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. We have provided you with a quick introduction to bonds, bond valuation and the concepts used in pricing bonds. The macaulay duration, dm, of a collection of cash flows, cfj,is a. A s another example, the m acaulay duration of a z erocoupon bond is. It is easier to immunize a bond portfolio when the duration of the portfolio is matched to the need for funds. Duration is thought of as a present value weighted measure of payback.
Before we use the modified duration formula, we first compute p. 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. The differential approach suggested here presents premiums. 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. The bond s cash flows consist of coupons paid periodically and principal repaid at maturity. Luckily, there are tools that can help you calculate the bond duration. The video uses a comprehensive example to demonstrate how macaulay duration is calculated, and it explains how macaulay duration is used to. 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. 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. For an 8% coupon annual pay fouryear bond with a yield. In practice a bond or other fixedincome security will often be valued off a yield curve, and we can extend the. Youll also learn how to come up with an appropriate discount rate to use in the bond value calculation.
Understanding bond duration can help investors determine how bonds fit in to a broader investment portfolio. Taylor s expansion implies the following approximation formula. Hicks 1939 derived the same formula as a measure of how sensitive a bonds price was to changes in interest rates. Duration present value of a bonds cash flows, weighted by length of time to receipt and divided by the bonds current market value.
In more technical terms, bond duration is measurement of interest rate risk. Bond price calculation on the ti ba ii pus and professional. Using duration and convexity to approximate change in present. Oct 02, 2008 bond covenants and other security features of revenue bonds cont. 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. Explicit sample calculations a for an 8% coupon annual pay fouryear bond with a. How to calculate modified duration the motley fool. Bond duration, yield to maturity and bifurcation analysis. So in the example above, investors can expect to see a 1. The calculation of macaulay duration is straightforward. Macaulay duration of a zerocoupon bond is equal to its maturity. Examples are callable bonds and mortgages without a prepayment penalty. Macaulay duration illinois institute of technology.
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. Chapter 33 valuing bonds the value of a bond is the present value of the expected cash flows on the bond, discounted at an interest rate that is appropriate to the riskiness of that bond. As the table below shows, the shorter a bonds duration, the less volatile it is likely to be. Duration is a measure of the sensitivity of the price the value of principal of a fixedincome investment to a change in interest rates.
Chapter 11 duration, convexity and immunization section 11. Pdf duration is significant in that it summarizes a bond or a portfolios sensitivity to interest rates. 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. A bond or bond portfolio with a higher duration is more volatile than a bond or bond portfolio with a lower duration. In contrast, a bond with a duration of 10 years would lose 10% if rates were to rise by that same 1%. Federal law limits the amount of taxexempt bond proceeds that can be used to fund the debt service reserve fund to the lesser of.
For example, a bond with a oneyear duration would only lose 1% in value if rates were to rise by 1%. 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. Bond duration is a way of measuring how much bond prices are likely to change if and when interest rates move. How to calculate the bond duration example included. Interest rates are 6% per annum with semiannual compounding. There is a consistency between the properties of bond price volatility and the properties of modified duration. Note that if the bond s price is not provided, you may refer to the following source that explains how to calculate the bond s price. 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. 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. For a standard bond the macaulay duration will be between 0 and the maturity of the bond. For a bond of face value usd1,000 with a semiannual coupon of 8. The bonds cash flows consist of coupons paid periodically and principal repaid at maturity. 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.
A guide to duration, dv01, and yield curve risk transformations. Let us take an example of a bond with annual coupon payments. Chapter 4 bond price volatility stanford university. Price change of a bond duration convexity cfa level 1. Duration formula definition, excel examples calculate. The duration of a bond is expressed as a number of years from its purchase date. Duration and convexity are important numbers in bond portfolio management, but it is far from obvious how to calculate them on the hp 12c. 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. From the example, it can be seen that the duration of a bond increases with the decrease in coupon rate. It is equal to the maturity if and only if the bond is a zerocoupon bond. Apr, 2018 modified duration is a measure of a bond price sensitivity to changes in its yield to maturity.
For example, a bond with a duration of 7 will gain about 7% in value if interest rates fall 100 bp. Fixed income calculating macaulay duration for a bond. Bond duration is a measure of how bond prices are affected by changes in interest rates. 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. Duration can be used to compare bonds with different issue and maturity dates, coupon rates, and yields to maturity. Thus, buying a coupon bond whose duration exactly matches the investment time horizon of the insurer also immunizes the insurer against interest rate changes. Thus, anyone who can calculate a bond price can quickly estimate the interest rate risk associated with a. For an 8% coupon annual pay fouryear bond with a yield to maturity of 10%, we have. Sep 06, 2019 the modified duration alone underestimates the gain to be 2. Determine how much money you would accumulate by investing a given amount of money at a fixed annual rate of return at recurring intervals.
Bond duration calculator macaulay duration and modified macaulay duration. The higher the bonds duration, the greater its sensitivity to changes in interest rates also known as volatility and vice versa. Duration can help predict the likely change in the price of a bond given a change in interest rates. Explicit sample calculations a for an 8% coupon annual pay fouryear bond with a yield to maturity of 10%. For zeroes, duration is easy to define and compute with a. If a bond has a duration of 6 years, for example, its price will rise about 6% if its yield drops by. I will work mostly with dv01 throughout this paper but the ideas apply equally well to modified duration. There is more than one way to calculate duration which well get to below, but the macaulay duration. 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.
Duration is an approximate measure of a bonds price sensitivity to changes in interest rates. Calculating bond duration can be a tedious task, especially if you have a bond with a maturity far into the future. The formula is complicated, but what it boils down to is. A simplified approach to measuring bond duration citeseerx. Bonds duration will be equal to its term to maturity if and only if it is a zero coupon bond. Duration is a measure of a bond s sensitivity to interest rate changes.