9.23 Duration Formal Mathematical Formulation, Definitions, And Summary Effects
The following formula represents the manner in which we calculated Duration.
[latex]Mac. D=\sum \frac{PVCF_{t}\times p}{PV_{Bond}}[/latex]
The following formula represents an alternate, more mathematical manner in which Duration may be calculated. We will not cover this formula herein.
[latex]Mac. D = \frac{1+y}{y}-\frac{(1+y)+n(c-y)}{c[(1+y)^{n}-1]+y}[/latex]
where,
c = annual coupon (%)
n = term to maturity (years)
y = YTM (reinvestment rate)
Note: Unless otherwise noted, “D” refers to Macaulay Duration.
Duration Definitions:
- The weighted average life of the present value of a bond’s cash flows.
- The point at which the Duration Analog’s Fulcrum balances the seesaw.
- The point in time at which price- and reinvestment-risks exactly offset one another.
- Duration is the tangent-point of a straight line along the Price-Yield curve.
- More definitions and characteristics will come!
Summary Duration and Other Effects:
| Coupon Increases | Price Increases |
| ALCF Decreases | |
| Duration Decreases |
| YTM Increases | Price Decreases |
| Duration Decreases |
Remember:
- Think of a Zero-coupon bond as your base case.
- Price and Duration are positively related, i.e., relative to YTM.
