Valuing Coupon Cash Flows with Spot Rates

Introduction

A coupon-bearing security makes payments on more than one date. Its price therefore depends on the value today of each promised payment, not just on its final redemption amount or on a single rate applied indiscriminately to the whole schedule. A spot curve supplies a discount factor for each maturity, allowing the payments to be valued consistently by date.

This tutorial develops that valuation method for a security with known, certain cash flows. It focuses on the pricing calculation: building the payment schedule, selecting the matching spot discount factor, and adding the present values. The same method applies to any collection of certain dated cash flows with the same valuation date and currency.

Definition: Let $D(t)$ be the time-0 spot discount factor for a certain payment due at time $t$. It is the time-0 price of one unit of currency payable at $t$. If the annual effective spot rate for maturity $t$ is $i_t$, then $D(t)=(1+i_t)^{-t}$. For a security with certain payments $C_1,\ldots,C_n$ at times $t_1,\ldots,t_n$, its spot value is $\sum_{k=1}^{n} C_kD(t_k)$.

Build the Cash-Flow Schedule First

A coupon is the periodic interest payment specified by the security’s terms. For a security with face amount $F$ and annual coupon rate $c$, paid once per year, the annual coupon is $cF$. If the security redeems at amount $R$ at the end of year $n$, its final payment is usually the final coupon plus redemption, $cF+R$. Do not treat redemption as a substitute for that last coupon unless the terms explicitly say the last coupon is omitted.

For annual coupons and redemption at year $n$, write the cash flows as $C_1,\ldots,C_n$, where the first $n-1$ payments are the coupons and $C_n$ includes both the final coupon and redemption. Then use the spot discount factor corresponding to each payment date. The coupon rate determines the payments; the spot rates determine how those payments are valued today. These are separate parts of the calculation.

Worked Example: Pricing a Three-Year Annual-Coupon Security

A hypothetical default-free security has face amount $1,000, pays a 5% annual coupon, and redeems at face amount at the end of year 3. The annual spot discount factors for years 1, 2, and 3 are 0.96, 0.91, and 0.84, respectively. Find its time-0 price. Report money to the nearest cent.

Method. List the promised payments by date, then multiply each by the discount factor for that date. This avoids applying the year-3 factor to earlier coupons.

Step 1: Find the annual coupon.

$$ 1{,}000(0.05)=50 $$

The payments are $50 at year 1, $50 at year 2, and $1,050 at year 3: the final $50 coupon plus $1,000 redemption.

Step 2: Discount each payment at its own maturity.

\begin{aligned} PV_1&=50(0.96)=48.00,\\ PV_2&=50(0.91)=45.50,\\ PV_3&=1{,}050(0.84)=882.00. \end{aligned}

Step 3: Add the present values.

P=48.00+45.50+882.00=\boxed{\$975.50}.

Check. The three undiscounted payments total $1,150, and each discount factor is less than 1, so a price below $1,150 is consistent with the data. The calculated $975.50 is also below the $1,000 redemption amount. That comparison is a reasonableness check, not a general rule for every spot curve.

Why the Cash-Flow Formula Holds

The pricing rule follows from the definition of a spot discount factor and the no-arbitrage principle. A certain payment of $C_k$ at time $t_k$ has time-0 value $C_kD(t_k)$: $D(t_k)$ is the price today of one unit payable at that date, so scaling the payment by $C_k$ scales its price by the same amount. If an investor purchases a zero-coupon claim paying $C_k$ at $t_k$ for every payment date, the resulting claims exactly reproduce the security’s promised cash flows.

Separate the payment dates.
Write down the amount due at each date, including any final coupon and redemption amount.

Value each dated payment.
Multiply the amount due at $t_k$ by $D(t_k)$, the time-0 price of one unit due at that same date.

Add the values.
The total is the price that matches the security’s full promised cash-flow schedule.

More formally, the portfolio of dated zero-coupon claims has the same cash flow as the security at every payment date. Under no arbitrage, two assets with identical certain cash flows must have the same price; otherwise, an investor could buy the cheaper set of cash flows and sell the more expensive one. Thus the security’s price is

P=\sum_{k=1}^{n}C_kD(t_k) =\sum_{k=1}^{n}\frac{C_k}{(1+i_{t_k})^{t_k}}.

The second expression applies when $i_{t_k}$ is the annual effective spot rate for maturity $t_k$, with time measured in years under that rate convention. The first expression works directly with spot discount factors, which can be supplied without separately stating the rates.

Assumptions: The payments are known and certain; the stated discount factors apply to the same valuation date, currency, and payment dates; and there are no taxes, transaction costs, or other features that change the cash flows. The formula values promised cash flows under these assumptions. It does not by itself adjust for default risk, embedded options, or uncertainty in payments.

Apply the Curve Date by Date

A spot rate is specific to a maturity. Consequently, a coupon due in one year and a coupon due in three years generally use different discount factors. Even if the coupon amount is the same each year, its present value need not be the same: both the maturity-specific factor and the payment amount determine that value.

When the data give spot rates rather than discount factors, convert each rate using its own maturity. With annual effective spot rates, a payment at time $t$ is discounted by $(1+i_t)^{-t}$. Do not use $i_1$ for every maturity merely because it is the one-year rate, and do not use the security’s coupon rate as a discount rate. The coupon rate sets the contractual coupon; the spot curve values the contractual payments.

Worked Example: A Final Coupon Plus a Larger Redemption

A hypothetical default-free note makes annual payments of $70 at year 1 and $90 at year 2. At year 3 it pays a $110 coupon and redeems for $1,000. The spot discount factors for years 1, 2, and 3 are 0.98, 0.94, and 0.89. Find its time-0 price to the nearest cent.

Method. Use the stated payment amounts rather than assuming equal coupons. Include the final coupon in addition to redemption.

Step 1: Confirm the year-3 cash flow. The payment at year 3 is $110+1{,}000=\$1{,}110$. The complete schedule is therefore $70, $90, and $1,110.

Step 2: Discount each amount with the factor for its date.

\begin{aligned} PV_1&=70(0.98)=68.60,\\ PV_2&=90(0.94)=84.60,\\ PV_3&=1{,}110(0.89)=987.90. \end{aligned}

Step 3: Sum the values.

P=68.60+84.60+987.90=\boxed{\$1{,}141.10}.

Check. The undiscounted payments total $1,270. All three factors are below 1, and the calculated price of $1,141.10 is below that total. The common mistake avoided here is to discount only the $1,000 redemption at year 3 and leave out the $110 coupon due on that same date.

Multiple Securities and Shared Payment Dates

The same rule applies to a portfolio. Value each security’s payments date by date, or first combine the amounts due on each date and then discount those combined amounts. The results agree because multiplication distributes over addition. Combining the cash flows can make a portfolio calculation shorter, but it does not justify combining payments from different dates before discounting them.

Worked Example: Valuing Two Coupon Securities Together

A hypothetical investor holds two default-free securities. Security A has a $1,000 face amount, pays a 4% annual coupon, and redeems at face after two years. Security B has a $500 face amount, pays a 6% annual coupon, and also redeems at face after two years. The spot discount factors for years 1 and 2 are 0.97 and 0.92. Find the portfolio value at time 0.

Method. Build both payment schedules, combine payments due on the same date, and apply the matching factor once to each combined amount.

Step 1: Calculate each security’s coupons.

\text{Coupon A}=1{,}000(0.04)=40,\qquad \text{Coupon B}=500(0.06)=30.

At year 1, the portfolio pays $40+30=\$70$. At year 2, Security A pays $1{,}000+40=\$1{,}040$, and Security B pays $500+30=\$530$. The combined year-2 payment is $1{,}040+530=\$1{,}570$.

Step 2: Discount the combined payments.

\begin{aligned} PV_1&=70(0.97)=67.90,\\ PV_2&=1{,}570(0.92)=1{,}444.40. \end{aligned}

Step 3: Add.

P_{\text{portfolio}}=67.90+1{,}444.40=\boxed{\$1{,}512.30}.

Check. The year-1 and year-2 portfolio payments sum to $1,640 before discounting. Both factors are below 1, so the value of $1,512.30 is plausible. The calculation keeps the two dates separate while combining only payments that are due together.

Common Trap: Reusing One Discount Factor

A frequent error is to apply the longest-maturity discount factor to every payment. That does not value the coupons at their own maturities. For example, in the first worked example, using 0.84 for all three payments would give

(50+50+1{,}050)(0.84)=1{,}150(0.84)=\$966.00,

instead of the correct $975.50. This calculation treats every payment as if it were due at year 3. The year-1 and year-2 coupons are due earlier, so their respective factors are 0.96 and 0.91, not 0.84. The correction is to match each payment to the spot factor for its own date before summing.

Other checks help catch related errors:

Check the last payment. Unless the terms state otherwise, it includes both the final coupon and redemption.
Keep coupon and spot rates distinct. A coupon rate determines the contractual payment; it is not automatically the rate used to discount that payment.
Match dates before adding. Payments due at different maturities need their own factors. Payments at the same date may be added first.
Use one valuation date. All discount factors must price payments relative to the same time-0 date.

Key takeaway: Write the complete payment schedule first. Then multiply each payment by the spot discount factor for its own maturity and add the resulting present values. This is the spot-curve price of the stated certain cash flows under the assumptions above.

Check Your Understanding

For each problem, use the specified spot factors or spot rates and report dollar amounts to the nearest cent.

A hypothetical security has face amount $1,000, a 4% annual coupon, and two years to redemption at face. Its year-1 and year-2 spot discount factors are 0.95 and 0.90. What is its time-0 price? (Answer: $974.00)
A hypothetical $2,500 security pays a 5% annual coupon for three years and redeems at face at the end of year 3. The spot discount factors are 0.98, 0.94, and 0.88 for years 1, 2, and 3. Find its price. (Answer: $2,550.00)
A hypothetical instrument pays $30 at year 1, $50 at year 2, and a $70 coupon plus $900 redemption at year 3. The corresponding spot discount factors are 0.99, 0.95, and 0.89. Find its price. (Answer: $940.50)
A portfolio has combined cash flows of $300 at year 1 and $1,500 at year 2. The spot discount factors are 0.96 and 0.90. Find the portfolio value. (Answer: $1,638.00)
A hypothetical default-free bond has face amount $1,000, pays a 6% annual coupon for two years, and redeems at face at year 2. The annual effective spot rates are 4% at year 1 and 5% at year 2. Find its price. (Answer: $1,019.14)

What You'll Learn