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10.3 Compound Interest

What SIMPLE INTEREST looks like: I have $500. I put it in an account earning 8% annual interest. After 10 years it is worth:

F =500(1 + .08(10))

𝑭 = 𝑷(𝟏 + 𝒓𝒕) F = 900

$900.00

What COMPOUND INTEREST looks like: I have $500. I put it in an account earning 8% interest compounded annually. Compounding annually means that I earn interest at the end of each year.

After 1 year: F =500(1.08) = $540

After 2 years: F =500(1.08)(1.08) = 540(1.08)=$583.20

After 3 years: F =500(1.08)(1.08)(1.08) = $583.20(1.08) =$629.86

After N years: F =500(1.08)N

After 10 years: F =500(1.08)10 = $1079.46

Because you are earning interest on both your principle and interest on your interest, compounding causes you to earn more money (or owe more money) in the long run than simple interest.

Periodic Interest Rate: p

Periodic interest is how much interest you are charged/you earn per compounding period. r = APR (annual percentage rate) m = number of compounding periods per year

𝒑 =𝒓

𝒎

Example: If your account has an 8% APR, you would earn: Compounded annually _______ , _______ times per year Compounded quarterly _______, _______ times per year Compounded monthly _______, _______ times per year Total Number of Compounding Periods: T

m = number of compounding periods per year Example: Compounded annually for 5 years T = _____

t = number of years

Compounded monthly for 3 years T = _____

T = mt Compounded daily for 2 years T = _____

Common Compounding Periods: Term Number of Times Interest

Paid/Charged Per Year annually 1 semiannually 2 quarterly 4 monthly 12 daily 365 or 360 continuously ?

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1) You invest $758 in an account earning 5.9% APR compounded annually. How much is the investment

worth after 10 years? 2) You invest $4000 in an account earning 6.4% APR compounded quarterly. How much is the investment

worth after 8 years? 3) You invest $350.82 in an account earning 9.5% APR compounded continuously. How much is the

investment worth after 5 years?

APY: Annual Percentage Yield (Effective Rate) Your friend offers to loan you money at 6% annual interest. The bank offers a loan at 5.8% APR compounded monthly. Which is a better deal? What would a dollar ($1.00) be worth at the end of year? Borrow from friend: Borrow from bank:

Compound Interest Formula P = principal m = number times compounded per year

F = future value

r = annual interest rate p = periodic interest rate = 𝑟

𝑚

t = time of investment, in years

T = total compounding periods = 𝑚𝑡

𝑭 = 𝑷(𝟏 + 𝒑)𝑻

“Continuously Compounded” Interest P = principal

F = future value

r = annual interest rate

t = time of investment, in years

𝑭 = 𝑷𝒆𝒓𝒕

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4) Find the APY a) 5.9% APR compounded annually

b) 6.4% APR compounded quarterly c) 9.5% APR compounded continuously

5) Which is the best investment? Option 1: 7% compounded annually Option 2: 6.8% compounded monthly Option 3: 6.5% compounded continuously?

Annual Percentage Yield (APY)/Effective Rate m = number times compounded per year

r = annual interest rate

p = periodic interest rate = 𝑟

𝑚

𝑨𝑷𝒀 = (𝟏 + 𝒑)𝒎 − 𝟏 or 𝑨𝑷𝒀 = 𝒆𝒓 − 𝟏

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10.4 Geometric Sequences

A Geometric Sequence starts with an initial term, P, and then multiplies every term that follows by a common ratio, c. Example: 5, 10, 20, 40, 80, …. Generically: 𝑃, 𝑃𝑐, 𝑃𝑐2, 𝑃𝑐3, 𝑃𝑐4, …

Example 1: 𝑃 = 500 𝑐 = 1.08 𝐺0 = 500 What is 𝐺10 =? 𝐺1 = 500(1.08) = 540 𝐺2 = 500(1.08)2 = 583.20 𝐺3 = 500(1.08)3 = 629.86 Example 2: Your rent is currently $700 per month. You read in a local newspaper that due to changes in the housing market, rents in the area are expected to increase by approximately 5% per year. If you stay in your apartment for 5 years, what should you expect to pay for rent? Example 3: You buy a new car for $21,000. On the way home you hear Dave Ramsey on the radio saying new cars are a bad purchase because they depreciate in value on average by 18% per year. If this is true, what will your car be worth in 6 years? Example 4: A new infectious disease X26 has been discovered and there is no current vaccine or treatment. The Center for Disease Control estimates that until a vaccine becomes available, that the virus will spread at a 25% annual rate of growth. The CDC currently is tracking 1000 known cases of X26. How many new cases will occur each year over the next 3 years? What if it takes 10 years to develop a vaccine

Geometric Sequence: 𝐺0 = 𝑃 the initial (starting) term 𝑐 the common ratio Recursive Formula: 𝐺𝑁 = 𝐺𝑁−1 ∙ 𝑐

Explicit Formula: 𝐺𝑁 = 𝑃 ∙ 𝑐𝑁

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Example 4 (CONTINUED): A new infectious disease X26 has been discovered

and there is no current vaccine or treatment. The Center for Disease Control estimates that until a vaccine becomes available, that the virus will spread at a 25% annual rate of growth. The CDC currently is tracking 1000 known cases of X26. How many new cases will occur

each year over the next 3 years? What if it takes 10 years to develop a vaccine? Example 3: Find the sum of the following geometric sequences

a) 2 3 36

$400 1.042 $400 1.042 $400 1.042 ... 400 1.042

b) 2 3 24

$250 0.85 $250 0.85 $250 0.85 ... 250 0.85

Sum of the Terms in Geometric Sequence: 𝐺0 = 𝑃 the initial (starting) term 𝑐 the common ratio

Sum = 1

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NcP

c