## Lecture Notes: Monday, October 26, 1998

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.

### Chapter 5 Section 3 (Compound Interest - continued)

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)^t

2. 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??

• \$21,000,000 is the Present value of \$1million per year forever at 5% interest.
• \$19,874,754 is the present value of \$1 million per year for 60 years at 5% interest.

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.

Lecture Notes Directory

Course Guide