Value a bond by discounting at spot rates
The idea
Discounting every cash flow at a single yield to maturity assumes the yield curve is flat, which it rarely is. Pricing with spot rates discounts each cash flow at the rate appropriate to its own date, which is the arbitrage-free value: any other price could be exploited by stripping the bond into its component payments or reconstituting it from them. The yield to maturity implied by that price is a complex weighted average of the spot rates, weighted toward the maturity where most of the present value sits, which is why two bonds of equal maturity but different coupons have different yields.
Formulas
Price of a bond from spot rates
PV = PMT ÷ (1 + z₁) + PMT ÷ (1 + z₂)² + (PMT + FV) ÷ (1 + z₃)³
- PMT
- the coupon each year
- FV
- the principal, repaid with the last coupon
- z₁, z₂, z₃
- the spot rates for one, two and three years
Each cash flow discounted at the rate for its own date, which is the arbitrage-free price. The single yield to maturity that prices the same bond is an average of these, weighted towards the dates carrying the most cash.
A practice question
What is a spot rate?
- The coupon rate on a newly issued bondWhy not: A coupon rate is set by the issuer to sell the bond near par. A spot rate is a market rate for a single payment at one date.
- The yield on a single payment to be received at a specified future date — correct
- The rate at which a bond can be repurchasedWhy not: That would be a repo rate, which concerns short-term secured financing rather than the discounting of future cash flows.
Each spot rate applies to one date. Pricing with them values every cash flow at the rate genuinely appropriate to its timing.
A harder one
A bond is priced below its arbitrage-free value. What would a trader do?
- Sell the bond and buy the stripsWhy not: That is the trade for a bond priced *above* its arbitrage-free value. Here the package is cheap relative to its parts.
- Nothing, since the difference is not tradableWhy not: Stripping and reconstitution are established mechanisms in government bond markets, which is what makes the relationship enforceable.
- Buy the bond and strip it into its component payments to sell separately — correct
The ability to separate and reassemble the cash flows is what forces the price back to the sum of its parts.
The hardest tier
Why do two bonds with the same maturity but different coupons have different yields to maturity on an upward-sloping curve?
- Because yield is a weighted average of spot rates — correct
- Because the higher-coupon bond has more credit riskWhy not: The coupon is set at issue and says nothing about the issuer's ability to pay. Two bonds from the same issuer carry the same credit risk.
- Because yields to maturity are quoted on different conventionsWhy not: Conventions can differ between markets, but here both bonds are quoted the same way. The difference is real, not a quotation artefact.
A high coupon shifts weight toward the nearer, lower rates. The weights are the present values of each cash flow, so a high coupon puts more present value early and pulls the average yield down, which is the coupon effect behind the difference between the par and spot curves.
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