Duration

The duration of a bond indicates the mean time to payment of its cash flows.

The higher the bond’s yield the shorter is duration.

In the following P, y, f, t, and CF(t) are price of bond, yield rate, frequency of payments per year, and time-weighted cash flows (present value).

Duration:

measures the bond price volatility.

$D = \frac{P(-\Delta y) - P(+\Delta y)}{2 P_0 \Delta y}$

The yield is changed by a given amount $\Delta y$ and the price of the bond evaluated.

Convexity:

2nd derivative of price to yield.

$C = \frac{P(-\Delta y) - P(+\Delta y) - 2 P_0}{P_0 {\Delta y}^2 }$

DV01:

$DV01 = | P(YTM_{0}) - P(YTM_{+1bp}) |$

The dollar value of a basis point is the difference of the price at given yield to maturity and the price at changed yield to maturity by 1 bp.

yield based method:

$DV01 = \frac{1}{10000} * \frac{1}{1+y/f} * \sum(PV(CF(t)) * t)$

The DV01 indicates the percentage change in bond price following a 1 basis point (1 bp = 1/10000) change in yield.

Modified duration:

$D^{*} = \frac{1}{P} * \frac{1}{1+y/f} * \sum(PV(CF(t)) * t)$

The modified duration can be interpreted as the approximate percentage change of a bond’s price from a change in yield. Or, the elasticity of bond price to yield.

Macaulay duration:

$D^{M} = (1+\frac{y}{f}) * D^{*}$

The Macaulay duration indicates the weighted average term to maturity of a bond’s cash flow.