Negative Convexity: Review, Real World Scenarios and Wealth Management

Bonds and financial markets:

Bonds are debt instruments that pay some periodic interest (coupon) and a par value (principal) at maturity. When a company wishes to raise funds for its operations or expansion, it will issue bonds to institutional or individual investors. The bonds are generally issued at a par of $1,000 and carry a periodic (mostly semi-annual) interest payment or coupon. The time period of the bond is decided at the time of issue and after its completion, the investors receive the par value. During this time period, investors also receive coupon.

Since the bonds are generally issued at par, the coupon has to match the prevailing interest rate. The prevailing interest rate depends majorly on the risk of the issuing company or entity. Over the course of the bond’s life, the risk might increase or decrease. Therefore, the interest required by the investors also increases and rises. However, since the coupon is usually fixed, adjustment in required return happens by change in the price of the bond. When risk decreases, the required return also decrease. But since the coupon are higher, to adjust for the return price increases. In case of risk increasing, investors require higher return but since coupon cannot increase, the price of bond decreases to adjust for a higher risk.

Mortgage is a debt on properties (usually homes), where the homeowner keeps the property as a collateral. The homeowner is the debtor whereas a bank or the financial institution is the lender. However, there is a limit to the amount a financial institution can lend. Therefore, some institutions sell off their mortgage/loans to other larger institutions. These larger institutions would club several mortgage loans in one pool and sell them as securities to the investors. Investors receive money from the repayment of mortgage loans. These securities are known as mortgage-backed securities or MBS. Due to the multi-layered clubbing of MBSs, they exhibit some characteristics of debt/bond more than other. One of these characteristics is negative convexity. Negative convexity MBS can be risky can also yield good returns.

Bond Duration

Duration is the percentage change in the price of bond for a unit percentage change in the interest rate. Using economic analogy, duration is the elasticity of the bond price due to the change in interest rates. This is the Modified Duration. Another way to calculate duration is by calculating the weighted average time period of the cash flows received by the bond. This is called Macaulay Duration and usually differs from the modified duration.

Bond Convexity

Bond convexity is the rate of change in the duration of the bond. In most cases, the price of the bond does not change equally for the same rise and fall in the interest rates. This can be attributed to the convexity. An example will clear this up.

A bond has the following characteristics:

• Coupon = $50

• Periods=10

• Par Value = $1,000

• Rate = 5%

• Price = $1,000

Price if the rate increases by 1%:

• Coupon = $50

• Periods=10

• Par Value = $1,000

• Rate = 6%

• Price = $926.40

Price if the rate decreases by 1%:

• Coupon = $50

• Periods=10

• Par Value = $1,000

• Rate = 4%

• Price = $1,081.11

We see that the change in price for a 1% increase is $73.60 ($1,000 - $926.40) whereas the change in price for a 1% decrease is $81.11 ($1,081.11 - $1,000).

Now for this example, average percentage change in the price of bond is nearly (($73.60+$81.11)/2)/$1,000 which comes out to be nearly 7.74%. This is an approximate measure of the duration of the bond and will differ at different price points as well as the percentage increase in prices. The convexity here is that that change is not same for increase and decrease in the market interest rate.