Chapter 11 - Compound Interest - notes · Chapter 11 - Compound Interest - notes Compounding involves the calculation of interest periodically over the life of the loan. After each

Chapter 11 - Compound Interest - notes Compounding involves the calculation of interest periodically over the life of the loan. After each calculation the interest is added to the principal. Future calculations are based on the old principal plus interest. Future Value (FV) is the final amount at the end of the last period Present Value (PV) is the amount you begin with Number of Periods - number of years times the number of times the interest is compounded per year. (annually, semi annually, quarterly, monthly)

Example: if you compound $1 for 4 years at 8% annually, you get the following periods: annually: 4 years X 1 = 4 periods semi annually: 4 years X 2 = 8 periods quarterly: 4 years X 4 = 16 periods Rate for Each period - annual interest rate divided by the number of times the interest is compounded per year. annually: 8% divided by 1 = 8% semi annually: 8% divided by 2 = 4% quarterly: 8% divided by 4 = 2%

Example: Bob deposited $80 in a savings account for 4 years at an annual interest rate of 8%. Calculate the simple interest and the compound interest. (For compound interest, presume that the interest is compounded annually) Simple Interest Interest = Principal X Rate X Time Interest = 80 X .08 (8%) X 4 years Interest = $25.60 MV = 80 + 25.60 = $105.60 at the end of 4 years Compound Interest Year 1: Interest = Principal X Rate X Time

Interest = 80 x .08 (8%) x 1 Interest = 6.40 Compounded Amount/new Principal (principal + interest) =

80 + 6.40 = $86.40 New Principal at the end of year #1 Year 2: Interest = Principal X Rate X Time

Interest = 86.40 x .08 (8%) x 1 Interest = 6.91 Compounded Amount (principal + interest) = 86.40 + 6.91 =

$93.31 New Principal at the end of year #2 Year 3: Interest = Principal X Rate X Time

Interest = 93.31 x .08 (8%) x 1 Interest = 7.46

Compounded Amount (principal + interest) = 93.31 + 7.46 = $100.77 New principal at the end of year #3 Year 4: Interest = Principal X Rate X Time

Interest = 100.77 x .08 (8%) x 1 Interest = 8.06 Compounded Amount (principal + interest) = 100.77 + 8.06 =

$108.83 Simple interest - 4 years at 5% $80 became $105.60 at the end of 4 years Compound Interest - 4 years at 5% - $80 became $108.83 at the end of 4 years

Calculating Compound Interest Amount Using Tables #1 - Find the periods - years multiplied by the number of times interest is compounded in 1 year Annual - 1 time per year Semiannual - 2 times per year Quarterly - 4 times per year 2 year loan compounded semiannually = ___ 4 __ periods # of years (2) X # times compounded in 1 year (2) = 2 x 2 = # of periods = 4 5 year loan compounded semiannually = how many periods? 5 years X 2 (semiannual) = 10 periods 10 year loan compounded annually = ? periods 10 years x 1 (annual) = 10 periods 6 year loan compounded quarterly = ? periods 6 years x 4 (quarterly) = 24 periods 10 year loan compounded semiannually = ? periods 10 years x 2 (semiannual) = 20 periods 8 year loan compounded quarterly = ? periods 8 years x 4 (quarterly) = 24 periods 8 year loan compounded annually = ? periods 8 years x 1 (annual) = 8 periods

#2 - Find the rate - annual rate divided by number of times interest is compounded in 1 year Stated rate is 4% compounded semiannually = ? rate 4% divided by 2 (semiannual) = 2% Stated rate is 10% compounded semiannually = ? rate

10% divided by 2 (semiannual) = 5% Stated rate is 12% compounded quarterly = ? rate 12% divided by 4 (quarterly) = 3% Stated rate is 5% compounded semiannually = ? rate 5% divided by 2 (semiannual) = 2.5% Stated rate is 6% compounded quarterly = ? rate 6% divided by 4 (quarterly) = 1.5% Stated rate is 8% compounded annually = ? rate 8% divided by 1 (annually) = 8%

#3 - Using the table (which table?) find the intersection of the periods and rate #4 - Multiply that number by the amount of the loan. This gives the compound amount (or FV)

Sam owes $250 8% interest compounded annually for 5 years Step #1) # of periods = 5 Step #2) Rate = 8% Step #3) - Chart look for the intersection between 5 periods and 8% 1.46933 number from the chart Step #4) 1.46933 x 250 = $367.33

Sam owes $250 8% interest compounded quarterly for 5 years Step #1) # of periods = 20 periods Step #2) Rate = 2% Step #3) Look for the intersection of 20 periods and 2% = 1.48595 Step #4) 1.48595 x 250 = $371.49

Try these: Find the interest on $6,000 at 10% compounded semiannually for 5 years (in the future). Step 1) # of periods = 10 Step 2) Rate = 5% Step 3) Chart = 1.62889 Step 4) 6000 x 1.62889 = $9,773.34

Pam deposits $8,000 in her savings account that pays 6% interest compounded quarterly. What will the balance of her account be at the end of 5 years? (in the future) Step #1) # of periods = 20 = 5 years X 4 (quarterly) = 20 Step #2) Rate = 1.5% = 6% divided by 4 (quarterly) = 1.5% Step #3) Chart = 1.34686 Step #4) 8,000 x 1.34686 = $10,774.88

Find the interest on $5,000 at 8% compounded semiannually (2 times per year) for 6 years. (in the future) Step 1) # of periods = 6 years X 2 (semiannual) = 12 periods Step 2) Rate = 8% divided by 2 (semiannual) = 4% Step 3) 1.60103 Step 4) 5,000 x 1.60103$8,005.15

Amy deposits $12,000 in her savings account that pays 10% interest compounded quarterly. What will the balance of her account be at the end of 7 years? (in the future) Step 1) # of periods = 7 years x 4 (quarterly) = 28 periods Step 2) Rate = 10% divided by 4 (quarterly) = 2.5% Step 3) chart = 28 periods/2.5% = 1.85394 Step 4) 12,000 x 1.85394 = $22,247.28

APY - Annual Percentage Yield A 6% interest rate will not yield the same results if one person is compounding annually and another is compounding quarterly or even daily, so instead we compare rates using the APY. To calculate the APY, divide the interest for 1 year by the principal.

APY = Interest for 1 year Principal What is the APY for $8,000 compounded quarterly at 8%? How much would you earn at the end of 5 years? Periods = 4 x 1 = 4 Percent = 8% divided by 4 = 2% Lookup 4 periods/2% on the chart = 1.0824 8000 x 1.0824 = 8659.20 I = 8659.20 - 8000 I = 659.20 659.20 divided by 8000 = APY = .0824 or 8.24% So compounded quarterly you are actually doing better than 8% (which is what you would make if compounded annually) Present Value We have learned how to compute the Future Value (FV) of an amount by calculating how much interest will be earned during a period of time.

Computing Present value (PV) is figuring out how much money needs to be the original principal in order to arrive at the desired outcome. For example: If you will need $10,000 in 5 years to buy a car, how much do you need to invest now at 8% compounded semiannually to get that amount? (present) Step 1) # of periods = 5 years x 2 (semiannual) = 10 periods Step 2) rate = 8% divided by 2 (semiannual) = 4% Step 3) Chart - present value PV = 10 periods/4% = .67556 Step 4) 10,000 x .67556 = $6.755.60

Uncle Joe wants to give you $12,000 at graduation. You will graduate in 4 years. How much does Uncle Joe need to invest today at 12% interest compounded quarterly for 4 years in order to have $12,000 for your graduation? (present) Step #1) # of periods = 4 years x 4 (quarterly) = 16 Step #2) rate = 12% divided by 4 (quarterly) = 3% Step #3) PV Chart 16 periods/3% = .62317 Step 4) 12,000 x .62317 = $7,478.04

Chapter 11 - Compound Interest - handout #1 - Lou loaned Amanda $25,000 to open a hair salon. After 6 years Amanda will repay Lou with 8% interest compounded quarterly. How much will Lou receive at the end of 6 years? #2 - Paul promised his grandson that he would give him $10,000 8 years from now when he graduates. How much should Paul invest today at 6% interest compounded quarterly to have enough money in 8 years? #3 - What is the APY for the year with a principal of $8,000 and an interest rate of 8% compounded quarterly? #4 - Shirley deposited $40,000 in a savings account at 6% interest compounded semiannually. At the beginning of year 4, Shirley deposits an additional $60,000 at 6% interest compounded semiannually. At the end of 6 years what is the balance in Shirley’s account? #5 - Mark has $12,000 to invest. He could put it into Bank A at 9% interest compounded annually or Bank B at 8% interest compounded quarterly. Find the APY of each to decide which bank gives Mark the best deal. (10 pts)