## 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** r_{eff} = simple interest
rate to give the same result over one year:

P(1+ r/m)^m = P(1 + r_{eff}) so r_{eff} =
(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 + r_{eff})^5. Solving
for r_{eff} gives

r_{eff} = 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:

- $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 Pe^{rt} 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.

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