1. Calculating Simple Interest

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 • t = Number of periods (usually years) the money is being borrowed • Simple Interest = Principal times interest times years • Simple Interest = P(r)(t) • Total Owed = P + P(r)(t)

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 = • t = • Total = P + P(r)(t) $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 = • t = • Total = P + P(r)(t) $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? • The coefficient is 6,500, so the y-intercept is 6,500. • What is the constant multiplication factor? • The base is 1.17, so 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 people living the Chinese city of Kunming: C(t) = 50,000(2)t • What is the y-intercept? • Coefficient is 50,000, so the y-intercept is 50,000. • What is the constant multiplication factor? • The base is 2, so 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: Henok 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 annually (n=1)? • 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: Henok 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 Henok’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

17. 6. Finding the Initial Value Of Exponential Growth/Interest • Compound Interest Formula with Periodic Compounding • A = P(1 + r/n)nt • To find theInitial Value, we need to solve for P • We will be given: A, r, n, t

18. 6. Finding the Initial Value Of Exponential Growth/Interest Ex 1:The future value of an investment at the end of five years is $25,000. What is the initial investment if you earned 10% interest, compounded annually? A = 25,000 P = ? r = 0.10 n = 1 (annually) t = 5 (years) 25000 = P(1 + 0.1/1)(1*5) 25000 = P(1.61051) 25000 = 1.61051P 1.61051 1.61051 $15,523.03 = P This initial value was $15,523.03 Check: 15523.03(1.1)5 = 25,000 Go to five decimal places

19. 6. Finding the Initial Value Of Exponential Growth/Interest Ex 2:The future value of an investment at the end of seven years is $35,000. What is the initial investment if you earned 5% interest, compounded quarterly? A = 35,000 P = ? r = 0.05 n = 4 (quarterly) t = 7 (years) 35000 = P(1 + 0.05/4)(4*7) 35000 = P(1.41599) 35000 = 1.41599P 1.41599 1.41599 $24,717.69 = P This initial value was $24,717.69 Check: 24717.69(1.0125)28 = 35,000 Go to five decimal places

20. 6. Finding the Initial Value Of Exponential Growth/Interest Ex 3:You decide you need $50,000 to go to graduate school in five years. You find an investment that pays 12% interest, compounded monthly. How much money will you need to invest today, to go to graduate school in five years ? A = 50,000 P = ? r = 0.12 n = 12 (monthly) t = 5 (years) 50000 = P(1 + 0.12/12)(12*5) 50000 = P(1.81670) 50000 = 1.81670P 1.81670 1.81670 $27,522.48 = P This initial value was $27,522.48 Check: 27522.48(1.01)60 = 50,000 Go to five decimal places

21. 7. What is Annual Percentage Yield? Comparing APY vs. APR • The Annual Percentage Rate (APR)is the rate of interest earned per year. • This is the rate that we’ve used in all of our problems so far • The Annual Percentage Yield (APY) is the actual annual percent earned when you calculate for compounding • APY ≥ APR • Both are rates per one year

22. 7. What is Annual Percentage Yield? Comparing APY vs. APR Ex 1: Annual Percentage Rate (APR) is 10% and interest is compounded annually. What is APY? r = APR = 0.10 n = 1 (annually) t = 1 (years) 1 + APY = (1 + r/n)(n*t) 1 + APY = (1 + 0.10/1)(1*1) 1 + APY = 1.10000 -1 -1 APY = 0.10 = 10% If annual compounding APY = APR Go to five decimal places

23. 7. What is Annual Percentage Yield? Comparing APY vs. APR Ex 2: Annual Percentage Rate (APR) is 10% and interest is compounded quarterly. What is APY? r = APR = 0.10 n = 4 (annually) t = 1 (years) 1 + APY = (1 + r/n)(n*t) 1 + APY = (1 + 0.10/4)(4*1) 1 + APY = 1.10381 -1 -1 APY = 0.10381 = 10.381% 10.38% > 10% APY > APR bcs of compounding Go to five decimal places

24. 7. What is Annual Percentage Yield? Comparing APY vs. APR Ex 3: Annual Percentage Rate (APR) is 10% and interest is compounded daily. What is APY? r = APR = 0.10 n = 365 (annually) t = 1 (years) 1 + APY = (1 + r/n)(n*t) 1 + APY = (1 + 0.10/365)(365*1) 1 + APY = 1.10516 -1 -1 APY = 0.10516 = 10.516% 10.52% > 10% APY > APR bcs of compounding Round to five decimal places

25. 7. What is Annual Percentage Yield? Comparing APY vs. APR • Why does it matter? • Loans will advertise APR (even though you pay higher APY because of compounding) • Investments will advertise APY (since it is higher than APR)