# Convexity (Bonds) - Explained

What is Convexity of a Bond?

# What is Convexity?

In the bond world, convexity is simply defined as a measure of the sensitivity of the bonds duration to change its yield. Convexity is believed to be a good measure for bond price changes that are accompanied by greater fluctuations in their interest rates. Mathematically, convexity is the second derivative of the formula for change in bond prices with a change in interest rates and a first derivative of the duration equation. The duration of the changes in a bond in relation to the changes in its interest rate can be demonstrated by using convexity. This has enabled the measurement and management of the portfolio's exposure to interest rate risk by portfolio managers who use convexity as a risk-management tool.

## How Does Bond Convexity Work?

To understand convexity, it's necessary to understand the relationship between bond prices and market interest rates. The lower the interest rate, the higher the bond price, and vice versa. As interest rates increases, the bond may suffer a decrement in the earnings they may offer a potential investor when compared to other securities. This can be said to be the cause of the opposite reaction between bond prices and interest rates. The earnings or returns expected to be made by an investor through the acquisition or a holding of a particular security is known as the bond yield. Several characteristics including the market interest rates which can change regularly are largely dependent on by bond prices.

## How Market Interest Rates and Bond Yields Relate

When it comes to yields and interest rates, as the interest rate increases, the price of bonds returning less than the increment rate attained by the interest rate will fall. A rise in market rates will lead to a rise in the yields of new bonds coming on the market as they are being issued at the new, higher rates. Also, investors demand a higher yield from the bonds they buy, as rates increases. If they expect a future rise in interest rates, they don't want a fixed-rate bond at current yields. Hence, the issuer of these debt vehicles must also raise their yields to remain competitive when interest rates increase.

## How Interest Rates and Bond Prices Relate

Bond prices and interest rates are inversely related. A rise in interest rates will lead to a fall in bond prices, and some degree of pain will be felt by bond investors, especially those who remain in bond funds. In a market experiencing rising rates, bondholders will look to sell their existing bonds and acquire newly-issued bonds that are paying higher yields. The presence of lower rates selling bonds on the market will cause a drop in the prices of these debt holdings. An investor may have to wait for a stop in the rising rates before buying the higher-yielding security. This results in an inverse relationship between the two.

## Bond Duration

Bond duration is an approximate measure of a bond's price sensitivity to changes in interest rates. An increase in a bond's duration implies that there will be a movement to a greater degree by the bond's price in the opposite direction of the rates. Alternately, a decrease in this figure will lead to a decreased movement in the debt instrument. If for instance, a bond has a duration of 6 years, its price will rise by about 6%, if its yield drops by a percentage point (100 basis points), and its price will fall by about 6%, if its yield rises by that amount.

## Convexity and Risk

Where duration assumes the existence of a linear relationship between bond prices and interest rates, convexity allows for the consideration of other factors and generates a slope. This has made convexity to be a better measurement tool for interest rate risk, concerning bond duration. Though duration is a good tool for measuring the effect of small and sudden fluctuations in interest rates on bond prices, convexity is a better tool because it measures the impact of large interest rate fluctuations on bond prices. The rate of exposure of the portfolio to systemic risk increases as convexity increases, and vice versa. Generally, the higher the yield, the lower the convexity of a bond.

## Negative and Positive Convexity

A bond is said to have a negative convexity if there is an increase in its duration as its yields increase. That is, there will be a decline in the bond price by a greater rate when there is a rise in yields than if yields had fallen. Therefore, a bond whose price falls with an increased duration is said to have negative convexity. A positive convexity bond, on the other hand, is one whose bond duration rises as yields fall. That is, there will be a rise in the bond price by a greater rate when there is a fall in yields than if yields had risen.

## Real World Example of Convexity

A negative convexity will be experienced by a majority of mortgage-backed securities (MBS) because their yield is generally higher than that of traditional bonds. As a result of this, to make an existing MBS holder have an unattractive yield compared to that in the current market, a significant rise in yields must occur. An example is an SPDR Barclays Capital Mortgage Backed Bond ETF (MBG) offering a 3.33% yielding as of the 26th of March, 2019. By comparing the ETF's yield to the current 10-year Treasury yield trading at roughly 2.45%, there will have to be a substantial increase well above 3.33% of the rates, in order to make the MBG risk losing out on higher yields.