Negative convexity is a characteristic of a loan which is best pictured by a notably unusual pattern in a yield curve. This characteristic reverses the normal situation that the longer a debt has to run, the higher the interest rate will be. Mortgage-backed securities are one of the most common forms of debts which can have negative convexity.
The yield curve is a graph which compares the length of time a debt has left before repayment, known as the time to maturity, with the prevailing interest rates. One example of this relationship's usual form comes with savings in a bank, which are effectively a loan from the customer to the bank. The bank will usually pay a higher interest rate for a savings account where the money must be left in the bank for a fixed period of time than it would pay for a checking account where the money can be taken out quickly. Similarly, a company borrowing money by issuing a bond will usually pay a higher annual interest rate if the bond has longer to run. In both examples, the higher rate is effectively the price for the guarantee of having the money for longer.
This relationship means that in most cases a yield curve graph shows a convex curve, meaning that the interest rate slopes upwards before leveling off. That's because, for example, the difference between a 1-year and 2-year loan is much more significant than between a 24-year and 25-year loan. In situations with negative convexity, the curve is partially or entirely concave. That is to say that at some points the interest rate slopes downwards as the time to maturity increases.
When looking at negative convexity for bonds, economists and investors normally approach the issue from the other perspective. Rather than looking at how the duration of a loan affects interest rates, they look at how interest rates affect the duration of a loan. It's generally believed that the more concave a yield curve is, the less sensitive the price people will pay for bonds is to changes in interest rates.
One common area where a debt can have negative convexity is in callable bonds. These are bonds where the issuing company, which is effectively the borrower, has the right to repay the bond before the agreed maturity date. If interest rates fall, the company may find it is better off taking out a new loan at a lower rate and using that money to repay the bond early.
Another area with negative convexity is mortgage-backed securities. This is because the mortgages themselves are often based on variable mortgage rates. When interest rates fall, homeowners become more likely to repay the loan quicker and pay it off in full earlier. This means that the cut in interest rates has shortened the duration of the loan. Owners of mortgage-backed securities will often aim to protect against this variation by buying or selling long-term assets such as treasury bonds which can't be repaid early.