# Approximate Modified Duration and Convexity Adjustment | CFA Level I Fixed Income

In this lesson, we’ll explore how to calculate the approximate modified duration and convexity adjustment of a bond. This will help us estimate the change in bond price due to a change in yield-to-maturity more accurately.

## Approximate Modified Duration

Remember that the modified duration can be used to estimate the change in bond price given a change in yield. However, the actual price to yield-to-maturity relationship is not a straight line but follows a convex curve. To simplify the process, we can use the approximate modified duration.

Steps to Calculate Approximate Modified Duration: Find the value of the bond at its current yield (V0) using the discounted cash flow method. Find the value of the bond if its yield drops by a small amount (V-). Find the value of the bond if its yield rises by the same magnitude (V+). Calculate the slope of the line that intersects V- and V+.

Note: For the approximation to be accurate, the change in yield-to-maturity should be very small, and the precision used in the calculations should be high.

Both the modified duration and approximate modified duration can be used to estimate the change in bond price due to a change in yield. However, as the actual relationship between price and yield-to-maturity is a convex curve, using a straight line to calculate is only an approximation. To improve the estimate, we can introduce a second term based on the bond’s convexity.

Convexity is a measure of the curvature of the price-yield relation. The more curved it is, the higher the convexity measure, and the greater the convexity adjustment to the duration-based estimate of the change in price.

EXAMPLE

Let’s illustrate this with an example where a fixed coupon bond has 15 years to maturity and pays an 8% coupon annually. The bond’s current yield-to-maturity is 7.4%.

We’ll follow the steps to calculate V0, V-, and V+ using a 1 basis point decrease and increase in yield. This will allow us to calculate the approximate modified duration and the approximate convexity.

With the modified duration and convexity of the bond, we can estimate the price change for a wider range of changes in the yield of the bond. We are no longer constrained to small changes in yield as we are now making convexity adjustments.

## Impact of Individual Yield Components

A bond’s yield-to-maturity is composed of a government benchmark yield and a spread over that benchmark. The benchmark yield curve’s interest rates have two components: the real rate of return and expected inflation. The benchmark rate is often considered the risk-free rate.

A bond’s spread to the benchmark curve can be decomposed into three components: a premium for the difference in tax treatment of the bond over the benchmark bonds, a premium for the lack of liquidity relative to the benchmark bonds, and a premium for credit risk.

We can estimate the impact of each individual yield component on the bond’s value using the same formula we introduced earlier.

EXAMPLE

In conclusion, by understanding the concepts of approximate modified duration and convexity adjustment, we can better manage our bond investments and estimate the impact of individual yield components on bond prices.

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