Duration is an estimate of a financial instrument’s price sensitivity in response to changes in interest rates. It can be defined as the net present value of weighted average cash flows and is measured in years. For example, the duration of a €1,000 8-year bond with a fixed annual coupon of 7% can be calculated by using the following formula:
\(Duration=\frac{\sum\limits_{t=1}^n\:t\frac{CF_t}{1 \: + \: i\:^t}}{\sum\limits_{t=1}^n\:\frac{CF_t}{1 \: + \: i\:^t}}\)
Where:
- t = time to cash flow
- CFt = cash flow (interest plus principal) at time t
- i = interest rate
- n = years to maturity
Duration can be explained as the sum of the net present values (PV) of all individual cash flows weighted by the time until the payment is made, divided by the net present value of all individual cash flows (PV).
Bearing in mind that the present value of a cash flow is calculated as
\(pv=\frac{CF_t}{1\:+\:i^t}\)
the duration formula can be simplified to:
\(Duration=\frac{∑ \:PVt}{∑ \:PV}\)
The given example of a 8-year €1,000-bond with a 7% annual coupon will result in:
\(Duration=\frac{65.42\:×\:1\:+\:61.14\:×\:2\:+\:57.14\:×\:3\:+\:53.40\:×\:4\:+\:49.91\:
×\:5\:+\:46.64\:×\:6\:+\:43.59\:×\:7\:+\:622.75\:×\:8}{65.42\:+\:61.14\:+\:57.14\:+\:53.40\:+\:49.91\:+\:46.64\:+\:43.59\:+\:622.75}=\frac{6389.24}{1000}=6.389 \: years\)
In order to calculate how much the price of the bond changes when interest rates change, it is possible to use modified duration.
\(\%\Delta P=–\:DUR\:\frac{\Delta i}{1\:+\:\frac{i}{n}}\)
Where:
- %∆P= price change in percent
- DUR= duration
- ∆i= change in interest rate
- i= interest rate
- n= number of coupon payments per year
If interest rates change by 2%, this would result in the following price change for the bond used in the above example:
\({-6.389}\times\frac{0.02}{1\:+\:{\frac{0.07}{1}}}=-\:11.94\:\%\)
A 2% price change in interest rates will change the price of the bond by 11.94%.
