Section 6.3 Compound Interest and Continuous Growth

Let’s say you just won $1000 that you would like to invest. You have the choice of three different accounts: – Account 1 pays 12% interest each year – Account 2 pays 6% interest every 6 months (this is called 12% compounded semi-annually) – Account 3 pays out 1% interest every month (this is called 12% compounded monthly) Do all the accounts give you the same return after one year? What about after t years? If not, which one should you choose? NOTE: In each case 1% is called the periodic rate

If an annual interest r is compounded n times per year, then the balance, B, on an initial deposit P after t years is For the last problem, figure out the growth factors for 12% compounded annually, semi- annually, monthly, daily, and hourly – We’ll put them up on the board – Also note the nominal rate versus the effective rate or annual percentage yield (APY) The nominal rate for each is 12%

n is the compounding frequency is called the periodic rate The growth factor is given by So to calculate the Annual Percentage Yield we have Now back to our table This is the base, b, from our exponential function y = ab x

Now let’s look at continuously compounded We get r is called the continuous rate The growth factor in this form is e r So to calculate the Annual Percentage Yield we have Find the APY for 12% – How does it compare to our previous growth rates? This is the base, b, from our exponential function y = ab x

Now 2 < e < 3 so what do you think we can say about the graph of Q(t) = e t ? – What about the graph of f(t) = e -t It turns out that the number e is called the natural base – It is an irrational number introduced by Lheonard Euler in 1727 – It makes many formulas in calculus simpler which is why it is so often used

Consider the exponential function Q(t) = ae kt – Then the growth factor (or decay factor) is e k So from y = ab t, b = e k – If k is positive then Q(t) is increasing and k is called the continuous growth rate – If k is negative then Q(t) is decreasing and k is called the continuous decay rate Note: for the above cases we are assuming a > 0

Example Suppose a lake is evaporating at a continuous rate of 3.5% per month. – Find a formula that gives the amount of water remaining after t months if it begins with 100,000 gallons of water – What is the decay factor? – By what percentage does the amount of water decrease each month?

Example Suppose that $500 is invested in an account that pays 8%, find the amount after t years if it is compounded – Annually – Semi-annually – Monthly – Continuously Find the APY for a nominal rate of 8% in each case From the chapter 6.3 – 11, 23, 25, 31, 43