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 interest rate you built in the previous module, now used as a discount rate.

Every bond fits one of three 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
  • TVM keys: N, I/Y, PMT, FV semiannual: double N, halve I/Y and PMT
• Perpetual bond Memorize coupons for ever, no face value to repay
  • PV = PMT ÷ r a perpetuity

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.

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

Discounting is always compound. The shortcut FV (future value) ÷ (1 + r × t) is simple interest and gives a different, wrong price.

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 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.

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.

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. With quarterly coupons of 1.20 and a stated annual yield of 5% compounded quarterly, the quarterly rate is 1.25%, so the price is 1.20 ÷ 0.0125 = 96.00. Dividing 1.20 by 0.05 would mix a quarterly coupon with an annual 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.