What is the value of a 2-year fixed-for-floating compounding...

What is the value of a 2-year fixed-for-floating compounding swap where the principal is $$\$ 100$$ million and payments are made semiannually? Fixed interest is received and floating is paid. The fixed rate is $8 \%$ and it is compounded at $8.3 \%$ (both semiannually compounded). The floating rate is LIBOR plus 10 basis points and it is compounded at LIBOR plus 20 basis points. The LIBOR zero curve is flat at $8 \%$ with semiannual compounding. The risk-free discount rate is $7.5 \%$ continuously compounded.

Key Concepts

Net Present Value (NPV) Calculation

The net present value of a swap is determined by calculating the present values of both the fixed leg and the floating leg, then taking the difference between them. This approach assesses which party benefits from the swap and quantifies the financial advantage, making it a central concept in determining whether a swap is fairly priced or if one side holds an inherent value advantage.

Compounding Conventions

Compounding conventions determine how interest accrues on investments or liabilities over time. In this context, semiannual compounding means that interest is calculated and added twice a year, and differences in the compounding assumptions—such as compounding at one rate for a payment vs. another rate for discounting—can significantly impact the calculation of cash flows and the overall valuation of the swap.

Zero-Coupon Yield Curve

A zero-coupon yield curve represents the yields of hypothetical default-free bonds that pay no coupons—only a lump sum at maturity. It provides the necessary discount factors for valuing future cash flows at various maturities. When the curve is flat, it implies uniform yields across all time horizons, simplifying the valuation process of swaps and other financial instruments.

Interest Rate Swap

An interest rate swap is a derivative contract in which two counterparties exchange streams of interest payments, typically one fixed rate versus one floating rate, on a predetermined notional amount. This instrument is used for managing interest rate risk or for speculative purposes by aligning the interest rate exposure with the party's financing strategies.

Fixed and Floating Legs

In a swap, the fixed leg consists of payments calculated using a predetermined fixed rate, while the floating leg consists of payments based on a reference rate, such as LIBOR, plus or minus a predetermined spread. The valuation of the swap involves comparing the present values of these two sets of cash flows to determine the swap's net value.

Discounting Cash Flows

Discounting cash flows involves converting future payments to their present value using a discount rate, which reflects the time value of money. In swap valuation, the risk-free discount rate, possibly based on a continuously compounded rate, is applied to future cash flows to determine their present worth, ensuring that valuation reflects prevailing market conditions.

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