Uncle Vern has just come into some money — $100,000 — and is thinking about putting this away into some investment accounts for a while.
A: Vern is a simple guy — so he goes to the bank and asks them what the easiest option for him is. They tell him he could put it into a savings account with a 10%interest rate (compounded annually).
(a) Vern quickly does some math to see how much money he’ll have 1 year from now, 5 years from now, 10 years from now and 25 years from now assuming he never makes withdrawals. He doesn’t know much about compounding — so he just guesses that if he leaves the money in for 1 year, he’ll have 10% more; if he leaves it in 5 years at 10% per year he’ll have 50% more; if he leaves it in for 10 years he’ll have 100% more and if he leaves it in for 25 years he’ll have 250% more. How much does he expect to have at these different times in the future?
(b) Taking the compounding of interest into account, how much will he really have?
(c) On a graph with years on the horizontal axis and dollars on the vertical, illustrate the size of Vern’s error for the different time intervals for which he calculated the size of his savings account.
(d) True/False: Errors made by not taking the compounding of interest into account expand at an increasing rate over time.
B: Suppose that the annual interest rate is r.
(a) Assuming you will put x into an account now and leave it in for n years, derive the implicit formula Vern used when he did not take into account interest compounding.
(b)What is the correct formula that includes compounding.
(c) Define a new function that is the difference between these. Then take the first and second derivatives with respect to n and interpret them.