A 10 -year 8 % coupon bond currently sells for 90 . A 10...

A 10 -year $8 \%$ coupon bond currently sells for $\$ 90 .$ A 10 -year $4 \%$ coupon bond currently sells for $\$ 80 .$ What is the 10 -year spot rate? (Hint: Consider taking a long position in two of the $4 \%$ coupon bonds and a short position in one of the $8 \%$ coupon bonds.)

Key Concepts

Spot Rate

A spot rate is the yield on a zero-coupon bond that pays no interim cash flows and only a single payment at maturity. It is the interest rate used to discount a cash flow that occurs at a specific future date and forms the foundation for constructing the yield curve.

Coupon Bond

A coupon bond is a type of debt security that pays periodic interest payments (coupons) along with the return of its face value at maturity. Its price is determined by discounting all future coupon payments and the principal repayment back to the present using appropriate discount rates.

Synthetic Zero-Coupon Bond

A synthetic zero-coupon bond is created by combining positions in coupon bonds in a way that cancels out all intermediate cash flows, leaving only a single payment at maturity. This technique helps in isolating the spot rate corresponding to that maturity.

Arbitrage Strategy

An arbitrage strategy in this context involves taking offsetting positions in different bonds to eliminate intermediate cash flows and exploit any mispricing. By designing a portfolio that mimics a zero-coupon bond, one can extract the market's implied spot rate without exposure to additional risk.

Bootstrapping

Bootstrapping is a sequential method used to derive spot rates from the prices of coupon bonds. Starting with shorter maturities, the technique involves solving for the spot rate that makes the present value of a bond’s cash flows equal to its market price, thereby building up a complete term structure of interest rates.

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