27  Convexity

27.1 Convexity of a General Cash Flow

27.1.1 Modified Convexity

The modified convexity of a series of cashflows measures the curvature of an instrument’s or a portfolio’s price-yield function: the higher the curvature, the higher the convexity and the higher the adjustment we need to make when we estimate the change in present value using the modified duration.

For the series of cash flows \(C_1, ..., C_n\) at time \(1,...,n\), the modified convexity is defined to be: \[C_{mod}(i) = \frac{\frac{d^2P(i)}{di^2}}{P(i)} = \frac{\sum_{t>0} t(t+1)v^{t+2}\times C_t }{P(i)}\]

27.1.2 Macaulay Convexity

For the series of cash flows \(C_1, ..., C_n\) at time \(1,...,n\), the Macaulay convexity is defined to be: \[C_{mac}(i) = \frac{\frac{d^2P(i)}{d\delta^2}}{P(i)} = \frac{\sum_{t>0} t^2 v^{t}\times C_t }{P(i)}\]

Relationship between \(C_{mod}\) and \(C_{mac}\): \[C_{mod} = v^2(C_{mac} + D_{mac})\] Note that, when the text refers to convexity, modified convexity is implied.

27.2 Second-Order Approximations

27.2.1 Second-Order Modified Approximation to Present Value

Modified convexity can be used to better approximate the change in present value for a change in interest rate. The second-order approximation is:

\[P(i) \approx P(i_0) \bigg( 1 - (i-i_0)\times D_{mod}(i_0) +\frac{(i-i_0)^2}{2}\times C_{mod}(i_0)\bigg)\] ### Second-Order Macaulay Approximation to Present Value

The main application of Macaulay convexity is for the Macaulay approximation of the change in present value:

\[P(i) \approx P(i_0) \bigg(\frac{1+i_0}{1+i}\bigg)^T \times \bigg(1 + \bigg(\frac{i-i_0}{1+i_0}\bigg)^2 \frac{C_{mac}(i_0)-T^2}{2}\bigg)\]

Let \(T = D_{mac}(i_0)\) and \(V(i) = P(i) \times (1+i)^{T}\). To derive the above formula, we will use the second-order Taylor expansion of \(V(i)\).

We calculate the first derivative of \(V(i)\): \[V'(i) = P(i)\times T \times (1+i)^{T-1} + P'(i) \times (1+i)^T\] It can be proved that \(V'(i_0)=0\); we used this result in the previous section.

The second derivative of \(V(i)\) is:

\[\begin{align}V''(i) &= P(i) \times T(T-1)\times (1+i)^{T-2} + 2 \times P'(i)\times T \times (1+i)^{T-1} + P''(i)\times (1+i)^T \\ &= P(i) \times (1+i)^{T-2} \bigg(T(T-1)-2D_{mod}(i)\times T \times (1+i) + C_{mod}(i) \times (1+i)^2\bigg) \\ &=P(i) \times (1+i)^{T-2} \bigg(T(T-1)-2D_{mac}(i)\times T + C_{mac}(i) + D_{mac}(i)\bigg)\end{align}\] In particular, for \(i=i_0\),

\[V''(i_0) = P(i_0) \times (1+i_0)^{T-2} (C_{mac}(i_0) - T^2 )\] Substituting to the Taylor’s expansion of \(V(i)\)

\[\begin{align}V(i) &\approx V(i_0) + (i-i_0)\times V'(i_0) + \frac{(i-i_0)^2}{2}\times V''(i_0) \\ P(i)(1+i)^T &\approx P(i_0) (1+i_0)^T + 0 + \frac{(i-i_0)^2}{2}\times \bigg(P(i_0) \times (1+i_0)^{T-2} (C_{mac}(i_0) - T^2 )\bigg) \end{align}\] we obtain the above formula.

27.3 Comparison Between Macaulay and Modified Approximations

Proposition: Approximation using Macaulay duration yields a lower error rate than using modified duration.

Proof: We denote: \[\begin{align} T &= D_{mac}(i_0) \\ F_{mod} &= P(i_0) \bigg( 1 - \frac{i-i_0}{1+i_0}T\bigg) \\ F_{mac} &= P(i_0) \bigg(\frac{1+i}{1+i_0}\bigg)^{-T}\end{align}\] so that \(F_{mod}\) and \(F_{mac}\) are the first-order modified and Macaulay approximations to \(P(i)\).

Lemma: \(F_{mod}(i) \le F_{mac}(i)\).
We have: \[\begin{align}F_{mac}(i) &= F_{mac}(i_0) + (i-i_0) F'_{mac}(i_0) + \frac{(i-i_0)^2}{2}F''_{mac}(i_0) \\ &\ge F_{mac}(i_0) + (i-i_0) F'_{mac}(i_0) \\ &= P(i_0) + (i-i_0)\times \frac{-T \times P(i_0)}{1+i_0} \\ &= F_{mod}(i) \end{align}\] since \(F''_{mac}(i_0) = T(T+1) P(i_0) \big(\frac{1+i}{1+i_0}\big)^{-T-2}\frac{1}{(1+i_0)^2} > 0\)

Let us define the present value, Macaulay duration, convexity and approximation functions in terms of \(\delta\) for simplicity:

\[\begin{align}P_\infty(\delta) &= P(i) (e^\delta-1) \\ D_\infty(\delta) &= D_{mac}(i) (e^\delta-1) \\ C_\infty(\delta) &= C_{mac}(i) (e^\delta-1) \\ F_\infty(\delta) &= F_{mac}(i) (e^\delta-1) \\ \end{align}\]

Lemma: \(C_\infty - D_\infty^2 \ge 0\)

We set \[q_t = \frac{C_t \times e^{-t\delta}}{P_\infty(\delta)}\] Note that, \(\sum_{t>0} q_t = 1\) and \[\begin{align}D_\infty(\delta) &= \sum_{t} (t \times q_t) \\ C_\infty (\delta) &= \sum_{t} (t^2 \times q_t)\end{align}\]

Then, \[\begin{align}C - D^2 &= C - 2\times D \times D + D^2 \times 1 \\ &= \sum_{t} (t^2 \times q_t) - 2\times D \times \sum_{t} (t \times q_t) + D^2 \times \sum_t q_t\ \\ &= \sum_t \bigg(\big(t - D\big)^2 \times q_t\bigg) \ge 0\end{align}\]

Lemma: \(D'_\infty \le 0\)

Indeed, \[D_\infty = \frac{-P'_\infty}{P_\infty} \implies D'_\infty = \frac{-P_\infty P''_\infty + P'_\infty P'_\infty}{P_\infty ^2} = -C_\infty + D_\infty^2 \le 0\] Lemma: \(F_\infty(\delta) \le P_\infty(\delta)\)

Consider the Taylor expansion of \(V_\infty(\delta) = P_\infty(\delta) \times e^{T\delta}\) with remainder, there exists \(j\) between \(\delta\) and \(\delta_0\) such that: \[\begin{align}P_\infty(\delta) \times e^{T\delta} &= P_\infty(\delta_0 ) \times e^{T\delta_0} + (\delta - \delta_0)P_\infty(\delta) \times e^{T\delta} \times (T - D_\infty (j)) \\ &= P_\infty(\delta_0 ) \times e^{T\delta_0} + (\delta - \delta_0)P_\infty(\delta) \times e^{T\delta} \times (D_\infty(\delta_0) - D_\infty (j))\end{align}\]

If \(\delta \le j \le \delta_0\), then \(\delta - \delta_0 \le 0\) and \(D_\infty(\delta_0) - D_\infty (j)) \le 0\) since \(D'_\infty \le 0\). We have the similar for \(\delta \ge \delta_0\). Thus: \[(\delta - \delta_0) (D_\infty(\delta_0) - D_\infty (j)) \ge 0\] Since \[F_\infty(\delta) = P_\infty (\delta_0) \times e^{-T(\delta - \delta_0)} \le P_\infty (\delta)\] we now can show that: \[F_{mod}(i) \le F_{mac}(i) = F_\infty (\ln(1+i))\le P_\infty(\ln(1+i)) = P(i)\]