Chapter 7 Interest Rate Risk

Durations

• Macaulay duration: The present-value-weighted average time to the cash flows

\[D_{mac}(\delta)=\frac{\sum_tt\cdot CF_t\cdot v^t}{\sum_t CF_t\cdot v^t}=\frac{\sum_tt\cdot CF_t\cdot v^t}{P(\delta)}=-\frac{P'(\delta)}{P(\delta)}\]

• Modified duration: The sensitivity of price to change in the interest rate

\[D_{mod}\left(i^{(m)}\right)=-\frac{P'\left(i^{(m)}\right)}{P\left(i^{(m)}\right)}=\frac{D_{mac}}{1+i^{(m)}/m}\]

\[\lim_{m\rightarrow\infty} D_{mod}\left(i^{(m)}\right)=D_{mac}(\delta)\]

• Effective duration for interest-sensitive cash flows

\[D_{eff}=\frac{P(i+\Delta i)-P(i-\Delta i)}{2\times\Delta i\times P(i)}\]

• The relative price changes with (small) changes in the nominal interest rate

\[\frac{\Delta P}{P\left(i^{(m)}\right)} \approx -\Delta i^{(m)}\times D_{mod}\left(i^{(m)}\right)\]

Convexity

• Macaulay convexity

\[C_{mac}=\frac{P''(\delta)}{P(\delta)}=\frac{\sum_tt^2\cdot CF_t\cdot v^t}{P(\delta)}\]

• Modified convexity

\[C_{mod}=\frac{P''(i^{(m)})}{P(i^{(m)})}=\frac{\sum_{t}t(t+1)v^{t+2}CF_t}{P(i^{(m)})}\]

• Effective convexity

\[C_{eff}=\frac{P(i+\Delta i)+P(i-\Delta i)-2P}{(\Delta i)^2P}\]

• The relative price changes with (small) changes in the nominal interest rate

\[\frac{\Delta P}{P(i^{(m)})} \approx - D_{mod}(i^{(m)})\times\Delta i^{(m)}+\frac{1}{2}C_{mod}(i^{(m)})\times(\Delta i^{(m)})^2\]

• The duration and convexity of a portfolio

\[D=\sum_{k=1}^n \frac{P_k}{P}D_k\]

\[C=\sum_{k=1}^n \frac{P_k}{P}C_k\]

Immunization

• Definition: immunization is a process of protecting a financial organization from changes in interest rates.

• Conditions for Redington immunization protecting against small changes in interest rates:

  1. \(P_A=P_L\)

  2. \(D_A=D_L\)

  3. \(C_A>C_L\).

• Full immunization protecting against any changes in interest rates:

  1. \(P_A=P_L\)

  2. \(D_A=D_L\)

  3. \(T_L\in (T_{A_1},T_{A_2})\).

• Cash flow matching: solve a system of simultaneous equations.