A company has a long position in a two-year bond and a...

A company has a long position in a two-year bond and a three-year bond as well as a short position in a five-year bond. Each bond has a principal of $\$ 100$ and pays a $5 \%$ coupon annually. Calculate the company's exposure to the 1 -year, 2 -year, 3 -year, 4 -year, and 5 -year rates. Use the data in Tables $16.3$ and $16.4$ to calculate a 20 -day $95 \% \mathrm{VaR}$ on the assumption that rate changes are explained by (a) one factor, (b) two factors, and (c) three factors. Assume that the zerocoupon yield curve is flat at $5 \%$.

Key Concepts

Factor Models in Interest Rate Risk Management

Factor models decompose interest rate changes into underlying independent components—commonly interpreted as level, slope, and curvature factors in a multi-factor model. The one, two, and three factor frameworks allow risk managers to capture different dimensions of yield curve movements, providing a more nuanced view of potential impacts on a bond portfolio’s value.

Value at Risk (VaR)

Value at Risk is a statistical technique used to quantify the potential loss in value of a portfolio over a specified time period with a given confidence level. In the context of interest rate risk, VaR calculations incorporate the sensitivity exposures of the bonds along the yield curve, allowing risk managers to estimate the risk of unfavorable rate movements on the bond portfolio.

Duration and Sensitivity Analysis

Duration measures the sensitivity of a bond’s price to changes in interest rates, essentially capturing the weighted average time until cash flows are received. In exposure analysis, the bond’s cash flows are attributed to specific periods, helping quantify the effect of rate changes at different maturities on the bond’s overall value.

Coupon Bond Pricing

Coupon bonds pay periodic interest payments as well as a principal repayment at maturity. Pricing these bonds involves discounting each cash flow by the prevailing interest rate for its respective maturity. The sensitivity of each payment to changes in the interest rate—especially when rates change differently across terms—is central to assessing overall risk exposure.

Yield Curve and Term Structure

The yield curve illustrates the relationship between interest rates and the maturities of debt securities, and the term structure represents how these rates change over time. When the yield curve is assumed to be flat, the same rate applies to all maturities, which simplifies discounting cash flows but still requires analysis on how rate changes at specific points affect overall bond values.

Interest Rate Risk Exposure

This concept refers to the sensitivity of a bond’s value to changes in interest rates. In fixed income, each cash flow of a bond is discounted at the appropriate rate, so even small movements in the yield curve can impact the price significantly. Understanding how different cash flows respond to shifts in different maturities is essential when measuring the risk associated with holding bonds.

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