Yield is the most important, but not the exhaustive criterion for choosing a bond. In particular, the indicator of attractiveness for an investor of one or another type of bonds is the length of the term to maturity. As the latter increases, the degree of financial risk for its owner increases. Undoubtedly, the risk of acquiring bonds with coupon income is much lower than the risk associated with bonds, payment of interest on which is made at the end of the term. In this regard, there are indicators that characterize the distribution of income in the period from the time of buying the bond until maturity. One of them is a duration indicator, which is necessarily calculated to make an informed decision on the appropriateness of acquiring specific bonds, for example, if the analyzed bonds have the same yield and/or maturities.
Duration - is the weighted average of the time until each payment (coupon and principal) for the relevant bond.
The present value of the relevant payment ( PV t) is considered as the weighting factors ( w,) to the price of this bond:
The formula for calculating the duration (measured in years) is as follows:
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or in a more compact form:
The figure calculated in this way is called the Macaulay duration, after the name of the scientist Frederico Macaulay, who in 1938 introduced this characteristic to measure the weighted average term of investing funds in bonds (instead of the maturity period).
Let's consider an example of calculating the duration indicator for a bond characterized by the following parameters: 9% coupon rate; maturity - 8 years; Yield to maturity - 10%; the nominal value is $ 1,000 (Figure 3.10).
The cash receipts for the bond under consideration in the first seven years are coupon payments of $ 90, and in the last year, a coupon is added to the payment of the face value of the (N = 1000 dollars .), so the total payment is $ 1090. The height of each column in the figure corresponds to the payment amount, and their shaded part is the present value of the payment (PV f ) with a discount rate of 10%. If we imagine that the time axis is a weighted board with a support, then the duration of the bond is the distance from the beginning of the time axis to the point of support, on which the balance is in equilibrium.
In the most general form, the formula for calculating the duration of a bond can be written as follows:
Fig. 3.10. Illustration of calculating the duration indicator
For the example in question, D = 5.97 years.
When a bond coupon is paid t once a year, the duration is calculated using the formula
The unit of duration measurement in this case is the coupon period.
If the bond is to be redeemed for many years, the duration can be calculated as
For the case when the coupon on the bond is paid t once a year, the formula for calculating duration is
Note: The last two formulas give the duration value in years.
The calculated duration of the duration can be effectively used to assess market risk:
1) a separate financial instrument: the higher the duration of the instrument, the higher its market risk;
2) a portfolio of various financial instruments.
From what has been said above, it also clearly follows that:
• duration of the financial instrument for which the coupon is not paid from the moment of issue or during the time remaining until maturity, for example, bonds with zero coupon, is equal to the time remaining until maturity;
• the duration of an ordinary bond is always less than its maturity. For example, a bond with a 10-year maturity for which a 10% coupon is paid, approximately has a duration of 7 years;
• The higher the coupon rate, the lower the duration, and the bond is less risky, and vice versa;
• The more coupons on a bond are paid, the less duration is, since more payments are placed to the starting point. At the same time, the discounting factor for par value is increasing and, correspondingly, its specific weight in the bond price is decreasing;
• The longer the time to maturity of a bond, the longer the duration;
• The higher the yield to maturity of the bond, the lower the duration, and vice versa. This is because at a higher rate of return, future cash flows are discounted to a greater extent than the nearest cash flows. When the yield of the bond falls, the equilibrium point shifts to the right.
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The duration indicator is used to estimate the change in the price of a bond when interest rates change. In order to measure the sensitivity (elasticity) of a bond, use the modified duration indicator
where D m is the modified duration; D - the duration of Macaulay; r - The market interest rate.
The modified duration is measured in percent. So, if the duration is 12.47 years, and the market rate is 10%, then the modified duration
Thus, if the market interest rate increases by 1%, then the bond price will decrease by 11.34% (since there is an inverse relationship between the bond rate and interest rates).
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