Bond duration

From Free net encyclopedia

In economics and finance, duration is the weighted average maturity of a bond's cash flows or of any series of linked cash flows. 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.

1 Price
2 Cash Flow
3 Macaulay duration
4 Modified duration
5 Embedded options and effective duration
6 Average duration
7 Convexity
8 PVO1
9 See also
- 9.1 Lists
10 External links

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Price

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%.

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Cash Flow

Duration is also a measure of how long it will take for an investor who is reinvesting bond coupons to recoup the initial price of the bond. For example, as interest rates rise, bond prices drop, but bond coupons can be reinvested at higher interest rates. The point at which the drop in price equals the money generated by reinvested bond coupons is the bond duration. Thus the higher the coupon rate from a bond, the shorter the duration. Duration is always less than the life (maturity) of a bond (with the exception of zero-coupon bonds where, by definition, the duration equals the life of the bond).

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Macaulay duration

Macaulay duration is the weighted average maturity of a bond where the weights are the relative discounted cash flows in each period.

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

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Modified duration

Modified duration is calculated as follows:

<math>\mbox{Modified duration}=\frac{\mbox{Macaulay duration}}{1+\frac{r}{n}}</math>

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

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Embedded options and effective duration

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 than 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.

<math>\mbox{Effective Duration} = \frac {V_{-\Delta y}-V_{+\Delta y}}{2(V_0)\Delta y} </math>

where <math>\Delta y</math> is the amount that yield changes, and <math>V_{-\Delta y} \mbox{and} V_{+\Delta y} </math> are the values that the bond will take if the yield falls by y or rises by y, respectively.

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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.

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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.

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