Bond duration

Duration is a measure of sensitivity denotes the modified duration of an investment in a fixed-interest securities. More precisely and in general terms, the duration of the weighted average of the time points at which the investor receives payments from a security.

Duration concept

The duration was designed in 1938 by Frederick R. Macaulay and is therefore called Macaulay duration. The duration represents the point in time occurs at the complete immunization against interest rate risk within the meaning of Endwertschwankungen. The concept is based on the fact that unforeseen changes in interest rates have two opposing effects on the future value of a fixed income security (eg bond): For example, a rise in interest rates leads though to a lesser present value of the bond; because of Reinvestitionsprämisse future payments ( coupons ), however, higher interest rates. Finally, a rise in interest rates leads to a higher final value. Conversely, it behaves when interest rates fell. The point in time up to which the market value of the bond interest has increased at least returned to the expected value because of reinvested coupons or to whom he has not fallen below the value expected for the lower interest rates because of the lower discounting is called duration.

Another term is Mean residual retention period. For the duration is a weighted average of the time points at which the investor receives payments from a security. The weighting factors of this average value, the respective proportions of the present value of the interest and principal payments will be taken at the time of the total present value of all payments.

More precisely corresponds to the duration of a Taylor series expansion of the change in value that is truncated after the first linear term. For practical results with the duration a simple formula that links the change in value of a bond with the interest rate. The value of coupon bonds with no special features, however, is convex in the level of interest rates. Due to the aforementioned linear approximation one therefore underestimated the change in value of bonds, with an estimate of the duration is therefore always pessimistic. The loss in value with rising interest rates is overestimated, the value added to a decreasing level of interest rates is underestimated. This effect is more pronounced the greater the change in interest rates. Reaches the approximation with the linear approximation, in practice no longer sufficient, the second term of the Taylor series development is taken into account. This approach leads to the concept of convexity.

Model assumptions

The following assumptions are made during the duration concept:

• Flat yield curve: This simplifying assumption term independent interest payments, which occur at different times, can be discounted at a uniform rate
• One-time change in market interest rates through parallel shift of the (flat) yield curve. This change takes place immediately after acquisition of the bond
• Reinvestment of coupon payments at market rates
• No transaction costs or Ganzzahligkeitsprobleme
• No taxes

Modified Duration

The ( Macaulay ) duration is measured in units of years. However, this complicates the practicality strong. Much more desirable it would be to be able to make a statement about the relative change in the bond price in response to a change in market interest rates. These tasks does the modified duration (English: modified duration ). It indicates by how much the price of a bond changes as market interest rates change by one percentage point, that is, it measures the impact of a marginal interest rate change rate effect and thus represents a kind of elasticity of the bond price on the market rate dar. Since this the very restrictive assumptions of Durationskonzeptes apply a practical application is given only at very low interest rates.

Modified duration is a measure of financial mathematics, which indicates how much the total return of a bond (consisting of the principal, coupon payments, as well as compounding effect of the reinvestment of repayments) changes when changes the interest rate on the market.

The modified duration, is as follows with the duration in the context:

Portfolio duration

To determine the duration of a portfolio is calculated in the first step, the durations of the bonds in the portfolio. The portfolio duration is calculated as the weighted by the share in total portfolio value of each bond sum of the individual Anleihedurationen:

With

Alternatively, the duration for a total cash flows calculated by the individual cash flows are added.

Derivation of Durationsformel

The present value of a bond can generally by discounting ( discounting ) of future payments (that is, the often incurred annual coupon payments and the coupon and principal payment at the time ) calculate:

With

Assuming that ( with for all time points) is a term independent interest rate and passes by from, one obtains:

This is the Euro - Duration. Division of derivation by the cash value in supplies:

The calculated expression represents the approximate relative price change in (small ) changes in interest rates represents a such a definition of the Macaulay duration has historical reasons.

Macaulay Duration:

Summary of formulas of Durationsberechnung

Immunization against interest rate risk

A position is then immunized against interest rate risk if the weighted with the market values ​​modified durations of the long and the short position correspond to each other.

D ( long) · Co ( long) = D ( short) · Co ( short) with Co as the price of the option and D as duration.

This process is called "duration matching". Such a secure position can be considered as zero -coupon bond.

Review of Durationskonzeptes

For the assessment of the interest rate sensitivity of a bond, it is not sufficient to consider only the total running time: For example, a zero-coupon bond with a single payment at maturity a far greater sensitivity to interest rates up as a "normal" bond with the same maturity, are made at the annual coupon payments.

In addition to the maturity of a bond is thus the temporal seizures of payments is important. The duration linked these two components relevant to multiplicative way, so weights the respective payment date with the relative contribution to the present value. A higher duration indicates a tendency High Interest Rate Sensitivity and shows how long the capital is tied up in the middle.

Duration is higher, the lower the coupon. For the extreme case of zero coupon bond, the duration coincides with the remaining term of the bond.

Since the interest is usually not steady, but gradual ( discrete) change, and the dependence of the bond rate on the interest rate is not a linear relationship, are the changes that calculates the duration, not quite exact. The decline is overestimated if the interest rises, and the price increase is underestimated when the interest rate falls. This error caused by the approximation of a non-linear relationship by a linear, falls with only minor changes in interest rates negligible. For larger changes in interest rates, however, this Konvexitätsfehler increases greatly alleviating this error provides the inclusion of convexity in the price estimate.

The existence of contributions condition is the existence of market imperfections.

Article

Buhler, Alfred / Hies, Michael: Key Rate Duration: A new instrument for the measurement of interest rate risk, in: The bank, Issue 2, 1995, pp. 112-118.