Wim Ruitenburg's Spring 2011 MATH 1300-101

Last updated: February 2011
Comments and suggestions: Email wimr@mscs.mu.edu

Despite all the money talk, chapter 10 is really about exponential growth, with examples from money management.
From the chapter we can learn:

Increasing an amount like 1000 by 5 percent is essentially the same as multiplying 1000 by (1 + 5/100) or by 105/100 or by 1.05, and the result equals 1050.
Decreasing an amount like 1000 by 5 percent is essentially the same as multiplying 1000 by (1 - 5/100) or by 95/100 or by 0.95, and the result equals 950.
Compound interest is very powerful. When we start with just 1000 (dollars, say), and increase the amount by 5 percent each year for 10 years, the new amount equals 1000 * 1.05 * 1.05 * 1.05 * 1.05 * 1.05 * 1.05 * 1.05 * 1.05 * 1.05 * 1.05, also written 1000 * 1.05^10, and which equals approximately 1628.8946 (a bank may round this off to 1628.89).
Suppose inflation neutral interest rates are 5 percent a year. What is a current fair price P so that we have 1500 in 10 years? We must solve P * 1.05^10 = 1500. The formula is P = 1500 / 1.05^10, which equals 920.86988 approximately.
Increasing an amount like 1000 by 5 percent and then reducing the new amount by 5 percent is NOT neutral. For first we multiply 1000 by 1.05 to get 1050, and then we multiply 1050 by 0.95 and get as final result only 997.5.

We spent some time on continuously compounded interest. We approximate this through the following example.

If we start with 1 dollar and get 100% interest for the year, we end the year with 2 dollars.
If we start with 1 dollar and get 50% interest for the first 6 months and 50% for the second 6 months, we end the year with 1 * (1 + 1/2) * (1 + 1/2) = (1 + 1/2)^2 dollars, which equals 2.25.
If we start with 1 dollar and get 25% interest for the first 3 months and 25% for the second 3 months and 25% for the third 3 months and 25% for the fourth 3 months, we end the year with 1 * (1 + 1/4) * (1 + 1/4) * (1 + 1/4) * (1 + 1/4) = (1 + 1/4)^4 dollars, which is approximately 2.4414.
And so on. There are 31556952 seconds in the average Gregorian calendar year. If we start with 1 dollar and get 1/31556952 interest each second, then we end the year with 1 * (1 + 1/31556952)^31556952 dollars, which is approximately 2.71828.
And so on. The farther we go, the closer we get to Euler's number 2.71828182845999988...