Duration formula definition, excel examples calculate. Apr, 2018 modified duration is a measure of a bond price sensitivity to changes in its yield to maturity. Apr 17, 2015 the video uses a comprehensive example to demonstrate how macaulay duration is calculated, and it explains how macaulay duration is used to compare the interest rate risk of different bonds. Thus, anyone who can calculate a bond price can quickly estimate the interest rate risk associated with a. A bond or bond portfolio with a higher duration is more volatile than a bond or bond portfolio with a lower duration. Calculating bond duration can be a tedious task, especially if you have a bond with a maturity far into the future. For zeroes, duration is easy to define and compute with a. It is important to understand the concept of duration as it is used by bond investors to check a bonds sensitivity to changes in interest rates. Bond duration is a measure of how bond prices are affected by changes in interest rates. If a bond has a duration of 6 years, for example, its price will rise about 6% if its yield drops by. The calculation of macaulay duration is straightforward. The duration of a bond is expressed as a number of years from its purchase date. I will work mostly with dv01 throughout this paper but the ideas apply equally well to modified duration.
The bond s cash flows consist of coupons paid periodically and principal repaid at maturity. Macauley duration modified duration effective duration percentage change in price for a 1% increase in the yield to maturity problem 6 consider a bond that has a coupon rate of 5. Chapter 4 bond price volatility stanford university. A guide to duration, dv01, and yield curve risk transformations. Oct 02, 2008 bond covenants and other security features of revenue bonds cont. Let us take an example of a bond with annual coupon payments. Bonds duration will be equal to its term to maturity if and only if it is a zero coupon bond. Bond duration is a way of measuring how much bond prices are likely to change if and when interest rates move. For an 8% coupon annual pay fouryear bond with a yield. From the example, it can be seen that the duration of a bond increases with the decrease in coupon rate. There is a consistency between the properties of bond price volatility and the properties of modified duration. It is equal to the maturity if and only if the bond is a zerocoupon bond. The macaulay duration named after frederick macaulay, an economist who developed the concept in 1938 is a measure of a bond s sensitivity to interest rate changes.
It is easier to immunize a bond portfolio when the duration of the portfolio is matched to the need for funds. We have provided you with a quick introduction to bonds, bond valuation and the concepts used in pricing bonds. In more technical terms, bond duration is measurement of interest rate risk. Duration and convexity are important numbers in bond portfolio management, but it is far from obvious how to calculate them on the hp 12c. A s another example, the m acaulay duration of a z erocoupon bond is. Modified duration, on the other hand, is a mathematical derivative rate of change of price and measures the percentage rate of change of price with respect to yield. Example 316 interest rates remain at 8 percent the cash flows received by the insurer on the bond if interest rates stay at 8 percent. Duration is thought of as a present value weighted measure of payback. Federal law limits the amount of taxexempt bond proceeds that can be used to fund the debt service reserve fund to the lesser of. Sep 06, 2019 the modified duration alone underestimates the gain to be 2.
A simplified approach to measuring bond duration 35 m number of payments per year n total number of payments remaining until maturity the choices for calculating duration boil down to the original procedure involving numerous weighted present value calculations or the above simplified formulas. Taylor s expansion implies the following approximation formula. Technically, duration is the weighed average number of years the investor must hold a bond until the present value of the bond s cash flows equals the amount paid for the bond. For example, a bond with a oneyear duration would only lose 1% in value if rates were to rise by 1%. Since the value of the bond portfolio is merely the present value of future cash flows, using the concepts of duration defined below, such approximations can be done quickly using nothing more than a handheld calculator. Examples are callable bonds and mortgages without a prepayment penalty. Duration is an approximate measure of a bonds price sensitivity to changes in interest rates. Luckily, there are tools that can help you calculate the bond duration. For a standard bond the macaulay duration will be between 0 and the maturity of the bond. Duration is a measure of a bond s sensitivity to interest rate changes. As a general rule, for every 1% increase or decrease in interest rates, a bonds price will change approximately 1% in the opposite direction for every year of duration. Effective duration incor porates a bonds yield, coupon, final maturity and call features into one number that indi cates how pricesensitive a bond or portfolio is for example, the price of a bond with an effective duration of two years will rise fall two percent for every one percent decrease increase in yield. A differential approach demonstrating duration and convexity tewahn hahn, david lange 1 abstract a traditional bond pricing scheme used in introductory finance texts is simple enough but not necessarily intuitive. This can help an investor understand a bond s potential interest rate risk.
In contrast, a bond with a duration of 10 years would lose 10% if rates were to rise by that same 1%. Macaulay duration of a zerocoupon bond is equal to its maturity. As the table below shows, the shorter a bonds duration, the less volatile it is likely to be. Understanding bond duration can help investors determine how bonds fit in to a broader investment portfolio. Duration present value of a bonds cash flows, weighted by length of time to receipt and divided by the bonds current market value. Pdf duration is significant in that it summarizes a bond or a portfolios sensitivity to interest rates. The formula is complicated, but what it boils down to is. Explicit sample calculations a for an 8% coupon annual pay fouryear bond with a. Duration the duration of a bond is a linear approximation of minus the percent change in its price given a 100 basis point change in interest rates.
Interest rates are 6% per annum with semiannual compounding. In practice a bond or other fixedincome security will often be valued off a yield curve, and we can extend the. The macaulay duration, dm, of a collection of cash flows, cfj,is a. Before we use the modified duration formula, we first compute p. Duration is a measure of the sensitivity of the price the value of principal of a fixedincome investment to a change in interest rates. The bonds cash flows consist of coupons paid periodically and principal repaid at maturity. Bond duration calculator macaulay duration and modified macaulay duration. Macaulay duration illinois institute of technology.
Duration can be used to compare bonds with different issue and maturity dates, coupon rates, and yields to maturity. Bond price calculation on the ti ba ii pus and professional. The video uses a comprehensive example to demonstrate how macaulay duration is calculated, and it explains how macaulay duration is used to. Using duration and convexity to approximate change in present. Price change of a bond duration convexity cfa level 1. Fixed income calculating macaulay duration for a bond. Chapter 11 duration, convexity and immunization section 11. Determine how much money you would accumulate by investing a given amount of money at a fixed annual rate of return at recurring intervals. Debt service reserve fund provides a cushion to make timely debt service payments in the event of temporary adversity. There is more than one way to calculate duration which well get to below, but the macaulay duration. What modified duration means the modified duration tells you how much the price of a bond will change for a given change in its yield. For an 8% coupon annual pay fouryear bond with a yield to maturity of 10%, we have. Macauley duration modified duration effective duration percentage change in price for a 1% increase in the yield to maturity problem 2 consider a bond that has a coupon rate of 7. Calculate the bond duration for the following annual coupon rate.
Also, lower coupon rates generally have greater macaulay and modified bond durations. Explicit sample calculations a for an 8% coupon annual pay fouryear bond with a yield to maturity of 10%. How to calculate modified duration the motley fool. The differential approach suggested here presents premiums. Since the cash flows on a straight bond are fixed at issue, the value of a bond is inversely related to the interest rate that investors demand for that bond. For a bond of face value usd1,000 with a semiannual coupon of 8. Hicks 1939 derived the same formula as a measure of how sensitive a bonds price was to changes in interest rates. Thus, buying a coupon bond whose duration exactly matches the investment time horizon of the insurer also immunizes the insurer against interest rate changes. Of course, there are formulas that you can type in see below, but they arent easy for most people to remember and are tedious to enter. My objective then is to determine the bifurcation set between families of bonds, according to the behaviour of their duration. Bond duration, yield to maturity and bifurcation analysis.
For example, a bond with a duration of 7 will gain about 7% in value if interest rates fall 100 bp. How to calculate the bond duration example included. Duration can help predict the likely change in the price of a bond given a change in interest rates. So in the example above, investors can expect to see a 1. A simplified approach to measuring bond duration citeseerx.
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