!!!Considerations of the Investment in Bonds
In an introductory corporate finance course, you may learn that the relationship between the interest rate and price of a bond is inverse (convex). But, this relationship does not tell us the ”degree” of bond price change with respect to the ”given” interest rate change.
For the standard type of coupon bonds, the cash flows spread out until the maturity date. For this reason, we need to know the effective maturity of the bond on top of the price sensitivity in relation to the interest rate to effectively manage our portfolio.
The metric that helps bond portfolio managers gauge uncertainty in these two areas is the concept of duration. There are two popular duration metrics: Macaulay Duration and Modified Duration. The term ”duration” has a special meaning in the context of bonds. It is extensively used as a measure of how long it takes for the coupon payments and principal to pay the market price of the bond. It is also an important estimate of price sensitivity of bonds to the interest rate. Bonds with a higher duration are riskier than bonds with lower duration.
Frederick Macaulay, in 1938, came up with the effective maturity concept on the duration of a bond. Graphically, Macaulay Duration is the fulcrum point of a group of present values of cash flows generated by a bond. It is also interpreted as the weighted average number of years for the cash flow from the bonds to equal the market price of the bond. Therefore, as a bond’s maturity increases, duration increases. Call provisions, on the other hand, decreases the duration of bonds.
Modified duration is slightly different from the Macaulay Duration in that it expresses the change in the value of a bond in response to a change in interest rates. This formula is used to determine the effect that a percentage change in interest rates will have on the bond price.
!!!What is Convexity?
Now let’s move on to the concept of convexity. A bond’s convexity is the rate of change of its duration. Since the concept of duration applies to the point measurement of price sensitivity with respect to interest rate, the convexity is, in calculus terms, the second derivative of the bond’s price with respect to its yield.
Negative convexity exists when the shape of a bond’s yield curve is concave instead of convex. Mortgage-backed bonds and callable corporate bonds show this characteristic.
!!!Examples of Negative Convexity
The relationship between interest rates and price of bonds is inverse in most cases. We call this the convexity of bonds. If a bond has a __negative convexity__ the price will decrease as interest rates fall. For example, callable corporate bonds’ price fall as interest rates fall and the incentive for the issuer to call the bond at par increases. This is why we say that a callable bond has negative convexity with respect to interest rates.
Mortgage-backed securities (MBS) are another example of negative convexity since those MBS typically lose value when interest rates fall. As in the case of a bond with a call option, with MBS, a decrease in interest rates increases the number of borrowers that will repay their loans and refinance at the lower rate. The property of __negative convexity__ brings down the value of the bonds when interest rates head south.
!!!Lesson Summary
Despite the common inverse relationship of the interest rate and price of bonds, some bonds’ prices go down when the interest rates fall. This is called the __negative convexity__ of bonds. Mortgage-backed securities and corporate bonds with call options are common examples.