Financial Mathematics → Theorem: Accumulation Factor

**Accumulation Factor and Force of Interest Theorem.**

If δ(t) and A(t_{0},t) are continuous functions of t and t≥t_{0}, and the principle of consistency holds, then, for t_{0}≤t_{1}≤t_{2}

**Example 1.**

Assume that δ(t), the force of interest per unit time at time t, is given by:

(a) δ(t) = δ

(b) δ(t) = a+bt

Find formulae for the the accumulation of a unit investment from time t_{1} to time t_{2} in each case.

**Solution:**

**(a)** When δ(t) = δ

As we know that

So,

**(b)** When δ(t) = a+bt

As we know that

So,

**Example 2.**

A bank credits interest on deposit using accumulation factors based on a variable force of interest. On 1 July 1999, a customer deposited $100,000 pounds with the bank. On 1 July 2001, his deposit has grown to $120,000. Assuming that the force of interest per annum was bt (1 July 1999 = 0, unit = year) during the period, find the force of interest per annum on 1 July 2000.

**Solution:**

As we know that

So,

So, the force of interest per annum on 1 July 2000 (t=1) is 0.091.

**Example 3.**

A bank credits interest on deposit using accumulation factors based on a variable force of interest. On 1 July 1999, a customer deposited $100,000 pounds with the bank. On 1 July 2001, his deposit has grown to $120,000. Assuming that the force of interest per annum was a+bt (1 July 1999 = 0, unit = year) during the period, find the force of interest per annum on 1 July 2000.

**Solution:**

As we know that

So,

So, the force of interest per annum on 1 July 2000 (t=1) is 0.091.