LOS c: Describe positive convexity, negative convexity, and their relation to bond price and yield.

Negative convexity illustrates how the relationship between the price of a bond and market yields changes as the bond price rises and approaches the call price. The convex curve that we generally see for non-callable bonds bends backward to become concave (i.e., exhibit negative convexity) as the bond approaches the call price.

 

As yields increase, bond prices fall, the price curve gets flatter, and changes in yield have a smaller effect on bond prices. As yields decrease, bond prices rise, the price curve gets steeper, and changes in yield have a larger effect on bond prices. Thus, the price increase when interest rates decline must be greater than the price decrease when interest rates rise (for the same basis point change). Remember that this applies to percentage changes as well.

Since the issuer of a callable bond has an incentive to call the bond when interest rates are very low in order to get cheaper financing, this puts an upper limit on the bond price for low interest rates and thus introduces the concave relationship between yields and prices.

A convex price/yield graph has a larger increase in price as yield decreases than the decrease in price when yields increase.  This comes from the definition of a convex graph.

A convex price/yield graph has a larger increase in price as yield decreases than the decrease in price when yields increase.  This comes from the definition of a convex graph.

There are three features that determine the magnitude of duration:

(1) The lower the coupon, the greater the bond price volatility.
(2) The longer the term to maturity, the greater the price volatility.
(3) The lower the initial yield, the greater the price volatility.

The bond with the 30-year maturity will have a greater price impact than the 10-year maturity. The bond selling at the greatest discount will have a large price impact, a discount means that the coupon payments are low or the initial yield is low. So, the bond with the 30-year maturity and selling at 70 will have the greatest price volatility.

When bond prices go up faster than they go down, it is called positive convexity.

When bond prices go up faster than they go down, it is called positive convexity.

Yes, fixed income securities can have a negative security. The only type of fixed income security with a negative convexity will be callable bonds. 

All noncallable bonds exhibit the trait of being positively convex and callable bonds have a negative convexity.  Callable bonds have a negative convexity because once the yield falls below a certain point, as yields fall, prices will rise at a decreasing rate, thus giving the curve a negative convex shape.

Negative convexity occurs with bonds that have prepayment/call features. As interest rates fall, the borrower/issuer is more likely to repay/call the bond, which causes the bond’s price to approach a maximum. As such, the bond’s price increases at a decreasing rate as interest rates decrease.

Positive convexity refers to the principle that for a given change in market yields, bond price sensitivity is lowest when market yields are high and highest when market yields are low.

Although the statements that begin, the graph of a callable bond . . . and the price of a fixed-coupon bond . . . are true, they are not the best choices to describe positive convexity.

There are three features that determine the magnitude of the bond price volatility: 

    (1) The lower the coupon, the greater the bond price volatility. 

    (2) The longer the term to maturity, the greater the price volatility. 

    (3) The lower the initial yield, the greater the price volatility.

Since both of these bonds are the same with the exception of the term to maturity, the bond with the longer term to maturity will have a greater price volatility.  Since bond value has an inverse relationship with interest rates, when interest rates decrease bond value increases. 

Negative convexity is the idea that as interest rates decrease they get to a certain point where the value of certain bonds (bonds with negative convexity) will start to increase in value at a decreasing rate.  

Interest rate risk is the risk of having to reinvest at rates that are lower than what an investor is currently receiving.

Mortgage backed securities (MBS) may have negative convexity because when interest rates fall mortgage owners will refinance for lower rates, thus prepaying the outstanding principle and increasing the interest rate risk that investors of MBS may incur.

Callable bonds are similar to MBS because of the possibility that the principle is being returned to the investor sooner than expected if the bond is called causing a higher level of interest rate risk.

When the option expires, the prices at the lower end of the curve will become lower. This will make the curve less convex.

The convexity of bond prices means that bond prices as a function of interest rates approach a floor as interest rates rise.

Modified duration is a good approximation of price changes for an option-free bond only for relatively small changes in interest rates. As rate changes grow larger, the curvature of the bond price/yield relationship becomes more prevalent, meaning that a linear estimate of price changes will contain errors. The modified duration estimate is a linear estimate, as it assumes that the change is the same for each basis point change in required yield. The error in the estimate is due to the curvature of the actual price path. This is the degree of convexity. If we can generate a measure of this convexity, we can use this to improve our estimate of bond price changes.