time 3. The maturity-specific annual effective spot rates are approximately 2.041%, 3.141%, and 4.348%, giving factors $0.98$, $0.94$, and $0.88$. (Answer: $3,190)

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