Bond Duration
Bond duration measures the sensitivity of a bond's price to changes in interest rates. It represents the weighted average time it takes to receive the bond's cash flows, including both coupon payments and the return of principal. A higher duration indicates greater interest rate risk, as the bond's price is more sensitive to changes in interest rates.
8 Key excerpts on "Bond Duration"
eBook - ePubBond Duration and Immunization
Early Developments and Recent Contributions
- Gabriel Hawawini, Gabriel Hawawini(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
...Suggestions for empirical research are offered in our concluding section. I. Duration and Risk In his study of bond yields Macaulay [15] defined and used duration as the measure of the length of a loan. Unlike maturity, which looks only at the last payment, duration gives some weight to the time at which each cash payment occurs. The weight assigned to each period is the present value of the cash flow for that period divided by the current price of the security. Following Fisher and Weil [10], duration measured at time t 0 can be expressed as where A t j is the present value measured at time t o of certain cash flows to be received in period t. Any investment which provides cash payments before maturity necessarily has duration less than maturity. Also, for a given current bond price, the larger the periodic coupon payments or the discount rate, the smaller the duration. 1 The link between the bond price volatility and duration is developed by Fisher [9] and extended in a recent paper by Hopewell and Kaufman [13]. Assuming continuous compounding at the yield to maturity it is shown that for small yield changes where dP it and P it are the price change and initial bond price respectively; D it is the duration of the bond at time t and dr it is the change in yield to maturity. 2 Duration is a constant of proportionality relating percentage bond price changes to changes in yield. Since the time to maturity, N, has a weight,, it will have some impact upon the duration. However, the size of that impact will depend upon the size of interim coupon payments, the yield to maturity, the time of maturity and the size of the principal payment. Clearly, two default-free bonds could have the same maturity but quite different durations; there need be no formal relationship between maturity and volatility. The question of how equation (2) can be incorporated into a risk measure for bonds remains to be considered...
eBook - ePub- Moorad Choudhry(Author)
- 2010(Publication Date)
- Wiley(Publisher)
...Chapter 3 BOND INSTRUMENTS AND INTEREST-RATE RISK Chapter 1 described the basic concepts of bond pricing. This chapter discusses the sensitivity of bond prices to changes in market interest rates and the key related concepts of duration and convexity. DURATION, MODIFIED DURATION AND CONVEXITY Most bonds pay a part of their total return during their lifetimes in the form of coupon interest. Because of this, a bond’s term to maturity does not reflect the true period over which its return is earned. Term to maturity also fails to give an accurate picture of the trading characteristics of a bond or to provide a basis for comparing it with other bonds having similar maturities. Clearly, a more accurate measure is needed. A bond’s maturity gives little indication of how much of its return is paid out during its life or of the timing and size of its cash flows. Maturity is thus inadequate as an indicator of the bond’s sensitivity to moves in market interest rates. To see why this is so, consider two bonds with the same maturity date but different coupons: the higher-coupon bond generates a larger proportion of its return in the form of coupon payments than does the lower-coupon bond and so pays out its return at a faster rate. Because of this, the higher-coupon bond’s price is theoretically less sensitive to fluctuations in interest rates that occur during its lifetime. A better indication of a bond’s payment characteristics and interest-rate sensitivity might be the average time to receipt of its cash flows. The cash flows generated during a bond’s life differ in value, however. The average time to receipt would be a more accurate measure, therefore, if it were weighted according to the cash flows’ present values...
eBook - ePubA simple approach to bond trading
The introductory guide to bond investments and their portfolio management
- Stefano Calicchio(Author)
- 2020(Publication Date)
- Stefano Calicchio(Publisher)
...Analysis of financial duration The financial duration is a fundamental parameter for measuring an obligation, as it shows its residual life. This value has direct implications for the volatility and value of a debt security, since instruments with a longer maturity will also be accompanied by greater risk, which must necessarily be taken into account when planning your investments. Let's omit the algebraic formula of the financial duration of a bond (which can be easily found in any financial mathematics manual), as its dissertation would not be suitable for an introductory guide. However, I advise you to take up the issue again if you plan to follow up this discussion with real investments in the bond market. This does not detract from the fact that we can give a theoretical definition of financial duration, such as the measure of the weighted average of the time left to a bond until its maturity. The values to be used to calculate the weighted average correspond to those of the interest and the final value at maturity, divided by the cost necessary to purchase the bond. This parameter is always between zero and the overall duration of the security (which can be traced back to its maturity). Obviously, different scenarios can be assumed depending on the type of bond: - bond zero coupon: in this case the yield of the security is incorporated in its value and liquidated at the date of expiration. - Bond with coupon payment: the financial duration is always lower than the value of its duration, as well as inversely proportional...
eBook - ePubAn Introduction to International Capital Markets
Products, Strategies, Participants
- Andrew M. Chisholm(Author)
- 2009(Publication Date)
- Wiley(Publisher)
...It is possible, for example, for yields to rise at the 10-year point but at the same time fall at the five-year maturity, so that the trader in the example could actually lose on both the long and the short position. This is sometimes known as curve risk. 5.17 CHAPTER SUMMARY The price of low coupon, long maturity bonds is highly sensitive to changes in market yields. This is because most of the cash flows occur far out into the future and cannot be reinvested at new levels of interest rates for a long time. The weighted average life of a bond’s cash flows is measured by Macaulay’s duration. A zero coupon bond has a Macaulay’s duration equal to its maturity. The duration of a coupon bond and its effective exposure to interest rate changes are less than its maturity because the coupons can be reinvested at the prevailing market rate. Modified duration is a related measure which can be used to estimate the change in the money value of the bond for a 0.01 % yield change. This is known as the price value of a basis point or basis point value. Duration can be used to put together a portfolio of bonds which matches expected future cash flows. This technique is called immunization. Duration can also be used to construct a hedge against the fall in the value of a bond portfolio. Duration is an approximate measure because the relationship between the price of a bond and its yield is nonlinear. Convexity is a measure of this curvature. It can be used to provide a better estimate of the actual change in the price of a bond for a substantial yield change....
eBook - ePubIntroduction to Fixed Income Analytics
Relative Value Analysis, Risk Measures and Valuation
- Frank J. Fabozzi, Steven V. Mann(Authors)
- 2010(Publication Date)
- Wiley(Publisher)
...A better measure of exposure of an individual issue or sector to changes in interest rates is in terms of its contribution to portfolio duration. Contribution to portfolio duration is computed by multiplying the percentage that the individual issues comprises of the portfolio by the duration of the individual issue or sector. Specifically, This exposure can also be cast in terms of dollar exposure. To accomplish this, the dollar duration of the issue or sector is used instead of the duration of the issue or sector. A portfolio manager who desires to determine the contribution to a portfolio of a sector relative to the contribution of the same sector in a broad-based market index can compute the difference between these two contributions. OTHER DURATION MEASURES Numerous duration measures are routinely employed by fixed income practitioners that relate to both fixed rate and floating rate securities. Furthermore, there are more sophisticated duration measures that allow for nonparallel yield curve shifts. We discuss these measures in this section. Spread Duration for Fixed Rate Bonds As we have seen, duration is a measure of the change in a bond’s value when interest rates change. The interest rate that is assumed to shift is the Treasury rate, which serves as the benchmark interest rate. However, for non-Treasury instruments, the yield is equal to the Treasury yield plus a spread to the Treasury yield curve. This is why non-Treasury securities are often called “spread products.” Of course, the price of a bond exposed to credit risk can change even though Treasury yields are unchanged because the spread required by the market changes. A measure of how a non-Treasury security’s price will change if the spread sought by the market changes is called spread duration. The problem is, what spread is assumed to change? There are three measures that are commonly used for fixed rate bonds: nominal spread, zero-volatility spread, and option-adjusted spread...
eBook - ePubFixed Income Securities
Concepts and Applications
- Sunil Kumar Parameswaran(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
...5 Duration, Convexity, and Immunization The duration of a plain vanilla bond can be defined as its average life. It is very easy to define duration in the case of securities that yield a single cash flow, like a zero coupon bond. In such cases there is no difference between the average time to maturity and the actual time to maturity, for we need concern ourselves only with the terminal payment. Consequently, in such cases, the duration of the security is nothing but its stated term to maturity. However the definition is not so clearcut in the case of a conventional coupon paying debt security. In such cases, the asset gives rise to a series of cash flows, usually on a semiannual basis, as well as a relatively large cash flow at the end that constitutes the principal repayment. The average life of such a security can be obtained only by taking cognizance of the times to maturity of the component cash flows. Because the cash flows occur at different points in time, we also need to factor in the issue of the time value of money. Convexity of a bond accounts for the fact that the price-yield relationship of a bond is convex and not linear. Whereas, duration accounts for a first order approximation to the price-yield relationship, convexity factors into the fact that the relationship is indeed convex. Immunization strategies protect a bond or a bond portfolio against interest rate risk. As discussed earlier, interest rate changes impact bonds in two ways. The higher the interest rate, the more the income from the re-investment of coupons. However the higher the rate, the lower the sale price of the bond at the end of the investment horizon. See Figure 5.1...
eBook - ePubDerivatives
Theory and Practice
- Keith Cuthbertson, Dirk Nitzsche, Niall O'Sullivan(Authors)
- 2019(Publication Date)
- Wiley(Publisher)
...The duration formula also assumes that yields at all maturities move by the same (absolute) amount – that is, a parallel shift in the yield curve. According to the duration formula, the change in value of the bond will be around 2.5% of its current market value of $1,000, that is an increase of $25, so the price of the bond at the end of the week will be close to $1,025. Clearly, duration is useful for fixed-income traders who speculate on changes in interest rates – the larger the duration of the bond, the greater the percentage change in the bond price and hence the greater the (‘market’) risk of the bond. The relationship between the ‘true’ change in bond price and the change in YTM is shown in Figure 10.2 by the curved or ‘convex’ line. The approximate change in price, given by the duration formula is represented by the ‘tangent line’ (at the current yield of 5%). Any actual price rise will exceed that given by the duration equation – and any actual price fall will be less than that calculated using duration. For small changes in yield, the actual price change and the approximated price change – that given using the duration formula – will be very close because the ‘curve’ and the ‘straight line’ coincide. FIGURE 10.2 Duration and price changes 10.2.1 Duration of a Portfolio of Bonds Suppose you hold -bonds. The duration of a portfolio of -bonds is simply a weighted average of the duration of the constituent bonds in the portfolio, where the weights are determined by the market value of the individual bonds: (10.2) where (assuming no short-selling) and. For example, if you hold $200m in bonds, each of which has a duration and $400m in bonds each with a duration of, then the duration of the bond portfolio is. This implies that if bond yields change by 1%, the value of your bond portfolio will change by (approximately) 9.33 per cent, hence: (10.3a) (10.3b) where, the change in the YTM, is expressed as a decimal (e.g. if the current YTM is 3% p.a...
eBook - ePubDemystifying Fixed Income Analytics
A Practical Guide
- Kedar Nath Mukherjee(Author)
- 2020(Publication Date)
- Routledge India(Publisher)
...Modified duration follows the concept that interest rates and bond prices move in opposite direction. It refers to the change in value of the security to one per cent (100 basis points) change in interest rates (yield). A formula that expresses the measurable change in the value of a security in response to a change in interest rates is expressed as. under: M D u r a t i o n = [ M a c a u l a y D u r a t i o n / (1 + Y T M / k) ] M o d i f i e d - D u r a t i o n = [ { ∑ 1 i = 1 n t i × (P V C F i ∑ i = 1 n P V C[--=PLGO. -SEPARATOR=--]F i) × 1 / k } × { 1 / (1 + Y T M / k) } ] (7.3) Mathematically, m-duration is the relative price change, i.e. the first order differentiation of the bond price function [ P=f (r) ], and can also be derived as: M − D u r a t i o n = 1 P × d P d r = 1 P × Σ i = 1 n - i × C (1 + r) (i + 1) (7.4) where P = Σ i = 1 n C i (1 + r) i Like Macaulay duration, modified duration can also be calculated in MS-Excel by using the following function230a: = MDURATION(Settlement, Maturity, Coupon, Y/d, Frequency, [Basis]) Where the inputs are exactly the same as. required for estimating the Macaulay duration described in the previous section. Example The above example may be used to estimate the m-duration of the bond (6.90 GS 2019)...
