A coupon bond makes a series of payments over its life, and so fixed-income investors need a measure of the average maturity of the bond's promised cash flow to serve as a summary statistic of the effective maturity of the bond. Such investors also need a measure that can be used as a guide to the sensitivity of a bond to interest rate changes, since price sensitivity tends to increase with time to maturity.

The statistic that aids investors in both areas is duration, which along with convexity can help fixed-income investors gauge uncertainty when managing their portfolios. (For background reading, check out our *Advanced Bond Concepts* tutorial.)

### Duration of a Bond

In 1938, Frederick Macaulay termed the effective-maturity concept the duration of the bond, and suggested that duration be computed as the weighted average of the times to each coupon or principal payment made by the bond. Macaulay's duration formula is as follows:

- D is the bond's duration
- C is the periodic coupon payment
- F is the face value at maturity (in dollars)
- T is the number of periods until maturity
- r is the periodic yield to maturity
- t is the period in which the coupon is received

### Duration in Fixed Income Portfolio Management

Duration is key in fixed-income portfolio management for the following three reasons:

- It is a simple summary statistic of the effective average maturity of a portfolio.
- It is an essential tool in immunizing portfolios from interest rate risk.
- It is an estimate of the interest rate sensitivity of a portfolio.

Because duration is so important to fixed-income portfolio management, it is worth exploring the following properties:

- The duration of a zero-coupon bond equals its time to maturity.
- Holding maturity constant, a bond's duration is lower when the coupon rate is higher. This rule is due to the impact of early higher coupon payments.
- Holding the coupon rate constant, a bond's duration generally increases with time to maturity. This property of duration is fairly intuitive; however, duration does not always increase with time to maturity. For some deep-discount bonds, duration may fall with increases in maturity.
- Holding other factors constant, the duration of a coupon bond is higher when the bond's yield to maturity is lower. This principle applies to coupon bonds. For zero-coupon bonds, duration equals time to maturity, regardless of the yield to maturity.
- The duration of a level perpetuity is (1 + y) / y. For example, at a 10% yield, the duration of perpetuity that pays $100 once a year forever will equal 1.10 / .10 = 11 years, but at an 8% yield it will equal 1.08 / .08 = 13.5 years. This principle makes it obvious that maturity and duration can differ substantially. The maturity of the perpetuity is infinite, whereas the duration of the instrument at a 10% yield is only 11 years. The present-value-weighted cash flow early on in the life of the perpetuity dominates the computation of duration. (For more information on portfolio management, read
*Equity Portfolio Management Mechanics*and*Preparing For A Career As A Portfolio Manager*.)

### Duration for Gap Management in Fixed Income

Many banks have a natural mismatch between asset and liability maturities. Bank liabilities are primarily the deposits owed to customers, most of which are very short-term in nature and of low duration. Bank assets by contrast are composed largely of outstanding commercial and consumer loans or mortgages. These assets are of longer duration and their values are more sensitive to interest rate fluctuations. In periods when interest rates increase unexpectedly, banks can suffer serious decreases in net worth if their assets fall in value by more than their liabilities.

To manage this risk, a technique called gap management became popular in the 1970s and early 1980s, with the idea being to limit the "gap" between asset and liability durations. Adjustable-rate mortgages (ARM) were one way to reduce the duration of bank-asset portfolios.

Unlike conventional mortgages, ARMs do not fall in value when market rates increase because the rates they pay are tied to the current interest rate. Even if the indexing is imperfect or entails lags, it greatly diminishes sensitivity to interest rate fluctuations.

On the other side of the balance sheet, the introduction of longer-term bank certificates of deposit (CD) with fixed terms to maturity served to lengthen the duration of bank liabilities, also reducing the duration gap. (Learn more about financial gaps in *Playing The Gap*.)

### Understanding Gap Management

One way to view gap management is as an attempt by the bank to equate the durations of assets and liabilities to effectively immunize its overall position from interest rate movements. Because bank assets and liabilities are roughly equal in size, if their durations are also equal, any change in interest rates will affect the value of assets and liabilities equally. Interest rate changes would have no effect on net worth. Therefore, net worth immunization requires a portfolio duration, or gap, of zero. (To learn more about bank assets and liabilities, read *Analyzing A Bank's Financial Statements*.)

Institutions with future fixed obligations, such as pension funds and insurance companies, are different from banks in that they think more in terms of future commitments. Pension funds, for example, have an obligation to provide workers with a flow of income upon retirement and must have sufficient funds available to meet this commitment. As interest rates fluctuate, both the value of the assets held by the fund and the rate at which those assets generate income fluctuate.

The portfolio manager, therefore, may want to protect (immunize) the future accumulated value of the fund at some target date against interest-rate movements. The idea behind immunization is that with duration-matched assets and liabilities, the ability of the asset portfolio to meet the firm's obligations should be unaffected by interest rate movements. (Read more about pension funds' obligations in *Analyzing Pension Risk*.)

### Convexity in Fixed Income Management

Unfortunately, duration has limitations when used as a measure of interest rate sensitivity. The statistic calculates a linear relationship between price and yield changes in bonds. In reality, the relationship between the changes in price and yield is convex.

In Figure 1, the curved line represents the change in prices given a change in yields. The straight line, tangent to the curve, represents the estimated change in price via the duration statistic. The shaded area shows the difference between the duration estimate and the actual price movement. As indicated, the larger the change in interest rates, the larger the error in estimating the price change of the bond.

Figure 1

Convexity, which is a measure of the curvature of the changes in the price of a bond in relation to changes in interest rates, is used to address this error. Basically, it measures the change in duration as interest rates change. The formula is as follows:

- C is convexity
- B is the bond price
- r is the interest rate
- d is duration

In general, the higher the coupon, the lower the convexity, becauseĀ a 5% bond is more sensitive to interest rate changes than a 10% bond. Due to the call feature, callable bonds will display negative convexity if yields fall too low, meaning the duration will decrease when yields decrease. Zero-coupon bonds have the highest convexity. These relationships are only valid when comparing bonds that have the same durations and yields to maturity.

A high convexity bond is more sensitive to changes in interest rates and should see larger fluctuations in price when interest rates move.

The opposite is true of low convexity bonds; their prices don't fluctuate as much when interest rates change. When graphed on a two-dimensional plot, this relationship should generate a long-sloping U shape (hence, the term "convex").

Low-coupon and zero-coupon bonds, which tend to have lower yields, have the highest interest rate volatility. In technical terms, this means that the modified duration of the bond requires a larger adjustment to keep pace with the higher change in price after an interest-rate moves. Lower coupon rates lead to lower yields, and lower yields lead to higher degrees of convexity.

(To read about some risks associated with callable and other bonds, read *Call Features: Don't Get Caught Off Guard* and *Corporate Bonds: An Introduction To Credit Risk*.)

### The Bottom Line

Interest rates are constantly changing and add a level of uncertainty to fixed-income investing. Duration and convexity allow investors to quantify this uncertainty and are useful tools in the management of fixed-income portfolios.

For further reading on fixed-income investing, see *Creating The Modern Fixed-Income Portfolio* and *Common Bond Buying Mistakes*.