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Interest

C ompound. Interest. Interest. Interest. Further Topics. &. Applications. Chapter 9. Calculate the…. Learning Objectives. After completing this chapter, you will be able to:. LO-1. … interest rate and term in c ompound i nterest applications. LO-2.

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Interest

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  1. Compound Interest Interest Interest Further Topics & Applications Chapter 9

  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. CPT ENTER Using your ON/OFF 2nd CF NPV IRR FV I/Y PV N QUIT SET DEL INS PMT % . 2 X xP/Y P/Y AMORT BGN CLR TVM . X Some key Keys! 1/ K x RAND x ( ) INV y X HYP SIN COS TAN X! 9 7 LN e X STAT BOND DATA 8 nPr - ROUND DEPR BRKEVN 6 % 4 nCr STO 5 + DATE ICONV PROFIT ANS 3 CLR WORK 1 RCL MEM FORMAT RESET Financial Calculator 2 +/- Advanced Business Analyst = CE/C 0 . . Review

  4. CPT CPT ENTER ON/OFF 2nd CF NPV IRR FV FV I/Y I/Y PV PV N N QUIT SET DEL INS PMT PMT % . 2 X xP/Y P/Y AMORT BGN CLR TVM . X 1/ K x RAND x ( ) INV y X HYP SIN COS TAN X! 9 7 LN e X STAT BOND DATA 8 nPr - ROUND DEPR BRKEVN 6 % 4 nCr STO 5 + DATE ICONV PROFIT ANS 3 CLR WORK 1 RCL MEM FORMAT RESET 2 +/- Advanced Business Analyst = CE/C 0 . . Find the following KEYS: • Number of compoundings (for lump payments) • Number of payments (for annuities) The nominal interest rate (Interest/Year) Present Value or initial(first) lump sum Represents the Periodic Annuity Payment (used in chapter 10) FutureValue or terminal(last) lump sum Tells the calculator to compute (CPT)

  5. CPT ENTER ON/OFF 2nd 2nd CF NPV IRR FV I/Y I/Y PV N QUIT SET DEL INS PMT % . 2 X xP/Y P/Y AMORT BGN CLR TVM . X 1/ K x RAND x ( ) INV y X HYP SIN COS TAN X! 9 7 LN e X STAT BOND DATA 8 nPr - ROUND DEPR BRKEVN 6 % 4 nCr STO 5 + DATE ICONV PROFIT ANS 3 CLR WORK 1 RCL MEM FORMAT RESET 2 +/- Advanced Business Analyst = CE/C 0 . . Find the following KEYS: Previously, it was noted that …it is rare for interest to be compounded only once per year! However, we can now input the number of compoundings per year into the financial calculator. This can be performed by using the symbol P/Y To access this symbol use: …and you will see

  6. The 12 is a default setting P/Y= 12 Appears automatically C/Y= To access use: 12 C/Y This display is referred to as “the worksheet”. … represents the number of Payments per year P/Y … represents the number of Compoundings per year C/Y Note: You can override these values by entering new ones! …more

  7. CPT ENTER ON/OFF 2nd 2nd CF NPV IRR FV I/Y PV N QUIT SET DEL INS PMT % . 2 X xP/Y P/Y AMORT BGN CLR TVM . X 1/ K x RAND x x ( ) INV y y X HYP SIN COS TAN X! 9 7 LN e X STAT BOND DATA 8 nPr - ROUND DEPR BRKEVN 6 % 4 nCr STO 5 + DATE ICONV PROFIT ANS 3 CLR WORK 1 RCL MEM FORMAT RESET 2 +/- +/- Advanced Business Analyst = CE/C 0 . . Useacalculator and algebraic sequencing Find the following KEYS: The Power function Key.Used to calculate the value of exponents. Usedto access symbols located “above” another key, i.e. its acts like the SHIFT key on a computer keyboard. Changes the sign of the data value of the number being displayed.

  8. CPT ENTER ON/OFF 2nd 2nd 2nd CF NPV IRR FV I/Y PV N QUIT SET DEL INS PMT % . 2 X xP/Y P/Y AMORT BGN CLR TVM . X 1/ K x RAND x x x ( ) INV y y y X HYP SIN COS TAN X! 9 7 LN e X STAT BOND DATA 8 nPr - ROUND DEPR BRKEVN 6 % 4 = = nCr STO 5 + DATE ICONV PROFIT ANS 3 CLR WORK 1 RCL MEM FORMAT RESET 2 +/- Advanced Business Analyst = CE/C 0 . . Useacalculator and algebraic sequencing Find the following KEYS: Some calculators have the yx symbol above the calculator key. The key stroke sequence to evaluate an EXPONENT that is… (1.04)8 is… 1.04 8 Positive 1.368569 Negative (1.04)-8 is… 1.04 8 0.73069

  9. CPT ENTER ON/OFF 2nd CF NPV IRR FV I/Y PV N QUIT SET DEL INS PMT % . 2 X xP/Y P/Y AMORT BGN CLR TVM . X 1/ K x RAND x ( ) INV y X HYP SIN COS TAN X! 9 7 LN e X STAT BOND DATA 8 nPr - ROUND DEPR BRKEVN 6 % 4 nCr STO STO 5 + DATE ICONV PROFIT ANS 3 CLR WORK 1 RCL RCL MEM FORMAT RESET 2 +/- Advanced Business Analyst = CE/C 0 . . Useacalculator and algebraic sequencing Find the following KEYS: UsedtoStore or save displayed values. UsedtoRecall the saved values. This calculator can store up to 10 values. Therefore, the calculator must be informed as to where the values are to be stored. Let’s Practise

  10. CPT ENTER ON/OFF 2nd Using the key CF NPV IRR FV I/Y PV N QUIT SET DEL INS PMT % . 2 X xP/Y P/Y AMORT BGN CLR TVM . X 1/ K x RAND x ( ) INV y X HYP SIN COS TAN X! ..choose from 0 - 9 9 7 LN e X STAT BOND DATA 8 nPr - ROUND DEPR BRKEVN 6 % 4 nCr STO STO STO 5 + DATE ICONV PROFIT ANS 3 CLR WORK 1 RCL RCL MEM FORMAT RESET 2 +/- Advanced Business Analyst = CE/C CE/C 0 0 0 . . Useacalculator and algebraic sequencing e.g. you want to store the value ’45’. The key stroke sequence ‘to store’ is: 45 …’clear’ display The key stroke sequence ‘to recall’ is: …where you stored the value!

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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%

  17. 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%

  18. 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

  19. 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

  20. 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

  21. CPT ENTER ON/OFF 2nd 2nd FV 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%

  22. 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

  23. 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)

  24. 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

  25. CPT ENTER ON/OFF 2nd 2nd FV 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

  26. 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

  27. 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

  28. 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

  29. CPT ENTER ON/OFF 2nd 2nd FV 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

  30. 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? FV = PV = $1 FV = $3 i = .072/ 2 =.036 Note: When no FV or PV given use the easiest number! $1

  31. 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

  32. CPT ENTER ON/OFF 2nd 2nd FV 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

  33. 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.

  34. CPT ENTER ON/OFF 2nd 2nd FV 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

  35. 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

  36. Nominal & Effective Rates LO-2 of Interest

  37. 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

  38. 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%

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

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

  41. Comparisons

  42. 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…

  43. 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

  44. 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%

  45. 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%

  46. 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%

  47. 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!

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

  49. Example Clarifying Terminology Compounded quarterly Refers tocompoundingfrequency …NOT ‘per quarter’! 4%compounded quarterly equals per quarter 1%for each quarter Refers to the compounding period!

  50. 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

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