Debt & Yield Curvesintermediate

Zero Rates

A zero rate (or spot rate) is the yield on a single payment due at a future date with no intervening cash flows. It is the rate that discounts a payment at time $T$ to its present value. Zero rates form the discount curve used to price every cash flow in a fixed-income portfolio. The discount curve is recovered from observable market data via bootstrapping of T-bills, coupon bond prices, and swap rates.

Why it matters

A coupon bond is a bundle of zero-coupon cash flows packaged into one instrument. Its yield to maturity is a weighted average of the individual zero rates that match each cash flow's date. If you want the pure time value of money for exactly 5 years out, you do not use a 5-year coupon bond yield. You use the 5-year zero rate. Bootstrapping pulls the 5-year zero out of a chain of shorter zeros and the 5-year coupon bond price, one maturity at a time.

Formulas

Zero-coupon bond price (continuous compounding)

P = FV \times e^{-r_T T}

r_T

is the continuously compounded zero rate for maturity

T

. Hull §4.4.

Zero-coupon bond price (annual compounding)

P = \frac{FV}{(1 + r_T)^T}

Equivalent under annual compounding. Converting between conventions uses

r_{\text{cont}} = m \ln(1 + r_{\text{disc}}/m)

for

m

compounding periods.

Bootstrapping 2-year zero from a 2-year coupon bond

P_{\text{bond}} = \frac{C}{1 + r_1} + \frac{C + FV}{(1 + r_2)^2}

Given the observable 1-year zero

r_1

and market price

P_{\text{bond}}

, solve for the unknown 2-year zero

r_2

Worked examples

Scenario

A 2-year zero-coupon bond trades at A$92.31 per A$100 face value. Compute the 2-year continuously compounded zero rate.

Solution

Solve $92.31 = 100 \times e^{-r_2 \times 2}$. Then $e^{-2r_2} = 0.9231$ , so $-2r_2 = \ln(0.9231) = -0.08004$ , giving $r_2 = 4.00\%$ . The 2-year zero rate is $4.00\%$.

Scenario

Bootstrap the 2-year zero from a 2-year 5% annual-coupon bond priced at A$101.93, given the 1-year zero $r_1 = 3.00\%$ .

Solution

$101.93 = 5/1.03 + 105/(1 + r_2)^2$. Then $5/1.03 = 4.854 $, so$ (1 + r_2)^2 = 105/(101.93 - 4.854) = 105/97.076 = 1.0817 $, giving$ r_2 = \sqrt{1.0817} - 1 = 4.01\%$. The 2-year zero rate is approximately $4.01\%$.

Common mistakes

✗The yield to maturity of a coupon bond equals the zero rate for that maturity. The YTM is a single weighted average of the zero rates for each cash flow date. Only for a par bond with a flat yield curve do the two coincide.
✗Zero rates can only be derived from zero-coupon bonds. Pure zero-coupon bonds (such as US STRIPS or Australian Treasury Notes) provide direct zero rates, but most zero curves are built by bootstrapping from coupon bond prices and swap rates, which is how AOFM and RBA published curves are constructed.
✗A higher zero rate always means a higher YTM on every coupon bond. The relationship depends on the curve shape. A steeply upward-sloping zero curve pulls long-dated coupon bond YTMs below the longest zero rate, because earlier coupons are discounted at lower short-end zeros.

Revision bullets

•Yield on a single future payment, no coupons
•Also called spot rate for that maturity
•Derived via bootstrapping from market instruments
•Used to discount every cash flow in fixed income
•Building block for forward rates
•Coupon bond YTM is a weighted average of zeros

Quick check

Zero rates are also commonly called

A 1-year zero rate is $4\%$ and a 2-year zero-coupon bond is priced at A$92.46 per A$100 face value (annual compounding). The 2-year zero rate is closest to

Connected topics

Forward Rates Bond Price

In learning paths

Fixed Income & Rates Path

Sources

Hull (2022), §4.4 to §4.6
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Defines zero rates under continuous compounding, sets out the bootstrapping algorithm, and links zeros to coupon bond pricing.
Fabozzi (2021), Ch. 5
Fabozzi, Frank J. Bond Markets, Analysis, and Strategies. 10th ed. MIT Press, 2021. ISBN 978-0-262-04627-3.
Detailed treatment of the spot-rate curve, bootstrapping methodology, and the relation to par and coupon yield curves.
RBA Statistical Table F17
Reserve Bank of Australia. Statistical Tables F17 Zero-Coupon Interest Rates - Analytical Series. RBA, monthly release.
Authoritative source for Australian zero-coupon government bond yields constructed by RBA staff from AGS prices.
Bank of England Yield Curve
Bank of England. Yield Curve Documentation. BoE Statistical Interactive Database, accessed 2026.
Methodological notes on zero-coupon yield curve estimation from coupon bond prices, including spline interpolation.
US Treasury STRIPS
U.S. Department of the Treasury. STRIPS Information. TreasuryDirect, accessed 2026.
Description of Separate Trading of Registered Interest and Principal Securities, the cleanest source of US zero rates.

How to cite this page

Dr. Phil's Quant Lab. (2026). Zero Rates. Derivatives Atlas. https://phucnguyenvan.com/concept/zero-rates

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