Duration

The duration is a measure of sensitivity to the average duration of a tied-up capital investment in a fixed income securities designated. More precisely and formulated in general terms, duration is the weighted average of the times at which the investor receives payments from a security.

Duration concept

The duration was introduced in 1938 by Frederick R. Macaulay and is therefore called Macaulay duration . The duration represents the point in time at which complete immunization against the risk of interest rate changes occurs in the sense of fluctuations in the terminal value. The concept is based on the fact that unforeseen changes in interest rates have two opposing effects on the final value of a fixed-income security (e.g. bond): For example, an increase in interest rates leads to a lower present value of the bond; However, due to the reinvestment premise, future payments (coupons) will bear higher interest. Ultimately, a rise in interest rates leads to a higher terminal value. The opposite is the case with an interest rate cut. The point in time up to which the market value of the bond has at least reached the expected value again due to the reinvested coupons due to increased interest rates or until which it has not fallen below the expected value due to lower discounting is called duration.

Another term is mean remaining bond time . This is because duration is the weighted average of the times at which the investor receives payments from a security. The respective shares of the present value of the interest and repayment payments at the respective point in time in the total present value of all payments are used as weighting factors for this mean value .

More precisely, the duration of a Taylor series development corresponds to the change in value that is cut off after the first linear term. For the practice, a simple formula that the change in value of a bond is the duration of the interest rate : links . The value of coupon bonds without special features is, however, convex in terms of interest rates . The aforementioned linear approximation therefore underestimates the change in value of bonds; an estimate using duration is therefore always pessimistic. The loss in value when interest rates rise is overestimated, while the increase in value when interest rates fall is underestimated. This effect becomes stronger the greater the change in the interest rate level. If the approximation with the linear approximation is no longer sufficient in practice, the second term of the Taylor series expansion must be taken into account. This approach leads to the concept of convexity . ${\displaystyle \Delta P}$ ${\displaystyle \Delta r}$ ${\displaystyle \Delta P=-D/(1+r)\cdot \Delta r}$

Model assumptions

The following assumptions are made in the duration concept:

• Flat yield curve : This simplified assumption of interest independent of maturity means that payments that occur at different points in time can be discounted using a uniform interest rate
• One-time change in the market interest rate level through parallel shifting of the (flat) interest structure curve. This change takes place immediately after the bond has been acquired
• The coupon payments are reinvested at the market interest rate ${\displaystyle r}$
• no transaction costs or integer problems
• no taxes

Modified duration

The (Macaulay) duration is measured in years. However, a particularly common question from practice is to be able to make a statement about the relative change in the bond price depending on a change in the market interest rate. This task is performed by the modified duration (English modified duration ). It indicates the percentage by which the bond price changes if the market interest rate changes by one percentage point; It thus measures the price effect triggered by a marginal change in interest rate and thus represents a kind of elasticity of the bond price from the market interest rate. Since the very restrictive assumptions of the duration concept also apply here, practical applicability is only given for very small changes in interest rates.

The modified duration is a key figure from financial mathematics that indicates how much the total return on a bond (consisting of the repayments, coupon payments and the compound interest effect when the repayments are reinvested) changes if the interest rate changes in the market.

The modified duration is related to the duration as follows: ${\displaystyle D_{\mathrm {MD} }}$

${\displaystyle D_{\mathrm {MD} }=D_{\mathrm {Mac} }\cdot {\frac {1}{1+r}}}$

Portfolio duration

To determine the duration of a portfolio , the first step is to calculate the durations of the bonds in the portfolio. The portfolio duration is the sum of the individual bond durations weighted with the share of the total portfolio value of each bond:

${\displaystyle D_{\mathrm {PF} }=\sum _{i=1}^{N}x_{i}\cdot D_{i}}$

With

${\displaystyle D_{\mathrm {PF} }}$ = Duration of the portfolio

${\displaystyle x_{i}}$= Share of the bond in the total portfolio value ${\displaystyle i}$

${\displaystyle D_{i}}$ = Duration of the bond ${\displaystyle i}$

${\displaystyle N}$ = Number of different bonds in the portfolio

Alternatively, the duration can be calculated for a total cash flow by adding up the individual cash flows.

Derivation of the duration formula

The present value of a bond can generally be calculated by discounting the future payments (i.e. the coupon payments that often occur annually as well as the coupon and principal payments at the time ): ${\displaystyle n}$

${\displaystyle P_{0}=\sum _{t=1}^{T}{\frac {C_{t}}{(1+r_{t})^{t}}}={\frac {C_{1}}{1+r_{1}}}+{\frac {C_{2}}{(1+r_{2})^{2}}}+\dotsb +{\frac {C_{T}}{(1+r_{T})^{T}}}}$

With

${\displaystyle P_{0}}$ = Present value at the time of observation ${\displaystyle t_{0}}$

${\displaystyle C_{t}}$= Payment at the time (in years) ${\displaystyle t}$

${\displaystyle r_{t}}$= Interest rate valid for the term ${\displaystyle t}$

${\displaystyle T}$ = End of term of bond (last payment)

If one assumes that there is a term-independent interest rate (with for all points in time ), and derives according to, one obtains: ${\displaystyle r}$ ${\displaystyle r=r_{t}}$ ${\displaystyle t}$ ${\displaystyle r}$

${\displaystyle {\frac {\partial P_{0}}{\partial r}}=\sum _{t=1}^{T}C_{t}\cdot (-t)\cdot (1+r)^{-(t+1)}=-\sum _{t=1}^{T}{\frac {t\cdot C_{t}}{(1+r)^{(t+1)}}}=-{\frac {1}{1+r}}\cdot \sum _{t=1}^{T}{\frac {t\cdot C_{t}}{(1+r)^{t}}}}$

This is the euro duration. Division of the derivative by the present value in yields: ${\displaystyle P_{0}}$ ${\displaystyle t_{0}}$

${\displaystyle {\frac {\frac {\partial P_{0}}{\partial r}}{P_{0}}}=-{\frac {1}{1+r}}\cdot \underbrace {\sum _{t=1}^{T}{\frac {t\cdot C_{t}}{(1+r)^{t}}}\cdot {\frac {1}{P_{0}}}} _{\mathrm {Macaulay-Duration} }}$

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

Macaulay duration : ${\displaystyle D_{\mathrm {Mac} }}$

${\displaystyle D_{\mathrm {Mac} }=\sum _{t=1}^{T}{\frac {t\cdot C_{t}}{(1+r)^{t}}}\cdot {\frac {1}{P_{0}}}}$

Summary of the formulas for calculating the duration

${\displaystyle D_{\mathrm {Euro} }=\sum _{i=1}^{n-1}t_{i}c_{i}(1+{\bar {y}})^{-(t_{i}+1)}+t_{n}(100+c_{n})(1+{\bar {y}})^{-(t_{n}+1)}}$

${\displaystyle D_{\mathrm {mod} }={\frac {\sum _{i=1}^{n-1}t_{i}c_{i}(1+{\bar {y}})^{-(t_{i}+1)}+t_{n}(100+c_{n})(1+{\bar {y}})^{-(t_{n}+1)}}{\sum _{i=1}^{n-1}c_{i}(1+{\bar {y}})^{-(t_{i}+1)}+(100+c_{n})(1+{\bar {y}})^{-(t_{n}+1)}}}}$

${\displaystyle D_{\mathrm {Mac} }={\frac {\sum _{i=1}^{n-1}t_{i}c_{i}(1+{\bar {y}})^{-(t_{i}+1)}+t_{n}(100+c_{n})(1+{\bar {y}})^{-(t_{n}+1)}}{\sum _{i=1}^{n-1}c_{i}(1+{\bar {y}})^{-(t_{i}+1)}+(100+c_{n})(1+{\bar {y}})^{-(t_{n}+1)}}}}$

Immunization against interest rate risks

A position is immunized against interest rate risks if the modified durations weighted with the market values ​​of the long and short positions correspond to one another.

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

This process is called "duration matching". Such a secured position can be viewed as a zero coupon bond .

Evaluation of the duration concept

When assessing the interest rate sensitivity of a bond, it is not sufficient to only consider the total term: For example, a zero coupon bond with only one single payment at the end of the term is far more sensitive to interest than a standard bond with the same term, for which coupon payments are made annually.

In addition to the term of a bond, the timing of the payments is important. The duration links these two relevant components in a multiplicative way, i.e. weights the respective time of payment with the relative contribution to the present value. A higher duration suggests a tendency towards high interest rate sensitivity and shows how long the capital is tied up on average. ${\displaystyle t}$

The duration is higher, the lower the coupon is. In the extreme case of the zero-coupon bond , the duration is the same as the bond's remaining term.

However, since interest rates do not change continuously , but rather gradually ( discretely ), and the bond price's dependence on the interest rate is not a linear relationship, the changes that the duration calculates are not exactly exact. The price decline is overestimated when the interest rate rises and the price increase is underestimated when the interest rate falls. This error, triggered by the approximation of a nonlinear relationship by a linear one, is of little consequence with only minor changes in interest rates. This convexity error increases sharply with larger interest rate changes; one alleviation of this error is to include convexity in the price estimation.

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

The assumption of a flat yield curve can be softened with the help of the key rate duration .

literature

• Alfred Bühler, Michael Hies: Key Rate Duration: A new instrument for measuring interest rate risk . In: The Bank . Volume 2, 1995, pp. 112-118 .

Web links

• Price, duration and yield to maturity calculator with graph

Individual evidence

1. ↑ Frederick R. Macaulay, "Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields, and Stock Prices in the United States since 1856" Online at: http://www.nber.org , accessed September 20, 2017 .