1. Dollar convexity and bond price prediction
Now let's see how convexity can be used in bond price prediction.
2. Dollar convexity
Just like dollar duration, we also have the concept of dollar convexity.
Convexity can be thought of as the percentage change in duration for a one percent change in yields.
If you are familiar with derivatives, convexity is the second derivative of bond prices with respect to yields, while duration is the first derivative.
Dollar convexity is the dollar change in duration for a one percent change in yields.
To find the dollar convexity of a bond we multiply the convexity of the bond by the bond price and then by zero-point-zero-one squared.
3. Dollar convexity example
Take a ten year bond paying a three percent annual coupon, with a five percent yield and face value of one hundred dollars. We want to find its dollar convexity.
First, we find the convexity of the bond in the usual way by finding the bond price, shifting yields up and down one percent, and repricing, then using our formula for the convexity of the bond.
Finally, to find the dollar convexity of the bond, we multiply the convexity of the bond by the bond price, and then by zero-point-zero-one squared, giving us sixty nine cents.
This means for a one percent change in yields, the dollar duration will change by sixty nine cents.
4. The convexity adjustment
The dollar convexity from the previous slide can be used to improve our estimates of bond price changes when yields change.
As dollar convexity measures the dollar amount our duration changes by for a one percent change in yields, we need to convert this into a quantity that predicts how bond prices change, not duration. This is known as the convexity adjustment.
The convexity adjustment is calculated as zero-point-five times dollar convexity times one hundred squared, times the change in yields squared.
5. Convexity adjustment example
Lets take the same bond from before and also calculate its convexity adjustment this time.
We begin by finding the convexity of the bond in the usual way and assign the result to convexity.
Next, we find the dollar convexity by multiplying the convexity by the bond price and then by zero-point-zero-one squared.
Finally, we find the convexity adjustment by multiplying one half by dollar convexity by one hundred squared, and then by the change in yields squared.
6. Combining duration and convexity
We now have all the tools we need to use both duration and convexity together to predict bond prices.
In the previous chapter, we multiplied the dollar duration of a bond by the change in yield to predict how bond prices would change, and saw that this is only an accurate estimate for small yield changes.
The convexity adjustment is what we add to our prediction of price changes from duration alone to improve our estimate of bond price changes.
Now our estimate of bond price changes takes into account both the duration and the convexity of the bond.
7. Duration and convexity example
Let's use our bond from earlier and use both duration and convexity to predict how its price will change.
We begin in the usual way, pricing up the bond at current yields, as well as yields one percent higher and lower.
Next, we find the duration and dollar duration of the bond.
Then we find the convexity, dollar convexity, and convexity adjustment.
Finally, we combine both the duration and convexity adjustment to get a prediction of our bond price change. This will be the amount we expect the bond's price to change for a 1% change in yields.
8. Let's practice!
Now let's practice using dollar convexity in bond price prediction.