Zero-Coupon Bond Pricing

Pure discounting, with no coupons to muddy it.

Zero coupon bond pricing is the cleanest problem in fixed income. There are no interim cash flows to discount, no reinvestment assumptions to argue over — just a single payment of face value at maturity and the question of what that payment is worth today. The pricing formula is a one-liner, yet zeros concentrate every insight about duration, rate sensitivity, and the structure of the discount curve into a single instrument. Whether you are using the bond pricing calculator to check a market quote or building a theoretical yield curve from scratch, understanding the zero is the right place to start.

The zero coupon bond pricing formula

A zero-coupon bond pays no periodic interest. It is issued at a discount to face value and redeems at par at maturity. Because there is exactly one cash flow, the fair price is just the present value of that payment:

P = F / (1 + y)ⁿ

where F is face value, y is the periodic yield (expressed per period), and n is the number of periods to maturity. Under continuous compounding — the convention used in most derivatives and curve-building work — the equivalent expression is:

P = F · e^−yT

where T is time to maturity in years. The two formulas are interchangeable once you match the compounding convention; the continuous form is analytically cleaner because discounting and compounding collapse into a single exponential. Either way, the mechanics are identical: pick a yield, discount the face value back, and the result is the fair price.

Worked example

Suppose you are evaluating a zero-coupon bond with the following terms:

• Face value: $1,000
• Maturity: 10 years
• Annual yield: 4.5%
• Compounding: annual

Plugging into the periodic formula:

P = 1,000 / (1.045)¹⁰ = 1,000 / 1.5530 ≈ $643.93

You pay roughly $644 today for the right to receive $1,000 in ten years. The $356 difference is the implied interest — it accretes into the price each year as the bond approaches maturity. Under the continuous convention at the same yield:

P = 1,000 · e^{−0.045 × 10} = 1,000 · e^−0.45 ≈ $637.63

The small difference reflects the compounding mismatch; the direction is always the same. Contrast this with pricing a coupon bond, where you must sum the present values of every coupon payment plus the final principal — the zero is that sum collapsed to its terminal term.

Duration and rate sensitivity

Duration measures how much a bond's price changes when yields move. For a coupon bond, duration is a weighted-average time to cash flows — coupons paid early pull it below maturity. For a zero-coupon bond there are no early cash flows, so the Macaulay duration equals the maturity exactly. A 10-year zero has duration of 10 years; a 10-year coupon bond paying 5% annually might have a duration of roughly 7.8 years.

Modified duration — the actual price sensitivity — follows directly:

Modified Duration = Macaulay Duration / (1 + y)

For the 10-year, 4.5% zero: Modified Duration ≈ 10 / 1.045 ≈ 9.57. That means a 100-basis-point rise in yield drops the price by approximately 9.57%. The same move on a 10-year coupon bond causes a smaller loss because those earlier coupon payments are less affected by the long-end discount rate. This makes zeros the most rate-sensitive instrument for any given maturity — useful if you want to express a strong duration view with a small notional, dangerous if you are on the wrong side of a rate move.

STRIPS: how the market creates zeros

The U.S. Treasury does not routinely issue zero-coupon bonds at long maturities. Instead, the market manufactures them through a program called STRIPS — Separate Trading of Registered Interest and Principal of Securities. A dealer purchases a coupon Treasury and strips it into its component cash flows: each semiannual coupon becomes a standalone zero maturing on that payment date, and the final principal repayment becomes a separate zero maturing on the bond's stated maturity. The result is a family of zeros spanning the maturity spectrum, all backed by Treasury credit.

Prices for STRIPS are quoted in the same way as any zero: a percentage of par, reflecting the present value of one dollar received at the relevant maturity. The 10-year principal STRIP quoted at 64.39 (in a 4.5% rate environment) is simply the worked example above restated on a per-$100 face basis.

Phantom interest and the tax treatment

There is a catch for U.S. taxable holders: the IRS treats the annual accretion of a zero's discount as ordinary income even though no cash is actually received. This "phantom interest" — technically called original issue discount (OID) — must be reported and taxed each year on a pro-rata accreted basis. The result is an annual tax liability with no corresponding cash inflow until maturity.

The practical implication is that zeros held in taxable accounts create a negative carry effect: you owe tax each year on income you have not yet collected. This is why zeros are frequently held in tax-deferred accounts such as IRAs or used by tax-exempt institutions. The phantom interest issue does not affect the pricing formula — price is still the discounted present value — but it materially affects the after-tax yield comparison between zeros and coupon bonds for a taxable investor.

Zeros as building blocks of the discount curve

The deepest reason to understand zero coupon bond pricing is that zeros are the atomic units of the discount curve. Any fixed cash flow, on any future date, can be priced by multiplying its amount by the zero-coupon discount factor for that date. A coupon bond's price is just a sum of those products. An interest rate swap can be priced the same way. A mortgage-backed security's expected cash flows can be discounted term by term using the same curve.

The discount factor at maturity T is simply Z(T) = 1 / (1 + y_T)^T, or e^−y_TT continuously — exactly the pricing formula for a $1 zero. Building the yield curve from observed market prices is fundamentally an exercise in reverse-engineering these discount factors: bootstrap short-dated instruments, interpolate across maturities, and extract the implied zero yield at each point. Every forward rate, every swap rate, every option on rates is ultimately derived from the same underlying set of discount factors. The zero is where fixed income arithmetic begins.

Face Value	Maturity (yr)	Yield	Price (annual)	Modified Duration
$1,000	5	3.0%	$862.61	4.85
$1,000	10	4.5%	$643.93	9.57
$1,000	20	5.0%	$376.89	19.05
$1,000	30	5.5%	$200.64	28.44