# Theorem: Accumulation Factor

Financial Mathematics → Theorem: Accumulation Factor

Accumulation Factor and Force of Interest Theorem.

If δ(t) and A(t0,t) are continuous functions of t and t≥t0, and the principle of consistency holds, then, for t0≤t1≤t2 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 t1 to time t2 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.

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