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

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$x today $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 $x in future? $x today or $x in future? A matter of Preference or Risk? Time Value of Money(TVM)

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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

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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 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 TVM capture the Opportunity Cost

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

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$100 today in a savings account that pays 5 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. Future Value on a Timeline 0 1 2 $100 FV 5% $105$110.25 PV

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FV n PVFV 2 $100 $110.25 FV n = PV (1+r) n FV 2 = $100 (1.05) 2 = $110.25 Future Value, General Formula Lets Put The Calculator to Work!

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

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

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$5,000 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%? Future Value Example 5 6 0 1 2 3 4 5 6 $5,000 FV 6 7%

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Calculator keystrokes : 1.07 y x 6 5000 = Future Value Solution FV n FV 5 $7,503.65 Calculation based on the formula: FV n = PV (1+r) n FV 5 = $5,000 (1+ 0.07) 6 = $7,503.65

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7 5000 I/Yr PV CPT, FV N6 Always Press 2 nd, then FV PressEnter 7,503.65 Future Value on TI BAII Plus

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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.

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

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PV 0 FV$10,000 $4,440.12 PV 0 = FV / (1+r) 12 = $10,000 / (1.07) 12 = $4,440.12 Present Value (Formula) 6 0 3 6 9 12 PV 0 $10,000 7%

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Present Value on TI BAII Plus 7 10000 I/Yr FV CPT, PV N12 Always Press 2 nd, then FV PressEnter 4,440.12 Calculator keystrokes: 1.07 y x 12 = 1/x 10000 =

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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?

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4000 6154.50 +/-, PV FV CPT, I/Y N5 Always Press 2 nd, then FV PressEnter 9.00% Present Value on TI BAII Plus

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General Formula: PV 0 FV n,m = PV 0 [1 + (r/m)] mn n: Number of Years m: Number of Compounding per Year r: Annual Interest Rate FV n,m : Future Value at Year n PV 0 PV 0 : Present value of amounts Frequency of Compounding

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Example: If your deposit of $3,000 in a savings account, paying monthly compounded 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

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

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Frequency of Compounding on (TI BAII Plus ) 3000 0.75 PV I/Y CPT, FV N72 Always Press 2 nd, then FV PressEnter $5,137.66

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Annuities 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 u An Annuity u An Annuity represents a series of equal payments (or receipts) over EQUAL intervals.

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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? FVA 3 FVA 3 = $4,000(1.07) 2 + $4,000(1.07) 1 + $4,000(1.07) 0 $12,610 = $12,610 Future Value of an Ordinary Annuity -- FVA 0 1 2 3 $4,000 $4,000 $4,000 $12,859.60 = FVA $12,859.60 = FVA 3 End of Year 7% $4,28 0 $4,579.60

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Future Value (TI BAII Plus) 4000 7 PMT I/Y CPT, FV N3 Always Press 2 nd, then FV PressEnter $12,859.60

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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? PVA 3 PVA 3 = $2,500/(1.06) 1 + $2,500/(1.06) 2 + $2,500/(1.06) 3 + $2,500/(1.06) 4 $8,662.76 = $8,662.76 Present Value of an Ordinary Annuity -- PVA $2,500 $2,500 3 0 1 2 3 4 Yearend 6% $8,662.76 = PVA 3 $2,358.49 $2,224.99 $2,099.05 $1,980.23

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Present Value on TI BAII Plus 2500 6 PMT I/Y CPT, PV N4 Always Press 2 nd, then FV PressEnter $8,662.76

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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? PV of Unequal Cash Flows 0 1 2 3 $3000 $5000 7,000 $3000 $5000 7,000 PV 0 12%

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

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

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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?

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

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

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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?

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

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

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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?

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Possible Answers - Problem 2 $40,020.76 $45,637.57 $100,890.11 HELP!

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TI BAII Plus Solution to #2 PMT PVA=10000/(1.12) 1 + 10000/(1.12) 2 +…+ 10000/(1.12) 7 10000 I/Y N PV 7 12 CPT Always Press 2 nd, then FV PressEnter 0123 4 … 7 10000100001000010000... 10000 $45,637.57 Click for Next Problem

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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%?

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Possible Solutions - Problem 3 $843.78 $785.18 $388.33 HELP!

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

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Excellent! A job well done! Click for Next Problem

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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

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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

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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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