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v • d • e

In finance, duration is the weighted average maturity of a bond's cash flows or of any series of linked cash flows. Thus the duration of a zero coupon bond with a maturity period of n years is n years. If there are coupon payments, the duration will be less than n years. The field of finance refers to the concepts of time, money and risk and how they are interelated. ... For alternative meanings, see bond (a disambiguation page). ... Zero coupon bonds are bonds which do not pay periodic coupons, or so-called interest payments. ...

This measure is closely related to the derivative of the bond's price function with respect to the interest rate (in terms of the Greeks, the Δ, where the underlying is the interest rate), and some authors consider the duration to be this derivative divided by the price (in terms of the Greeks, the λ), with the weighted average maturity simply being an easy method of calculating the duration for a non-callable bond.

The duration is often confused with other notions, as detailed below.

1 Price
2 Basics
3 Cash flow
4 Dollar duration and applications to VaR
5 Macaulay duration
6 Modified duration
7 Embedded options and effective duration
8 Average duration
9 Bond duration closed-form formula
10 Convexity
11 PV01 and DV01
12 Confused notions
13 See also
- 13.1 Lists
14 References
15 External links

Price

Duration is useful as a measure of the sensitivity of a bond's price to interest rate movements. It is approximately proportional 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 that the value of the bond will lose for a 1% increase in interest rates. So a 15-year bond with a duration of 7 would fall approximately 7% in value if the interest rate increased by 1%. The duration mentioned here is Modified Duration, while Macaulay Duration will not predict the bond price's exact reaction to an interest-rate change.^[1] An interest rate is the price a borrower pays for the use of money he does not own, and the return a lender receives for deferring his consumption, by lending to the borrower. ...

Basics

The standard definition of duration:

$D = sum_{i=1}^{n}frac {P(i)t(i)}{V}$

Where P(i) is the present value of coupon i or the final principal payment, t(i) is the payment date from now, V is the bond Price and D is the duration. The present value of a single or multiple future payments (known as cash flows) is the nominal amounts of money to change hands at some future date, discounted to account for the time value of money, and other factors such as investment risk. ...

Cash flow

As stated at the beginning, the duration is the weighted average maturity time of a bond cash flow. For a zero-coupon the duration will be $Δ T = T f - T 0$ , where $T f$ is the maturity date and $T 0$ is the starting date of the bond. If there are different cash flows $C i$ the duration of every cash flow is $Δ T i = T i - T 0$ . From the current market price of the bond $V$ , one can calculate the yield to maturity of the bond $r$ using the formula

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:

$D = sum_i Delta T_i frac{C_i e^{-rDelta T_i}}{V}$

The higher the coupon rate from a bond, the shorter the duration. Duration is always less than or equal to the life (maturity) of a coupon 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

$frac{partial V}{partial r} = - sum_i Delta T_i C_i e^{-rDelta T_i} = -D cdot V$

then for small variation $partial r$ of the rate of the bond we have

$frac{partial V}{V} = -D partial r + O(partial r^2)$

That means that the duration gives the negative of the relative variation of the value of a bond respect to a variation of the rate of the bond, forgetting the quadratic terms. The quadratic terms are taken in account in the Convexity. This article does not cite any references or sources. ...

Dollar duration and applications to VaR

The Dollar duration is defined as the product of the Duration and the price (value). It gives then the variation of a bond value for a small variation of the interest rate. 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 $r_1, ldots, r_n$ as risk factors, and let

denote the value of such portfolio. Then the exposure vector $boldsymbol{omega} = (omega_1, ldots, omega_n)$ has components

Accordingly, the change in value of the portfolio can be approximated as
$Delta V = sum_{i=1}^n omega_i Delta r_i + sum_{1 leq i,j leq n} O(Delta r_i Delta r_j)$
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. Var is a dÃ©partement of southeastern France. ... Var is a dÃ©partement of southeastern France. ... Var is a dÃ©partement of southeastern France. ... This article does not cite any references or sources. ... Var is a dÃ©partement of southeastern France. ...

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.

$mbox{Macaulay duration} = frac {sum (mbox{cash flow discounted with yield to maturity}timesmbox{time to cash flow})}{mbox{price of the bond}}.$

Macaulay showed that an unweighted average maturity is not useful in predicting interest rate risk. He gave two alternative 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.

With the use of computers, both forms may be calculated, but the Macaulay duration is still widely used. Yield to maturity (YTM) is the yield promised by the bondholder on the assumption that the bond will be held to maturity, that all coupon and principal payments will be made and coupon payments are reinvested at the bonds promised yield at the same rate as invested. ...

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 $frac{delta V}{V} = -D delta r + O(delta r^2)$ is not valid anymore. That is why the modified duration $D *$ is used instead:

$D^* = frac{text{Macaulay duration}}{1+frac{r}{n}}$

where r is the yield to maturity of the bond, and n is the number of cashflows per year. Yield to maturity (YTM) is the yield promised by the bondholder on the assumption that the bond will be held to maturity, that all coupon and principal payments will be made and coupon payments are reinvested at the bonds promised yield at the same rate as invested. ...

Let us prove that the relation

$frac{delta V}{V} = -D^* delta r + O(delta r^2)$

is valid. We will analyze the particular case n = 1. The value (price) of the bond is

$V = sum_i frac{C_i}{(1+r)^i}$

where i is the number of years after the starting date the cash flow $C i$ will be paid. The duration, defined as the weighted average maturity, is then

$D=frac{1}{V}sum_i frac{C_i}{(1+r)^i} cdot i$

The derivative of V with respect to r is:

$frac{partial V}{partial r} = - sum_i frac{C_i}{(1+r)^{i+1}}cdot i$

multiplying by $frac{(1+r)}{V}$ we obtain

$frac{partial V}{partial r} cdot frac{1+r}{V} = -D$

$frac{partial V}{partial r} = -V cdot D^*$

from which we can deduce the formula

$frac{delta V}{V} = -D^* delta r + O(delta r^2)$ wdsd which is valid for yearly compounded yield.

Embedded options and effective duration

For bonds that have embedded options, such as puttable and callable bonds, Macauley duration and modified 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 (properly, lambda), which is the duration. The effective duration is a discrete approximation to this latter, and depends on an option pricing model. It has been suggested that this article or section be merged into option. ... The Greeks redirects here. ... The Greeks redirects here. ...

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 points before the bond's maturity. No matter how high interest rates become, the price of the bond will never go below $1,000. This bond's price sensitivity to interest rate changes is different from a non-puttable bond with identical cashflows. Bonds that have embedded options should 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.

$text{Effective Duration} = frac {V_{-Delta y}-V_{+Delta y}}{2(V_0)Delta y}$

where $Δ y$ is the amount that yield changes, and $V - Δ y and V + Δ y$ are the values that the bond will take if the yield falls by y or rises by y, respectively.

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. In finance, a portfolio is a collection of investments held by an institution or a private individual. ... This article deals with U.S. mutual funds. ... Yield to maturity (YTM) is the yield promised by the bondholder on the assumption that the bond will be held to maturity, that all coupon and principal payments will be made and coupon payments are reinvested at the bonds promised yield at the same rate as invested. ...

Bond duration closed-form formula

$Dur=frac{C}{P}frac{(1+ai)(1+i)^m-(1+i)-(m-1+a)i}{i^2(1+i)^{(m-1+a)}}+frac{100(m-1+a)}{(1+i)^{(m-1+a)}}$ A single arithmetic formula obtained to simplify an infinite sum in a general formula. ...

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

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. This article does not cite any references or sources. ... For other uses, see Linear (disambiguation). ... In mathematics, convex function is a real-valued function f defined on an interval (or on any convex subset C of some vector space), if for any two points x and y in its domain C and any t in [0,1], we have Convex function on an interval. ... This article does not cite any references or sources. ... For other uses, see Derivative (disambiguation). ...

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.)

PV01 and DV01

PV01 (pronounced "Pee-Vee-oh-one") 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). It is also known as DV01 (Dollar Value of 1 basis point, (pronounced "Dee-Vee-oh-one"). A basis point (often denoted as bp, bps or ; rarely, permyriad) is a unit that is equal to 1/100th of 1%. It is commonly used to denote the change in a financial instrument, or the difference (spread) between two interest rates; although it may be used in any case...

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. This quantity is simply $frac{1}{r}$ , assuming the tenor is this long, or the tenor otherwise (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), and is the duration of a perpetual bond, assuming a flat yield curve at the coupon. Note the absurdity of this definition: given a bond paying 5% per annum with a tenor of 5 years, the duration will be approximately 2.5, while the price of the bond will not be repaid in full until maturity (at 5 years). A perpetual bond, which is also known as a Perpetual or just a Perp, is a bond with no maturity date. ... The US dollar yield curve as of 9 February 2005. ...

The Weighted-Average Life is the weighted average of the principal repayments of an amortizing loan, and is longer than the duration. In banking and finance, an amortizing loan is a loan where the principal of the loan is paid down over the life of the loan, typically through equal payments. ...

References

^ "Macaulay Duration" by Fiona Maclachlan, The Wolfram Demonstrations Project.

External links

Bond Duration Defined & Simplified
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

v • d • e Bond market

Fixed income · Bond · Debenture The bond market, also known as the debit, credit, or fixed income market, is a financial market where participants buy and sell debt securities usually in the form of bonds. ... This article does not cite any references or sources. ... For alternative meanings, see bond (a disambiguation page). ... In finance, a debenture is a long-term debt instrument used by governments and large companies to obtain funds. ...

Types of bonds by issuer	Government bond · Sovereign bond · Agency bond · Municipal bond · Corporate bond (Senior debt, Subordinated debt) · Emerging market debt A government bond is a bond issued by a national government denominated in the countrys own currency. ... A sovereign bond is a bond issued by a national government. ... Agency debt (sometimes referred to in plural as Agencies) is a type of bond issued by a corporation that is nominally independent of the government - though ownership may be public or private - but considered to be backed by the government, usually on a de facto basis. ... In the United States, a municipal bond (or muni) is a bond issued by a state, city or other local government, or their agencies. ... A corporate bond is a bond issued by a corporation. ... Senior debt refers to debt secured by collateral on which the lender has put in place a first lien. ... A loan or security that, in the case of default, would only be paid out after other, more senior loans were paid in full. ... Emerging Market Debt (EMD) is a term used to encompass bonds issued by less developed countries. ...

Types of bonds by payout	Fixed rate bond · Floating rate note · Zero coupon bond · Inflation-indexed bond · Commercial paper · Accrual bond · Auction rate security · High-yield debt · Convertible bond · Mortgage-backed security · Asset-backed security In finance, a fixed rate bond is a bond with a fixed coupon (interest) rate, as opposed to a floating rate note. ... Floating rate notes (FRNs) are bonds that have a variable coupon, equal to a money market reference rate, like LIBOR or federal funds rate, plus a spread. ... Zero coupon bonds are bonds which do not pay periodic coupons, or so-called interest payments. ... Inflation-indexed bonds (also known as linkers) are bonds whose principal are indexed to inflation, cutting out inflation risk. ... Commercial paper is a money market security issued by large banks and corporations. ... An accrual bond is a fixed-interest bond that is issued at its face value and repaid at the end of the maturity period together with the accrued interest. ... An auction rate security (ARS) typically refers to a debt instrument (corporate or municipal bonds) with a long-term nominal maturity for which the interest rate is reset through a dutch auction. ... In finance, a high yield bond (non-investment grade bond, speculative grade bond or junk bond) is a bond that is rated below investment grade at the time of purchase. ... A convertible bond, or convertible debenture, is a type of bond that can be converted into shares of stock in the issuing company, usually at some pre-announced ratio. ... In finance, a mortgage-backed security (MBS) is an asset-backed security whose cash flows are backed by the principal and interest payments of a set of mortgage loans. ... An asset-backed security is a type of bond or note that is based on pools of assets, or collateralized by the cash flows from a specified pool of underlying assets. ...

Derivatives	Bond option · Credit derivative · Credit default swap · Collateralized debt obligation · Collateralized mortgage obligation A bond option is similar to a stock option with the difference that the underlying asset is a bond. ... // A credit derivative is a financial instrument or derivative (finance) whose price and value derives from the creditworthiness of the obligations of a third party, which is isolated and traded. ... A credit default swap (CDS) is a bilateral contract under which two counterparties agree to isolate and separately trade the credit risk of at least one third-party reference entity. ... For other subjects with the same abbreviation, see CDO. In financial markets, collateralized debt obligations (CDOs) are a type of asset-backed security and structured credit product. ... A Collateralized Mortgage Obligation (CMO) is a type of Mortgage Backed Security, which has been divided up into tranches. ...

Pricing	Bond valuation · Par value · Coupon · Clean price · Dirty price · Accrued interest · Day count convention Bond valuation is the process of determining the fair price of a bond. ... Par value has several meanings depending on the context, whether used in the equities market, or in the bond markets, and partially also dependent on where in the world the par value term is used. ... In finance, coupons are attached to bonds, either physically, as with old bonds (with a stapler), or electronically. ... The price quoted for a bond excluding accrued interest. ... Dirty Price A bond price that includes accrued interest. ... In finance, accrued interest is the interest that has accumulated since the principal investment, or since the previous interest payment if there has been one already. ... In finance, a day count convention determines how interest accrues over time for a variety of investments, including bonds, notes, loans, medium-term notes, swaps, and FRAs. ...

Yield analysis	Nominal yield · Current yield · Yield to maturity · Yield curve · Bond duration · Bond convexity Nominal yield is the income received from a fixed income security in one year divided by its par value. ... This article or section does not cite its references or sources. ... Yield to maturity (YTM) is the yield promised by the bondholder on the assumption that the bond will be held to maturity, that all coupon and principal payments will be made and coupon payments are reinvested at the bonds promised yield at the same rate as invested. ... The US dollar yield curve as of 9 February 2005. ... This article does not cite any references or sources. ...

Credit and spread analysis	Credit analysis · Credit risk · Credit spread · Yield spread · Z-spread · Option adjusted spread Credit analysis is the method by which one calculates the creditworthiness of a company according to the numbers made available by the audited financials for the financial year. ... Credit risk is the risk of loss due to a debtors non-payment of a loan or other line of credit (either the principal or interest (coupon) or both). ... In finance, a credit spread is the difference in yield between different securities due to different credit quality. ... In finance the Yield spread is the difference between the quoted rates of return on two different investments; a way of comparing any two financial products. ... Option adjusted spread (OAS) is the flat spread over the treasury yield curve required to discount a mortgage-backed securitys volatile coupon payments to match its market price. ...

Interest rate models	Short rate models · Rendleman-Bartter · Vasicek · Ho-Lee · Hull-White · Cox-Ingersoll-Ross · Chen · Heath-Jarrow-Morton · Black-Derman-Toy · Brace-Gatarek-Musiela In the context of interest rate derivatives, a short rate model is a mathematical model that describes the future evolution of interest rates by describing the future evolution of the short rate. ... The Rendleman-Bartter model in finance is a short rate model describing the evolution of interest ratess. ... A trajectory of the short rate and the corresponding yield curve In finance, the Vasicek model is a mathematical model describing the evolution of interest rates. ... In financial mathematics, the Ho-Lee model is a Short rate model of future interest rates. ... In financial mathematics, the Hull-White model is a model of future interest rates. ... The Cox-Ingersoll-Ross model in finance is a mathematical model describing the evolution of interest rates. ... [edit] The model The first stochastic mean and stochastic volatility model was described by Lin Chen in 1996. ... Heath-Jarrow-Morton framework is a general framework to model the evolution of interest rates (forward rates in particular). ... Black-Derman-Toy, or BDT, in finance, is a model of the evolution of the yield curve, sometimes referred to as an short rate model. ... // The LIBOR Market Model, also referred to as the BGM Model in industry, is an interest rate model used for the pricing of interest rate derivatives, especially exotic derivatives like Bermudan swaptions. ...

Categories: Fixed income analysis

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Results from FactBites:

Bond duration - Wikipedia, the free encyclopedia (1090 words)
	Duration is useful as a measure of the sensitivity of a bond's price to interest rate movements.
	That means that the duration gives the opposite of the relative variation of the value of a bond respect to a variation of the rate of the bond, forgetting the quadratic terms.
	Bonds that have embedded options should 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.

Zero coupon bond - Wikipedia, the free encyclopedia (738 words)
	Zero coupon bonds are bonds which do not pay periodic coupons, or so-called "interest payments." Zero coupon bonds are purchased at a discount from their value at maturity.
	The impact of interest rate fluctuations on strip bonds, known as the bond duration, is higher than for a coupon bond.
	This high duration means that these bonds' prices are particularly sensitive to changes in the interest rate, and therefore offset, or immunize the interest rate risk of these firms' long-term liabilities.

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