Bond Price Sensitivity Basics
The price of a fixed coupon bond with face value $F$, coupon $C$, maturity $n$, and yield-to-maturity $y$ is:
\[P = \sum_{t=1}^n \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n} = C\,\ddot{a}_{\overline{n}|y} + F\,v_y^n\]The dollar value of a basis point (DV01) measures the price change per 1 bp (0.01%) change in yield:
\[\text{DV01} = -\frac{dP}{dy} \cdot 0.0001\]Macaulay Duration
Macaulay duration is the time-weighted average of cash flow present values, interpreted as the effective maturity of the bond:
\[D_{\text{Mac}} = \frac{\sum_{t=1}^n t \cdot PV(CF_t)}{P} = \frac{\sum_{t=1}^n t \cdot CF_t \cdot v^t}{P}\]where $v = 1/(1+y)$.
For a zero-coupon bond maturing at $T$: $D_{\text{Mac}} = T$ (all cash flow at maturity).
For a perpetuity with yield $y$: $D_{\text{Mac}} = (1+y)/y$.
Modified duration adjusts for the compounding convention:
\[D^* = \frac{D_{\text{Mac}}}{1+y}\]The first-order price sensitivity is:
\[\frac{dP}{dy} = -D^* \cdot P \implies \frac{\Delta P}{P} \approx -D^* \Delta y\]Portfolio duration: For a portfolio of bonds with values $P_i$ and durations $D_i^*$:
\[D_{\text{port}}^* = \frac{\sum_i P_i D_i^*}{\sum_i P_i}\]For a liability stream (annuity or pension): replace bond cash flows with liability cash flows in the Macaulay duration formula.
Convexity
Duration is only a first-order approximation. Convexity captures the curvature:
\[\text{Conv} = \frac{1}{P}\frac{d^2P}{dy^2} = \frac{\sum_{t=1}^n t(t+1) \cdot CF_t \cdot v^{t+2}}{P}\]The second-order price approximation is:
\[\frac{\Delta P}{P} \approx -D^*\,\Delta y + \tfrac{1}{2}\,\text{Conv}\,(\Delta y)^2\]Dollar convexity $= \text{Conv} \times P$, so the DV01 change per 1 bp shift is:
\[\Delta(\text{DV01}) \approx \tfrac{1}{2}\,\text{Conv}\,P\,(0.0001)\]Convexity is always positive for standard bonds (no embedded options), meaning the price rises more than linearly as yields fall — a benefit to holders. Callable bonds or mortgage-backed securities can exhibit negative convexity in some yield environments.
Immunization Theory
Redington immunization (1952) seeks a portfolio of assets that is insensitive to small parallel yield shifts. For assets (A) and liabilities (L) with the same present value:
Conditions for immunization:
- $PV(\text{Assets}) = PV(\text{Liabilities})$ (value matching)
- $D^_A = D^_L$ (duration matching)
- $\text{Conv}_A > \text{Conv}_L$ (convexity matching — actually exceeds)
Condition 3 ensures that for large yield shifts, assets outperform liabilities: the asset price curve is more convex than the liability price curve, so the surplus is a convex function of yield with a minimum at zero (the current position).
Proof sketch: Define surplus $V(\Delta y) = A(\Delta y) - L(\Delta y)$. Taylor expand:
\[V(\Delta y) = V(0) + V'(0)\Delta y + \tfrac{1}{2}V''(0)(\Delta y)^2 + \ldots\]| Conditions 1 and 2 give $V(0) = 0$ and $V’(0) = 0$. Condition 3 gives $V’‘(0) > 0$, so $V(\Delta y) > 0$ for small $ | \Delta y | $. |
Limitations: Immunization assumes parallel yield shifts; non-parallel shifts (twists, butterflies) require key rate durations.
Key Rate Durations
Key rate duration (KRD) at tenor $\tau$ measures price sensitivity to a localized shift in the yield curve at maturity $\tau$:
\[\text{KRD}_\tau = -\frac{1}{P}\frac{\partial P}{\partial r_\tau}\]where $r_\tau$ is the $\tau$-year spot rate and the shift is a hat-shaped bump of height 1 affecting only nearby maturities.
Portfolio KRDs (typically at 5-10 key maturities: 1, 2, 3, 5, 7, 10, 20, 30 years) allow bucket hedging: match each KRD of liabilities with assets at the corresponding maturity. This neutralizes non-parallel curve risk.
DV01 by key rate: $\text{KRD-DV01}\tau = \text{KRD}\tau \times P \times 0.0001$.
The full vector of KRDs hedges the portfolio against all curve shape changes that can be expressed as combinations of shifts at the key rate tenors.
Interest Rate Models for ALM
Vasicek model — short rate $r_t$ satisfies the Ornstein-Uhlenbeck SDE:
\[dr_t = \kappa(\theta - r_t)\,dt + \sigma\,dW_t\]Parameters: $\kappa$ (mean-reversion speed), $\theta$ (long-run mean), $\sigma$ (volatility). Under the risk-neutral measure $\tilde{Q}$:
\[P(t, T) = A(t,T)\,e^{-B(t,T)r_t}\] \[B(t,T) = \frac{1-e^{-\kappa(T-t)}}{\kappa}, \quad \ln A(t,T) = \left(\theta - \frac{\sigma^2}{2\kappa^2}\right)(B-T+t) - \frac{\sigma^2}{4\kappa}B^2\]Cox-Ingersoll-Ross (CIR) model — ensures positive rates:
\[dr_t = \kappa(\theta - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t\]Zero-coupon bond prices have the same affine form $P = A(t,T)e^{-B(t,T)r_t}$ with different coefficient functions. The Feller condition $2\kappa\theta > \sigma^2$ prevents rates hitting zero.
ALM application — liability duration under stochastic rates: The market value of a life annuity under Vasicek is:
\[V(r) = \sum_{t=1}^{\omega-x} {}_t p_x \cdot P(0, t; r)\]where $P(0, t; r) = A(0,t)e^{-B(0,t)r}$. The stochastic duration is:
\[D^{\text{stoch}} = \frac{\sum_{t} {}_t p_x \cdot B(0,t) \cdot P(0,t)}{\sum_{t} {}_t p_x \cdot P(0,t)}\]This differs from the deterministic duration because the discount factors are no longer $e^{-rt}$ — the mean reversion structure flattens long-dated sensitivities.
Cash Flow Matching vs Duration Matching
Cash flow matching (dedication) constructs an asset portfolio whose cash flows exactly replicate liability cash flows period by period:
\[\min_{\mathbf{x}} \sum_j c_j x_j \quad \text{subject to} \quad \sum_j CF_j(t)\,x_j \ge L(t) \; \forall t, \quad x_j \ge 0\]This is a linear program (no residual interest rate risk once matched) but requires a specific bond for each liability maturity — often impossible or expensive.
Duration matching relaxes this: any bond can be used, provided the portfolio duration matches the liability duration. It is cheaper and more liquid but leaves residual curve risk (it immunizes only against parallel shifts).
In practice, insurers use a hybrid: cash flow match the near-term liabilities (where specific bonds exist) and duration match the long end (where matching bonds are scarce), supplemented by interest rate swaps and inflation swaps to fine-tune the sensitivity profile.