Section 6.3Compound 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%

nt

n

rPB

1

• 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

nt

n

rPB

1

r

n1

nr

n

This is the base, b, from our exponential function y = abx

1 1n

rAPY

n

• Now let’s look at continuously compounded• We get• r is called the continuous rate• The growth factor in this form is er

• So to calculate the Annual Percentage Yield we have

• Find the APY for 12%– How does it compare to our previous growth rates?

rtPeB

This is the base, b, from our exponential function y = abx

1rAPY e

• Now 2 < e < 3 so what do you think we can say about the graph of Q(t) = et?–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) = aekt – Then the growth factor (or decay factor) is ek

• So from y = abt, b = ek

– 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 chapter6.3 – 11, 23, 25, 31, 43