Reference no: EM131005111
Suppose that a sum S0 is deposited into a bank with an annual interest rate, denoted r. Denote S(t) to be the investment at time t, where t is the number of years since we've made the deposit. If the interest is compounded d times per years, then after t years, the investment value is given by
S(t) = S0(1 + r/d)dt.
(a) Next, suppose that interests are compounded continuously. Compute S(t) in that case by letting d tend to infinity in the previous equation
(1 + r/d)dt = eIn((1 + r/d)^dt)
(b) We can alternatively derive this formula by solving a differential equation: Express S(t + 1/d) in terms of just S(t). With this expression, substitute it into the difference quotient
{S(t + 1/d) - S(t) } / {1/ d},
and let d tend to infinity, so show that S(t) satisfies the differential equation dS/dt = rS. Finally, solve this differential equation to confirm your result in part (a.).
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