Lesson 8.3: The Term Structure and Interest-Rate Risk

Introduction

Understanding the term structure of interest rates is essential for anyone involved in fixed-income securities. This lesson will delve into the concepts of spot rates, forward rates, and the yield curve. Additionally, we will explore duration and convexity—two critical measures of a bond's price sensitivity to interest rate changes. By the end of this lesson, students will be able to compute and interpret these essential concepts, enhancing their skills in bond investment and risk assessment.

Learning Objectives

• Define and analyze spot, forward, and par rates and their relationship to the yield curve.
• Explain the concepts of duration and convexity and how they measure price sensitivity.
• Relate spot and forward rates while interpreting the 정보 with respect to the yield curve.
• Calculate and make sense of duration and convexity in practical terms for fixed-income securities.
• Estimate the change in a bond's price based on a change in yield using duration and convexity.

The Term Structure of Interest Rates

The term structure of interest rates describes the relationship between interest rates (or bond yields) and the time to maturity of the debt. This structure is illustrated through the yield curve, which can take on various shapes: normal, inverted, or flat.

Spot and Forward Rates

• Spot Rate: The current interest rate available for a zero-coupon bond maturing at a specific point in the future.
• Forward Rate: The expected interest rate between two future periods. It can be viewed as the rate agreed upon today for a loan that will occur in the future.

Examples of Spot and Forward Rates

Suppose the following spot rates are observed:

• $S_1 = 2\%$ for 1 year
• $S_2 = 3\%$ for 2 years
• $S_3 = 4\%$ for 3 years

Using this information, we can calculate the 1-year forward rate starting in Year 1 ($F_{1,1}$):

$$

F_{1,1} = $\frac{(1 + S_2)^2}{(1 + S_1)}$ - 1 = $\frac{(1 + 0.03)^2}{(1 + 0.02)}$ - 1 = $\frac{1.0609}{1.02}$ - $1 \approx 0$.0379 \text{ or } 3.79\%

$$

This means that investors expect a 3.79% interest rate for the loan from Year 1 to Year 2.

The Yield Curve

The yield curve plots the yields of bonds of the same credit quality but different maturities. A typical yield curve is upward sloping, reflecting higher yields for longer maturities due to increased risk and time value of money.

Types of Yield Curves

• Normal Yield Curve: Upward sloping, indicating that long-term rates are higher than short-term rates.
• Inverted Yield Curve: Downward sloping, indicating short-term rates are higher than long-term rates, often predicting an economic downturn.
• Flat Yield Curve: Indicates that there is little difference between short-term and long-term rates.

Duration and Convexity

Duration measures a bond's sensitivity to interest rates. Specifically, it indicates the percentage change in price for a 1% change in yield. There are different types of duration:

• Macaulay Duration: This is a weighted average time to receive the bond's cash flows.
• Modified Duration: This is the Macaulay Duration adjusted for yield, reflecting price sensitivity to changes in yield.

Gift of Calculation: Duration Example

Consider a 3-year bond with:

• Face value $F = 1000$
• Coupon rate $C = 5\%$ (annual payments)
• Yield $Y = 4\%$

The cash flows are:

1. Year 1: $50
2. Year 2: $50
3. Year 3: 1050

Using Macaulay Duration:

$$

D = \frac{\sum \frac{t \cdot C_t}{(1 + Y)^t}}{\text{Current Price}}

$$

With the calculated present value of cash flows, we could determine the duration.

For Modified Duration:

$$\text{Modified Duration} = \frac{D}{1 + Y}$$

This yields the bond's price sensitivity. If the modified duration is 2.5, then for a 1% increase in yield, the bond's price would decrease by approximately 2.5%.

Price Sensitivity: Understanding Convexity

Convexity further adjusts duration to account for the curvature of price changes in response to interest rate changes. It illustrates how bonds react to changes in interest rates beyond the linear approximation given by duration.

Convexity Calculation Example

Assuming we calculate the convexity of our bond:

$$\text{Convexity} = \frac{1}{P}\sum \frac{C_t}{(1 + Y)^{t + 2}} \cdot t(t + 1)

$$

where $P$ is the current price of the bond, and $C_t$ is the cash flow at time $t$.

Price Change Estimates Using Duration and Convexity

The estimated price change of a bond can be calculated by the following approximation formula:

$$\Delta P \approx - (D \cdot \Delta Y) + \frac{1}{2} \cdot \text{Convexity} \cdot (\Delta Y)^2$$

This formula shows the contribution of both duration and convexity to price changes due to yield shifts.

Conclusion

In summary, understanding the term structure of interest rates is crucial for analyzing fixed-income securities. students has learned how to relate spot and forward rates, interpret yield curves, compute duration and convexity, and estimate a bond's price change based on yield shifts. This knowledge is essential for making informed investment decisions in the fixed-income market.

Study Notes

• The term structure illustrates the relationship between interest rates and different maturities.
• Spot rates are current interest rates for zero-coupon bonds, while forward rates are agreed-upon rates for future loans.
• The yield curve can be normal, inverted, or flat, with each shape indicating different economic conditions.
• Duration measures price sensitivity, while convexity adjusts this measure for greater accuracy regarding price movements.
• The price change of a bond can be estimated using duration and convexity for varying interest rate changes.