Monday, 14 November 2011

Bond Mathematics for Fixed Income Securities

Although the name is Fixed Income securities, there are different types of risks while investing in fixed income securities. The major type of risks associated with fixed income securities are credit risks, interest rate risks, yield curve risks, liquidity risks and basis risks.

The most common risks which an investor attaches to fixed income investments is credit risk. Having said that, there is one more risk which is as important as credit risk i.e. interest rate risk. Few people understand this risk. For example, a long term Government of India Security (GSec) may have zero credit risk (because the Government can literally “print notes” and pay back the loan), but it has one of the highest interest rate risks. Interest rate risk is directly related with the maturity of a security. I will cover two very important contributors of interest rate risks in this note – duration and convexity.


I have seen many investors confuse duration with maturity. However, they both are distinctly different. Maturity is simply when the fixed income security will mature and pay back the principal. For example, the maturity of a 10-year paper will be 10 years at the time of issue. On the other hand, duration is the time within which the investor receives back all the cash flows related to the security i.e. interest and principal. For example, if there is a 10-year maturity paper paying yearly coupon at the interest rate of 8.0% p.a. issued at par (Rs.100) will have the following cash flows, 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 108 which will be paid at the end of every year for the next 10 years till it matures.  The Rs.8 is the interest at 8.0% p.a. on Rs.100 par value. Kindly note that at the end of the 10th year the investor will receive Rs.108 i.e. Rs.100 of principal + Rs.8 of the 10th year’s interest. This example clearly shows that although the maturity of the security is after 10-years, the investor receives cash flows frequently at regular intervals much before the final maturity of the security. That brings me to the concept of duration. The duration of a bond is defined as the “weighted average term to maturity of a security’s cash flows”. Since the cash flows on a security are received piecemeal before the actual maturity of the security, the duration of all coupon paying bonds will be less than its maturity. And as a Zero coupon bond does not pay any interest during its life, its duration = maturity.

There are different forms of duration. The basic one is the Macaulay or unadjusted duration. The one which we use for our calculation is the adjusted or Modified Duration. I would not go into the mathematical formulae of computing these but will explain the concepts which are necessary for understanding interest rate risks associated with fixed income securities.

Duration is useful primarily as a measure of the sensitivity of a bond's market price to interest rate (i.e. yield) movements. It is approximately equal 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 by which the value of the bond will fall for a 1% per annum increase in market interest rate. So a 10-year bond with a duration of 7 years would fall approximately 7% in value if the interest rate increased by 1% per annum. In other words, duration is the elasticity of the bond's price with respect to interest rates.

The summary of Duration characteristics is as follows:

Ø       The duration of a zero coupon bond will equal to its term to maturity.
Ø       The duration of a coupon paying bond will always be less than its term to maturity.
Ø       There is an inverse relationship between coupon and duration. The higher the coupon of a bond the lesser its duration and vice versa. The logic is simple because the higher the coupon, the sooner will the cash flows accrue to the investor and hence the lesser the interest rate risk associated with future cash flows.
Ø       There is generally a positive relationship between duration and term to maturity. Note that the duration of a coupon bond increases at a decreasing rate with maturity and the shape of the duration / maturity curve will depend on the coupon and the yield-to-maturity (YTM) of the bond.
Ø       There is an inverse relationship between YTM and duration.
Ø       Sinking fund and call provisions can cause dramatic change in the duration of a bond.


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. Convexity deals with the curvature of the price / yield relationship or chart. 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.

Convexity also gives an idea of the spread of future cashflows. Just as the duration gives the discounted mean term, so convexity can be used to calculate the discounted standard deviation, say, of return.

Note that duration can be either negative or positive depending on the way the interest rates move but Convexity is always a positive feature of the bond. The exception to this rule is in the case of “callable bonds” where the convexity is a negative feature. By positive feature of convexity I mean that for a given change in interest rates and the modified duration of a bond, the change is price of the bond will be in favour of the investor. For example, because of the positive feature of convexity, when interest rates rise, the price of the bond will fall less than that indicated by the duration and when interest rates fall, the price of the bond will rise more than that indicated by the duration. This is because when we study the price / yield relationship of a coupon paying option free bond, the larger the increase in the YTM, the greater the magnitude of the error by which the modified duration will overestimate the bond’s price decline; the larger the decrease in the YTM, the greater the magnitude of the error by which the modified duration will underestimate the bond’s price rise.

As the YTM changes, the bond’s duration changes as well. Thus, modified duration is an accurate predictor of price change only for vanishing small changes in YTM.


Without making this note too long and complicated, the change in price of a fixed income security is duration times the change in yield i.e. for a security having a modified duration of 7 years, for every 1 % (100 bps) decrease in yield (interest rates), the price will go up by 1 * 7 = 7% and vice versa. Due to convexity of the bond, the gain will be little more than 7% in case of interest rate decline and the loss will be little less than 7% in case of interest rate increase.

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