# Convexity and duration relationship counseling

### Duration & Convexity - Fixed Income Bond Basics | Raymond James

Jul 1, In this article, we will cover the Duration and Convexity measure. Duration assumes a linear relationship between bond price and yield. Similarly, as the yield increases, the slope of the curve will decrease, as will the duration. Convexity is a measure of the amount of “whip” in the bond's price yield . Our goal is to help investors understand why it's important to consider both duration and convexity the relationship between bond prices and interest rate changes. . Reference. Vanguard Investment Counseling & Research, Investing.

As highlighted in the figure 1.

## What Is Duration? Macaulay Duration, Modified Duration and Convexity

As evident from figures 1. This is demonstrated as duration effect and is given below.

Duration effect captures both the direction and magnitude of change in bond price due to a change in yield. Modified duration is very similar to Macaulay duration, but improvises by discounting the Macaulay duration back by one payment time period.

## FRM-I “Duration” Tutorial: Master the Art of Calculating Duration & Convexity

For example, if the bond is a semi-annual bond, the Macaulay duration should be discounted back by six months using the current yield. Duration works only for small changes in the yield. When the change in the yield is high, say basis points, duration becomes inaccurate due to the convexity effect. Convexity Duration assumes a linear relationship between bond price and yield, which is not true especially when there is high volatility in yields.

Convexity takes off where duration stops in a sense that it gives a more accurate price of a bond based on interest rate changes than Duration. The formula for measuring convexity is: T he second term in the above equation is the second derivative of the bond price and yield functions.

The first derivative measure that is duration measures how price changes with yields, while second derivative is a measure of changes in the first derivative with changes in the yields. The relationship between yields and bond prices is more of a curved line rather considering a straight line as estimated by duration.

This curvature in bond price can be best estimated when we combine Duration effect and the Convexity effect which results in a more accurate price of the bond.

Previous Jobs in Finance. In plain-terms — think of it as an approximation of how long it will take to recoup your initial investment in the bond. There are two types of duration: Macaulay duration and modified duration.

Macaulay duration is useful in immunization, where a portfolio of bonds is constructed to fund a known liability.

### What Is Duration? Macaulay Duration, Modified Duration and Convexity

Macaulay Duration The calculation of Macaulay Duration is shown below: Graphically, Macaulay Duration is the point of balance in years for the cash flows from the bond see below. Modified Duration Modified duration is a measure of the price sensitivity of a bond to interest rate movements. It is calculated as shown below: The percentage change applies to the price of the bond including accrued interest.

The bond was repriced for an increase and decrease in rates of 2. The Modified Duration for this bond will be: Since the bond was initially priced at par, the estimated prices are: The actual prices were: The discrepancy between the estimated change in the bond price and the actual change is due to the convexity of the bond, which must be included in the price change calculation when the yield change is large.

**Investopedia Video: The Basics Of Bond Duration**