Determine the time necessary for P dollars to double when it is invested at interest rate r compounded annually, monthly, daily, and continuously. (Round your answers to two decimal places.)

Question

r = 5.7%

(a) annually

 yr

(b) monthly

 yr

(c) daily

 yr

(d) continuously

 yr

Answer 1

P(t) is the value at time t (in years).

For a change after time increment i (1 yr, 1/12 yr, 1/365 yr etc.) at rate r (in yr⁻¹), the general equation is

P(t) = (1 + r i)P(t – i) = (1 + r i)²P(t – 2i) = ... = (1 + r i)^t/iP(0).

To derive this, just observe that the number of (1 + r i) factors corresponds to by how many steps of i is the t argument subtracted.

Also note that in the limit, P(t) _i→0→ exp(r t)P(0).