Use spot rates to price a bond theoretically.

FRM® Part I · Financial Markets and Products · Properties of Interest Rates · FRM1.FM.16.d

The idea

A coupon bond is a bundle of zero-coupon bonds, one for each coupon and the principal. The theoretical price discounts each cash flow at the spot (zero) rate for its own maturity. Here CF(t) is the cash flow at time t and z(t) is the spot rate for that maturity. With annual compounding, Price = Σ CF(t) ÷ (1 + z(t))^t. Continuous compounding changes the discount factor to an exponential. With continuously compounded rates, Price = Σ CF(t) × e^(−z(t) × t). For semiannual spot rates, halve the rate and count half-year periods. This differs from discounting everything at the yield to maturity, which is a single average rate backed out from the price. If a bond's market price differs from its spot-rate value, the gap is an arbitrage opportunity between the bond and its stripped cash flows: buy the cheap side and sell the other. Common errors are applying one spot rate to all cash flows and mismatching the compounding convention.

Formulas

Bond price from annually compounded spot rates

P = Σ CF(t) ÷ (1 + z(t))ᵗ

CF(t): the cash flow at time t
z(t): the spot rate for maturity t
t: the time of each cash flow, in years

Each cash flow uses the spot rate for its own date. For semiannual spot rates, halve the rate and count half-years.

Bond price from continuously compounded spot rates

P = Σ CF(t) × e^(−z(t) × t)

CF(t): the cash flow at time t
z(t): the continuously compounded spot rate for maturity t
t: the time of each cash flow, in years

If the market price differs from this value, the bond and its stripped cash flows offer an arbitrage.

A practice question

From the question bank, with the reason each wrong option is wrong.

According to the spot rate (zero rate) approach, how is the theoretical price of a coupon bond obtained?

All cash flows are discounted at the bond's yield to maturity quoted in the marketWhy not: The yield to maturity is a single rate backed out of the price; it cannot be the starting input of a theoretical price built from the spot curve.
All cash flows are discounted at the spot rate for the bond's final maturityWhy not: Uses one long-maturity rate for every cash flow and ignores that earlier coupons have shorter spot rates.
Each cash flow is discounted at the bond's coupon rateWhy not: Discounting at the coupon rate gives par, not a value tied to the spot curve.
Each coupon and the principal is discounted at the spot rate that matches its own payment date, and the present values are summed - correct

A coupon bond is a package of zero-coupon bonds, one for each cash flow. Each is valued with the spot rate for its own maturity, and the sum is the theoretical price. Discounting everything at one rate is only a shortcut.

A harder one

A 3-year bond with face value 100 pays a 6% annual coupon. The continuously compounded spot rates are 3.0% for 1 year, 4.5% for 2 years and 6.5% for 3 years. The theoretical price is closest to:

99.07Why not: Applies the spot rates with annual compounding instead of continuous compounding.
98.11Why not: Discounts every cash flow at the 3-year spot rate.
98.53 - correct
108.35Why not: Discounts every cash flow at the 1-year spot rate.

Price = 6e^-0.03 + 6e^-0.09 + 106e^-0.195 = 98.53.

The hardest tier

A 1.5-year bond with face value 100 pays a 6% coupon semiannually. The spot rates, quoted with semiannual compounding, are 2% for 0.5 years, 5% for 1.0 year and 8% for 1.5 years. The theoretical price is closest to:

87.43Why not: Forgets to halve the spot rates and also counts the three payment dates as 1, 2 and 3 years.
105.88Why not: Discounts all three cash flows at the 0.5-year spot rate.
103.03Why not: Uses the semiannual periodic rate but with exponents in years (0.5, 1, 1.5) instead of in half-year periods.
97.39 - correct

Periodic rates are 1%, 2.5% and 4%. Price = 3/1.01 + 3/1.025² + 103/1.04³ = 97.39.

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