# Metrics of The Duration of a Bond

The duration of a fixed rate financial instrument, as an obligation, is the average lifespan of its financial flows weighted by their value. All things being equal, the longer duration, the greater the risk. More precisely, it is generally the duration of the weighted average of the time at which the investor receives payments from an investment.

The duration was developed in 1938 by Frederick R. Macaulay and is therefore also called Macaulay Duration. The duration represents that point of time, occuring while full immunity against interest rate risk is enjoyed. The concept is based on the fact that unforeseen changes in interest rates have two opposing effects on the final value of fixed-income securities.

For example, a rise in interest rates does lead to a lower present value of the bond. Ultimately, a rise in interest rates leads to a higher final value, the opposite is true for a rate cut.

If a loss in value is overestimated whenever interest rates rise, the value added to falling interest rates will be underestimated. And if this effect is substantial, the greater the change in interest rates.

The (Macaulay) duration is measured in units of years, however, this complicates the practical applicability of it. Hence, it would therefore be more desirable to take a statement about the relative change in the bond’s price as a function of a shift in market interest rates. This task adopts the modified duration.

It indicates the percentage change of the bond as regards its course in the event that the market interest rate alters by one percentage point. That is, it measures the marginal change in interest rates triggered by a price effect and thus represents a kind of elasticity of the bond’s price from the market interest rate.

The modified duration is a measure of financial mathematics, which shows how much the total yield of a bond (consisting of the redemptions, coupon payments, and compounding effect of reinvestment of repayments) changes whenever the interest rate shifts in the market.

The modified duration, is as follows in connection with the duration: D_ (\ mathrm (MD)) = D_ (\ mathrm (Mac)) \ cdot \ frac (1) (1 + r). The duration has to be linked with the market value of the loan at higher interest rates because of the reinvestment of coupons.

Another term is the mean residual commitment period. For the duration is the weighted average of the times at which the investor receives payments from an investment. As a weighting factor of the average actuarial value of the interest and principal payments of the respective investment of funds may be used.

To assess the interest rate sensitivity of a bond, it is not sufficient to consider only the total running time. The extreme case of a zero coupon bond is that the duration matches the remaining life of the loan.

Since the interest is usually not continuous but stepwise, it represents the dependence of the bond rate on the interest rate. If the decline is overestimated, the interest rate rises and the price increase is underestimated. This is caused by the approximation of a non-linear relationship.

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