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Interest

Interest. Unit 3. What is Interest?. Only your dumbest or kindest friends and family members will be willing to lend you large amounts of money without expecting something in return. Most people borrow money for large purchases from banks. Banks will charge you interest.

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Interest

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  1. Interest Unit 3

  2. What is Interest? • Only your dumbest or kindest friends and family members will be willing to lend you large amounts of money without expecting something in return. • Most people borrow money for large purchases from banks. • Banks will charge you interest. • Interest is the price you pay to borrow money. • Paying interest is not the same as paying back the loan. • Interest is the money that you pay above and beyond what you borrowed. • Ex. If you borrow $10 and pay back the lender $11 in a week. $1 is the interest earned by the lender.

  3. What is Interest? • Interest (i) is the cost of borrowing money. • Principal (p) is the amount of borrowed money. • Interest Rate (r) is the annual interest listed as a percentage of the principal amount. • Balance (b) is the amount of money in the account after interest is earned. (Ending Balance) • Typically listed as an annual percentage rate. • Examples – • If $100 is borrowed for a year at a 5% interest rate, the borrower must pay $105 at the end of one year. • The p = $100, r = .05, i = $5 • i = pr, $5 = $100 x (.05) • p + pr = B • $100 + $100 x .05 = $105 • p(1+r) = b (The formula is easier by factoring out “p”) • $100(1+.05) = $105

  4. Simple Interest • Simple Interest is simply the money earned on the principal only. • Simple Interest Formula • i = prt t = time • For example, you deposit $100 in a bank account at 5% annual interest. Each year they send you a check for the interest. How much interest have you earned after 10 years? • i = prt i = $100 x .05 x 10 i = $50

  5. Compound Interest • Compound Interest means that you not only earn interest on the principal, but also on the accumulated interest that has been earned. • For Example - $100 is deposited into a savings account earning 10% interest compounded annually. • After 1 year the account has $110, ($100 + $100 x .10) • After 2 years the account has $121, ($110 + $110 x .10) • The interest earned the first year becomes part of the principal for the 2nd year. • After 3 years the account has $133.10, ($121 + $121 x .1) • One can use the following formula to calculate compound interest. B = p (1+r)t • $100 (1.1)3 = $133.10

  6. The Power of Compounding • Two people Joe and Schmoe have $100 and have the same investment opportunity. They can both earn 10% interest each year for as long as they live. • Joe earns the 10% each year and then reinvests the interest in the same investment. • Schmoe earns the 10% and then spends it. • How much will Joe and Schmoe earn over time?

  7. Joe and Schmoe • In year 1, they each earned $10. • In year 2, Joe earned $11 and Schmoe earned $10. • Let’s see how the rest of their future looks. After 10 years Schmoe earned $100 on top of his original $100. Joe earned $159 on top of his original $100. An extra $59 after 10 years might not seem like much.

  8. Joe and Schmoe • After 40 years Joe’s $100 grows to $4500 while Schmoe only has $500 • Joe has 9x the wealth of Schmoe

  9. Joe and Schmoe after 70 years After 70 years Joe’s wealth is more than 80x of Schmoe

  10. Compound Interest 2 • Interest can be compounded annually, semi-annually, quarterly, daily or even continuously. • Example • Jose has $100 in a savings account which pays 10% interest compounded quarterly. How much money will he have after 1 quarter? 1 year? 3 years? • $100 (1+ .1/4) = $100 (1.025) = $102.50 • $100 (1+ .1/4) 4= $100 (1.025) 4 = $110.38 • $100 (1+ .1/4) (4*3)= $100 (1.025) (4*3) = $134.49 • When compounded annually at 10% it was only $133.10

  11. Compound Interest Formula • B = ending balance • p = principal • r = interest rate • n = number of times interest is compounded annually • t = number of years • B = p(1 + r/n)nt • For Example - $500 is earning 4% interest compounded daily. What is the account balance in 2 years? • B = $500(1 + .04/365) (365 x 2) • B = $541.64 • The compound interest formula just turns an annual interest rate into a rate over a different period of time and then compounds it over the appropriate # of time periods.

  12. Continuous Compounding • Imagine instead of daily compounding, you compound by the hour, minute, or second. • Continuous Compounding is compounding every fraction of every second. • This does not result in an infinite balance. • For example what if you deposit $1 at 100% interest compounded continuously, how much would the balance be in 1 year? • p = $1 • r = 1 • t = 1 • n = x (We will see what happens as x approaches infinity) • B = $1 (1 + 1/x)x • Fin Alg p 152-153

  13. Continuous Compounding 2 • Focus on the (1 + 1/x)x • Let’s see what various numbers give us. • As x gets larger the balance approaches a limit. • This limit can be rounded to 2.718, also known as “e” • e is known as the exponential base and is an important number in mathematics.

  14. Continuous Compounding 3 • In our previous example, the $1 would grow to $2.71828. • Continuous Compound Formula • b = pert • Someone who deposits $100 at a 4.5% interest rate for 3 years would earn…. • b = $100(2.718) (.045 x 3) = $114.45

  15. Future Value of Periodic Investments • Instead of investing a lot of money all at once, it is common for investors to save a little bit of their income on a regular basis. • For example, if you save $100 each month and place it in a savings account earning 5% compounded monthly. How much would you have at the end of 10 years? • The long way to do this would be to figure out the future balance of each of your individual $100 investments. • Clearly this would take a while.

  16. Future Value of Periodic Investments • A much quicker was to solve the question is to use the formula below. • This formula only works when the investment period is the same as the compounding period. • n = compounding per year and investments per year. b = p[(1+r/n)nt-1] _____________________________ r/n

  17. Future Value of Periodic Investments • To answer our previous question… substitute p = $100, r = 5%, n = 12, t = 10 b = $100[(1+.05/12)12*10-1] _____________________________ .05/12 • b = $15,531

  18. Annual Percentage Yield • The annual percentage yield (APY) tells you how much you will earn in a year. • APY is different than APR (Annual Percentage Rate) because APR does not incorporate compounding within the year. • APR is what we call (r) in our equations. • For example - $100 is invested for one year at 6% compounded monthly. • b = 100(1+ .06/12)12 = $106.17 • The APR is 6%. The APY is 6.17%. • APY = (1+r/n)n -1

  19. Review • Balance = Principal + Interest b = p + i • b = ending balance p = principal • i = interest r = interest rate • t = number of years e = 2.718 • n = number of times interest is compounded annually • Simple Interest i=prt, b=p+prt • Key words – simple interest • For Annual Compoundingb = p (1+r)t • Key words – Annual compounding • For Periodic Compounding b = p(1 + r/n)nt • Key words – compounding quarterly, daily, monthly, etc… • For Continuous Compounding b = pert • Key words - Compound continuously • For Periodic Investments b = p[(1+r/n)nt-1] / (r/n) • Key words – every week, each month, etc….

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