Pricing bonds: zero-coupon, coupon and perpetual

How do you turn a bond’s promised payments into a price?

A bond is a promise of cash flows. Its price today is the present value of those cash flows, each discounted at the rate the market demands for bonds of that risk and maturity. That rate is the market discount rate , quoted as the yield to maturity . It is the required return from Learning Module 1, now used as a discount rate. The required rate of return and the discount rate are two names for the same number here.

Every bond fits one of four cash-flow patterns, and each has its own shortcut.

A bond pricethe present value of every promised payment, at the market yield

• Zero-coupon (discount) bond Memorize one payment, the face value at maturity
  • PV = FV ÷ (1 + r)ᵗ always below face value unless the yield is negative
• Coupon bond Memorize coupons every period, then the face value
  • Keys: N, I/Y, PMT, FV semiannual: double N, halve I/Y and PMT
• Floating-rate bond Use it coupon resets to a reference rate plus a margin
  • Coupon = FV × (reference + margin) ÷ payments a year
• Perpetual bond Memorize coupons forever, no face value to repay
  • PV = PMT ÷ r a perpetuity

How does money move through time?

Memorize

Pricing a bond is moving money along a timeline, and one idea does it in both directions. The future value (FV) of an amount today is what it grows to at the interest rate. The present value (PV) of a later amount is that growth undone.

$$FV_t = PV_0 \times (1 + r)^{t}$$

$$PV_0 = FV_t \times (1 + r)^{-t}$$

Interest is often added more than once a year. How often is the compounding frequency . A rate quoted per year and compounded m times a year earns r ÷ m in each of m × t periods, so money grows a little faster the more often it compounds. The limit is continuous compounding : FV = PV × e^(r × t) and PV = FV × e^(−r × t).

$$FV_t = PV_0 \times \left(1 + \frac{r}{m}\right)^{m \times t}$$

Here r is the annual (stated) rate, m is the number of compounding periods a year and r ÷ m is the rate for each period. The annual rate the money really earns is $(1 + r/m)^{m} - 1$.

Annualizing a return earned over days, weeks or months works the same way and is covered in Learning Module 1.

One payment: the zero-coupon bond

A zero-coupon bond pays nothing until maturity, when it repays the face value. It is also called a discount instrument , because it is sold below face value and the investor’s whole return is the gap. Its price is one present-value calculation.

$$PV = \frac{FV}{(1 + r)^{t}}$$

A bond quoted with an annual yield is discounted with compounding: one division by (1 + r) for every year. The simple-interest shortcut FV ÷ (1 + r × t) gives a different, wrong price. The exception is short money-market bills, below.

Time and yield move the price

With the yield unchanged, a zero’s price climbs toward face value as maturity nears (it accretes): three years later the same zero, now with 5 years left, is worth 100 ÷ 1.045⁵ = 80.25. If instead the yield jumps at once from 4.5% to 5.5%, the price falls to 100 ÷ 1.055⁸ = 65.16, 5.16 lower. Price and yield move in opposite directions. A basis point (bp) is 0.01%, so that move was 100 bp.

Semiannual discounting

Discounting the same 8-year payment at 4.5% compounded semiannually (2.25% for 16 half-years) gives 100 ÷ 1.0225¹⁶ = 70.05, a little below 70.32. With a positive rate, more frequent compounding means a lower present value.

Bills priced with simple interest

Bills of a year or less are often quoted on a simple-interest basis: price = face value ÷ (1 + yield × days ÷ 365). With a negative yield the price is above face value: the buyer accepts a small loss for safety. The Fixed Income lesson on money-market rates covers the other quoting conventions.

Go deeper Continuous discounting and negative yields

With a continuously compounded yield, the price is PV = FV × e^(−rt). Eight years at a continuously compounded 4.5% gives 100 × e^(−0.36) = 69.77, a little below the annual-compounding price because continuous compounding grows money faster.

A yield can also be negative (a negative yield ). A 2-year zero at a yield of −0.4% prices at 100 ÷ 0.996² = 100.80, above face value. The investor pays more than they will get back, in return for parking money safely. Nothing in the formula changes; the growth factor is just below 1.

$$PV = FV \times e^{-rt}$$

The negative-yield zero

TI BA II Plus HP 12C

1. 2NDCLR TVM
2. 2N0.4+|-I/Y100FV
3. CPTPV −100.80

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · I/Y interest rate per year (per period when P/Y is 1) · FV future value · CPT compute · PV present value

1. fCLEAR FIN
2. 2n0.4CHSi100FV
3. PV −100.80

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · i interest rate per period · FV future value · PV present value

Many payments: the coupon bond

A coupon instrument pays a fixed coupon every period and the face value at maturity. Discount each coupon and the face value, then add. That is a stack of zero-coupon prices.

$$PV = \sum_{k=1}^{N} \frac{PMT}{(1 + r)^{k}} + \frac{FV}{(1 + r)^{N}}$$

Nobody adds these by hand. Use the time value of money (TVM) keys: N is the number of periods, I/Y the yield per period in percent, PMT the coupon per period, FV the face value. The price is the PV you compute, shown with a negative sign.

Exam tip Spreadsheets do the same job. In Excel or Google Sheets the 5-year bond below is =PV(0.05, 5, 3.5, 100), which returns −93.51: like the calculator, it shows money you pay as a negative number. The fifth argument, type, is 0 (the default) for payments at the end of each period.

Memorize

The rule that decides premium or discount compares the coupon rate with the yield.

Compare the coupon rate with the market yield

• Coupon rate above yield Memorize price above par: a premium
• Coupon rate equal to yield Memorize price equals par
• Coupon rate below yield Memorize price below par: a discount

Exam tip Premium or discount is a comparison of coupon with yield. A bond with a 2% yield and a 5% coupon is a premium bond, however low the yield looks.

Go deeper Pull to par and stripped bonds

A premium bond’s price falls toward 100 as it ages, and a discount bond’s price rises toward 100, because the remaining payments shrink in number. The 5-year 3.5% bond at a 5% yield costs 93.51. One year later, with four years left and the same yield, it is worth 94.68. The fixed-income modules build on this.

The same bond a year later

TI BA II Plus HP 12C

1. 2NDCLR TVM
2. 4N5I/Y3.5PMT100FV
3. CPTPV −94.68

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

1. fCLEAR FIN
2. 4n5i3.5PMT100FV
3. PV −94.68

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

A government coupon bond can be split (“stripped”) into separate zero-coupon bonds, one for each coupon date and one for the principal. Each strip is a single payment, so it is priced with the zero formula off its own yield. Adding the strip prices gives the coupon bond’s price, which is the starting point of cash flow additivity later in this module.

What happens to your return when the yield moves?

Use it

A bond’s price moves with its yield, so an investor who sells before maturity earns something different from the yield they paid. The one-year return is the holding period return from Learning Module 1, with the bond’s price worked out from its yield: (new price − old price + coupon) ÷ old price. The new price is the price of the same bond with one period fewer to run.

Coupons that float

Use it

A floating-rate bond resets its coupon each period to a reference rate (a short-term market rate, such as a 3-month interbank rate) plus a fixed margin that reflects the issuer’s risk.

$$PMT = FV \times \frac{\text{reference rate} + \text{margin}}{\text{payments a year}}$$

Because the coupon follows market rates, a floater’s price stays close to face value; pricing it is a Fixed Income topic.

Payments forever: the perpetual bond

A perpetual bond pays a fixed coupon every period with no maturity and no face value. Its payments form a perpetuity , and the infinite sum collapses to one line.

$$PV = \frac{PMT}{r}$$

Match the coupon and the rate to the same period. Dividing the quarterly coupon of 1.20 by the annual 5% would give 24.00, mixing a quarterly coupon with an annual rate. The formula needs a positive rate.

Go deeper Why bond pricing shows up everywhere

The same present-value stack prices a loan (next lesson) and a share (the lesson after). Fixed Income develops it in full: bond valuation off the whole yield curve, how price moves when yield moves, and how spreads are read. What you need here is the habit of listing each payment, putting it on a date, and discounting it at the right periodic rate.