Duration convexity and key rate sensitivities

Definition Consider fixed payments $C_1, \dots, C_n$ at times $0 < t_1 < \dots < t_n$ in years, with no default and no dependence of the payments on rates, and the price-yield function under annual compounding

for $y > -1$, as in Bond cash flows and yield mathematics. At a yield $y$ with $V(y) > 0$, the present-value weights and the Macaulay duration are

measured in years. The weights always add up to $1$. If every $C_k \ge 0$ and at least one is positive, the weights are nonnegative, so $D_{\text{Mac}}$ is a weighted average of payment times and lies between the earliest and the latest date with a positive payment. It equals $T$ for a zero-coupon bond maturing at $T$. If some payments are negative, some weights are negative, and $D_{\text{Mac}}$ can lie outside the range of payment dates.

Definition The modified duration is the negative logarithmic derivative of the price in the yield,

The second equality follows from $V'(y) = -(1+y)^{-1} \sum_k t_k C_k (1+y)^{-t_k}$ under annual compounding. With $m$ compounding periods per year, $V(y) = \sum_k C_k (1 + y/m)^{-m t_k}$ and $D_{\text{mod}} = D_{\text{Mac}}/(1 + y/m)$, with the weights of that convention. Under continuous compounding, $V(y) = \sum_k C_k e^{-y t_k}$ and the two durations coincide. Modified duration is the relative price change per unit of decimal yield, to first order, and the DV01 of a long position is

in currency units, positive for positive payments. After a rise of $1$ bp the price changes by approximately $-\operatorname{DV01}$. These formulas assume that the payments do not change when the yield changes. A callable bond, whose issuer may repay early when rates fall, needs an effective duration computed from a model of that decision, which this lesson does not build.

Definition Under annual compounding, the convexity of fixed payments is the normalized second derivative

measured in years squared. If every $C_k \ge 0$ and at least one is positive, then $V'' > 0$, so the price-yield function is strictly convex on $(-1, \infty)$. Under continuous compounding the convexity is $\sum_k t_k^2 w_k$ with the weights of that convention. For a zero-coupon bond maturing at $T$, annual compounding gives $D_{\text{mod}} = T/(1+y)$ and $\mathcal{C} = T(T+1)/(1+y)^2$, while continuous compounding gives duration $T$ and convexity $T^2$. Each pair is correct in its own rate coordinate, and the change of variables $e^{z} = 1 + y$ connects them.

Definition For fixed payments the function $V$ is infinitely differentiable on $(-1, \infty)$. For a yield change $\Delta y$ with $y + \Delta y > -1$, Taylor’s theorem with the Lagrange remainder gives

with $R_3 = \tfrac16 V'''(\xi)\,(\Delta y)^3$ for some $\xi$ between $y$ and $y + \Delta y$. Dividing by $V(y)$ gives the second-order price approximation

whose error is $R_3/V(y)$. If every $C_k \ge 0$ and at least one is positive, then $V''' < 0$, so the remainder is negative after a rise in the yield and positive after a fall. The quadratic estimate then overstates the new price after a rise and understates it after a fall, while the first-order estimate understates the new price in both directions because $V$ is strictly convex. The approximation is local, and for a large move the exact repriced value is the scenario answer.

Definition A parallel shift moves every rate of a chosen rate coordinate by the same amount $\Delta$. For one bond described by its own yield it is the move from $y$ to $y + \Delta$, which is a parallel shift of the flat curve at level $y$. For a zero curve with continuously compounded zero rates $z(t)$ it is the move to $z(t) + \Delta$ at every maturity, which multiplies each discount factor $P(0, t)$ by $e^{-\Delta t}$. Durations and convexities are derivatives along such a direction, so they are defined only once the coordinate is named. A shift that is parallel in one coordinate is generally not parallel in another. Since $1 + y_a(t) = e^{z(t)}$ for the annually compounded zero rate $y_a(t)$, a shift of every $z(t)$ by $\Delta$ moves $y_a(t)$ by $(1 + y_a(t))(e^{\Delta} - 1)$, an amount that depends on $t$ unless the curve is flat.

Definition Represent the curve by continuously compounded zero rates $z_1, \dots, z_m$ at chosen nodes, and let the zero rate $z(t)$ at each payment date be a fixed differentiable function of the node rates, for example through linear interpolation between nodes. The price is $V = \sum_k C_k e^{-z(t_k)\, t_k}$, and the key rate duration of node $j$ is

the relative price sensitivity to a bump of node $j$ alone with every other node held fixed. By the chain rule,

When the nodes are the payment dates themselves and each node rate moves alone, only one term is left and $\operatorname{KRD}_j = t_j w_j$, with the continuous-compounding weights $w_j = C_j e^{-z_j t_j}/V$. The key rate durations add up to the duration for a parallel shift of the continuous zero curve, $\sum_k t_k w_k$, whenever $\sum_j \partial z(t_k)/\partial z_j = 1$ at every payment date, as with linear interpolation between nodes. That sum need not equal a modified duration in annual yield. A key rate duration is incomplete without its nodes, its rate coordinate and its bump rule.