Research & Publications  
BacktoBasics 
The following article is reprinted from the Nov/Dec, 1998 issue of On the Edge, the Interactive Data Fixed Income Analytics bimonthly newsletter.
Back to Basics: ConvexityPositive, Negative or Both?
Teri Geske In a previous BacktoBasics article we discussed how to interpret convexity, including the fact that negative convexity indicates the presence and importance of embedded optionsı. After reading that column, one of our BondEdge clients suggested we do a followup article describing why the convexity of a callable bond or mortgagebacked security can change from negative to positive (or vice versa) as interest rates change. With the ongoing rally in interest rates we have seen this year, it seemed like an appropriate time to examine this issue, so with thanks to the client for the suggestion, we offer this month's column. When confronted with a question about fixed income securities, I always find it useful to keep in mind that the measures used to describe these securities, e.g. price, yieldtomaturity, duration, OAS, etc., ultimately depend upon the security's expected future cash flows, and convexity is no exception. Therefore, when we see a bond's convexity changing from positive to negative and wonder why this is so, we should think about what is happening to the security's expected future cash flows as interest rates change; therein lies the answer. With this in mind, let us review why bonds have this characteristic called convexity, and why it can be positive or negative or both. If we plot the percentage change in a bond's price under various interest rate shifts (e.g., +/ 300 bps) and connect these points, we will typically draw a curve, not a straight line. If we then draw a line that is tangent to this price curve at the point representing the current price and the curve sits on top of the tangent line, the bond has positive convexity; if the curve lies below the tangent line, the bond has negative convexity. Okay, but why does a price curve curve? Well, this is due in part to the nature of discounting cash flows to compute present value. The price of a bond today is, of course, equal to the present value of expected future cash flows, discounted at the appropriate interest rate(s). This discounting function involves an exponent, and plotting a series of points described by an exponential function creates a curve. "Convexity" is a term that describes the degree and type of curvature observed. For bonds with fixed cash flows (such as a noncallable, fixed rate bonds), the price curve will have some degree of positive convexity simply because of the discounting function. For callable bonds and mortgagebacked securities, convexity is often negative. This is an indication that the cash flows for the security are not fixed, but rather are expected to change adversely as interest rates change. With a callable bond, as interest rates rally it becomes more likely that the issuer will call the bond, thereby providing the investor with a set of cash flows to the call date that are worth less than the cash flows to the maturity date. This change in expected future cash flows limits the potential increase in the bond's price as rates rally, causing the bond's price curve to display negative convexity. If the call is deep inthemoney (i.e. virtually 100% certain to occur) and the call date is near, a further decline in rates may produce almost no increase in the bond's price and the price curve will be flat, i.e. with zero convexity. Prepayments on mortgages are like partial calls (at par). When rates decline, prepayments increase and the investor's principal is repaid sooner than originally expected. Faster prepayments reduce expected future coupon payments at a time when the coupon rate on the mortgage is increasingly attractive to the investor, and force reinvestment of a greater amount of principal at current (lower) market rates. When rates rise and prepayments slow down, the investor is forced to wait longer than originally expected to be repaid, and is earning a coupon rate that is now below market rates. Changes in prepayments limit a mortgagebacked security's potential price appreciation in a rally, and accelerates the price decline in a rising rate environment compared to a security with fixed (unchangeable) cash flows. This causes mortgagebacked securities to display negative convexity. So, how can a security have both positive and negative convexity? This can occur if expected future cash flows are fixed (stable) within a certain range of interest rates, but unstable under other rate environments. The following graph shows the price curve of a CMO PAC with an average life of 3.75 years and convexity of 0.50 (effective duration is 2.23). Notice the line tangent to the curve at the 0shift point  the curve is below the tangent line, i.e., it is negatively convex at that point. But at another point, e.g. at the "150bp" area, the tangent line lies below the curve; there, the bond displays a small amount of positive convexity. This is because of the security's cash flow profile, which can be summarized by looking its average life under rate shifts of +/ 300bps: The bond's average life of 3.75 declines to 3.33 if rates rally 100bps, but remain at 3.33 under any further decline in rates. In other words, the cash flows become fixed, and the process of discounting the fixed set of cash flows produces positive convexity. However, if rates rise 100bps the average life extends, and it continues to do so for +200bp and +300bp rate shifts, producing negative convexity as the bond's cash flows extend. We hope this discussion has been helpful in understanding why convexity can change from positive to negative, depending upon a security's expected future cash flows. If you have a suggestion for a future BacktoBasics column, we'd like to hear from you  please contact marketing at (310) 4799715 or via email, fia.marketing@interactivedata.com.  

