6  Interest Accumulation

6.1 Accumulation Functions

Definition: The accumulation function, \(a(t)\), is the accumulated value of an investment of \(\$1\) that is made at time \(0\). Suppose that a deposit is made at time \(t\) and accumulated to time \((t+k)\). The ratio of the accumulated value of the deposit to its initial value is equal ot the ratio of the accumulation function’s values at those times:

\[\frac{AV_{t+k}}{AV_t} = \frac{a(t+k)}{a(t)}\]

Example: The current rate of interest quoted by a bank on its savings account is 9% per annum (per year), with interest credited annually. Smith opens an account with a deposit of 1000. Assuming that there are no transactions on the account other than the annual crediting of interest, determine the account balance just after interest is credited at the end of 3 years and the accumulation function \(a(t)\)

Solution: After one year the interest credited will be \(1000\times.09 = 90\), resulting in a balance (with interest) of \(1000+1000 \times.09 = 1000(1.09) = 1090\). It is standard practice that this balance is reinvested and earns interest in the second year, producing an interest amount of \(1090\times.09 = 98.10\), and a total balance of \(1090 + 1090 \times.09 = 1090(1.09) = 1000(1.09)^2 = 1188.10\). The accumulation function \(a(t)\) can be determined from: \[a(t) = \frac{AV_t}{PV_0} = \frac{1000(1.09)^t}{1000}=1.09^t\] ### Accumulation Function as a function of Force of Interest

In place of the current/accumulated value in the formula for force of Interest, we can also use the accumulation function. Conversely, we can also express the force of interest in terms of the accumulation function:

\[a(t) = \int_0^t \delta_s ds \iff \delta_t = \frac{a'(t)}{a(t)}\]