### Term Structure of Yield Volatility and Interest Rate Risk

Time-horizon is a very important aspect in understanding interest rate risk and the return characteristics of a fixed-rate investment. The primary concern for an investor is the change in the price of a bond given a sudden change in its yield-to-maturity. Reinvestment of coupon interest is a key factor in the investor’s horizon yield.

Prices of fixed income securities are affected not just by the level of interest rates but also the volatility of interest rates. The risk of a default-free security emanates from two sources: interest rate shift and the risk of changes in interest rate volatility.

## Risk of Interest Rate Shift

To manage this type of risk, analysts have to measure it first. In this regard, duration becomes a particularly important tool since it helps the analyst establish the sensitivity of a portfolio to changes in the level of interest rate changes.

## Risk of Changes in Interest Rate Volatility

This type of risk is less obvious but nonetheless represents a major component of the total risk of a fixed income security. For a given duration and current value of a position, interest rate increases as the expected yield volatility increases.

## Investment Horizon and Interest Rate Risk

When the government par-curve is shifted up or down by some amount to calculate effective duration and effective convexity, it is described as “parallel” yield curve shifts. Yield curves are rarely straight lines, so this shift may also be described as “shape-preserving” shift to the yield curve.

Given a parallel shift to the yield curve, the yield-to-maturity and coupon reinvestment rates are assumed to change by the same amount and in the same direction. Macaulay duration is particularly important because it identifies the investment horizon such that losses (gains) from coupon reinvestment offset gains (losses) from changes in market prices.

The term structure of yield volatility is the relationship between the volatility of bond yields-to-maturity and times-to-maturity. The term structure of bond yields (also called the “term structure of interest rates”) is typically upward sloping. However, it can be different depending on some factors, including government policy.

For instance, a central bank engaging in an expansionary monetary policy might cause the yield curve to steepen by reducing short-term interest rates. However, this policy might result in greater volatility in short-term bond yields to maturity than in longer-term bonds. Longer-term bond yields are mainly determined by future inflation and economic growth expectations. Such expectations are often less volatile.

## Question

An investment bank needs to rank 3 bonds in terms of interest rate risk.

$$\begin{array}{l|c|c|c} \textbf{Bond} & \textbf{Modified Duration} & \textbf{Convexity} & \textbf{Yield} \\ \hline \text{A} & \text{3.50} & \text{100} & \text{25} \\ \text{B} & \text{5.25} & \text{350} & \text{15} \\ \text{C} & \text{5.30} & \text{250} & \text{10} \\ \end{array}$$

The modified duration and convexity statistics are annualized. Given the predicted change in yield in each of these bonds, how should the bonds be ranked in terms of interest rate risk (from the highest risk to the lowest)?

A. Bond C has the highest degree of interest rate risk, followed by Bond A, and Bond B

B. Bond B has the highest degree of interest rate risk, followed by Bond C, and Bond A

C. Bond C has the highest degree of interest rate risk, followed by Bond B, and Bond A

Solution

Based on the assumed changes in the yields and given measures of modified duration and convexity, Bond C has the highest degree of interest rate risk followed by Bond B, and then Bond C.

Bond A: %ΔPVFULL ≈ (-3.50×0.0025) + (1/2×100×(0.0025)2) ≈ -0.0084375

Bond B: %ΔPVFULL ≈ (-5.25×0.0015) + (1/2×350×(0.0015)2) ≈ -0.0074813

Bond C: %ΔPVFULL ≈ (-5.30×0.0010) + (1/2×250×(0.0010)2) ≈ -0.005175

Describe how the term structure of yield volatility affects the interest rate risk of a bond

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