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 zero 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

Coupon Bond Pricing

Coupon bonds are debt securities that pay periodic interest (coupons) and return the principal at maturity. Their price is computed as the sum of the present values of all future coupon payments and the final principal repayment, discounted using an appropriate yield or discount rate. This foundational concept is critical in understanding how differing coupon rates affect bond prices when yields or discount rates change.

Zero Coupon Bond and Zero Rate

A zero coupon bond does not pay periodic coupons but is sold at a discount and pays its face value at maturity. The zero rate, or spot rate, is the interest rate used to discount a single future payment to its present value, reflecting the time value of money for that specific maturity. It serves as a benchmark for discounting cash flows and is central to the valuation of other bonds.

Replicating Portfolio and Arbitrage Strategy

A replicating portfolio is a combination of instruments that mimics the cash flows of another asset—in this case, creating a synthetic zero coupon bond by strategically taking long and short positions in coupon bonds. This arbitrage-based approach eliminates coupon payments by offsetting flows from different bonds, allowing for the extraction of the underlying zero rate from observed bond prices.

Cash Flow Matching

Cash flow matching involves designing a portfolio where the cash flows of various instruments exactly match the cash flows of a target security. In the context of the problem, it means aligning the coupons and maturity values of the chosen bonds to replicate those of a zero coupon bond, thereby ensuring that the derived yield accurately reflects the market's 10-year discount rate.

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