 Bond duration

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Public financeIn finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, is the weighted average of the times until those fixed cash flows are received. Duration also measures the price sensitivity to yield, the percentage change in price for a parallel shift in yields. ^{[1]} ^{[2]}
The dual characterization of duration, as both the weighted average time until repayment and as the percentage change in price, often causes confusion. Strictly speaking, Macaulay duration is the name given to the weighted average time until cash flows are received, and is measured in years. Modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield. When yields are continuouslycompounded Macaulay duration and modified duration will be numerically equal. When yields are periodicallycompounded Macaulay and modified duration will differ slightly, and in this case there is a simple relation between the two. Modified duration is used more than Macaulay duration.
Macaulay duration and modified duration are both termed "duration" and have the same (or close to the same) numerical value, but it is important to keep in mind the conceptual distinctions between them. Macaulay duration is a time measure with units in years, and really makes sense only for an instrument with fixed cash flows. For a standard bond the Macaulay duration will be between 0 and the maturity of the bond. It is equal to the maturity if and only if the bond is a zerocoupon bond.
Modified duration, on the other hand, is a derivative or price sensitivity and measures the percentage rate of change of price with respect to yield. (Price sensitivity with respect to yields can also be measured in absolute (dollar) terms, and the absolute sensitivity is often referred to as dollar duration, DV01, PV01, or delta (δ or Δ) risk.) The concept of modified duration can be applied to interestrate sensitive instruments with nonfixed cash flows, and can thus be applied to a wider range of instruments than can Macaulay duration.
For everyday use, the equality (or nearequality) of the values for Macaulay and modified duration can be a useful aid to intuition. For example a standard tenyear coupon bond will have Macaulay duration somewhat but not dramatically less than 10 years and from this we can infer that the modified duration (price sensitivity) will be somewhat but not dramatically less than 10%. In contrast, a twoyear coupon bond will have Macaulay duration somewhat below 2 years, and modified duration somewhat below 2%. (For example a tenyear 5% par bond has modified duration 7.8% while a twoyear 5% par bond has modified duration 1.9%.)
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Macaulay duration
Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows. Consider some set of fixed cash flows. The present value of these cash flows is:
Macaulay duration is defined as:^{[1]}^{[2]} ^{[3]}
 (1)
where:
 i indexes the cash flows,
 PV_{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.
In the second expression the fractional term is the ratio of the cash flow PV_{i} to the total PV. These terms add to 1.0 and serve as weights for a weighted average. Thus the overall expression is a weighted average of time until cash flow payments, with weight being the proportion of the asset's present value due to cash flow i.
For a set of allpositive fixed cash flows the weighted average will fall between 0 (the minimum time), or more precisely t_{1} (the time to the first payment) and the time of the final cash flow. The Macaulay duration will equal the final maturity if and only if there is only a single payment at maturity. In symbols, if cash flows are in order (t_{1},...,t_{n}), then:
with the inequalities being strict unless it has a single cash flow. In terms of standard bonds (for which cash flows are fixed and positive), this means the Macaulay duration will equal the bond maturity only for a zerocoupon bond.
Macaulay duration has the diagrammatic interpretation shown in figure 1.
This represents the bond discussed in the example below, two year maturity with a coupon of 20% and continuouslycompounded yield of 3.9605%. The circles represents the PV of the payments, with the coupon circles getting smaller the further out they are and the final large circle including the final principal repayment. If these circles were put on a balance beam, the fulcrum of the beam would represent the weighted average distance (time to payment), which is 1.78 years in this case.
For most practical calculations, the Macaulay duration is calculated using the yield to maturity to calculate the PV(i):
 (2)
 (3)
where:
 i indexes the cash flows,
 PV_{i} is the present value of the ith cash payment from an asset,
 CF_{i} is the cash flow of the ith payment from an asset,
 y is the yield to maturity (continuouslycompounded) for 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.
Macaulay gave two alternative measures:
 Expression (1) is Macaulay–Weil duration which uses zerocoupon bond prices as discount factors, and
 Expression (3) 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 second form is based on a constant value of the yield y, not varying by term to payment. With the use of computers, both forms may be calculated but expression (3), assuming a constant yield, is more widely used because of the application to modified duration.
Modified duration
In contrast to Macaulay duration, modified duration (sometimes abbreviated DM) is a price sensitivity measure, defined as the percentage derivative of price with respect to yield:
It turns out that when the yield is expressed continuouslycompounded, Macaulay duration and modified duration are numerically equal, while for periodicallycompounded yields the relation between Macaulay duration and modified duration is:
where k is the compounding frequency (1 for annual, 2 for semiannual, etc.) We can derive this expression from the derivative definition of modified duration above.
First, consider the case of continuouslycompounded yields. If we take take the derivative of price or present value, expression (2), with respect to the continuouslycompounded yield y we see that:
In other words, for yields expressed continuouslycompounded, ModD = MacD.^{[1]}
In financial markets yields are usually expressed periodicallycompounded (say annually or semiannually) instead of continuouslycompounded. Then expression (2) becomes:
and expression (3) becomes:
where k is the compounding frequency (1 for annual, 2 for semiannual, etc.) In this case, when we take the derivative of the value V with respect to the periodicallycompounded yield we find ^{[4]}
This gives the wellknown relation between Macaulay duration and modified duration quoted above. It should be remembered that, even though Macaulay duration and modified duration are closely related, they are conceptually distinct. Macaulay duration is a weighted average time until repayment (measured in units of time such as years) while modified duration is a price sensitivity measure, the percentage change in price with respect to yield.
Units
For modified duration the common units are the percent change in price per one percentage point change in yield (for example yield going from 8% (y = 0.08) to 9% (y = 0.09)) – the above formulas for the modified duration has to be divided by 100 to make the unit change in yield equal to 0.01 (1%) instead of 1 (100%) which would mean the yield changes from 0.08 to 1.08. This will give modified duration close to the value of Macaulay duration (and the same when rates are continuouslycompounded).
Formally, modified duration is a semielasticity, the percent change in price for a unit change in yield, rather than an elasticity, which is a percentage change in output for a percentage change in input. Thus modified duration has units of years, since the top is nondimensional (a percentage), and the bottom has units of 1/years; by contrast, an elasticity is nondimensional, being the ratio of two nondimensional quantities (two percentages).
In derivatives pricing ("The Greeks"), the closest analogous quantity is the λ or Lambda, which is the price elasticity (percentage change in price for percentage change in input), and, unlike modified duration, is an actual elasticity.
NonFixed Cash Flows
Modified duration can be extended to instruments with nonfixed cash flows, while Macaulay duration applies only to fixed cash flow instruments. Modified duration is defined as the logarithmic derivative of price with respect to yield, and such a definition will apply to instruments that depend on yields, whether or not the cash flows are fixed.
Finite Yield Changes
Modified duration is defined above as a derivative (as the term relates to calculus) and so is based on infinitesimal changes. Modified duration is also useful as a measure of the sensitivity of a bond's market price to finite interest rate (i.e., yield) movements. For a small change in yield, Δy,
Thus modified duration is approximately equal to the percentage change in price for a given finite change in yield. So a 15year bond with a modified duration of 7 would fall approximately 7% in value if the interest rate increased by one percentage point (say from 7% to 8%).^{[5]}
Bond Duration Formulas
For a standard bond with fixed, semiannual payments the bond duration closedform formula is:^{[citation needed]}
 FV = par value
 C = coupon payment per period (halfyear)
 i = discount rate per period (halfyear)
 a = fraction of a period remaining until next coupon payment
 m = number of full coupon periods until maturity
 P = bond price (present value of cash flows discounted with rate i)
For a bond with coupon frequency k but an integer number of periods (so that there is no fractional payment period), the formula simplifies to: ^{[6]}
where
 y = Yield (per year, in decimal form),
 c = Coupon (per year, in percent),
 m = Number of coupon periods.
Example
Consider a 2 year bond with a 20% semiannual coupon and a yield of 4% semiannually compounded. The total PV will be:

 = 9.804 + 9.612 + 9.423 + 9.238 + 92.385 = 130.462
The Macaulay duration is then
 .
The simple formula above gives (y/k =.04/2=.02, c/k = 20/2 = 10):
The modified duration, measured as percentage change in price per one percentage point change in yield, is:
 .
The DV01, measured as dollar change in price for a $100 nominal bond for a one percentage point change in yield, is
 ($ per 1 percentage point change in yield)
where the division by 100 is because modified duration is the percentage change.
Dollar duration, DV01
The dollar duration or DV01 is defined as the derivative of the value with respect to yield:
so that it is the product of the modified duration and the price (value):
 ($ per 1 percentage point change in yield)
or
 ($ per 1 basis point change in yield)
The DV01 is analogous to the delta in derivative pricing (The Greeks) – it is the ratio of a price change in output (dollars) to unit change in input (a basis point of yield). Dollar duration or DV01 is the change in price in dollars, not in percentage. It gives the dollar variation in a bond's value per unit change in the yield. It is often measured per 1 basis point  DV01 is short for "dollar value of an 01" (or 1 basis point). The names BPV (basis point value) or PV01 (present value of an 01) are also used, although PV01 more accurately refers to the value of a one dollar or one basis point annuity. (For a par bond and a flat yield curve the DV01, derivative of price w.r.t. yield, and PV01, value of a onedollar annuity, will actually have the same value.^{[citation needed]})
DV01 or dollar duration can be used for instruments with zero upfront value such as interest rate swaps where percentage changes and modified duration are less useful.
Application to VaR
Dollar duration D_{$} is commonly used for VaR (ValueatRisk) calculation. 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 (multivariate) 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.
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 nonputtable 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, with weights proportional to the bond prices.^{[1]} 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.
Convexity
Main article: Bond convexityDuration 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  i.e. 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  i.e. 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 reissue a new bond at a lower rate, thus providing the issuer with valuable optionality.
Mortgagebacked securities (passthrough mortgage principal prepayments) with USstyle 15 or 30 year fixed rate mortgages as collateral are examples of callable bonds.
See also
 Bond convexity
 Bond valuation
 Immunization (finance)
 Stock duration
 Bond duration closedform formula
 Yield to maturity
Lists
Notes
References
 ^ ^{a} ^{b} ^{c} ^{d} Hull, John C. (1993), Options, Futures, and Other Derivative Securities (Second ed.), Englewood Cliffs, NJ: PrenticeHall, Inc., pp. 99–101
 ^ ^{a} ^{b} Brealey, Richard A.; Myers, Stewart C.; Allen, Franklin (2011), Principles of Corporate Finance (Tenth ed.), New York, NY: McGrawHill Irwin, pp. 50–53
 ^ Marrison, Chris (2002), The Fundamentals of Risk Measurement, Boston, MA: McGrawHill, pp. 57–58
 ^ Berk, Jonathan; DeMarzo, Peter (2011), Corporate Finance (Second ed.), Boston, MA: Prentice Hall, pp. 966–969
 ^ "Macaulay Duration" by Fiona Maclachlan, The Wolfram Demonstrations Project.
 ^ Bodie; Kane; Marcus (1993), Investments (Second ed.), p. 478
External links
 Investopedia’s duration explanation
 Hussman Funds  Weekly Market Comment: February 23, 2004  BuyandHold For the Duration?
 Online realtime 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
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