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Mathematics of Finance

Mathematics of Finance. The solutions to the examples in this presentation are based on using a Texas Instruments BAII Plus Financial calculator. $x today ? BUT WHY?

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Mathematics of Finance

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  1. Mathematics of Finance The solutions to the examples in this presentation are based on using a Texas Instruments BAII Plus Financial calculator.

  2. $x today? BUT WHY? Postponement of today’s opportunities for investments or consumption to the future would result in OPPORTUNITY COST. TVM captures and explains such lost opportunities. $x today or $x in future? A matter of Preference or Risk? Time Value of Money(TVM)

  3. Time Value of Money (TVM) “Time Value of Money” works through • Compounding • Future Value of a single amount • Future Value of an annuity • Future Value of uneven cash flows • Discounting • Present Value of a single amount • Present Value of an annuity • Present Value of uneven cash flows

  4. TVM capture the Opportunity Cost Through: • Compounding or determining the Future Values based on present $s, and • Discounting or determining the Present values based on future $s

  5. Compounding is paying interest both on principle and interest. For a 2-year savings commitment, the FV1 = PV + (PV x r) = PV (1 + r) FV2 = PV (1 + r) + PV (1 + i) x r = PV (1 + r) (1 + r) = PV (1 + r)2 FV1 = 100 + (100 x .05) = 100 (1 + .05) = 105 FV2 = 100 (1 + .05) + 100 (1 + .05) x .05 = 100 (1 + .05)2 = 110.25 Note: Present Value = Principal Compound Interest

  6. Future Value on a Timeline An investment of $100 today in a savings account that pays 5% interest, with interest compounded annually, will result in $110.25 at the end of year 2. 0 1 2 5% $100 $105 $110.25 FV PV

  7. Future Value, General Formula FVn = PV (1+r)nFV2 = $100(1.05)2 = $110.25 Lets Put The Calculator to Work!

  8. Future Value on TI BAII Plus Turn the calculator on and change the default setting by: Press 2nd I/Y Enter Press 1 ENTER These keystrokes will change the frequency of compounding to once per year

  9. Future Value on TI BAII Plus Always Press 2nd, then FV Enter Press 2 N 5 I/Y 100 PV CPT, FV $110.25

  10. Future Value Example How much money would be in your savings account after 6 years if you deposit $5,000 today and the bank pay an annual compound interest rate of 7%? 0 1 2 3 4 5 7% $5,000 FV5

  11. Future Value Solution • Calculator keystrokes: 1.07 yx 6  5000 = • Calculation based on the formula:FVn = PV (1+r)nFV5= $5,000 (1+ 0.07)6 = $7,503.65

  12. Future Value on TI BAII Plus Always Press 2nd, then FV Enter Press 6 N 7 I/Yr 5000 PV CPT, FV 7,503.65

  13. Present Value • Having FV = PV(1 + r)n then: • This represents the Discounting process or the process of determining the present value of a single future cash flow.

  14. Present Value (Graphic) If you need to have a $10,000down payment on a house 12 years from now, how much must you save today in an account that pays 7% interest, compounded annually? 0 3 6 9 12 7% $10,000 PV0

  15. Present Value on TI BAII Plus Always Press 2nd, then FV Enter Press Calculator keystrokes: 1.07 yx 12 = 1/x  10000 = 12 N 7 I/Yr 10000 FV CPT, PV 4,440.12

  16. Computing “n” or “i” knowing PV and FV • If John lends Linda $4,000 today for a return of $6,154.50 after 5 years, what rate of annual compound interest does he earn?

  17. Present Value on TI BAII Plus Always Press 2nd, then FV Enter Press 5 N 4000 +/-, FV 6154.50 PV CPT, I/Y 8.26%

  18. Frequency of Compounding General Formula: FVn,m = PV0[1 + (r/m)] mn n: Number of Years m: Number of Compounding per Year r: Annual Interest Rate FVn,m: Future Value at Year n PV0: Present value of amounts

  19. Frequency of Compounding Example: If your deposit of $3,000 in a savings account, paying monthlycompounded interest based on a 9% annual rate, is maintained for six years how much will be in the account at that time? PV = $3,000 r = 9%/12 = 0.75% per month n = 6 x 12 = 72 months

  20. Solution, based on formula: FV= PV (1 + r)n = 3,000(1.0075)72 = 5,137.66 Calculator Keystrokes: 1.0075 yx 72  3000 =

  21. Frequency of Compounding on (TI BAII Plus ) Always Press 2nd, then FV Enter Press 72 N 3000 PV 0.75 I/Y CPT, FV $5,137.66

  22. Annuities • An Annuity represents a series of equal payments (or receipts) over EQUAL intervals. • Annuities Can Be: • Ordinary (starting at the end of each period) or • Due (starting at the beginning of each period) • Example of Annuities Are: • Any kind of installment payment for retiring a loan • Insurance Premiums • Savings for Retirement

  23. A plan to save $4,000 a year at the end of each year for three years would result in how much savings, considering that your savings account pays 7% interest, compounded annually? FVA3 = $4,000(1.07)2 + $4,000(1.07)1 + $4,000(1.07)0 =$12,610 Future Value of an Ordinary Annuity -- FVA End of Year 0 1 2 3 7% $4,000$4,000$4,000 $4,280 $4,579.60 $12,859.60 = FVA3

  24. Future Value (TI BAII Plus) Always Press 2nd, then FV Enter Press 3 N 4000 PMT 7 I/Y CPT, FV $12,859.60

  25. Jamshid was approved for a business loan, which required $2,500 annual payment at the end of each next 4 years. The loan carried an annual interest rate of 6%. What was the amount of this loan? PVA3 = $2,500/(1.06)1 + $2,500/(1.06)2 + $2,500/(1.06)3 + $2,500/(1.06)4 =$8,662.76 Present Value of an Ordinary Annuity -- PVA Yearend 0 1 2 3 4 6% $2,500 $2,500$2,500 $2,500 $2,358.49 $2,224.99 $2,099.05 $1,980.23 $8,662.76 = PVA3

  26. Present Value on TI BAII Plus Always Press 2nd, then FV Enter Press 4 N 2500 PMT 6 I/Y CPT, PV $8,662.76

  27. PV of Unequal Cash Flows Your investment advisor recommends a security that provides $3,000, $5,000, and $7,000 respectively at the end of each of the next 3 years. If you require 12% return on this security, how much would you be willing to pay for it? 0 1 2 3 12% $3000 $5000 7,000 PV0

  28. Unequal Cash Flow Solution 0 1 2 3 12% $3,000 $5,000 $7,000 $2,678.57 $3,985.97 $4,982.46 $11,647.00 = PV0

  29. Unequal Cash Flow Solution (TI BAII Plus) Press CF 2nd, then CE/C Enter Press 0 ENTER 3000 ENTER 1 ENTER 5000 ENTER 1 ENTER 7000 ENTER 1 ENTER NPV 12 ENTER CPT $11,647.00 Frequency of the cash flows

  30. Computing Yield to Maturity DXL Industries bond is currently selling for $932.50. This bond is having a coupon interest rate of 11%, and will mature in 20 years. Considering that the bond’s face value is $1,000 and pays interest semiannually, what is the yield to maturity (YTM) on this bond?

  31. 0 1 2 ……….… 40 55 55 55 1000 YTM Solution on TI BAII Plus Enter Press Always Press 2nd, then FV 932.50 +/-, PV (.11  1000)  2= PMT 1000 FV 20  2 = N CPT, I/Y 5.945% for 6 months or 11.89% annually

  32. Check your command of the Concepts Click one of the following problems 1 2 3

  33. Problem #1 Morgan deposited $25,000 in a new savings account that is paying 9% annual interest rate compounded monthly. She will not be able to withdraw her deposit within the next 3 years. What will be the size of deposits in her account in 3 years?

  34. Problem 1 - Select one • $32,716.13 • $32,375.73 • $556,280.63 HELP!

  35. TI BAII Plus Solution to #1 Always Press 2nd, then FV Enter Press 3  12 = N 9  12 = I/Y 25,000 PV CPT, FV 32,716.13 FV = 25000 (1 + .0075)36 Click for Next Problem

  36. Problem #2 You currently receive $10,000 per year on a contract. You expect it to run another 7 years. Someone wants to buy the contract from you. If you can earn 12% on other investments of this quality, how much would you be willing to sell the contract for?

  37. Possible Answers - Problem 2 • $40,020.76 • $42,243.29 • $100,890.11 HELP!

  38. 0 1 2 3 4 … 7 10000 10000 10000 10000 ... 10000 TI BAII Plus Solution to #2 Always Press 2nd, then FV Enter Press 10000 PMT 7 N 12 I/Y CPT PV $45,637.57 PVA=10000/(1.12)1 + 10000/(1.12)2 +…+ 10000/(1.12)7 Click for Next Problem

  39. Problem #3 Thompson Corp. has issued a bond with a face value of $1,000. The bond carries a coupon interest rate of 6%, pays interest semi-annually, and will mature in 25 years. How much would you pay for this bond if your required return on similar investments is 8%?

  40. Possible Solutions - Problem 3 • $843.78 • $785.18 • $388.33 HELP!

  41. Enter Press 30 PMT 1000 FV 4 I/Y 50 N CPT, PV 0 1 2 ……….… 50 30 30 30 1000 TI BAII Plus Solution to #3 Always Press 2nd, then FV Click for Next Problem PVb

  42. Excellent! A job well done! Click for Next Problem

  43. Calculating the Future Value • When the frequency of compounding is more than once per year you should adjust both the discount rate, and the time. • Determine the future value of single amount. Click to return

  44. The Worth of a Contract • The worth of any asset is the present value of its future cash flows. • Terms such as “per year”, “annually”, “every year” are indications that the cash flows are annuities. Click to return

  45. Valuing a Bond • Consider the coupon payments as annuity and the face value of the bond as a single cash flow at maturity. • Remember that you should adjust the time, the discount rate, and the interest payments to reflect the semi-annual compounding. Click to return

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