# The Principle of Convexity between Bond Price and Yield

A bond is a debt security with fixed rules. The issuer promises the bond holder to pay interest upon the bond at the coupon rate, and to repay the principal when the bond reaches maturity. Thus, the issuer gives the bond holder a creditor stake in the company which has issued the bond.

The bond price is different from the bond yield. The bond yield to maturity (YTM) is the bond’s annual interest, added to the difference between the bond’s purchase price and the bond’s face value. If the bond’s purchase price was below the face value of the bond, the difference is positive.

The bond purchase price can be the same as the face value of the bond. However, bonds are often purchased on the secondary market when they are partway to maturity. In this case, the market assigns a present value to the bond.

The present value of a bond is based on the bond’s estimated future value, discounted according to distance from maturity. In a simple case with no allowance for credit or other risks, a bond purchased on the secondary market prior to maturity would be discounted solely based on the expected future loss in value due to inflation and the amount of interest yet to be received.

The inflationary link is why bond prices generally rise whenever interest rates fall and vice versa, although there are exceptions. This is not a linear function. Duration measures the change in bond prices relative to interest rate changes, which is a measure of price sensitivity.

Convexity is used to measure the rate of curvature, which is a measure of a bond’s duration sensitivity. In calculus terms, duration is the 1st derivative of the price sensitivity relative to interest rate changes, while convexity is the 2nd derivative.

The higher the convexity, the more sensitive bond price is to decreases in interest rates, but at the same time, the less sensitive bond price is to increasing rates. Lower convexity bonds are less sensitive to interest rate decreases, but more sensitive to interest rate increases. Knowing the convexity of a bond can give an idea of the spread of future cash-flows.

The highest price sensitivities are usually found in zero-coupon bonds, which do not derive any of their yield from future interest payments. The lowest price sensitivities are usually found in amortizing bonds.

Convexity can be positive or negative. Bonds without call features or other embedded options always have positive convexity.

Negative convexity is possible only with bonds which have embedded options. For example, an issuer can choose to redeem a bond before its maturity date. These kinds of bonds are said to have “call” features. Bonds with negative convexity will usually fall in price when interest rates fall, because the issuer is likely to call in old, high-interest bonds early and issue new, lower-interest bonds to replace them.