What Are Duration and Convexity?

Duration and convexity are two tools used to manage the risk exposure of fixed-income investments. Duration measures the bond's sensitivity to interest rate changes. 澳洲幸运5官方开奖结果体彩网:Convexity relates to the interaction between a bond's price and its yield as it experienc🤡e𒐪s changes in interest rates.

With 澳洲幸运5官方开奖结果体彩网:coupon bonds, investors rely on a metric known as duration to measure a bond's price sensitivity to changes in interest rates. Because a coupon bond makes a series of payments over its lifetime, fixed-income investors need ways to measure the average maturity of a bond's promised cash flow, to serve as a summary statistic of the bond’s effective maturity.𓂃 The duration accomplishes this, letting fixed-income investors more effectively gauge uncertainty when managing their portfolios.

Duration of a Bond

In 1938, Canadian economist Frederick Robertson Macaulay dubbed the effective-maturity concept the “duration” of the bond. In doing so, he suggested that this duration be computed as the weighted avera🦄ge of the times to maturity of each coup⛎on, or principal payment, made by the bond. 澳洲幸运5官方开奖结果体彩网:Macaulay's duration formula is as follows:

$\begin{aligned}&D=\frac{\sum\limits^N_{t=1}\frac{t\cdot C_t}{(1+r)^t}}{\sum\limits^N_{t=1}\frac{C_t}{(1+r)^t}}\\&\textbf{where:}\\&D=\text{The bond's MacAulay duration}\\&T=\text{The number of periods until maturity}\\&i=\text{The $i^{th}$ time period}\\&C=\text{The periodic coupon payment}\\&r=\text{The periodic yield to maturity}\\&F=\text{The face value at maturity}\end{aligned}$​D=t=1∑N​(1+r)tCt​​t=1∑N​(1+r)tt⋅Ct​​​where:D=The bond’s MacAulay durationT=The number of periods untiꦏl maturityi=The ith time periodC=The periodic coupon paymentr=The🧜 periodic yield to maturityF=The face value at maturity​

Duration in Fixed Income Management

Duration is critical to managing fixed-income 澳洲幸运5官方开奖结果体彩网:portfolios, for the following reasons:

1. It’s a simple summary statistic of the effective average maturity of a portfolio.
2. It’s an essential tool in 澳洲幸运5官方开奖结果体彩网:immunizing portfolios from 澳洲幸运5官方开奖结果体彩网:interest rate risk.
3. It estimates the 澳洲幸运5官方开奖结果体彩网:interest rate sensitivity of a portfolio.

The dur🅰ation metric carr🃏ies the following properties:

• The duration of a 澳洲幸运5官方开奖结果体彩网:zero-coupon bond equals time to maturity.
• Holding maturity constant, a bond's duration is lower when the 澳洲幸运5官方开奖结果体彩网:coupon rate is higher, because of the impact of early higher coupon payments.
• Holding the 澳洲幸运5官方开奖结果体彩网:coupon rate constant, a bond's duration generally increases with time to maturity. But there are exceptions, as with instruments such as 澳洲幸运5官方开奖结果体彩网:deep-discount bonds, where the duration may fall with increases in maturity timetables.
• Holding other factors constant, the duration of coupon bonds is higher when the bonds’ yields to maturity are lower. However, for zero-coupon bonds, duration equals time to maturity, regardless of the yield to maturity.
• The duration of level 澳洲幸运5官方开奖结果体彩网:perpetuity is (1 + y) / y. For example, at a 10% yield, the duration of perpetuity that pays $100 annually will equal 1.10 / .10 = 11 years. However, at an 8% yield, it will equal 1.08 / .08 = 13.5 years. This principle makes it obvious that maturity and duration may differ widely. Case in point: the maturity of the perpetuity is infinite, while 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 duration computation.

Duration for Gap Management

Many banks exhibit mismatches between asset and liability maturities. Bank liabilities, which are primarily the deposits owed to customers, are generally short-term in nature, with low duration statistics. By contrast, a bank's assets mainly comprise outstanding 澳洲幸运5官方开奖结果体彩网:commercial and consumer loans or 澳洲幸运5官方开奖结果体彩网:mortgages. These assets tend to be of longer duration, and their values are more sensitive to interest rate fluctuations. In periods 🎀when interest rates spike unexpectedly, banks may suffer drastic decreases in net worth, if their assets drop further in value than their liabilities.

A technique called gap management is a widely used risk management tool, where banks attempt to limit the "gap" between asset and liability durations. Gap management heavily relies on 澳洲幸运5官方开奖结果体彩网:adjustable-rate mortgages (ARMs) as key components in reducing the duration of bank-asset portfolios. Unlike 澳洲幸运5官方开奖结果体彩网:conventional mortgages, ARMs don’t decline in value when market rates increase, because the rates they pay are tied to the current interest rate.

On the other side of the 澳洲幸运5官方开奖结果体彩网:balance sheet, the introduction of longer-term bank 澳洲幸运5官方开奖结果体彩网:certificates of deposit (CDs) with 澳洲幸运5官方开奖结果体彩网:fixed terms to maturity serves to lengt꧅hen the duration of bank liabilities, likewise contributing to the reduction of the duration 🍎gap.

Understanding Gap Management

Banks employ gap management to equate the durations of assets and liabilities, effectively immunizing their overall position from 澳洲幸运5官方开奖结果体彩网:interest rate movements. In theory, a bank’s assets and liabilities are roughly equal in size. Therefore, if their durations are also equal, any change in interest rates will affect the value of assets and liabilities to the same degree, and interest rate changes would consequently have little or no final effect on net worth. Therefore, net worth immunizatioꦕn requires a portfolio duration, or gap, of zero.

Institutions with future fixed 澳洲幸运5官方开奖结果体彩网:obligations, such as 澳洲幸运5官方开奖结果体彩网:pension funds and 澳洲幸运5官方开奖结果体彩网:insurance companies, differ from banks in that they operate with an eye towards future commitments. For example, pension funds are obligated to maintain sufficient funds to provide workers with a flow of income upon retirement. As interest rates fluctuate, so do the value of the assets held by the fund and the rate at which those assets generate income. Therefore, 澳洲幸运5官方开奖结果体彩网:portfolio managers may wish to protect (immunize) the future 澳洲幸运5官方开奖结果体彩网:accumulated value of the fund at some target date, against interest rate movements. In other words, immunization safeguards duration-matcꦇhed assets and liabilities, so a bank can meet its obligations, regardless of interest rate movements.

Convexity in Fixed Income Management

Unfortunately, duration has limitations when used as a measure of interest rate sensitivity. While the statistic calculates a linear relationship between price andജ yield changes in b🐻onds, in reality, the relationship between the changes in price and yield is convex.

In the image below, the 澳洲幸运5官方开奖结果体彩网:calculation of 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 reveals 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 澳洲幸运5官方开奖结果体彩网:price change of the bond.

Convexity, a measure of the curvature of the changes in the price of a bond, in relation to changes in interest rates, addresses this error, by measurin𝐆g the change in duration, as interest rates fluctuate.🎐 The formula is as follows:

$\begin{aligned}&C = \frac {f'' \big ( B ( r ) \big ) }{ B Dr ^ 2 } \\&\textbf{where:} \\&f'' = \text{Second order derivative} \\&B = \text{Bond price}\\&r = \text{Interest rate}\\&D = \text{Duration}\\ \end{aligned}$​C=BDr2f′′(B(r))​where:f′′=Second order derivativeB=Bond pricer=Interest rateD=Duration​

In general, the higher the coupon, the lower the convexity, because a 5% bond is more sensitive to interest rate changes than a 10% bond. Due to the 澳洲幸运5官方开奖结果体彩网:call feature, 澳洲幸运5官方开奖结果体彩网:callable bonds will display 澳洲幸运5官方开奖结果体彩网:negative convexity if yields fall too low, meaning the duration wil❀l decrease when yields decrease. Zero-coupon bonds have the highest convexity, where relationships are only valid when the compared bonds have the same duration and yields to maturity. Pointedly: a high convexity bo🌱nd is more sensitive to changes in interest rates and should consequently witness larger fluctuations in price when interest rates move.

The opposite is true of low convexity bonds, whose prices don't fluc🧜tuate as much when interest rat🐼es 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, show the highest interest rate volatility. In technical terms, this means that the 澳洲幸运5官方开奖结果体彩网:modified duration of the bond requires a larger 澳洲幸运5官方开奖结果体彩网:adjustment to ke♎ep pace with the higher change in price after interest rate moves. Lower coupon rates lead to lower yields, and lower yields lead to higher degrees of convexity.

The Bottom Line

Ever-chang🍒ing interest rates introduce uncertainty in fixed-income investing. Duration and convexity let 🐬investors quantify this uncertainty, helping them manage their fixed-income portfolios.