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Home > Asset Management Calculations > Portfolio Analysis: Duration, Convexity, and Immunization

Asset Management Calculations

# Portfolio Analysis: Duration, Convexity, and Immunization

## What They Measure

Duration is a measure of how sensitive the price of bonds are to changes in interest rates (otherwise known as interest rate risk). For example, if interest rates rise 1%, a bond with a two-year duration will fall about 2% in value. Convexity is a measure of how prices rise when yields fall, and can also be used to measure interest rate risk.

## Why They Are Important

Using a combination of duration and convexity allows traders to hedge investments to minimize or offset the impact of changes in interest rates—a process known as immunization.

## How They Work in Practice

Duration is a weighted average of the present value of a bond’s payments. It provides an insight into how sensitive a bond or portfolio is to changes in interest rates. The longer the duration, the longer the average maturity and, therefore, the bond’s sensitivity to interest rate changes. Securities with the same duration have the same interest rate risk exposure. Duration can be expressed in years (to average maturity) or as a percentage (the percentage change in price for a 1% change in its yield to maturity).

Duration = (P- - P+) ÷ (2 × P0 × ?y)

where:

P0 = bond price

P- = bond price when interest rates are incremented

P+ = bond price when rates are decremented

?y = change in interest rates (decimal form)

Convexity is a measure of the rate at which duration changes as yields fall, and is expressed in squared time (t + 1). To estimate the convexity of a bond or portfolio, we can use the following formula:

Convexity approximation = (P+ + P- - 2P0) ÷ 2P0(?y)2

Immunization: To immunize a portfolio, you need to know the duration of the bonds and adjust the portfolio so the duration is equal to the investment time horizon. For example, you might select bonds that you know will return \$10,000 in five years’ time regardless of interest rate changes.

Normally, when interest rates go up, bond prices go down. But if a portfolio is immunized, the investor receives a specific rate of return over time regardless of what happens to interest rates, because the portfolio’s duration is equal to the investor’s time horizon. This means any changes to interest rates will affect the bond’s price and reinvestment at the same rate, keeping the rate of return steady. Maintaining an immunized portfolio means rebalancing the portfolio’s average duration every time interest rates change, so that the average duration continues to equal the investor’s time horizon.

• The concept of duration was first developed by Frederick Macaulay in 1938, as a tool for measuring bond price volatility in relation to the length of a bond. However, there are other formulae for calculating duration, including “effective duration” and “modified duration.”

• Convexity is usually a positive term, but sometimes the term is negative, such as occurs when a callable bond is nearing its call price. In this case, traders use modified convexity, which is the measured convexity when there is no expected change in future cash flows, or effective convexity, which is the convexity measure for a bond for which future cash flows are expected to change.

• The notion of bond convexity should not be confused with the convexity of the yield curve (see Term Structure of Interest Rates). The latter can assume an arbitrary shape (although a normal yield curve has negative convexity), and complex stochastic models have been proposed for its evolution.