The yield curve is flat at 10 % per annum with annual...

The yield curve is flat at $10 \%$ per annum with annual compounding. Calculate the value of an instrument where, in five years' time, the two-year swap rate (with annual compounding) is received and a fixed rate of $10 \%$ is paid, Both are applied to a notional principal of $\$ 100$ Assume that the volatility of the swap rate is $20 \%$ per annum. Explain why the value of the instrument is different from zero.

Key Concepts

Volatility and Convexity Adjustment

Even when the yield curve is flat and the fixed rate matches the current par swap rate, the instrument’s value can be nonzero due to the effects of interest rate volatility. Because the instrument’s payoff depends on the future realized swap rate—a convex function of the underlying interest rates—volatility introduces a convexity bias. This means the average payoff when receiving the floating rate versus paying a fixed rate is not zero, but instead reflects the additional value from the convexity inherent in the swap rate’s distribution.

Forward Swap Rate

The forward swap rate is the fixed rate that would make a future swap contract (beginning at a later date) have zero initial value given the current yield curve. In this problem, the two?year swap rate received in five years is a forward rate determined by the current flat yield curve. Although the forward swap rate is set for par value at the time the swap commences, its actual observed value in five years will depend on the realized path of interest rates.

Yield Curve

A yield curve represents the relationship between interest rates and the different maturities of debt. In this context, a flat yield curve at 10% per annum implies that the same interest rate is used to discount cash flows at all time horizons. This simplifies the discounting process because every future cash flow is discounted by the same compounding rate, but it also impacts the forward rates derived from the curve.

Interest Rate Swap

An interest rate swap is a financial derivative in which two parties exchange cash flows, typically one fixed rate for one floating rate, based on an agreed notional principal. The swap is typically structured so that, at inception, the value is zero because the fixed rate is set equal to the 'par' swap rate. This instrument, however, can acquire value after inception if there is a change in the underlying rates or due to exposure to volatility.

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