# Continuous Interest

Financial Mathematics → Interest → Simple Interest → Compound Interest → Continuous Interest

The basis for continuous compound interest. See the following example to know the basis for continuous interest.

Example 1.
Suppose \$1,000 is deposited in a savings account that earns compounded interest at a rate of 10% per year. How much will be in the account after 1 year if interest is compounded:
(a) Yearly
(b) Semi-annually
(c) Quarterly
(d) Monthly
(e) Weekly
(f) Daily
(g) Hourly
(h) Every minute

Solution:
(a) yearly
$A = C[1 + hi_h(t)] \\ = 1000[1 + (1)(0.1)] \\ = \1100$ Answer

(b) Semi-annually
$A = C[1 + hi_h(t)]^2 \\ = 1000[1 + (1/2)(0.1)]^2 \\ = \1102.5$ Answer

(c) Quarterly
$A = C[1 + hi_h(t)]^4 \\ = 1000[1 + (1/4)(0.1)]^4 \\ = \1103.81$ Answer

(d) Monthly
$A = C[1 + hi_h(t)]^{12} \\ = 1000[1 + (1/12)(0.1)]^{12} \\ = \1104.7$ Answer

(e) Weekly
$A = C[1 + hi_h(t)]^{52} \\ = 1000[1 + (1/52)(0.1)]^{52} \\ = \1105.06$ Answer

(f) Daily
$A = C[1 + hi_h(t)]^{365} \\ = 1000[1 + (1/365)(0.1)]^{365} \\ = \1105.155$ Answer

(g) Hourly
$A = C[1 + hi_h(t)]^{8760} \\ = 1000[1 + (1/8760)(0.1)]^{8760} \\ = \1105.17$ Answer

(h) Every minute
$A = C[1 + hi_h(t)]^{525600} \\ = 1000[1 + (1/525600)(0.1)]^{525600} \\ = \1105.1709$ Answer

In the above example, note that as h goes smaller and smaller (daily, hourly every minute etc.) the value of accumulated amount takes a limiting value.

Next: Accumulation Factor

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