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Basic Definitions TVM is one of the basics of any type of finance Used in security analysis to determine the value of an investment today, based upon its expected returns Also used to report returns, determine investment strategy Used in corporate finance extensively to assess potential projects and financing activities

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Basic Definitions Opportunity cost – the cost of foregone opportunities. I lend my friend $1000 for a year instead of investing it at 5%. My opportunity cost for being a good friend is $50. Present Value – earlier money on a time line Future Value – later money on a time line Interest rate – “exchange rate” between earlier money and later money Number – of compounding periods, can be years, months, days, weeks

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TI BA II Plus Almost every key has 2 functions You can toggle between these functions using the 2 nd key Primary functions are listed on the key, secondary are listed in white above the key To access the secondary function, hit 2 nd and then the key You will notice that when you press the 2 nd key, a small “2 nd ” symbol will appear in the upper left corner

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TI BA II Plus – Clearing the memory CE/C – clears the display 2 nd MEM 2 nd CLR Work – clears all 10 memory locations and the display 2 nd QUIT 2 nd CLR TVM – clears the TVM worksheet

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TI BA II Adjustments Factory setting is 12 I/Y – we want to start with annual compounding or 1 I/Y. Press 2 nd I/Y I enter 2 nd CPT Changing decimal places 2 nd. (decimal point) # of digits enter I hit 2 nd,., 4, enter

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Texas Instruments BA-II Plus Calculator Keys FV = future value PV = present value I/Y = period interest rate P/Y must equal 1 for the I/Y to be the period rate Interest is entered as a percent, not a decimal N = number of periods PMT = payment Remember to clear the registers (CLR TVM on TI) after each problem !

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Steps to Solve Time Value of Money Problems 1. Read problem thoroughly 2. Determine which variable is missing A minimum of 3 are needed to solve 3. Create a time line 4.Determine if solution involves a single CF, annuity stream(s), or mixed flow 5. Solve the problem 6.Check your calculation

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Inflows & out flows The calculator is “smart”, but you must tell it the direction that funds are going Each set-up problem, must have positive and negative cash flows.

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Solving for FV If you invest $1000 at 7% what will it be worth in 2 years ? What is the missing variable? What variables are given? What does the time line look like?

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What is the FV of $1000 invested for 2 yrs @ 7%? N:2 periods (enter as 2) I/Y:7% interest rate per period (enter as 7 NOT.07) PV:$1,000 (enter as negative as you have “less”) PMT:Not relevant in this situation (enter as 0) FV:Compute (Resulting answer is positive) 2 7 -1,000 0 1,144.90

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FV example $10,000 5 years Julie Miller wants to know how large her deposit of $10,000 today will become at a compound annual interest rate of 10% for 5 years????

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Solution The result indicates that a $10,000 investment that earns 10% annually for 5 years will result in a future value of $16,105.10. Solving the FV Problem 5N5N5N5N 10 I/Y -10,000 PV 0 PMT CPT FV = $16,105.10

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Future Value Suppose your company expects to increase unit sales of widgets by 15% per year for the next 5 years. If you currently sell 3 million widgets in one year, how many widgets do you expect to sell in 5 years? 5 N 15 I/Y 3,000,000 PV CPT FV = -6,034,072 units (remember the sign convention)

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Solving for PV $1,000 2 years. Assume that you need $1,000 in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 1000 FV 2 N 7 I CPT PV -$873.44

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Present Value –Example Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today? PV = 10,000 / (1.07) 1 = 9345.79 Calculator 1 N 7 I/Y 10,000 FV CPT PV = -9345.79

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A Zero-coupon Bond You can purchase a zero-coupon bond today for $200. It will mature for $1000 in 14 years. What rate of return will you earn if you buy this bond? 200 +/- PV 1000 FV 14 N CPT I/Y 12.1828 %

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Checking your solution The best way to proof your solutions is to use your answer and solve for one of the given variables. In this case: 12.1828 I/Y 14 N 1000 FV CPT PV 200

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Types of Annuities What is an annuity? Ordinary Annuity Ordinary Annuity: Payments or receipts occur at the end of each period. Annuity Due An Annuity Annuity Due: Payments or receipts occur at the beginning of each period.An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

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Adjusting calculator for beginning & end The default setting is end ( or ordinary annuity) To change to beginning (or annuity due) Hit 2 nd BGN 2 nd SET Good idea to always go back to END when you are done!

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Annuity solution Suppose you were offered an investment which will pay you $1000 per year for 10years. If you can earn a rate of 9% per year on investments of similar risk, how much should you be willing to pay for this annuity? What variable are we solving for? What have we been given? What does the time line look like?

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Annuity example – car loan You are going to buy a car for $35,000. You will make 48 monthly payments, with the first payment occurring TODAY. If the loan has an 8.5% APR, how much are your payments?

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Annuity Solution – car loan 2 nd BGN 2 nd SET 35000 PV 48 N 8.5/12 I/Y CPT PMT $856.62

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What if payments were due at the end of the month and NOT starting today? What would the PMT be? If you are making a payment, which option is better? If you are receiving a payment which option is better?

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Present Values – Example 2 You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today? N = 17 I/Y = 8 FV = 150,000 CPT PV = -40,540.34 (remember the sign convention)

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Present Values – Example Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest? N = 10 I/Y = 7 FV = 19,671.51 CPT PV = -10,000

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Present Value – Important Relationship ! For a given interest rate – the longer the time period, the lower the present value What is the present value of $500 to be received in 5 years? 10 years? The discount rate is 10% 5 years: N = 5; I/Y = 10; FV = 500 CPT PV = -310.46 10 years: N = 10; I/Y = 10; FV = 500 CPT PV = -192.77

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Present Value – Important Relationship ! For a given time period – the higher the interest rate, the smaller the present value What is the present value of $500 received in 5 years if the interest rate is 10%? 15%? Rate = 10%: N = 5; I/Y = 10; FV = 500 CPT PV = -310.46 Rate = 15%; N = 5; I/Y = 15; FV = 500 CPT PV = -248.58

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Quick Quiz Suppose you need $15,000 in 3 years. If you can earn 6% annually, how much do you need to invest today? If you could invest the money at 8%, would you have to invest more or less than at 6%? How much?

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Solutions #1: 15000 FV 3 N 6 I CPT PV $12,594.29 #2: You would have to invest less – to be exact your investment was be $11,907.48 Just enter 8 I/Y and hit CPT PV

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Discount Rate – Example You are looking at an investment that will pay $1200 in 5 years if you invest $1000 today. What is the implied rate of interest? Calculator – the sign convention matters!!! N = 5 PV = -1000 (you pay 1000 today) FV = 1200 (you receive 1200 in 5 years) CPT I/Y = 3.714%

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Discount Rate – Example Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest? N = 6 PV = -10,000 FV = 20,000 CPT I/Y = 12.25%

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Discount Rate – Example Suppose you have a 1-year old son and you want to provide $75,000 in 17 years towards his college education. You currently have $5000 to invest. What interest rate must you earn to have the $75,000 when you need it? N = 17 PV = -5000 FV = 75,000 CPT I/Y = 17.27%

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Number of Periods – Example You want to purchase a new car and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car? I/Y = 10 PV = -15,000 FV = 20,000 CPT N = 3.02 years

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Number of Periods – Example Suppose you want to buy a new house. You currently have $15,000 and you figure you need to have a 10% down payment plus an additional 5% in closing costs. If the type of house you want costs about $150,000 and you can earn 7.5% per year, how long will it be before you have enough money for the down payment and closing costs?

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Number of Periods – Example Continued How much do you need to have in the future? Down payment =.1(150,000) = 15,000 Closing costs =.05(150,000 – 15,000) = 6,750 Total needed = 15,000 + 6,750 = 21,750 Compute the number of periods PV = -15,000 FV = 21,750 I/Y = 7.5 CPT N = 5.14 years

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Quick Quiz When might you want to compute the number of periods? Suppose you want to buy some new furniture for your family room. You currently have $500 and the furniture you want costs $600. If you can earn 6%, how long will you have to wait if you don’t add any additional money?

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QQ solution #1 For many reasons: At the rate I am saving, when can I retire? When can I afford to buy something at my current savings rate? #2 500 -/+ PV 600 FV 6 I/y CPT N 3.129 years

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Solving the Frequency Problem (Quarterly) The result indicates that a $1,000 investment that earns a 12% annual rate compounded quarterly for 2 years will earn a future value of $1,266.77. (Quarterly) N I/Y PV PMT FV Inputs: Compute 2(4) 12/4 -1,000 0 1266.77

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Uneven cash flows Solving for the IRR & NPV Introducing the CF key Hitting the CF key brings you into the CF^0 This is used for a lump sum at the beginning of a cash flow It is often “0” Sometimes a $ amount

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Finding the PV of cash flows If an investment has the following distributions and your required return is 10%, how much should you pay for the investment? AKA what is the NPV?? Year 1 - $50 Year 2 - $100 Year 3 - $150 Year 4 - $200

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Solution Step 1:PressCF key Step 2:Press2 nd CLR Workkeys Step 3: For CF0 Press0Enter keys Step 4: For C01 Press50Enter keys Step 5: For F01 Press1Enter keys Step 6: For C02 Press100Enter keys Step 7: For F02 Press1Enter keys Step 8: For C03 Press150Enter keys Step 9: For F03 Press1Enter keys

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Step 10: For C04 Press400Enter keys Step 11: For F04 Press1Enter keys NPV 10 ENTER CPT DRUMROLL!……

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Solution NPV = $377.40 So, this cash flow is worth $377.40 today Significance of NPV Positive result? You have earned more than your required return & the cash flow has a value today. Negative Result? You have earned less than your required return & the cash flow has no value today in fact, it has a negative value. Result = 0? You have earned exactly your required return. So, this is a toss up – the CF has a zero value today, but the cash flows will give you your required return over the life of the investment.

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Calculating IRR Or “How much did I make?” Assume we invest $1000 in a mutual fund and we expect to receive dividends in the future (uneven CF) of: Yr 1- $300 Yr 2 - $400 Yr 3 - $200 Yr 4 - $600 What is our rate of return (ROR)?

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IRR Solution Step 1:PressCF key Step 2:Press2 nd CLR Work keys Step 3: For CF0 Press- 1,000Enter keys Step 4: For C01 Press300Enter keys Step 5: For F01 Press1Enter keys Step 6: For C02 Press400Enter keys Step 7: For F02 Press1Enter keys Step 8: For C03 Press200Enter keys Step 9: For F03 Press1Enter keys

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Step 10: For C04 Press600Enter keys Step 11: For F04 Press1Enter keys IRR CPT 16.7053%

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If your required return is 8%, we can see that this investment would exceed that – but by how much? I/YR 8 ENTER CPT SOLUTION IS NPV, WHICH FURTHER CONFIRMS THAT THE INVESTMENT EXCEEDED OUR REQUIRED RETURN BY $220.50

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Calculating IRR You are going to receive $10,000 in one year, $20,000 in year two and year three and $40,000 in year four. If the interest rate is 9.25%, what is the PV of these cash flows?

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Solution CF^0 = 0 CF^1 = 10000 CF^2 = 20,000 ( for 2 yrs, so FO2 = 2) CF^3 = 40,000 NPV 9.25 ENTER CPT $69,326.38…this means that this investment has a value today of $69,326.38 based upon your required return of 9.25%.

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Finding the IRR Joe invested $10,000 in a mutual fund He added $1000 in year 1 & 2 Rec’d a dividend of $300 in year 3 Sold the fund in year 4 for $13,300 What was his IRR??

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Solution 0 = -10,000 1 = -1000 2 = -1000 3 = +300 4 = + 13,300 IRR CPT 3.1842 %

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Congratulations!!! You have completed the tutorial and are on your way to being a TVM guru!! Please contact your instructor with any questions you might have. Best of luck !!!

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