DEFINITION

Force of interest refers to a nominal interest rate or a discount rate compounded infinite number of times (or continuously) per time period.

Consider a nominal interest rate(or even a discount rate) compounded half-yearly and another rate compounded quarterly, another rate compounded monthly, compounded weekly, compounded daily, compounded every second and so on until you can imagine an interest rate that is compounded every smallest fraction of a second(continuously). This interest rate compounded continuously is the force of interest.

If \(i^{\left(p\right)}\) is the interest rate compounded \(p\) times a year, then the limit of \(i^{\left(p\right)}\) as p tends to infinity, would be the force of interest.

Mathematically,

\[Force\ of\ interest=\delta ={\mathop{\mathrm{lim}}_{p\to \infty } i^{\left(p\right)}\ }={\mathop{\mathrm{lim}}_{p\to \infty } d^{\left(p\right)}\ }\]

**FORMULA AND DERIVATION**

For an effective interest rate \(i\), if \(i^{\left(p\right)}\) is the corresponding nominal interest rate compounded \(p\) times per time period, and if we go on increasing the value of \(p\), \(i^{\left(p\right)}\) will tend to a particular limit. This limit is known as the force of interest, denoted by \(\delta \) (Greek letter ‘Delta’).

\[\delta ={\mathop{\mathrm{lim}}_{p\to \infty } i^{\left(p\right)}\ }\]

The relation between the interest rate \(i\) and the force of interest \(\delta \) can be derived by the following equation explaining the relation between effective and nominal rates.

\[1+i={\left(1+\frac{i^{\left(p\right)}}{p}\right)}^p\]

The RHS can be simplified using Euler’s rule, which states:

\[1+i={\left(1+\frac{i^{\left(p\right)}}{p}\right)}^p\]

\[\therefore 1+i={\mathop{\mathrm{lim}}_{p\to \infty } {\left(1+\frac{i^{\left(p\right)}}{p}\right)}^p\ }\]
\[1+i=e^{i^{\left(\infty \right)}}\]
\[1+i=e^{\delta }(i^{\left(\infty \right)}=\delta \ is\ defined\ above)\]
\[\therefore i=e^{\delta }-1\ and\ \delta ={\mathrm{ln} \left(1+i\right)\ }\]

Similarly, \({\mathop{\mathrm{lim}}_{p\to \infty } d^{\left(p\right)}\ }=\delta \) , where \(d\)is a discount rate. The relation between a discount rate and its corresponding force of interest can be defined as follows:-

\[1-d=e^{-\delta }\]
\[\therefore d=1-e^{-\delta }\ and\ \delta ={\mathrm{-}\mathrm{ln} \left(1-d\right)\ }\]

**EXAMPLES**

Example 1

If \(10\ \%\) p.a. is the annual effective rate, then the corresponding force of interest would be

\[\delta ={\mathrm{ln} \left(1+i\right)\ }\]
\[\ \ ={\mathrm{ln} \left(1+0.1\right)\ }={\mathrm{ln} \left(1.1\right)\ }\]
\[\ \ =0.09531=9.531\%p.a.\]

This means that an amount will accumulate to the same value at a rate of interest of \(10\ \%\) p.a. and at a force of interest of \(9.531\ \%\) p.a.

Example 2

Consider the force of interest obtained in the above example \(\left(\delta =9.531\ \%\ p.a\ \right)\) . An investor wishes to find out the corresponding discount rate.

\[d=1-e^{-\delta }\]
\[\ \ =1-e^{-0.09531}\]
\[\ \ =0.0909=9.09\%\ p.a.\]

This means that using a discount rate of \(9.09\ \%\) p.a. would be the same as using an interest rate of \(10\ \%\) p.a. or a force of interest of \(9.531\ \%\) p.a.