Bond duration

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In economics and finance, duration is the weighted average maturity of a bond's cash flows or of any series of linked cash flows. Then the duration of a zero coupon bond with a maturity period of n years is n years. In case there will be coupon payments, the duration will be less than n years. This measure is closely related to the derivative of the bond's price function with respect to the interest rate, and some authors consider the duration to be this derivative, with the weighted average maturity simply being an easy method of calculating the duration for a non-callable bond. It is sometimes explained in inaccurate terms 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.

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 PVO1
12 DVO1
13 See also
- 13.1 Lists
14 External links

Duration is useful as a measure of the sensitivity of a bond's price to interest rate movements. It is approximately inversely 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 years would fall approximately 7% in value if the interest rate increased by 1%.

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, t(i) is the future payment date, V is the bond Price and D is the duration.

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 $Δ T i = T i - T 0$ . Being $r$ the rate of the bond, continuously compounded, the price of the bond is

$V = \sum_i C_i e^{-r\Delta T_i}$

To compute the duration, each duration of every cash flow is weighted with its value over the total value of the bond (n.b. the sum of the weights is 1):

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

Thus 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

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

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

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

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. The quadratic terms are taken in account in the Convexity.

The Dollar duration is defined as the product of the Duration and the price (value). It gives then the variation of a bond 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
$D_$ := -\frac{\partial V}{\partial r}$ .
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
$V = V(r_1, \ldots, r_n)$
denote the value of such portfolio. Then the exposure vector $\boldsymbol{\omega} = (\omega_1, \ldots, \omega_n)$ has components
$\omega_i = - D_{$,i} := \frac{\partial V}{\partial r_i}$
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 porfolio (cf. below) and it can be genearalized to include risk factors beyond interest rates.

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.

$\mbox{Macaulay duration} = \frac {\sum\ (\mbox{present value of cash flow}\times\mbox{time to cash flow})}{\mbox{price of the bond}}.$

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 Macauley-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.

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.

In case of yearly 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:

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

where r is the yield to maturity of the bond, and n is the number of cashflows per year.

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

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

which is valid for yearly compounded yield.

For bonds that have embedded options, Macauley duration and modified duration will not correctly approximate the price move for a change in yield. 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.

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

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.

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.

$Dur=\frac{C}{P}\frac{(1+ai)(1+i)^m-(1+i)-(m-1+a)i}{i^2(1+i)^{(m-1+a)}}+\frac{100(m-1+a)}{(1+i)^{(m-1+a)}}$

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

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.)

PVO1 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).

DVO1 (Dollar Value of 1 basis point) is the same as PV01