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Duration: a measure of bond price volatility
How do maturity and coupon rate affect volatility? Both determine how quickly you, the bondholder, get your money back. The longer you must wait to get your principal back, the more the price of your bond will fluctuate with a given change in interest rates. This effect is lessened by the receipt of coupon payments over the life of the bond. A higher coupon rate means you get a higher portion of your total return prior to maturity in the form of interest payments.
One of the first things one learns about bonds is that their prices increase when interest rates decline and decrease when interest rates rise. The extent to which a bond’s price fluctuates due to changing interest rates is called its volatility.
A bond’s volatility depends on two factors: its coupon rate and when it will be retired (at maturity or call date). Other things being equal, the general rule is that:
- The longer the time until retirement, the greater the price volatility.
- The lower the coupon rate, the greater the price volatility.
So, if two bonds have the same maturity (assuming no call options), the one with the lower coupon will be more volatile. On the other hand, if two bonds have the same coupon rate, the one with the longer maturity will be more volatile.
But how do we compare the volatility of two bonds with different coupon rates and maturities? We use duration.
In the calculation of duration, each payment to the bondholder is multiplied by the amount of time that will elapse until that payment is received. But there is a twist — duration is based not on the time-weighted cash amount of each payment, but the time-weighted present value of each payment.*
Once you have added up all the time-weighted present values of the payments, you divide that number by the price of the bond (which is the sum of the unweighted present values) and the result is duration, a measure of time. Duration, then, is the average amount of time it takes to receive the present value of your investment.
(present value of each payment x
time until that payment is received)
Duration = ____________________________
price of bond
For example, a par bond due in 10 years with a 3.00% coupon would have a duration of 8.71 years, calculated as follows:
______________ = 8.71 years
The 871 dollar-years in the numerator is the time-weighted present value of the various coupon payments and the final principal payment. Alternatively, if that bond had a 4.00% coupon and a dollar price of 108.58, the duration would be lower:
______________ = 8.42 years
Duration as a predictor of price changes
Duration has a very useful quality. With a slight modification (dividing by one plus the semi-annual yield of the bond) the duration number can estimate how much a bond’s price will change in response to changing interest rates. If you multiply the “modified duration” by the assumed change in interest rates, you can approximate the percentage change that will occur in the bond’s price.
For example, consider again the 3.00% par bond due in 10 years (with no call options). That bond would have a modified duration of 8.58 years (8.71 divided by 1.015). On the basis of duration, we would expect a decline of 10 basis points (0.10%) in yields to increase the value of the bond by 0.858% to 100.858 In fact, the bond’s value would rise to 100.863. In this case, the difference between the pure duration-based estimate and the actual price change was quite small. A greater change in interest rates would produce a less accurate prediction based on duration. The difference between the actual change in price and the expected change in price due to duration is the result of a characteristic of bonds known as convexity.
The effects of coupon and maturity
In our previous example, the par bond had a modified duration of 8.58 years and a maturity of 10 years. If a bond with the same coupon rate and price had a maturity of 11 years, its modified duration would be 9.31 years. If the coupon were raised to 4.00%, but with the same 3.00% yield, the modified duration would be reduced back to 8.98 years. As a general rule, bonds that pay interest prior to maturity will have a duration less than maturity, and the larger the coupon, the shorter the duration.
At the other extreme, reducing the coupon rate to zero makes the duration equal to maturity. This is logical since the full amount of one’s investment return will be received in one lump sum when the bond matures. There are no payments to reduce the average time it takes to recover the present value of the investment.
The following graph shows the relationship between a bond’s effective maturity (when the bond is expected to be retired) and its modified duration. As you can see, duration does not increase as quickly as maturity
Call provisions can also cause a bond’s duration to be less than its maturity. Typically duration is calculated based on the date to which the bond is priced. A premium bond, which is redeemable at par sometime before maturity, will be priced to a call date. So the modified duration will correspond to the call date, not the maturity date. Thus, in a market where prices are rising, the volatility of a portfolio will tend to decline as more and more bonds are priced to their call dates rather than to maturity. On the other hand, when prices fall, volatility and average duration tend to increase because of the increase in the number of bonds that are priced to maturity.
This also means that modified durations of callable bonds priced near par may shift rapidly as prices move above and below par – and duration shifts from the call date to the more distant maturity date.
An alternative measure of duration – known as “option-adjusted duration” or “effective duration” – takes into account the effect of the call option on the expected life of a bond. It weighs the probability that the bond will be called based on the spread between its coupon rate and its yield, as well as the volatility of interest rates. Generally speaking, option-adjusted duration (OAD) will be longer than modified duration when a bond is priced to a call date, and shorter than modified duration when a bond is priced to maturity. OAD is typically used to report the duration of portfolios containing mortgage-backed securities.
Call options limit the potential price appreciation of a bond, but do not limit the downside, when the bond is priced to maturity. As a result, callable bonds typically have negative convexity since the change in price in a rising market is not as great as the change in price in a falling market.
Duration and portfolio management
In any case, since duration reflects bond price volatility, a portfolio’s average duration is more meaningful than its average maturity. By comparing a bond’s duration with an existing portfolio’s average duration, a portfolio manager tries to anticipate the effect that buying or selling that bond would have on the portfolio’s volatility. Further, by exploring how the duration of a bond might change in different market environments, a portfolio manager can better evaluate the relative value of call protection.
Using duration analysis, the portfolio manager may elect, for example, to buy high-coupon, premium bonds to reduce downside risk, or buy deep discount bonds to try to maximize potential price appreciation. Or the manager may decide that intermediate-term, current coupon bonds offer the best value. Whatever the approach, duration analysis helps the portfolio manager evaluate the effects of various trading strategies in an effort to better achieve the goal of minimizing price volatility while maximizing total return.
*The present value of a sum yet to be received is the amount of money required today to have the desired amount at a specified future date after compounding interest earnings at an assumed rate.Endnotes
This material is not intended to be a recommendation or investment advice, does not constitute a solicitation to buy, sell or hold a security or an investment strategy, and is not provided in a fiduciary capacity. The information provided does not take into account the specific objectives or circumstances of any particular investor, or suggest any specific courseof action. Investment decisionsshould be made based on an investor’sobjectives and circumstances and in consultation with hisor her financial professionals.
The views and opinions expressed are for informational and educational purposes only as of the date of production/writing and may change without notice at any time based on numerousfactors,such as marketorotherconditions, legal and regulatory developments, additional risks and uncertainties and may notcometo pass.This material maycontain “forward-looking” information that is not purely historical in nature. Such information may include, among other things, projections, forecasts, estimates of market returns, and proposed or expected portfolio composition. Any changes to assumptions that may have been made in preparing this material could have a material impact on the information presented herein by way of example. Performance data shown represents past performance and does not predict or guarantee future results. Investing involves risk; principal loss is possible.
All information has been obtained from sources believed to be reliable, but its accuracy is not guaranteed. There is no representation or warranty as to the current accuracy, reliability or completeness of, nor liability for, decisions based on such information and it should not be relied on as such. For term definitions and index descriptions, please access the glossary on nuveen.com. Please note, it is not possible to invest directly in an index.Important information on risk
Investing involves risk; principal loss is possible. Investing in municipal bonds involves risks such as interest rate risk, credit risk and market risk. The value of the portfolio will fluctuate based on the value of the underlying securities. There are special risks associated with investments in high yield bonds, hedging activities and the potential use of leverage. Portfolios that include lower rated municipal bonds, commonly referred to as “high yield” or “junk” bonds, which are considered to be speculative, the credit and investment risk is heightened for the portfolio. Bond insurance guarantees only the payment of principal and interest on the bond when due, and not the value of the bonds themselves, which will fluctuate with the bond market and the financial success of the issuer and the insurer. No representation is made as to an insurer’s ability to meet their commitments.
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