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