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1. Calculating Simple Interest

1. Calculating Simple Interest. A dollar today is worth more than a dollar tomorrow Because of this cost, money earns interest over time If you are borrowing, you will pay interest If you are lending/investing, you will earn interest Simple Interest

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1. Calculating Simple Interest

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  1. 1. Calculating Simple Interest • A dollar today is worth more than a dollar tomorrow • Because of this cost, money earns interest over time • If you are borrowing, you will pay interest • If you are lending/investing, you will earn interest • Simple Interest • interest on an investment that is calculated once per period, usually annually • on the amount of the capital alone • interest that is not compounded

  2. 1. Calculating Simple Interest • Principal is the initial amount invested or borrowed (the loan amount or how much you save) • Simple Interest Formula: • P = Principal • r = Annual Interest Rate • n = Number of periods (usually years) the money is being borrowed • Simple Interest = Principal times interest times years • Simple Interest = P(r)(n) • Total Owed = P + P(r)(n)

  3. 1. Calculating Simple Interest • Ex 1: • Mr. Vasu invests $5,000. His annual interest rate is 4.5% and he invests his money for 5 years. What is the total in his account after this time? • P = • r = • n = • Total = P + P(r)(n) $5,000 0.045 5 5000 + 5000(0.045)(5) 5000 + 1125 = $6,125

  4. 1. Calculating Simple Interest • Ex 2: Trayvond saves $10,000 to pay for a car. His earns 6% on his investment and invests his money for 7 years. What is the total in his account after this time? • P = • r = • n = • Total = P + P(r)(n) $10,000 0.06 7 10000 + 10000(0.06)(7) 10000 + 4200 = $14,200

  5. 2. Calculating Compound Interest • Constant Multiplication Factor and Interest Rate • The constant multiplication factor = (1 + r) • r = annual interest rate (as a decimal) • Annual interest rate and growth rate are the same thing • Ex 1: If you earn 6%, what is the constant multiplication factor: (1 + 0.06) = (1.06) • Ex 2: If the CMF is 1.5, what is the growth rate? • 1.5 = 1 + r; r=0.50, which is 50%

  6. 2. Calculating Compound Interest • Ex 3: Mr. Vasu invests $10,000 in an account that earns 6% annual interest that compounds annually. How much will he have in 2 years: Mr. Vasu has $11,236 after two years.

  7. 2. Calculating Compound Interest • Ex 3: Mr. Vasu invests $10,000 in an account that earns 6% annual interest that compounds annually. How much will he have in 2 years: • Ex 4: Mr. Vasu invests $10,000 in an account that earns 6% annual interest that compounds annually. How much will he have in 7 years? 10,000(1.06)7= $15,036.30 Mr. Vasu has $15,036.30 after seven years.

  8. 2. Calculating Compound Interest • Compound Interest Formula • (Exponential Growth Function) • A = P(1 + r)t • A = Future Value or Final/Ending Value • P = Principal/Initial Value and Y-Intercept • r = Annual Interest Rate/Growth Rate • t = Years

  9. 2. Calculating Compound Interest • Ex 5: Aaliyah invests $6,000 and earns 5% per year. • Write an exponential growth equation for how much money Aaliyah has after t years? • A = ? • P = 6,000 • r = 0.05 • t = ? • A = 6000(1.05)t • How much will she have after six years if interest is compounded annually? • t = 6 years • A = 6000(1.05)6 • A = $8,040.57

  10. 2. Calculating Compound Interest • Ex 6: Ganiu invests $24,000 for ten years at 4.5%. • How much does he have in his account after the ten years? • A = ? • P = 24,000 • r = 0.045 • t = 10 • A = 24000(1.045)10 • A = $37,271.27 • Ganiu has $37,271.27 after 10 years. • How much did he earn in interest alone? • $37,271.27 – 24,000 = • Ganiu earned $13,271.27 in interest.

  11. 3. Analyzing Compound Interest Formula • Ex 7: The following function represents how much money Lashawn has in her account after t years: A(t) = 6,500(1.17)t • What is the y-intercept? • A(t) = b(a)x The y-intercept is 6,500. • What is the constant multiplication factor? • A(t) = b(a)x The CMF is 1.17. • How much money does Lashawn invest at the beginning into her account? • The y-intercept is where t=0, the initial value. So, she started with $6,500. • What is the annual interest rate? • CMF = (1+r) = 1.17, so r = 0.17 or 17% • How much Lashawn have after twelve years? • A(t) = 6,500(1.17)12 = $42,770.44.

  12. 3. Analyzing Compound Interest Formula • Ex 8: The following function represents the number Chinese people living the city of Kunming: C(t) = 50,000(2)t • What is the y-intercept? • A(t) = b(a)x The y-intercept is 50,000. • What is the constant multiplication factor? • A(t) = b(a)x The CMF is 2. • How many people were initially in Kunming? • The y-intercept is where t=0, the initial value. So, the initial population was 50,000 people. • What is the annual growth rate in population? • CMF = (1+r) = 2, so r = 1 or 100% growth • How many people in Kunming after 10 years? • C(t) = 50,000(2)10 = 51,200,000 people

  13. 4. Calculating Compound Interest w Periodic Compounding Semiannual Quarterly Monthly Daily • Compound Interest Formula • with Periodic Compounding • A = P(1 + r/n)nt • A = Future Value or Final/Ending Value • : • P = Principal/Initial Value and Y-Intercept • r = Annual Interest Rate/Growth Rate • t = Years • n = Periods per Year (1, 2, 4, 12, 365)

  14. 4. Calculating Compound Interest w Periodic Compounding Semiannual Quarterly Monthly Daily • Ex 9: Devin invests $6,000 and earns 5% per year. How much will he have after six years • A(t) = 6000(1 + .05/n)(n●6) • if interest is compounded annually (n=1)? • A = 6000(1.05/1) (1●6) • A = 6000(1.05) 6 • A = $8,040.57 • if interest is compounded semi-annually (n=2)? • A = 6000(1 + 0.05/2)(2●6) • A = 6000(1.025)12 • A = $8,069.33 • if interest is compounded quarterly(n=4)? • A = 6000(1 + 0.05/4)(4●6) • A = 6000(1.0125)24 • A = $8,084.11

  15. 4. Calculating Compound Interest w Periodic Compounding Semiannual Quarterly Monthly Daily • Ex 9: Devin invests $6,000 and earns 5% per year. How much will he have after six years • A(t) = 6000(1 + .05/n)6n • if interest is compounded monthly (n=12)? • A = 6000(1 + 0.05/12)(12*6) • A = $8,094.11 • if interest is compounded daily (n=365)? • A = 6000(1 + 0.05/365)(365●6) • A = $8,098.99 Devin’s investment gets bigger if interest compounds more frequently

  16. 5. Simple vs. Compound Interest Linear vs. Exponential Functions Ex 10: Homer invests $1,000 at 10% for nine years P = 1,000 r = 0.10 t = 9 Simple Interest Compound Interest (annual) Asimple = P + Prt A = 1000 + 1000(0.10)(9) Asimple = $1,900 Acompound = P(1+r)t A = 1000(1.10)9 Acompound = $2,357.95

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