4 Force of Interest
4.1 Derivation of the Force of Interest
Suppose that the accumulated value of an investment at time \(t\) is \(AV_t\), where time is measured in years.
- The amount of interest earned by the investment in the \(\frac{1}{m}\)-year period from time \(t\) to time \(t + \frac{1}{m}\) is \(AV{t+\frac{1}{m}}- AV_t\), and
- The \(\frac{1}{m}\)-year interest rate for that period is \(\frac{AV_{t+\frac{1}{m}}- AV_t}{AV_t}\).
- The nominal annual interest rate compounded m-thly (\(i^{(m)}\)) is \(m\times \frac{AV_{t+\frac{1}{m}}- AV_t}{AV_t}\)
If \(m\) is increased, the time interval \([t, t+\frac{1}{m}]\) decreases, and we are focusing more and more closely on the investment performance during an interval of time immediately following time \(t\). Taking the limit as \(m\to \infty\) results in:
\[ i^{(\infty)} = \lim_{m\to \infty} m\times \frac{AV_{t+\frac{1}{m}}- AV_t}{AV_t} = \frac{1}{AV_t}\lim_{h\to 0} \frac{AV_{t+h}- AV_t}{h} = \frac{1}{AV_t} \frac{d}{dt} (AV_t) \]
4.2 Force of Interest for Compound Interest
Definition: The force of interest is defined as the instantaneous change in the accumulated value per unit of the accumulated value. We call the force of interest \(\delta\):
\[\delta = \frac{\frac{d(AV_t)}{dt}}{AV_t} = i^{(\infty)}\]
4.3 The Force of Interest and Equivalent Rates
4.3.1 Converting Between Different Nominal Rates
The nominal interest rates and the nominal discount rates are related as follows:
\[e^\delta = 1 + i = \bigg(1+\frac{i^{(m)}}{m}\bigg)^m = \bigg(1-\frac{d^{(p)}}{p}\bigg)^{-p} = (1-d)^{-1}\] We can use the force of interest to find an accumulated value: \[AV_t = PV_0 \times (1+i)^t= PV_0 \times e^{\delta t}\] Where \(i\) is the equivalent effective rate of interest.
4.3.2 Force of Interest as a Continuously Compounded Rate
By the derivation above, we see that for a fixed nominal interest rate, the force of interest increases as the compounding frequency increases. \[\delta = \lim_{m\to\infty}i^{(m)}\]
Example: \(\$100\) is lent for 5 years and repaid in a single payment at the end of the 5 years. Calculate the amount of the payment and the equivalent force of interest if the interest rate is:
- an annual efective interest rate of \(7\%\).
- compounded twice per year and \(i^{(2)} = 7\%\).
- convertible monthly and \(i^{(12)} = 7\%\).
- the force of interest and \(\delta = 7\%\).
Solution:
- \(AV_5 = 100(1.07)^5 = 140.26\) and \(\delta = \ln(1.07) = 0.06766\)
- \(AV_5 = 100(1 + 0.07 / 2)^{2 \times 5} = 141.06\) and \(\delta = 2\ln(1+0.07/2) = 0.06880\)
- \(AV_5 = 100(1 + 0.07/12)^{12\times 5} = 141.76\) and \(\delta = 12\ln(1+0.07/12)=0.06980\)
- \(AV_5 = 100\times e^{0.07\times 5} = 141.91\) and \(\delta = 0.07\)