Short note: continuous compounding

Continuous compounding is the limiting case of compounding in which the number of compounding periods per year (m) tends to infinity, so interest is added at every instant.

Formulas

Let r = nominal annual rate and n = number of years.

• Effective annual rate: i = e^r − 1, because lim (1 + r/m)^m = e^r as m → ∞
• Future worth: F = P·e^(rn)
• Present worth: P = F·e^(−rn)
• Single payment factors: (F/P, r, n) = e^(rn), (P/F, r, n) = e^(−rn)
• Uniform series: (F/A, r, n) = (e^(rn) − 1)/(e^r − 1), (A/F, r, n) = (e^r − 1)/(e^(rn) − 1), (P/A, r, n) = (e^(rn) − 1)/[e^(rn)(e^r − 1)]

Example

Rs 10,000 at 12% nominal for 5 years.

• Continuous: F = 10,000 × e^(0.12 × 5) = 10,000 × 1.8221 = Rs 18,221; effective rate = e^0.12 − 1 = 12.75%
• Monthly compounding: F = 10,000 × (1.01)^60 = Rs 18,167; effective rate = 12.68%
• Annual compounding: F = 10,000 × (1.12)^5 = Rs 17,623

Remarks

• It gives the highest return for a given nominal rate, but only slightly more than daily compounding.
• Used for theoretical analysis, flows that occur steadily (such as oil production or continuous revenue) and some bank products.
• Most real banks compound monthly, quarterly or yearly, so it is mainly a limiting model.