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Interest

C ompound. Interest. Interest. Interest. Further Topics. &. Applications. Chapter 9. PowerPoint Presentation Prepared by C Quinn, Seneca College. Calculate the…. Learning Objectives. After completing this chapter, you will be able to:. LO-1.

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Interest

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  1. Compound Interest Interest Interest Further Topics & Applications Chapter 9 PowerPoint Presentation Prepared by C Quinn, Seneca College.

  2. Calculate the… Learning Objectives After completing this chapter, you will be able to: LO-1 … interest rate and term in compound interest applications LO-2 …effective interest rate when given a nominal interest rate LO-3 … equivalent interest rate at another compounding frequency when given a nominal interest rate

  3. PV = FV(1 + i)-n Formula Q Solve… Extract necessary data... Step Step LO-1 What amount must you invest now at 6%compounded daily to accumulate to$5000 after 1 year? n = 1 x 365 = 365 i= .06/365 FV = $5000

  4. PV = FV(1 + i)-n Formula n = 1x 365=365i= .06/365FV = $5000 Solve… . . x y X Step = = = = + +/- Financial .06 365 1 4,708.85 = $4708.85 365 5000

  5. CPT n = 1x 365=365i= .06/365FV = $5000 ON/OFF 2nd 2nd FV I/Y PV N PMT X = P/Y QUIT 6 365 1 4,708.85 5000 0 365 = $4708.85

  6. FV = PV(1 + i)n Formula Q Extract necessary data... Step See next slide for steps in developing this formula Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. Whatrate of interest do you need to accomplish your goal? Assume interest is compounded annually. Solving for i this becomes… i = (FV/PV)1/n - 1 PV = $5000 n = 7 x 1 = 7 FV = $9000

  7. i = (FV/PV)1/n - 1 Formula Formula Developing the Back to Solving the problem FV =(1 + i)n PV PV PV FV FV FV PV PV PV Step 1/n 1/n 1. Base formula FV = PV(1 + i)n FV = PV(1 + i)n 2. Divide both sides by PV 3. To removen as exponent raise both sides to the power 1/n 1/n =(1 + i)n 1/n 4. Complete reducing =1 + i i= - 1 i = (FV/PV)1/n - 1

  8. i = (FV/PV)1/n - 1 Formula PV = $5000 n = 1 x 7 = 7 FV = $9000 . 1/ x Solve… . x y Step - = = = STO RCL 0 0 Financial 7 .0876 1.0876 Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. What rate of interest do you need to accomplish your goal? Assume interest is compounded annually. 9000 1 5000 8.76%

  9. CPT ENTER ON/OFF 2nd FV PV N PMT P/Y QUIT PV = $5000 n = 1 x 7 = 7 FV = $9000 5000 Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. What rate of interest do you need to accomplish your goal? Assume interest is compounded annually. I/Y = 8.7596 7 9000 1 0 8.76%

  10. Q Find the Nominal Rate if the Periodic Rate is: Note The formula gives the Periodic Rate (i) but in practice we usually refer to the Nominal(annual)Rate 1.3 * 12= 1.3% per month 15.6% per year 1.9% per quarter 1.9 * 4= 7.6% per year 0.05041 * 365= 0.05041%daily 18.39965 = 18.4% per year

  11. i = (FV/PV)1/n - 1 Formula Q Solve… Extract necessary data... Step Step Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. Whatrate of interest do you need to accomplish your goal? Assume interest is compounded monthly. PV = $5000 FV = $9000 n = 7 x 12 = 84

  12. i = (FV/PV)1/n - 1 Formula Solve… . Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. What rate of interest do you need to accomplish your goal? Assume interest is compounded monthly. . 1/ x x y X Step - = = = = STO RCL 0 0 Financial PV = $5000 n = 7 x 12 = 84 FV = $9000 84 1 .0070 .0070 .0843 1 9000 .0070 per month 5000 12 8.43% nominal rate

  13. CPT ENTER ON/OFF 2nd 2nd FV I/Y PV N PMT Suppose you have $5000 and you want it to grow to $9000 in 7 years so you could buy a car. What rate of interest do you need to accomplish your goal? Assume interest is compounded monthly. X = P/Y QUIT Financial 5000 I/Y = 8.4264 12 7 12 9000 0 8.43%

  14. FV = PV(1 + i)n Formula Q Extract necessary data... Step See next slide for steps in developing this formula Suppose you have $19,500 and you want it to grow to $23,000 so you could buy a car. You know that you can get3% interestcompounded daily. How long will it take to reach your goal? Solving for n this becomes… n = ln(FV/PV) / ln(1+ i) PV = $19500 i = .03/ 365 =.0001 FV = $23000

  15. Formula n = ln(FV/PV) / ln(1+ i) Formula Developing the Back to Solving the problem FV =(1 + i)n PV PV PV FV FV PV PV Step ln (1 + i) ln 1. Base formula FV = PV(1 + i)n 2. To isolate n, divide both sides by PV FV = PV(1 + i)n 3. Take the natural logarithm (LN or ln) of both sides =n * ln(1 + i) ln 4. Complete reducing n = n = ln(FV/PV) / ln(1+ i)

  16. n = ln(FV/PV) / ln(1+ i) Formula PV = $19500 i = .03/ 365 =.0001 FV = $23000 Solve… . . . Suppose you have $19,500 and you want it to grow to $23,000 so you could buy a car. You know that you can get 3% interest compounded daily. How long will it take to reach your goal? . . . Step LN LN = = = = STO + RCL 0 0 … Financial .03 23000 0.0001 0.1651 2008.55 365 19500 1 or 2009days

  17. CPT ENTER ON/OFF 2nd 2nd FV I/Y PV N PMT Suppose you have $19,500 and you want it to grow to $23,000 so you could buy a car. You know that you can get 3% compounded daily. How long will it take to reach your goal? P/Y QUIT Conversion of Days Financial 19500 N = 2008.55 3 23000 365 0 2009days

  18. Conversion of Days . 5 Years . X - = = = 6 Months 2008.55 365 0.50288 2008.55 6.0346 5.50288 5.50288 (years) (years) 5 12 6.0346 Months 5 Years 6 months

  19. n = ln(FV/PV) / ln(1+ i) Formula Q Solve… Extract necessary data... Step Step How long will it take at 4.2%compounded monthly for $800 to grow to$1100? PV = $800 FV = $1100 i = .042/ 12 =.0033

  20. n = ln(FV/PV) / ln(1+ i) Formula Solve… . . . How long will it take at 4.2%compounded monthly for $800 to grow to$1100? . . . Step LN LN = = = = STO + RCL 0 0 Financial PV = $800 FV = $1100i = .042/ 12 =.0033 .042 1100 12 800 0.0035 .3185 91.1459 1 91.1months 7 years 7.1 months

  21. CPT ENTER ON/OFF 2nd 2nd FV I/Y PV N PMT How long will it take at 4.2%compounded monthly for $800 to grow to$1100? P/Y QUIT Financial 800 N = 91.1459 4.2 12 1100 0 91.1months 7 years 7.1 months

  22. n = ln(FV/PV) / ln(1+ i) Formula Q Solve… Extract necessary data... Step Step How long will it take for money to triplewhen invested at 7.2%compounded semi-annually? PV = $1 FV = $3 i = .072/ 2 =.036 Note: When no FV or PV given use the easiest number! $1

  23. n = ln(FV/PV) / ln(1+ i) Formula Solve… . . . How long will it take for money to triple when invested at 7.2%compounded semi-annually? . . . Step LN LN = = = = STO + RCL 0 0 Financial PV = $1 FV = $3 i = .072/ 2 =.036 .072 3 31.0631 1.0986 0.0354 2 1 1 31.1sa periods 15 years 3.2 months

  24. CPT ENTER ON/OFF 2nd 2nd FV I/Y PV N PMT How long will it take for money to triple when invested at 7.2%compounded semi-annually? P/Y QUIT Financial 1 N = 31.0631 7.2 2 3 0 31.1sa periods 15 years 3.2 months

  25. n = ln(FV/PV) / ln(1+ i) Formula Q Solve… Extract necessary data... Step Step How long will it take for money to doublewhen invested at 10.0%compounded annually? Use $1 FV = $2 i = .010/ 2 =.005 PV = $1 Note: This results in a “rule ofthumb” that you can use for comparisons or ballparking your other solutions.

  26. CPT ENTER ON/OFF 2nd 2nd FV I/Y PV N PMT How long will it take for money to double when invested at 10.0%compounded annually? P/Y QUIT Financial 1 N = 7.2725 10 2 1 0 7.3 Years

  27. Rule of 72 …Rule of thumb to quickly estimate the time needed to double your money! 72 Doubling time (in years) = % annual rate of return … investment earning 9%compounded annually will double in approx… … investment earning 12%compounded annually will double in approx… … investment earning 4%compounded annually will double in approx… 72/9 = 8 years 72/12 = 6 years 72/4 = 18 years

  28. Nominal & Effective Rates LO-2 of Interest

  29. Nominal Rates Effective Rates Principal j … or Stated Rate is the rate on which the bank calculates interest f … is the annually compounded rate that produces the same Future Value as the given nominal rate (j) Effective Rate = Interest for 1 year

  30. NominalRatesof Interest Compared Invested for One Year Beginning Balance Nominal Rate Compounding Period Ending Balance Effective Rate APY of Interest Annual Semiannual Quarterly Daily 6.00% $1,060.00 6.09% $1,060.90 $1,000 + 6% 6.14% $1,061.36 $1,061.83 6.18%

  31. f = (1 + i)m - 1 Formula Q Solve… Step Step Financial What is the effective rate of interest corresponding to 9%compounded daily? Extract necessary data... j= .09 m= 365 i = .09/ 365 =.0002 Solve… f = (1 + .09/365)365 – 1 = .0942 = 9.42%

  32. CPT ENTER ON/OFF 2nd 2nd FV I/Y PV N Solve… PMT What is the effective rate of interest corresponding to 9%compounded daily? Step P/Y QUIT Financial 100 FV = 109.4162 Use $100 365 9 365 0 109.4162 - 100 = 9.42%

  33. Comparisons

  34. Q Financial Comparisons Which is the most attractive of the following interest rates offered on 3 year GICs? A. 5.8%compounded semi-annually B. 5.68%compounded monthly C. 5.4%compounded daily Tip Faster if you use Comparing…

  35. Comparisons Approach Invest $100at each of the rates for 1 year and compare A. 5.8%compounded semi-annually B. 5.68%compounded monthly C. 5.4%compounded daily

  36. CPT ENTER ON/OFF 2nd 2nd FV I/Y PV N PMT P/Y QUIT Comparisons 100 Invest $100 for 1 year FV = 105.88 2 A. 5.8%compounded semi-annually 5.8 2 0 105.88 - 100 = 5.88%

  37. CPT ENTER ON/OFF 2nd 2nd FV I/Y N P/Y QUIT Comparisons Invest $100 for 1 year 5.68 FV = 105.83 12 B. 5.68%compounded monthly 12 105.83 - 100 = 5.83%

  38. CPT ENTER ON/OFF 2nd 2nd FV I/Y N P/Y QUIT Comparisons Invest $100 for 1 year 5.4 FV = 105.55 365 C. 5.4%compounded daily 365 105.55 - 100 = 5.55%

  39. Comparisons Results 5.88% A. 5.8%compounded semi-annually 5.83% B. 5.68%compounded monthly 5.55% C. 5.4%compounded daily the5.8%compounded semi-annually provides for the best rate of return on investment!

  40. LO-3 Interest rates that produce the same future value after one year Equivalent Interest Rates

  41. Q Explanation Equivalent Interest Rates For a given interest rate of 8%compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding? Formula i2 = (1+i1)m1/m2 - 1 i1andm1represent thegivenvalues ofiandm i2andm2represent theequivalentvalues ofiandm

  42. Represented by Value Given Equivalent Explanation Equivalent Interest Rates i i1 For a given interest rate of 8% m1 m compounded quarterly what is the equivalent nominal rate of interest i2 m2 with monthly compounding?

  43. Q i2 = (1+ ) - 1 .08 /4 4 / 12 Note… Equivalent Interest Rates For a given interest rate of 8% pa, compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding? Formula i2 = (1+i1)m1/m2 - 1 i2 = .0066 nominal rate = j = .0066 * 12 = 7.95%

  44. Formula / 4 12 i2 = (1+ ) - 1 .08 /4 4 / 12 Solve… Equivalent Interest Rates Note: is a fraction, change it to a decimal number and STORE it before continuing. Since the exponent = 0.3333

  45. . . For a given interest rate of 8%, compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding? . . x y i2 = (1+ ) - 1 .08 /4 4 / 12 X - = = = = = STO + RCL 0 0 Financial Equivalent Interest Rates 1.0066 0.0066 0.0795 0.3333 1.02 4 12 .08 1 4 12 1

  46. Calculate the FV of $100 invested for 1 year as given CPT ENTER ON/OFF 2nd 2nd FV I/Y PV N For a given interest rate of 8%, compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding? Step P/Y QUIT 100 FV = 108.24 4 4 8 FV = 108.24 - 100 = 8.24%

  47. CPT Compute the value of i2that also makes $100 grow to the same FV in m2 compoundings ENTER 2nd 2nd I/Y N For a given interest rate of 8%, compounded quarterly, what is the equivalent nominal rate of interest with monthly compounding? Step P/Y QUIT 12 I/Y = 7.9473 12 7.95% compounded monthly = 8% compounded quarterly

  48. This completes Chapter 9

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