Sixth Homework Assignment - for discussion at recitations Tuesday October 20, and Thursday October 22. The sixth homework also includes your first MAPLE ASSIGNMENT.
Seventh Homework Assignment - for discussion at recitations Tuesday October 27, and Thursday October 29.
r = interest rate per year (as a decimal).
P = initial principal
Simple Interest: The amount after t years is given by
A(t) = P(1+rt)
Compound Interest:
1. compounded annually for t years.A(t) = P(1+r)^t2. compounded m times per year Now r is called the nominal interest rate.
A(t) = P(1+(r/m))^(mt)
effective annual rate reff = simple interest rate to give the same result over one year:
P(1+ r/m)^m = P(1 + reff) so reff = (1 + r/m)^m - 1
Ex. Say a company has growth rates of 8%, 20%, 7%, 11%, and 3% in 5 successive years. What is the average growth rate (i.e., the effective annual rate that would give the same result after five years)?
(1.08)(1.20)(1.07)(1.11)(1.03) = (1 + reff)^5. Solving for reff gives
reff = 9.655%. Note that this is not the same as the average of the growth rates (the sum over 5 which is 9.8%).
Present Value: Assume nominal interest rate r, compounded m times per year. Then the present value of an amount A, t years in the future is:
A(1 + r/m)^(-mt)
Example: Lottery payoff of $1,000,000 per year for 60 years, or forever. Say the interest rate is 5%. What is the present value of this income stream? Do you think it is infinite??
answer:
To get these answers we did a brief review of geometric series.
Continuous Compounding:
P(1 + r/m)^(mt) converges to Pert as m goes to infinity.
Example. r = 5%. Start with $100. Result after 10 years if the interest is
simple (not compounded)
$150
compounded annually
$162.89
compounded daily
$164.86
compounded continuously
$164.87
We stepped through MAPLE Demonstration #7. This is a very good demonstration of many of these notions.