8/3/2019 Portfolio Analysis Duration Convexity and Immunization

1/2

Portfolio Analysis: Duration, Convexity, and Immunization 1 of 2www.qfinance.com

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 knownas interest rate risk). For example, if interest rates rise 1%, a bond with a two-year duration will fall about2% in value. Convexity is a measure of how prices rise when yields fall, and can also be used to measureinterest rate risk.

Why They Are Important

Using a combination of duration and convexity allows traders to hedge investments to minimize or offset theimpact of changes in interest ratesa process known as immunization.

How They Work in Practice

Duration is a weighted average of the present value of a bonds payments. It provides an insight into howsensitive a bond or portfolio is to changes in interest rates. The longer the duration, the longer the averagematurity and, therefore, the bonds sensitivity to interest rate changes. Securities with the same durationhave the same interest rate risk exposure. Duration can be expressed in years (to average maturity) or as apercentage (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 squaredtime (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 portfolioso the duration is equal to the investment time horizon. For example, you might select bonds that you knowwill 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 investorreceives a specific rate of return over time regardless of what happens to interest rates, because theportfolios duration is equal to the investors time horizon. This means any changes to interest rates willaffect the bonds price and reinvestment at the same rate, keeping the rate of return steady. Maintaining animmunized portfolio means rebalancing the portfolios average duration every time interest rates change, sothat the average duration continues to equal the investors time horizon.

Tricks of the Trade

The concept of duration was first developed by Frederick Macaulay in 1938, as a tool for measuringbond 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 whena callable bond is nearing its call price. In this case, traders use modified convexity, which is the

http://www.qfinance.com/http://www.qfinance.com/

8/3/2019 Portfolio Analysis Duration Convexity and Immunization

2/2

Portfolio Analysis: Duration, Convexity, and Immunization 2 of 2www.qfinance.com

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 TermStructure of Interest Rates). The latter can assume an arbitrary shape (although a normal yield curvehas negative convexity), and complex stochastic models have been proposed for its evolution.

More Info

Article:

Radcliffe, Brent. Immunization inoculates against interest rate risk. Investopedia. Online at:www.investopedia.com/articles/financial-theory/09/bond-interest-rate-immunization.asp

Website:

This Matter on duration and convexity: thismatter.com/money/bonds/duration-convexity.htm

See AlsoChecklists

The Bond Market: Its Structure and FunctionCalculations

Current Price of a Bond

To see this article on-line, please visithttp://www.qfinance.com/asset-management-calculations/portfolio-analysis-duration-convexity-and-immunization

http://www.qfinance.com/http://www.qfinance.com/http://www.qfinance.com/asset-management-calculations/portfolio-analysis-duration-convexity-and-immunizationhttp://www.qfinance.com/asset-management-calculations/current-price-of-a-bondhttp://www.qfinance.com/asset-management-checklists/the-bond-market-its-structure-and-functionhttp://thismatter.com/money/bonds/duration-convexity.htmhttp://www.investopedia.com/articles/financial-theory/09/bond-interest-rate-immunization.asphttp://www.qfinance.com/asset-management-calculations/term-structure-of-interest-rateshttp://www.qfinance.com/asset-management-calculations/term-structure-of-interest-rateshttp://www.qfinance.com/http://www.qfinance.com/