bond duration

Bond durationFrom 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 tointerest rate movements. It broadly corresponds to the length of time before the asset is due to be repaid. There arevarious 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 theredemption yield of the bond. This equation is valid for small changes in those quantities only. Duration is known in thecontext 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 equalto 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: whencalculating 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.

Price

Duration is useful as a measure of the sensitivity of a bond's market price to interest rate (i.e., yield) movements. It isapproximately equal to the percentage change in price for a given change in yield. For example, for small interest ratechanges, the duration is the approximate percentage by which the value of the bond will fall for a 1% per annum increasein 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:

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

of 7Bond duration - Wikipedia, the free encyclopedia

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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-couponbond). 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 = Tf − T0, where Tf is the maturity date and T0 is the starting date of the bond. If there

are additional cash flows Ci at times Ti, the duration of every cash flow is ΔTi = Ti − T0. 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 issubstituted by ln(1+j) and this is equivalent of assuming a continuously compound interest. At the end of ther calculationswe 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 thisexpression 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 isalways 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 variationin the redemption yield on the bond, forgetting the quadratic and higher-order terms. The quadratic terms are taken into



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, notin percentage, and has units of Dollar-Years (Dollars times Years). It gives the dollar variation in a bond's value for asmall 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 ratesas 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 formulacan be used to calculate the VaR of the portfolio by ignoring higher order terms. Typically cubic or higher terms aretruncated. Quadratic terms, when included, can be expressed in terms of (multi-variate) bond convexity. One can makeassumptions about the joint distribution of the interest rates and then calculate VaR by Monte Carlo simulation or, insome special cases (e.g., Gaussian distribution assuming a linear approximation), even analytically. The formula can alsobe used to calculate the DV01 of the portfolio (cf. below) and it can be generalized to include risk factors beyond interestrates.

Macaulay duration

Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of abond 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 twoalternative measures that are useful:



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 ofthe price of the bond with respect to the yield—as shown above. In case of yearly compounded yield, the modifiedduration 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 Ci. The duration, defined as the weighted average maturity, is then

The derivative of V with respect to r is:

multiplying by we obtain

or



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 correctlyapproximate 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 computeits 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 atany time before the bond's maturity (i.e. an American put option). No matter how high interest rates become, the price ofthe bond will never go below $1,000 (ignoring counterparty risk). This bond's price sensitivity to interest rate changes isdifferent from a non-puttable bond with otherwise identical cashflows. Bonds that have embedded options can beanalyzed using "effective duration". Effective duration is a discrete approximation of the slope of the bond's value as afunction 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 varydepending 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 yieldcurve, 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. Theaverage duration of the bonds in the portfolio is often reported. The duration of a portfolio equals the weighted averagematurity of all of the cash flows in the portfolio. If each bond has the same yield to maturity, this equals the weightedaverage of the portfolio's bond's durations. Otherwise the weighted average of the bond's durations is just a goodapproximation, but it can still be used to infer how the value of the portfolio would change in response to changes ininterest rates.

Bond duration closed-form formula

FV = par valueC = coupon payment per period (half-year)i = discount rate per period (half-year)a = fraction of a period remaining until next coupon paymentm = number of coupon dates until maturityP = bond price (present value of cash flows discounted with rate i)



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 rateschange, the price does not change linearly, but rather is a convex function of interest rates. Convexity is a measure of thecurvature of how the price of a bond changes as the interest rate changes. Specifically, duration can be formulated as thefirst derivative of the price function of the bond with respect to the interest rate in question, and the convexity as thesecond derivative.

Convexity also gives an idea of the spread of future cashflows. (Just as the duration gives the discounted mean term, soconvexity 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 - iethe 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 negativeconvexity, which is to say its price should fall as rates fall. This is because the issuer can redeem the old bond at a highcoupon 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 ratemortgages 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 ofbonds 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 infull until maturity (at 5 years).

The Weighted-Average Life is the weighted average of the principal repayments of an amortizing loan, and is longer thanthe duration.

See also

Bond convexity Bond valuation Immunization (finance) Stock duration Bond duration closed-form formula Yield to maturity

Lists

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 generallyperpetual.



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 "http://en.wikipedia.org/wiki/Bond_duration"Categories: Fixed income analysis

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