1. Convexity
As we saw in the prior video,
2. Convexity measure
As we saw in the prior video, duration does a poor job when the change in yield is large. This is because duration does not capture the "convexity" or "curvature" of the relationship between bond prices and yields. As such, we need to calculate an adjustment to duration's estimate using what is called the "convexity measure."
3. Calculating the convexity measure
As with the full duration formula, the full convexity formula is a bit too complicated to explore in this course.
We thus again use an alternative used in practice called the "Approximate Convexity" formula. In the numerator, we have the bond price when yields go down plus the bond price when yields go up minus 2 times the current price. The denominator then is the current bond price times the expected change in yield squared.
4. Estimating effect on price
Once you calculate the convexity measure, you can once again estimate the percentage price change based on convexity as 0.5 times the convexity measure multiplied by the change in yield squared.
As with duration, the Dollar Change in price is then the percentage change calculated above multiplied by the bond's current price.
Note that the Convexity Measure is invariant to the direction of the yield change because you SQUARE the change in yield.
5. How do you use these formulas?
How then do we calculate the effect of convexity?
We can continue our example from the duration section. As you can see, applying the Convexity formula to our example yields a convexity of 77.57.
6. How do you use these formulas?
With this value in mind, we can then calculate a Percentage Change, which equals 0.38%. This is equivalent to a Dollar Change of 42 cents.
7. Effect of duration and convexity
So now we bring the effects of both duration and convexity together.
Recall that in our example, the Duration Dollar Change is NEGATIVE $8.53. The Convexity Measure adjustment adds another 42 cents to the mix,
8. Effect of duration and convexity
which yields a total effect of NEGATIVE $8.11.
Remember that the current price of the bond was $108.11, so the estimated price if yields go up by 1% is $100.
Now, how do we know this is a good estimate? Well, a property of bond prices is that if the coupon rate equals the yield, the bond price is equal to its par value. So recall that our bond has a 5% coupon and the initial yield was 4%. So a 1% increase results in a yield of 5%. Our bond has 5% coupon and it now has a 5% yield, so the bond's price will equal its par value of $100. This is exactly the estimate we get from the above!
9. Convexity in a chart
Before moving on, it's helpful to visualize the effect of convexity on our calculations. We now graphically show how convexity improves the estimate of the bond price compared to duration alone.
In this plot, the blue line is the estimate based on duration plus convexity. As we can see, our estimate including convexity is much closer to the black line - the full valuation bond price line - EVEN for large changes in yields. Although there could still be differences, the estimate is much closer.
10. Let's practice!
Now, it's time to practice what you learned.