Investing in the bond markets involves a certain level of knowledge about different factors that influence the pricing and returns of bonds.

Two key measures that help predict how bond prices are likely to change when interest rates fluctuate are bond duration and bond convexity.

These two measures are integral components of fixed-income risk management.

Understanding these two concepts is essential for any investor or financial professional.

## Bond Duration

Bond duration measures the sensitivity of a bond’s price to changes in interest rates.

In other words, it tells you how much the price of a bond would change for a one percent change in interest rates.

Technically, bond duration is the weighted average of the present value of a bond’s cash flows, which are its regular interest payments and the repayment of principal at maturity.

A bond’s duration will increase with its term to maturity, other things being equal.

However, it will decrease with an increase in the bond’s coupon rate or yield to maturity.

For example, a 10-year zero-coupon bond has a duration of 10 years, but a 10-year bond with a 5% annual coupon will have a duration of less than 10 years.

## Duration and Convexity Formula

The Macaulay duration formula calculates bond duration as the weighted average of the times until each payment is received, using the present value of the cash flow divided by the bond price.

The formula for convexity, on the other hand, is more complex.

It involves second derivatives and the sum of the present value of all cash flows multiplied by the square of the time until payment, divided by the product of the bond price and (1 + yield)^2.

While the duration is the first derivative of the bond price-yield function, convexity is the second derivative.

### Bond Duration and Bond Convexity Explained

## What Is Bond Convexity?

Bond convexity is a measure of the curvature, or the degree of the curve, in the relationship between bond prices and bond yields.

Convexity is an important measure of a bond’s price sensitivity to changes in interest rates.

If a bond’s price-yield curve is steep, it has high convexity; if it’s flat, it has low convexity.

## Convexity Definition

In bond market terminology, convexity is defined as the rate of change of the duration of a bond with respect to changes in interest rates.

The larger the convexity, the more sensitive the bond price is to changes in interest rates.

Convexity can be positive or negative, which indicates how the bond price will change with fluctuations in interest rates.

## Positive Convexity

A bond exhibits positive convexity when its price increases at an increasing rate as yields fall, and its price decreases at a decreasing rate as yields rise.

This is beneficial for investors because it means that bond prices are more sensitive to decreases in yields than to increases in yields.

Most ordinary bonds, such as treasury bonds and corporate bonds, exhibit positive convexity.

## Negative Convexity

A bond exhibits negative convexity when its price increases at a decreasing rate as yields fall, and its price decreases at an increasing rate as yields rise.

This is typically not desirable for investors, as it means the bond’s price is more sensitive to increases in yields than to decreases in yields.

Certain types of bonds, like mortgage-backed securities, exhibit negative convexity because the potential for prepayment increases when interest rates fall.

## Convexity of Zero Coupon Bond

Zero-coupon bonds provide a fascinating case.

These are bonds that do not pay any interest but are sold at a discount and redeemed at face value.

They have the highest possible duration for a given maturity because the entire cash flow, the face value, is received at maturity.

For the same reason, zero-coupon bonds also have higher convexity than similar coupon bonds.

## What Happens to Duration and Convexity as Maturity Increases?

Both duration and convexity increase as the maturity of a bond increases.

The longer the maturity of a bond, the higher its duration, meaning it is more sensitive to changes in interest rates.

Similarly, a longer maturity also results in higher convexity, meaning the bond price is more sensitive to changes in yields.

However, the relationship between maturity, duration, and convexity isn’t linear.

For example, doubling the maturity of a bond more than doubles its duration and convexity.

## How Does Duration Convexity Impact Bond Price?

Duration and convexity both impact bond prices, but they do so in different ways.

The duration of a bond directly relates to the percentage change in price for a 1% change in yield.

If a bond has a duration of 6 years, for example, its price would decrease by approximately 6% for a 1% increase in yield and increase by approximately 6% for a 1% decrease in yield.

Convexity, however, relates to the change in duration for a change in yield, giving a more accurate prediction of price changes, particularly for larger yield changes.

The higher a bond’s convexity, the more its duration changes as yields change, resulting in larger price changes.

By understanding these factors and how they impact bond prices, investors can make informed decisions about which bonds to invest in, based on their expectations of future interest rate changes.

## FAQs – Duration and Convexity in Bond Markets

### What is a bond’s duration, and why is it important?

Duration is a measure of a bond’s sensitivity to changes in interest rates.

It is expressed in years, and represents the weighted average time it takes for a bond’s cash flows (both the periodic interest payments and the final principal payment) to be received.

The longer a bond’s duration, the greater its sensitivity to interest rate changes.

Therefore, understanding duration is important for assessing the potential price risk associated with bonds in the context of changing interest rates.

### What is convexity in relation to bond markets?

Convexity is a measure that demonstrates how the duration of a bond changes as the interest rate changes.

It is used to better estimate the price change of bonds when there are large changes in interest rates.

High convexity means the bond’s price increases more when rates fall, and decreases less when rates rise, compared to bonds with low convexity.

### What is the difference between Macaulay Duration, Modified Duration, and Effective Duration?

Macaulay Duration is the weighted average time it takes to receive the bond’s cash flows.

Modified Duration adjusts the Macaulay Duration to measure the price sensitivity of a bond to interest rate changes.

Effective Duration, on the other hand, takes into account the fact that the cash flows of some bonds (like those with embedded options) can change as interest rates change.

Thus, Effective Duration provides a more accurate measure of interest rate risk for these types of bonds.

### What is the relationship between bond prices, duration, and interest rates?

Bond prices and interest rates move in opposite directions, a relationship that is quantified by duration.

When interest rates rise, bond prices fall, and vice versa.

The longer the duration of a bond, the more its price will change for a given change in interest rates.

### How can I use duration and convexity to manage interest rate risk?

Duration and convexity can be used to construct bond portfolios that are immunized against interest rate risk.

For example, a portfolio manager could match the duration of the portfolio’s assets to its liabilities to ensure that the portfolio’s value doesn’t change when interest rates change.

Additionally, a manager might seek to increase the portfolio’s convexity in order to take advantage of potential price increases should interest rates fall, and mitigate potential price decreases should rates rise.

### Why do bonds with higher coupon rates have lower durations?

The duration of a bond is affected by the timing and size of the cash flows it provides to investors.

Bonds with higher coupon rates provide larger cash flows in the near term, which reduces their duration because the weighted average time to receive the bond’s cash flows is shorter.

Conversely, lower coupon bonds provide smaller near-term cash flows and larger final cash flows, which increases their duration.

### Does a bond’s convexity change over time?

A bond’s convexity can change over time as it moves closer to its maturity date.

This is particularly true for callable and putable bonds, where the probability of the option being exercised changes as the bond ages, affecting the bond’s cash flow pattern and, in turn, its convexity.

But for plain vanilla bonds, the change in convexity over time is relatively minimal.

### What is negative convexity, and which bonds exhibit it?

Negative convexity occurs when a bond’s price decreases more when interest rates fall, and increases less when interest rates rise, compared to a bond with constant convexity.

This is usually the case for callable bonds.

The issuer of a callable bond has the right to redeem the bond before maturity when interest rates fall, which limits the bond’s potential price increase and gives the bond negative convexity.

Understanding duration and convexity is crucial for managing interest rate risk in a bond portfolio. Both measures provide investors with a way to compare bonds and bond portfolios, enabling better-informed investment decisions.