5.2 Compound Interest

Recall the function for compound interest: P is the principal amount r is the annual interest rate m is the number of times interest is compounded per year t is the number of years

Suppose $1000 is invested at 6% for 1 year. t = 1 year If interest is compounded annually (m = 1), then the amount in the account at the end of the year is A = P(1 + r/m)mt = 1000(1 + .06/1)(1)(1) = 1060

If interest is compounded quarterly, then the amount in the account at the end of the year is A = P(1 + r/m)mt = 1000(1 + .06/4)(4)(1) = 1061.36

The following table contains the results for different compounding periods

Would the account balance be much more if the interest were compounded more frequently? Every hour? Every minute? Every second? It turns out that no matter how frequently interest is compounded, the balance will not exceed 1061.84. Why?

Start with Let h = r/m, then 1/h = m/r and As the frequency of compounding increases, m gets larger, and h = r/m approaches 0.

Consider It can be shown that So, then gets closer to Pert as the number of interest periods per year is increased.

When the formula A = Pert is used to calculate the compound amount, we say that the interest is compounded continuously. Now, when $1000 is invested at 6% for 1 year with the interest compounded continuously, we have A = 1000e.06(1) which is approximately 1061.84

In many computations it is simpler to use the formula for interest compounded continuously as an approximation to ordinary compound interest.

When interest is compounded continuously, the compound amount A(t) is an exponential function of the number of years t that interest is earned A(t) = Pert and A(t) satisfies the differential equation A’(t) = r A(t) The rate of growth of the compound amount is proportional to the amount of money present.

Problem One thousand dollars is invested at 5% interest compounded continuously. Give the formula for A(t), the compounded amount after t years. How much will be in the account after 6 years? After 6 years, at what rate will A(t) be growing? How long is required to double the initial investment?

Problem Pablo Picasso’s “The Dream” was purchased in 1941 for a war distressed price of $7000. The painting was sold in 1997 for $48.4 million, the second highest price ever paid for a Picasso painting at auction. What rate of interest compounded continuously did the investment earn?

If P dollars are invested today, the formula A = Pert gives the value of this investment after t years (assuming continuously compounded interest). P is called the present value of the amount A to be received in t years. If we solve for P in terms of A, we obtain

Problem Find the present value of $5000 to be received in 2 years if the money can be invested at 12% compounded continuously.