Convexity
Convexity remains a preoccupying phenomenon for bond investors. The best way to define the term is to explain its origins, a task that entails a brief primer in what are known as callable bonds. A callable bond permits a borrower to extinguish debt prior to a bond's stated maturity by paying all monies owed to the lender. Because most mortgages permit prepayment, mortgage-backed securities (MBS) tend to be callable, as do many but not all corporate bonds and some longer-dated Treasuries. The key point is that investors in callable bonds never know with certainty when they will get their money back: as interest rates rise, borrowers (e.g., homeowners) have less incentive to refinance, and prepayments fall. As prepayments fall, the average time it takes lenders to recover their money gets extended. As this average recovery time � often referred to as a portfolio's duration � gets extended, the portfolio's sensitivity to interest rate changes increases. When interest rates are rising rapidly, holders of convex portfolios suffer ever-larger capital losses with each incremental increase in prevailing interest rates. [1] The reason bond specialists use the term convexity to describe this phenomenon is because graphing the price of a convex bond under varying interest rate conditions produces a convex line.
Market Value of a Convex Bond under Changing Rate Scenarios
As can be seen, the pain that holders of convex portfolios sustain when rates rise is not offset by equivalent pleasure when rates fall: note that the capital loss resulting from an increase in rates from B to C is much larger than the capital gain resulting from an equivalent decrease in rates from B to A. [2] Falling rates induce borrowers to refinance, thus reducing the effective maturity or duration of a highly convex portfolio. As the graph demonstrates, each incremental reduction in prevailing rates produces ever-smaller capital gains. Given these ugly odds, why would astute investors ever hold highly convex bonds? For the same reason that such investors might buy the securities of a bankrupt company: because the price is sufficiently low, i.e., the expected return � taking all risks into account, including convexity � is sufficiently high. In graphical terms, the line depicting a compellingly cheap convex bond would fall higher up the vertical axis than the line depicting a more richly priced non-convex issue.
Endnotes
1. Bond prices fall when interest rates rise because the future interest and principal payments that bonds generate are worth less in today's dollars. Investors intent on holding bonds until maturity may choose not to realize losses caused by rising rates, but they are losses nonetheless.
2. A rate increase from B to C may appear larger than a rate decrease from B to A but it is not: point B exactly bisects points A and C on the horizontal axis.