Compound interestEffective interest ratesDoubling periods and half lifesThe Consumer Price IndexMonthly payments on loans

A lot of the math of "the real world" is linear, but some of it involves exponents. The best known interest rate situations are those of a savings account. Here is the equation; instead of using superscripts for exponents, I use the symbol ^, which is the symbol for exponentiation on most of the TI calculators, though some have an "x rasied to the y" button.

Compound interest rate equation

A(t) = P(1+r)^t

A(t) is the amount in the bank at time t
P is the original principle invested 
r is the rate of interest, usually given as a percent


Effective rate compound interest rate equation

Some saving institutions will give the rate as "4% compounded quarterly". This is slightly different from 4% a year. If we use n as the number of compounding periods in a year, the equation changes to 

A(t) = P(1+r/n)^nt

4% compounded quarterly

In the stated example (1+.04/4)^4 = (1.01)^4 = 1.04060401, so the rate r = .04060401, usually rounded to the nearest hundredth of a percent at 4.06%. The difference is small but noticeable over long stretches of time.


$1000 invested at 4% over 20 years is $2,191.12
$1000 invested at 4% compounded quarterly over 20 years is $2,216.72

Doubling periods and half lifes

How long will it take at 4% to turn $1000 into $2000. This amount of time is called the doubling period and is found by the formula log(2)/log(1 + r)

In our case, that would be log(2)/(1 + .04) = 17.6729.....

If the bank is actually compounding annually, the amount only changes on the anniversary of your putting the money in.

After 17 years you have (1.04)^17 = $1,947.90
After 18 years you have (1.04)^18 = $2,025.82

The Consumer Price Index

This week, students got a green handout sheet with the consumer price index number for the years 1950 to 2015.  The simplest way to use it goes as follows.

How did prices change from 1960 to 1980? We use the two CPI values.

CPI(1960) = 29.6
CPI(1980) = 82.4

82.4/29.6 = 2.78378.....

29.6/82.4 = .35922...

What these numbers mean is that on average, a $10 item in 1960 sold for 10*2.78378... = $27.84 in 1980, while a $10 item in 1980 would have sold for 10*.35922... = $3.59.

We can use this to figure out the cost of living increase in any given year by dividing the CPI for that year by the CPI for the previous year.  For example, the rate in 1975 would be

CPI(1975)/CPI(1974) = 53.8/49.3 = 1.09127789... This is 1 + rate, so rounded to the nearest tenth of a percent we would have 9.1% and to the nearest hundredth of a percent it would be 9.13%.

Let's look at the number CPI(1980)/CPI(1960) = 2.783783... This gives us how much prices increased in the 20 year period from 1961 to 1980. To get the average increase over those 20 years, we take (2.783783)^(1/20) = 1.05252... This is 1 + rate, so the average rate = .05252... or 5.3% if rounded to the nearest tenth of a percent and 5.25% rounded to the nearest hundredth.



Monthly payments on loans

If you know the interest rate r, the number of years y you will take to pay off a loan and the amount you wish to borrow A. The formula for the monthly payment is

Monthly payment = A(1 + ry)/(12y)

We can use this to figure out monthly payments or, if we know how much we are willing to pay a month, the rate and the numbers of years we can get the loan for, conversely we can get the amount A with the formula

A = 12y(Monthly payment)/(1 + ry)

Practice problems

a) If you put $1,000 in the bank invested at 4.5%, how much will you have in 10 years?


b) If you put $1,000 in the bank invested at 4.5%, how much will you have in 15 years?


c) what is the effective rate of 5.5% compounded monthly?

d) what is the effective rate of 5.5% compounded daily?

e) A radioactive isotope loses 0.1% of its radioactivity every year. What is the half-life, rounded to the nearest tenth of a year?

f) The worst of inflation in last century in the United States happened in the 1970s and early 1980s. Find the average inflation rates for the following presidential terms. (Remember to start with the year prior to the start of the term.

1. Nixon (1969-1974)
2. Ford (1974-1976)
3. Carter(1977-1980)
4. Reagan(1981-1988)

g) If you borrow $20,000 at 6% and have a ten year loan, what is the monthly payment?


Answers in the comments.