Discounting Cash Flows with Spot Rates

Why Spot Rates Matter

An irregular stream may include payments at several dates, with each date having a different market rate for borrowing or investing over that exact maturity. Applying one rate to every payment can therefore give a different value from discounting each payment at its maturity-specific rate. The distinction matters whenever the rate curve is not flat.

This tutorial uses annual payment dates and annual effective rates. A spot rate is the annual effective rate for a single investment from the valuation date to one specified maturity. It is not the rate for only the final year of that investment, nor is it automatically a rate that applies to every payment date.

Definition: Let $s_t$ denote the annual effective spot rate from today to time $t$, where $t$ is measured in years. Its discount factor for a payment at time $t$ is $D_t=(1+s_t)^{-t}$. For deterministic cash flows $C_t$ at dates $t=1,\ldots,n$, their present value is the sum of each cash flow multiplied by the discount factor for its own date: $$ PV=\sum_{t=1}^{n} C_tD_t =\sum_{t=1}^{n}\frac{C_t}{(1+s_t)^t}. $$

Here, positive $C_t$ represents money received and negative $C_t$ represents money paid. A date with no payment contributes zero. A spot rate is quoted for a maturity, so the exponent is the time to that payment: a year-3 cash flow uses the year-3 spot rate raised to the third power. Rates and cash flows must use consistent currency, valuation date, and annual compounding conventions.

The formula assumes known cash-flow amounts and dates, and rates that are appropriate for valuing those cash flows. The examples assume deterministic cash flows, annual effective rates, and no mortality or probability distribution. If cash flows are uncertain, or if default, taxes, or other risks matter, additional assumptions and a suitable valuation framework are required; the deterministic formula alone does not resolve those issues.

Why Each Payment Uses Its Own Rate

A zero-coupon investment pays one dollar at a single maturity and nothing before then. If its annual effective spot rate to time $t$ is $s_t$, its price today is $D_t=(1+s_t)^{-t}$. This follows from the accumulated value of a present investment: $D_t(1+s_t)^t=1$. A cash flow of $C_t$ dollars at that same date is therefore worth $C_tD_t$ today. Adding the values of the separate dated cash flows gives the full stream's present value.

Match each payment to its maturity.
For every cash flow, record its date $t$ and the spot rate $s_t$ that applies from today to that date.

Find the date-specific discount factor.
Calculate $D_t=(1+s_t)^{-t}$, using the annual effective rate and the payment's full time to maturity.

Discount and add signed amounts.
Multiply $C_t$ by $D_t$ for each date, then sum the resulting present values, retaining negative amounts as negative.

This derivation also explains why replacing a non-flat curve with a single rate is not generally equivalent. To use one annual effective rate $i$ for every date, the discount factor would have to be $(1+i)^{-t}$ at every maturity. That agrees with spot-rate discounting at all payment dates only if the relevant spot rates equal that same $i$, or if the particular cash flows happen to make the different valuations coincide.

Worked Applications

Worked Example: Comparing a Spot Curve With a Flat Rate

A hypothetical project is expected to return $1,030 at the end of year 1 and $2,163.20 at the end of year 2. The annual effective spot rates are 3% for year 1 and 4% for year 2. Find the present value using the spot rates, then compare it with a valuation using a constant annual effective rate of 3%. Assume the amounts and dates are certain; no mortality or probability distribution is assumed.

Method. Match each payment to its own maturity-specific spot rate. For the comparison, use 3% at both dates.

At year 1, the spot-rate present value is:

\frac{\$1{,}030}{1.03}=\$1{,}000.00.

At year 2, the year-2 spot rate is 4%, so its two-year discount factor is $(1.04)^{-2}$:

(1.04)^2=1.0816,\qquad \frac{\$2{,}163.20}{1.0816}=\$2{,}000.00.

Adding the two present values gives the spot-rate value:

\$1{,}000.00+\$2{,}000.00=\$3{,}000.00.

Now use a flat 3% annual effective rate. The year-1 value remains $1,000.00. At year 2:

(1.03)^2=1.0609,\qquad \frac{\$2{,}163.20}{1.0609} =\$2{,}039.02347\ldots.

The flat-rate total is $\$1{,}000+\$2{,}039.02347\ldots=\$3{,}039.02$, rounded to the nearest cent. The flat-rate value is higher because the year-2 payment is discounted at 3% rather than its 4% spot rate. The common mistake is to use the year-1 rate for both payments without being asked to assume a flat rate.

Worked Example: Valuing Three Dated Receipts

A hypothetical contract will pay $1,040 at the end of year 1, $2,205 at the end of year 2, and $3,573.048 at the end of year 3. The annual effective spot rates are 4%, 5%, and 6% at years 1, 2, and 3, respectively. Find the spot-rate present value and compare it with discounting all three payments at a flat 5%. Assume all payments are certain; no mortality or probability distribution is assumed.

Method. Discount each receipt over its full time to payment using the rate at that maturity. Keep the unrounded present values until the total is calculated.

The year-1 present value is:

\frac{\$1{,}040}{1.04}=\$1{,}000.00.

The year-2 present value is:

(1.05)^2=1.1025,\qquad \frac{\$2{,}205}{1.1025}=\$2{,}000.00.

For year 3, the maturity-specific spot rate is 6%:

(1.06)^3=1.191016,\qquad \frac{\$3{,}573.048}{1.191016}=\$3{,}000.00.

Thus the spot-rate present value is $\$1{,}000+\$2{,}000+\$3{,}000=\$6{,}000.00$.

For the flat 5% comparison, the year-1 value is $\$1{,}040/1.05=\$990.476190\ldots$, and the year-2 value is $\$2{,}205/(1.05)^2=\$2{,}000.00$. At year 3:

(1.05)^3=1.157625,\qquad \frac{\$3{,}573.048}{1.157625} =\$3{,}086.533204\ldots.

Using the unrounded amounts, the flat-rate total is $\$990.476190\ldots+\$2{,}000+\$3{,}086.533204\ldots=\$6{,}077.009394\ldots$, or $6,077.01 to the nearest cent. The flat-rate value exceeds the spot-rate value here because the assumed 5% rate is below the year-3 spot rate but above the year-1 spot rate; the net effect depends on all payments and dates. A common calculation trap is to round an intermediate present value or miscalculate the year-3 quotient. Retaining precision gives the stated total.

Worked Example: Netting Receipts and Payments

A hypothetical arrangement has cash flows of +$1,020 at year 1, −$1,060.90 at year 2, and +$3,374.592 at year 3. The annual effective spot rates are 2%, 3%, and 4% at the respective maturities. Find the net present value of this stream. Assume the cash flows are certain; no mortality or probability distribution is assumed.

Method. Discount each signed cash flow separately, then add. Do not discard the year-2 payment just because it is an outflow.

At year 1:

\frac{\$1{,}020}{1.02}=\$1{,}000.00.

At year 2, the negative sign is retained:

(1.03)^2=1.0609,\qquad \frac{-\$1{,}060.90}{1.0609}=-\$1{,}000.00.

At year 3:

(1.04)^3=1.124864,\qquad \frac{\$3{,}374.592}{1.124864}=\$3{,}000.00.

The net present value is $\$1{,}000-\$1{,}000+\$3{,}000=\$3{,}000.00$. The result checks: the first two discounted amounts offset exactly, leaving the year-3 receipt's $3,000 present value. The common mistake is to treat every listed amount as a receipt or to discount the net cash flow at one date's rate. Each amount has its own date and sign.

Reading the Rate Curve Correctly

The spot-rate method is a date-by-date valuation, not an instruction to average the rates. A weighted average rate generally cannot reproduce the sum of maturity-specific discounted values, because each rate is compounded over a different number of years and applies to a different cash-flow amount.

Feature	Spot-rate valuation	Single-rate valuation
Rate used	The spot rate for each payment's maturity	One chosen annual effective rate at every maturity
Discount factor at date $t$	$(1+s_t)^{-t}$	$(1+i)^{-t}$
When the methods agree	They agree if the spot rates at all relevant payment dates equal $i$, or if the particular stream happens to produce equal totals.
Main risk of misapplication	Using the wrong maturity's rate	Assuming a flat curve without stating that assumption

The spot rate $s_t$ is an annual effective yield from the valuation date to time $t$, so its exponent is $t$. It is not a one-year forward rate for the interval immediately before year $t$. For example, a year-3 spot rate is applied as $(1+s_3)^{-3}$, not as a one-year discount factor. Do not substitute a nominal annual rate or a continuously compounded rate into this formula unless it has first been converted consistently to an annual effective basis.

For payments at non-integer dates, the time measure and compounding convention must be specified consistently with the rates. This tutorial's calculations use integer annual dates, so no fractional-year convention or interpolation between maturities is needed. If a rate is not quoted for a payment's exact maturity, the method for obtaining an appropriate rate must be supplied; silently choosing a nearby maturity is an extra assumption.

Common Exam Traps

Using one rate without justification. A flat rate is a separate assumption. If a spot curve is given, use the rate corresponding to each payment date.
Using the wrong exponent. The exponent is the full time from the valuation date to payment, not one for every cash flow.
Confusing spot and forward rates. The spot rate applies from today to its maturity; a rate for a later interval is a different quantity.
Ignoring the sign of a payment. Outflows reduce the value of the stream, so their present values are negative when cash flows are stated from the recipient's perspective.
Rounding too early. Keep sufficient precision for each discounting calculation and round the final dollar total to cents.
Mixing rate conventions. The exponent and rate must use compatible time and compounding conventions.

Key takeaway: For a deterministic irregular stream, match each dated cash flow to its maturity-specific annual effective spot rate, discount it over that full maturity, and sum the signed present values. A constant-rate calculation is a distinct comparison that assumes the same rate at every relevant maturity.

Check Your Understanding

For each problem, use annual effective spot rates and retain precision until the final answer. Unless stated otherwise, cash flows are deterministic, with no mortality or probability distribution assumed.

A fund receives $2,080 at year 1 and $4,410 at year 2. The spot rates are 4% and 5%, respectively. Find the present value. (Answer: $6,000.00)
A service contract pays $1,545 at year 1 and $2,315.25 at year 3. The corresponding spot rates are 3% and 5%. Find the spot-rate present value. (Answer: $3,500.00)
A project has receipts of $5,100 at year 1, $5,304.50 at year 2, and $7,019.15136 at year 4. The spot rates at those maturities are 2%, 3%, and 4%. Find the present value. (Answer: $16,000.00)
A stream pays $1,030 at year 1 and $2,163.20 at year 2. Its spot rates are 3% and 4%. Find the spot-rate value and the value using a flat 3% annual effective rate. (Answer: spot-rate value $3,000.00; flat-rate value $3,039.02)
A hypothetical agreement has cash flows of +$2,040 at year 1, −$2,121.80 at year 2, and +$4,499.456 at year 3. Spot rates are 2%, 3%, and 4%, respectively. Find the net present value. (Answer: $4,000.00)

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