Page 1 of 7 Bond duration From Wikipedia, the free encyclopedia In finance, the duration of a financial asset, specifically a bond, is a measure of the sensitivity of the asset's price to interest rate movements. It broadly corresponds to the length of time before the asset is due to be repaid. There are various definitions of duration and derived quantities, discussed below. However if not otherwise qualified, "duration" generally means the Macaulay duration, as defined below. This duration is equal to the ratio of the percentage reduction in the bond's price to the percentage increase in the redemption yield of the bond. This equation is valid for small changes in those quantities only. Duration is known in the context of "The Greeks" used for derivative pricing as the λ or Lambda. In contrast, the absolute change in a bond's price with respect to interest rate (Δ or Delta) is referred to as the dollar duration. The units of duration are years, and duration is generally between 0 years and the time to maturity of the bond. It is equal to the time to maturity if and only if the bond is a zero-coupon bond. One way to follow this is that the value of more distant cash flows is more sensitive to the interest rate, or yield: when calculating the present value of the cash flows under a bond, one divides each future cash flow by the (yield plus one) to the power of the number of years until that cash flow occurs: (1 + y) n thus the present value of more distant future cash flows is more sensitive to changes in yield. Contents 1 Price 2 Definition 3 Cash flow 4 Dollar duration 4.1 Application to VaR 5 Macaulay duration 6 Modified duration 7 Embedded options and effective duration 8 Spread duration 9 Average duration 10 Bond duration closed-form formula 11 Convexity 12 PV01 and DV01 13 Confused notions 14 See also 14.1 Lists 15 Notes 16 References 17 External links Price Duration is useful as a measure of the sensitivity of a bond's market price to interest rate (i.e., yield) movements. It is approximately equal to the percentage change in price for a given change in yield. For example, for small interest rate changes, the duration is the approximate percentage by which the value of the bond will fall for a 1% per annum increase in market interest rate. So a 15-year bond with a duration of 7 would fall approximately 7% in value if the interest rate increased by 1% per annum. [1] Definition The standard definition of duration, D, is Macaulay duration, the PV-weighted time to receive each cash flow, defined as: where:
Page 2 of 7 i indexes the cash flows, P(i) is the present value of the ith cash payment from an asset, t(i) is the time in years until the ith payment will be received, V is the present value of all cash payments from the asset until maturity, Both these definitions give a weighted average (weights sum to 1) of time to receive cash flows, and thus fall between 0 (the minimum time), or more precisely t(1) (the time to the first payment) and the time to maturity of the bond (the maximum time), with equality if and only if the bond only has a single payment at maturity (i.e., if it is a zero-coupon bond). In symbols, if cash flows are in order: with the inequalities being strict unless it has a single cash flow. Cash flow As stated above, the duration is the weighted average term to payment of the cash flows on a bond. For a zero-coupon bond, the duration will be ΔT = T f T 0, where T f is the maturity date and T 0 is the starting date of the bond. If there are additional cash flows C i at times T i, the duration of every cash flow is ΔT i = T i T 0. From the current market price of the bond V, one can calculate the yield to maturity of the bond r using the formula Note that in this and subsequent formulae, the symbol r is used for the force of interest, i.e. the logarithm of (1+j) where j is the interest yield expressed as an annual effective yield.to make this clear, imagine that if the YTM is j it is substituted by ln(1+j) and this is equivalent of assuming a continuously compound interest. At the end of ther calculations we will apply the inverse transformation to reveal a relationship involving the real YTM. In a standard duration calculation, the overall yield of the bond is used to discount each cash flow leading to this expression in which the sum of the weights is 1: The higher the coupon rate of a bond, the shorter the duration (if the term of the bond is kept constant). Duration is always less than or equal to the overall life (to maturity) of the bond. Only a zero coupon bond (a bond with no coupons) will have duration equal to the maturity. Duration indicates also how much the value V of the bond changes in relation to a small change of the rate of the bond. We see that so that for a small variation in the redemption yield of the bond we have That means that the duration gives the negative of the relative variation of the value of a bond with respect to a variation in the redemption yield on the bond, forgetting the quadratic and higher-order terms. The quadratic terms are taken into
Page 3 of 7 account in the convexity. As we have seen above, r = ln(1 + j). If (which could be defined as the Modified Duration) is required, then it is given by: and this relationship holds good whatever the frequency of convertibility of j. Dollar duration The dollar duration is defined as the product of the duration and the price (value): it is the change in price in dollars, not in percentage, and has units of Dollar-Years (Dollars times Years). It gives the dollar variation in a bond's value for a small variation in the yield. Application to VaR Dollar duration D $ is commonly used for VaR (Value-at-Risk) calculation. If V = V(r) denotes the value of a security depending on the interest rate r, dollar duration can be defined as To illustrate applications to portfolio risk management, consider a portfolio of securities dependent on the interest rates as risk factors, and let denote the value of such portfolio. Then the exposure vector has components Accordingly, the change in value of the portfolio can be approximated as that is, a component that is linear in the interest rate changes plus an error term which is at least quadratic. This formula can be used to calculate the VaR of the portfolio by ignoring higher order terms. Typically cubic or higher terms are truncated. Quadratic terms, when included, can be expressed in terms of (multi-variate) bond convexity. One can make assumptions about the joint distribution of the interest rates and then calculate VaR by Monte Carlo simulation or, in some special cases (e.g., Gaussian distribution assuming a linear approximation), even analytically. The formula can also be used to calculate the DV01 of the portfolio (cf. below) and it can be generalized to include risk factors beyond interest rates. Macaulay duration Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of a bond where the weights are the relative discounted cash flows in each period. It will be seen that this is the same formula for the duration as given above. Macaulay showed that an unweighted average maturity is not useful in predicting interest rate risk. He gave two alternative measures that are useful:
Page 4 of 7 The theoretically correct Macaulay Weil duration which uses zero-coupon bond prices as discount factors, and the more practical form (shown above) which uses the bond's yield to maturity to calculate discount factors. The key difference between the two is that the Macaulay Weil duration allows for the possibility of a sloping yield curve, whereas the algebra above is based on a constant value of r, the yield, not varying by term to payment. With the use of computers, both forms may be calculated, but the Macaulay duration is still widely used. In case of continuously compounded yield the Macaulay duration coincides with the opposite of the partial derivative of the price of the bond with respect to the yield as shown above. In case of yearly compounded yield, the modified duration coincides with the latter. Modified duration In case of n times compounded yield, the relation is not valid anymore. That is why the modified duration D * is used instead: where r is the yield to maturity of the bond, and n is the number of cashflows per year. Let us prove that the relation is valid. We will analyze the particular case n = 1. The value (price) of the bond is where i is the number of years to the cash flow C i. The duration, defined as the weighted average maturity, is then The derivative of V with respect to r is: multiplying by we obtain or
Page 5 of 7 from which we can deduce the formula which is valid for yearly compounded yield. Embedded options and effective duration For bonds that have embedded options, such as puttable and callable bonds, Macaulay duration will not correctly approximate the price move for a change in yield. In order to price such bonds, one must use option pricing to determine the value of the bond, and then one can compute its delta (and hence its lambda), which is the duration. The effective duration is a discrete approximation to this latter, and depends on an option pricing model. Consider a bond with an embedded put option. As an example, a $1,000 bond that can be redeemed by the holder at par at any time before the bond's maturity (i.e. an American put option). No matter how high interest rates become, the price of the bond will never go below $1,000 (ignoring counterparty risk). This bond's price sensitivity to interest rate changes is different from a non-puttable bond with otherwise identical cashflows. Bonds that have embedded options can be analyzed using "effective duration". Effective duration is a discrete approximation of the slope of the bond's value as a function of the interest rate. where Δ y is the amount that yield changes, and V Δy and V + Δy are the values that the bond will take if the yield falls by y or rises by y, respectively. However this value will vary depending on the value used for Δ y. Spread duration Sensitivity of a bond's market price to a change in Option Adjusted Spread (OAS). Thus the index, or underlying yield curve, remains unchanged. Average duration The sensitivity of a portfolio of bonds such as a bond mutual fund to changes in interest rates can also be important. The average duration of the bonds in the portfolio is often reported. The duration of a portfolio equals the weighted average maturity of all of the cash flows in the portfolio. If each bond has the same yield to maturity, this equals the weighted average of the portfolio's bond's durations. Otherwise the weighted average of the bond's durations is just a good approximation, but it can still be used to infer how the value of the portfolio would change in response to changes in interest rates. Bond duration closed-form formula FV = par value C = coupon payment per period (half-year) i = discount rate per period (half-year) a = fraction of a period remaining until next coupon payment m = number of coupon dates until maturity P = bond price (present value of cash flows discounted with rate i)
Page 6 of 7 Convexity Main article: Bond convexity Duration is a linear measure of how the price of a bond changes in response to interest rate changes. As interest rates change, the price does not change linearly, but rather is a convex function of interest rates. Convexity is a measure of the curvature of how the price of a bond changes as the interest rate changes. Specifically, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question, and the convexity as the second derivative. Convexity also gives an idea of the spread of future cashflows. (Just as the duration gives the discounted mean term, so convexity can be used to calculate the discounted standard deviation, say, of return.) Note that convexity can be both positive and negative. A bond with positive convexity will not have any call features - ie the issuer must redeem the bond at maturity - which means that as rates fall, its price will rise. On the other hand, a bond with call features - ie where the issuer can redeem the bond early - is deemed to have negative convexity, which is to say its price should fall as rates fall. This is because the issuer can redeem the old bond at a high coupon and re-issue a new bond at a lower rate, thus providing the issuer with valuable optionality. Mortgage-backed securities (pass-through mortgage principal prepayments) with US-style 15 or 30 year fixed rate mortgages as collateral are examples of callable bonds. PV01 and DV01 PV01 is the present value impact of 1 basis point move in an interest rate. It is often used as a price alternative to duration (a time measure). When the PV01 is in USD, it is the same as DV01 (Dollar Value of 1 basis point). Confused notions Duration, in addition to having several definitions, is often confused with other notions, particularly various properties of bonds that are measured in years. Duration is sometimes explained inaccurately as being a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows. [note 1] This quantity is the duration of a perpetual bond (assuming a flat yield curve at the coupon), and is simply. For instance, if a bond pays 5% per annum and was issued at par, it will take 20 years of these payments to repay its price. Note the absurdity of interpreting duration this way: given a bond paying 5% per annum with a term of 5 years, the duration is approximately 4.37, whereas the price of the bond will not be repaid in full until maturity (at 5 years). The Weighted-Average Life is the weighted average of the principal repayments of an amortizing loan, and is longer than the duration. See also Lists Bond convexity Bond valuation Immunization (finance) Stock duration Bond duration closed-form formula Yield to maturity List of finance topics Notes 1. ^ This may be a confusion with the Price/Dividend Ratio or P/E ratio, which can be so interpreted, as stocks are generally perpetual.
Page 7 of 7 References 1. ^ "Macaulay Duration" by Fiona Maclachlan, The Wolfram Demonstrations Project. External links Investopedia s duration explanation Hussman Funds - Weekly Market Comment: February 23, 2004 - Buy-and-Hold For the Duration? Online real-time Bond Price, Duration, and Convexity Calculator, by Razvan Pascalau, Univ. of Alabama Riskglossary.com for a good explanation on the multiple definitions of duration and their origins. Modified duration calculator Retrieved from "" Categories: Fixed income analysis This page was last modified on 26 October 2010 at 01:05. Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. See Terms of Use for details. Wikipedia is a registered trademark of the Wikimedia Foundation, Inc., a non-profit organization.