0
$\begingroup$

I am having trouble understanding the convexity of bonds and the relationship among bonds with different convexities. Exactly what is convexity and what is a simple way to

For instance, how is it possible for two bonds with the same price and same duration to have different convexities? Is convexity independent of price and duration?

For instance, if two bonds are both $100 with a duration of 20, but Bond A has a convexity of 500 and Bond B has a convexity of 100, how will each price be affected if interest rates go up or down? Shouldn't in theory the price of the Bond with higher convexity always will be worth more than the Bond with lower convexity.

Or would Bond A be worth more than Bond B when interest rates go down because it has a higher convexity (vice versa, as in Bond A would be worth less than Bond B if interest rates go up)?

$\endgroup$

2 Answers 2

3
$\begingroup$

Assume we are using continuously compounding rates, and that discount factors are given by the ZCBs $P(0, t_i) = e^{-y_i \cdot t_i}$.

The price of a fixed bond is given by

$$B = \sum_1^n N \cdot \delta \cdot K \cdot e^{-y_i t_i} + N \cdot e^{-y_nt_n},$$ where $\delta = t_i-t_{i-1}$, and $K$ is the coupon.

Now bump all discount rates (yields) by $\epsilon$

$$B(\epsilon) = \sum_1^n N \cdot \delta \cdot K \cdot e^{-(y_i+\epsilon) t_i} + N \cdot e^{-(y_n+\epsilon)t_n},$$

perform a Taylor expansion around $\epsilon=0$: $$B(\epsilon) = B(0) + \frac{dB}{d\epsilon}(0) \cdot \epsilon + \frac{d^2B}{d\epsilon^2}(0) \cdot \epsilon^2 + \mathcal{O}(\epsilon^3).$$

Now since duration and convexity are defined as \begin{align} \text{duration}:&= -\frac{1}{B(0)}\frac{dB}{d \epsilon}(0), \\ \text{convex}:&= \frac{1}{B(0)}\frac{d^2 B}{d \epsilon^2}(0), \end{align}

we can rewrite the Taylor expansion as \begin{align} B(\epsilon) = B(0) \left(1 - \text{duration} \cdot \epsilon + \text{convexity} \cdot \epsilon^2 \right) + \mathcal{O}(\epsilon^3). \end{align}

Here we can see that with a positive duration and convexity, a positive change in rates $\epsilon>0$ decreases the bond price from the duration, and increase the bond price from the convexity.

Using your provided examples

\begin{align} B_A(\epsilon) &= 100 \left(1 - 20 \cdot \epsilon + 500 \cdot \epsilon^2 \right) + \mathcal{O}(\epsilon^3) \\ B_B(\epsilon) &= 100 \left(1 - 20 \cdot \epsilon + 100\cdot \epsilon^2 \right) + \mathcal{O}(\epsilon^3). \end{align} Which is explaining how the bonds behave with change in interest rates (up to second order).

$\endgroup$
1
$\begingroup$

Salomon, "Convexity Bias and the Yield Curve - Understanding The Yield Curve: Part 5" is a good reference. You can find it online.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.