Ross, Chapter 5: Discounted Cash Flow Valuation

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  • Point out that you can find the value of a set of cash flows at any point in time; all you have to do is get the value of each cash flow at that point in time and then add them together. The students can read the example in the book. It is also provided here. You think you will be able to deposit $4,000 at the end of each of the next three years in a bank account paying 8 percent interest. You currently have $7,000 in the account. How much will you have in three years? In four years? Point out that there are several ways that this can be worked. The book works this example by rolling the value forward each year. The presentation will show the second way to work the problem. Calculator: Today (year 0 CF): 3 N; 8 I/Y; -7,000 PV; CPT FV = 8,817.98 Year 1 CF: 2 N; 8 I/Y; -4,000 PV; CPT FV = 4,665.60 Year 2 CF: 1 N; 8 I/Y; -4,000 PV; CPT FV = 4,320 Year 3 CF: value = 4,000 Total value in 3 years = 8,817.98 + 4,665.60 + 4,320 + 4,000 = 21,803.58 Value at year 4: 1 N; 8 I/Y; -21,803.58 PV; CPT FV = 23,547.87 I entered the PV as negative for two reasons. (1) It is a cash outflow since it is an investment. (2) The FV is computed as positive and the students can then just store each calculation and then add from the memory registers, instead of writing down all of the numbers and taking the risk of keying something back into the calculator incorrectly.
  • Calculator: Year 0 CF: 2 N; -500 PV; 9 I/Y; CPT FV = 594.05 Year 1 CF: 1 N; -600 PV; 9 I/Y; CPT FV = 654.00 Total FV = 594.05 + 654.00 = 1,248.05
  • Calculator: First way: Year 0 CF: 5 N; -500 PV; 9 I/Y; CPT FV = 769.31 Year 1 CF: 4 N; -600 PV; 9 I/Y; CPT FV = 846.95 Total FV = 769.31 + 846.95 = 1,616.26 Second way – use value at year 2: 3 N; -1,248.05 PV; 9 I/Y; CPT FV = 1,616.26
  • FV = 100(1.08) 4 + 300(1.08) 2 = 136.05 + 349.92 = 485.97 Year 1 CF: 4 N; -100 PV; 8 I/Y; CPT FV = 136.05 Year 3 CF: 2 N; -300 PV; 8 I/Y; CPT FV = 349.92 Total FV = 136.05 + 349.92 = 485.97
  • The students can read the example in the book. You are offered an investment that will pay you $200 in one year, $400 the next year, $600 the next year and $800 at the end of the next year. You can earn 12 percent on very similar investments. What is the most you should pay for this one? Point out that the question could also be phrased as “How much is this investment worth?” Calculator: Year 1 CF: N = 1; I/Y = 12; FV = 200; CPT PV = -178.57 Year 2 CF: N = 2; I/Y = 12; FV = 400; CPT PV = -318.88 Year 3 CF: N = 3; I/Y = 12; FV = 600; CPT PV = -427.07 Year 4 CF: N = 4; I/Y = 12; FV = 800; CPT PV = - 508.41 Total PV = 178.57 + 318.88 + 427.07 + 508.41 = 1,432.93 Remember the sign convention. The negative numbers imply that we would have to pay 1,432.93 today to receive the cash flows in the future.
  • Calculator: N = 1; I/Y = 10; FV = 1,000; CPT PV = -909.09 N = 2; I/Y = 10; FV = 2,000; CPT PV = -1,652.89 N = 3; I/Y = 10; FV = 3,000; CPT PV = -2,253.94
  • The next example will be worked using the cash flow keys. Note that with the BA-II Plus, the students can double check the numbers they have entered by pressing the up and down arrows. It is similar to entering the cash flows into spreadsheet cells. Other calculators also have cash flow keys. You enter the information by putting in the cash flow and then pressing CF. You have to always start with the year 0 cash flow, even if it is zero. Remind the students that the cash flows have to occur at even intervals, so if you skip a year, you still have to enter a 0 cash flow for that year.
  • Click on the tabs at the bottom of the worksheet to move from a future value example to a present value example.
  • You can also use this as an introduction to NPV by having the students put –100 in for CF 0 . When they compute the NPV, they will get –8.51. You can then discuss the NPV rule and point out that a negative NPV means that you do not earn your required return. You should also remind them that the sign convention on the regular TVM keys is NOT the same as getting a negative NPV.
  • The easiest way to work this problem is to use the uneven cash flow keys and find the present value first and then compute the others based on that. CF 0 = 0; C01 = 100; F01 = 1; C02 = 200; F02 = 2; C03 = 300; F03 = 2; I = 7; CPT NPV = 874.17 Value in year 5: PV = 874.17; N = 5; I/Y = 7; CPT FV = 1,226.07 Value in year 3: PV = 874.17; N = 3; I/Y = 7; CPT FV = 1,070.90 Using formulas and one CF at a time: Year 1 CF: FV 5 = 100(1.07) 4 = 131.08; PV 0 = 100 / 1.07 = 93.46; FV 3 = 100(1.07) 2 = 114.49 Year 2 CF: FV 5 = 200(1.07) 3 = 245.01; PV 0 = 200 / (1.07) 2 = 174.69; FV 3 = 200(1.07) = 214 Year 3 CF: FV 5 = 200(1.07) 2 = 228.98; PV 0 = 200 / (1.07) 3 = 163.26; FV 3 = 200 Year 4 CF: FV 5 = 300(1.07) = 321; PV 0 = 300 / (1.07) 4 = 228.87; PV 3 = 300 / 1.07 = 280.37 Year 5 CF: FV 5 = 300; PV 0 = 300 / (1.07) 5 = 213.90; PV 3 = 300 / (1.07) 2 = 262.03 Value at year 5 = 131.08 + 245.01 + 228.98 + 321 + 300 = 1,226.07 Present value today = 93.46 + 174.69 + 163.26 + 228.87 + 213.90 = 874.18 (difference due to rounding) Value at year 3 = 114.49 + 214 + 200 + 280.37 + 262.03 = 1,070.89 Point out that C03 is in years 4 and 5. The “03” indicates the third cash inflow quantity, and will not be the same as the time period whenever a preceding cash flow frequency (e.g. F01) is set to something other than “1.”
  • Other calculators also have a key that allows you to switch between Beg/End.
  • The students can read the example in the book. After carefully going over your budget, you have determined you can afford to pay $632 per month towards a new sports car. You call up your local bank and find out that the going rate is 1 percent per month for 48 months. How much can you borrow?
  • Calculator: 30 N; 5 I/Y; 333,333.33 PMT; CPT PV = 5,124,150.29
  • You might point out that you would probably not offer 154,501. The more likely scenario would be 154,500. Calculator: 30*12 = 360 N .5 I/Y 840 PMT CPT PV = 140,105
  • Calculator PMT = 5,000; N = 25*12 = 300; I/Y = .75; CPT PV = 595,808 Formula PV = 5,000[1 – 1 / 1.0075 300 ] / .0075 = 595,808
  • Calculator: 4(12) = 48 N; 20,000 PV; 8/12 = .666666667 I/Y; CPT PMT = 488.26
  • You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000. This is an excellent opportunity to talk about credit card debt and the problems that can develop if it is not handled properly. Many students don’t understand how it works and it is rarely discussed. This is something that students can take away from the class, even if they aren’t finance majors. Calculator: 1.5 I/Y 1,000 PV -20 PMT CPT N = 93.111 MONTHS = 7.75 years
  • Sign convention matters!!! 5 I/Y 2,000 PV -734.42 PMT CPT N = 3 years
  • The next slide talks about how to do this without a financial calculator.
  • Q1: 5(12) = 60 N; .75 I/Y; 5000 PMT; CPT PV = -240,867 PV = 5,000(1 – 1 / 1.0075 60 ) / .0075 = 240,867 Q2: -200,000 PV; 60 N; 5,000 PMT; CPT I/Y = 1.439% Trial and error without calculator Q3: -200,000 PV; .75 I/Y; 5,000 PMT; CPT N = 47.73 (47 months plus partial payment in month 48) 200,000 = 5,000(1 – 1 / 1.0075 t ) / .0075 .3 = 1 – 1/1.0075 t 1.0075 t = 1.428571429 t = ln(1.428571429) / ln(1.0075) = 47.73 months Q4: -200,000 PV; 60 N; .75 I/Y; CPT PMT = 4151.67 200,000 = C(1 – 1/1.0075 60 ) / .0075 C = 4,151.67
  • FV = 2,000(1.075 40 – 1)/.075 = 454,513.04 Remember the sign convention!!! 40 N 7.5 I/Y -2,000 PMT CPT FV = 454,513.04
  • Note that the procedure for changing the calculator to an annuity due is similar on other calculators. Calculator 2 nd BGN 2 nd Set (you should see BGN in the display) 3 N -10,000 PMT 8 I/Y CPT FV = 35,061.12 2 nd BGN 2 nd Set (be sure to change it back to an ordinary annuity) What if it were an ordinary annuity? FV = 32,464 (so receive an additional 2,597.12 by starting to save today.)
  • If you use the regular annuity formula, the FV will occur at the same time as the last payment. To get the value at the end of the third period, you have to take it forward one more period.
  • This is a good preview to the valuation issues discussed in future chapters. The price of an investment is just the present value of expected future cash flows. Example statement: Suppose the Fellini Co. wants to sell preferred stock at $100 per share. A very similar issue of preferred stock already outstanding has a price of $40 per share and offers a dividend of $1 every quarter. What dividend will Fellini have to offer if the preferred stock is going to sell?
  • Q1: 35(12) = 420 N; 1,000,000 FV; 1 I/Y; CPT PMT = 155.50 1,000,000 = C (1.01 420 – 1) / .01 C = 155.50 Q2: Set calculator to annuity due and use the same inputs as above. CPT PMT = 153.96 1,000,000 = C[(1.01 420 – 1) / .01] ( 1.01) C = 153.96 Q3: PV = 1.50 / .03 = $50
  • Where m is the number of compounding periods per year Using the calculator: The TI BA-II Plus has an I conversion key that allows for easy conversion between quoted rates and effective rates. 2 nd I Conv NOM is the quoted rate, ENTER up arrow to C/Y (compounding periods per year), ENTER up arrow to EFF (the effective rate) and then CPT (compute). You can compute either the NOM or the EFF by entering the other two pieces of information, then going to the one you wish to compute and pressing CPT.
  • Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.
  • Remind students that rates are quoted on an annual basis. The given numbers are APRs, not daily or semiannual rates. Calculator: 2 nd I conv 5.25 NOM ENTER up arrow, 365 C/Y ENTER up arrow, CPT EFF = 5.39% 5.3 NOM ENTER up arrow, 2 C/Y ENTER up arrow, CPT EFF = 5.37%
  • It is important to point out that the daily rate is NOT .014, it is .014383562 First Account: 365 N; 5.25 / 365 = .014383562 I/Y; 100 PV; CPT FV = 105.39 Second Account: 2 N; 5.3 / 2 = 2.65 I/Y; 100 PV; CPT FV = 105.37
  • On the calculator: 2 nd I conv down arrow 12 EFF ENTER down arrow, 12 C/Y ENTER down arrow, CPT NOM
  • 2(12) = 24 N; 16.9 / 12 = 1.408333333 I/Y; 3,500 PV; CPT PMT = -172.88
  • 35(12) = 420 N 9 / 12 = .75 I/Y 50 PMT CPT FV = 147,089.22
  • 3(365) = 1095 N 5.5 / 365 = .015068493 I/Y 15,000 FV CPT PV = -12,718.56
  • APR = period rate * # of compounding periods per year EAR is the rate we earn (or pay) after we account for compounding We should use the EAR to compare alternatives We need the period rate and we have to use the APR to get it
  • Remind students that the value of an investment is the present value of expected future cash flows. 1 N; 10,000 FV; 7 I/Y; CPT PV = -9,345.79
  • 4 N 8 I/Y 5,000 PV CPT PMT = -1,509.60
  • The reason that the loan balance does not decline to exactly zero is because of the rounding of the interest payments. Technically, the last payment would be 1,509.61 so that the loan balance would be zero after the last payment. This is a common issue.
  • This is another example, similar to the one in the book.
  • Present value problems: End of the year: 10 N; 8 I/Y; 100 PMT; CPT PV = -671.00 Beginning of the year: PV = $671.00 X 1.08 = $724.69 Future value problems: 10 N; 8 I/Y; -100 PMT; CPT FV = 1,448.66 10N; 8 I/Y; -1,000 PV; -100 PMT; CPT FV = 3,607.58
  • Ross, Chapter 5: Discounted Cash Flow Valuation

    1. 1. Chapter 5 Discounted Cash Flow Valuation
    2. 2. Key Concepts and Skills <ul><li>Be able to compute the future value of multiple cash flows </li></ul><ul><li>Be able to compute the present value of multiple cash flows </li></ul><ul><li>Be able to compute loan payments </li></ul><ul><li>Be able to find the interest rate on a loan </li></ul><ul><li>Understand how loans are amortized or paid off </li></ul><ul><li>Understand how interest rates are quoted </li></ul>
    3. 3. Chapter Outline <ul><li>Future and Present Values of Multiple Cash Flows </li></ul><ul><li>Valuing Level Cash Flows: Annuities and Perpetuities </li></ul><ul><li>Comparing Rates: The Effect of Compounding Periods </li></ul><ul><li>Loan Types and Loan Amortization </li></ul>
    4. 4. Multiple Cash Flows – FV Example 5.1 <ul><li>Find the value at year 3 of each cash flow and add them together. </li></ul><ul><ul><li>Today (year 0): FV = $7,000(1.08) 3 = $8,817.98 </li></ul></ul><ul><ul><li>Year 1: FV = $4,000(1.08) 2 = $4,665.60 </li></ul></ul><ul><ul><li>Year 2: FV = $4,000(1.08) = $4,320 </li></ul></ul><ul><ul><li>Year 3: value = $4,000 </li></ul></ul><ul><ul><li>Total value in 3 years = $8,817.98 + 4,665.60 + 4,320 + 4,000 = $21,803.58 </li></ul></ul><ul><li>Value at year 4 = $21,803.58(1.08) = $23,547.87 </li></ul>
    5. 5. Multiple Cash Flows – FV Example 2 <ul><li>Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years? </li></ul><ul><ul><li>FV = $ 500(1.09) 2 + $ 600(1.09) = $ 1,248.05 </li></ul></ul>
    6. 6. Example 2 Continued <ul><li>How much will you have in 5 years if you make no further deposits? </li></ul><ul><li>First way: </li></ul><ul><ul><li>FV = $ 500(1.09) 5 + $ 600(1.09) 4 = $ 1,616.26 </li></ul></ul><ul><li>Second way – use value at year 2: </li></ul><ul><ul><li>FV = $ 1,248.05(1.09) 3 = $ 1,616.26 </li></ul></ul>
    7. 7. Multiple Cash Flows – FV Example 3 <ul><li>Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%? </li></ul><ul><ul><li>FV = $ 100(1.08) 4 + $ 300(1.08) 2 = $ 136.05 + $ 349.92 = $ 485.97 </li></ul></ul>
    8. 8. Example 3 Time Line $ 100 0 1 2 3 4 5 $ 300 $ 136.05 $349.92 $485.97
    9. 9. Multiple Cash Flows – PV Example 5.3 <ul><li>Find the PV of each cash flow and add them </li></ul><ul><ul><li>Year 1 CF: $ 200 / (1.12) 1 = $ 178.57 </li></ul></ul><ul><ul><li>Year 2 CF: $ 400 / (1.12) 2 = $ 318.88 </li></ul></ul><ul><ul><li>Year 3 CF: $ 600 / (1.12) 3 = $ 427.07 </li></ul></ul><ul><ul><li>Year 4 CF: $ 800 / (1.12) 4 = $ 508.41 </li></ul></ul><ul><ul><li>Total PV = $ 178.57 + 318.88 + 427.07 + 508.41 = $ 1,432.93 </li></ul></ul>
    10. 10. Example 5.3 Time Line 0 1 2 3 4 200 400 600 800 178.57 318.88 427.07 508.41 1,432.93
    11. 11. Multiple Cash Flows – PV Another Example <ul><li>You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay? </li></ul><ul><ul><li>PV = $1,000 / (1.1) 1 = $909.09 </li></ul></ul><ul><ul><li>PV = $2,000 / (1.1) 2 = $1,652.89 </li></ul></ul><ul><ul><li>PV = $3,000 / (1.1) 3 = $2,253.94 </li></ul></ul><ul><ul><li>PV = $909.09 + 1,652.89 + 2,253.94 = $4,815.92 </li></ul></ul>
    12. 12. Multiple Uneven Cash Flows – Using the Calculator <ul><li>Another way to use the financial calculator for uneven cash flows is to use the cash flow keys </li></ul><ul><ul><li>Texas Instruments BA-II Plus </li></ul></ul><ul><ul><ul><li>Clear the cash flow keys by pressing CF and then 2nd CLR Work </li></ul></ul></ul><ul><ul><ul><li>Press CF and enter the cash flows beginning with year 0. </li></ul></ul></ul><ul><ul><ul><li>You have to press the “Enter” key for each cash flow </li></ul></ul></ul><ul><ul><ul><li>Use the down arrow key to move to the next cash flow </li></ul></ul></ul><ul><ul><ul><li>The “F” is the number of times a given cash flow occurs in consecutive years </li></ul></ul></ul><ul><ul><ul><li>Use the NPV key to compute the present value by [ENTER]ing the interest rate for I, pressing the down arrow, and then computing NPV </li></ul></ul></ul>
    13. 13. Example: Spreadsheet Strategies <ul><li>You can use the PV or FV functions in Excel to find the present value or future value of a set of cash flows </li></ul><ul><li>Setting the data up is half the battle – if it is set up properly, then you can just copy the formulas </li></ul><ul><li>Click on the Excel icon for an example </li></ul>
    14. 14. Decisions, Decisions <ul><li>Your broker calls you and tells you that he has this great investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment? </li></ul><ul><ul><li>Use the CF keys to compute the value of the investment </li></ul></ul><ul><ul><ul><li>CF 2 nd ClrWork; CF 0 = 0; C01 = 40; F01 = 1; C02 = 75; F02 = 1 </li></ul></ul></ul><ul><ul><ul><li>NPV; I = 15; CPT NPV = 91.49 </li></ul></ul></ul><ul><ul><li>No! The broker is charging more than you would be willing to pay. </li></ul></ul>
    15. 15. Saving For Retirement <ul><li>You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%? </li></ul><ul><ul><li>Use cash flow keys: </li></ul></ul><ul><ul><ul><li>CF; CF 0 = 0; C01 = 0; F01 = 39; C02 = 25,000; F02 = 5; NPV; I = 12; CPT NPV = 1,084.71 </li></ul></ul></ul>
    16. 16. Saving For Retirement Time Line 0 1 2 … 39 40 41 42 43 44 0 0 0 … 0 25K 25K 25K 25K 25K Notice that the year 0 cash flow = 0 (CF 0 = 0) The cash flows years 1 – 39 are 0 (C01 = 0; F01 = 39) The cash flows years 40 – 44 are 25,000 (C02 = 25,000; F02 = 5)
    17. 17. Quick Quiz: Part 1 <ul><li>Suppose you are looking at the following possible cash flows: Year 1 CF = $100; Years 2 and 3 CFs = $200; Years 4 and 5 CFs = $300. The required discount rate is 7% </li></ul><ul><li>What is the value of the cash flows at year 5? </li></ul><ul><li>What is the value of the cash flows today? </li></ul><ul><li>What is the value of the cash flows at year 3? </li></ul>
    18. 18. Annuities and Perpetuities Defined <ul><li>Annuity – finite series of equal payments that occur at regular intervals </li></ul><ul><ul><li>If the first payment occurs at the end of the period, it is called an ordinary annuity </li></ul></ul><ul><ul><li>If the first payment occurs at the beginning of the period, it is called an annuity due </li></ul></ul><ul><li>Perpetuity – infinite series of equal payments </li></ul>
    19. 19. Annuities and Perpetuities – Basic Formulas <ul><li>Perpetuity: PV = C / r </li></ul><ul><li>Annuities: </li></ul>
    20. 20. Annuities and the Calculator <ul><li>You can use the PMT key on the calculator for the equal payment </li></ul><ul><li>The sign convention still holds </li></ul><ul><li>Ordinary annuity versus annuity due </li></ul><ul><ul><li>You can switch your calculator between the two types by using the 2 nd BGN 2 nd Set on the TI BA-II Plus </li></ul></ul><ul><ul><li>If you see “BGN” or “Begin” in the display of your calculator, you have it set for an annuity due </li></ul></ul><ul><ul><li>Most problems are ordinary annuities </li></ul></ul>
    21. 21. Annuity – Example 5.5 <ul><li>You borrow money TODAY so you need to compute the present value. </li></ul><ul><ul><li>48 N; 1 I/Y; -632 PMT; CPT PV = 23,999.54 ($24,000) </li></ul></ul><ul><li>Formula: </li></ul>
    22. 22. Annuity – Sweepstakes Example <ul><li>Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today? </li></ul><ul><ul><li>PV = $ 333,333.33[1 – 1/1.05 30 ] / .05 = $ 5,124,150.29 </li></ul></ul>
    23. 23. Buying a House <ul><li>You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house? </li></ul>
    24. 24. Buying a House - Continued <ul><li>Bank loan </li></ul><ul><ul><li>Monthly income = $36,000 / 12 = $3,000 </li></ul></ul><ul><ul><li>Maximum payment = .28($3,000) = $840 </li></ul></ul><ul><ul><li>PV = $840[1 – 1/1.005 360 ] / .005 = $140,105 </li></ul></ul><ul><li>Total Price </li></ul><ul><ul><li>Closing costs = .04($140,105) = $5,604 </li></ul></ul><ul><ul><li>Down payment = $20,000 – 5,604 = $14,396 </li></ul></ul><ul><ul><li>Total Price = $140,105 + 14,396 = $154,501 </li></ul></ul>
    25. 25. Example: Spreadsheet Strategies – Annuity PV <ul><li>The present value and future value formulas in a spreadsheet include a place for annuity payments </li></ul><ul><li>Click on the Excel icon to see an example </li></ul>
    26. 26. Quick Quiz: Part 2 <ul><li>You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value? </li></ul><ul><li>You want to receive $ 5,000 per month in retirement. If you can earn .75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement? </li></ul>
    27. 27. Finding the Payment <ul><li>Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = .666666667% per month). If you take a 4-year loan, what is your monthly payment? </li></ul><ul><ul><li>$ 20,000 = C[1 – 1 / 1.0066667 48 ] / .0066667 </li></ul></ul><ul><ul><li>C = $ 488.26 </li></ul></ul>
    28. 28. Example: Spreadsheet Strategies – Annuity Payment <ul><li>Another TVM formula that can be found in a spreadsheet is the payment formula </li></ul><ul><ul><li>PMT(rate,nper,pv,fv) </li></ul></ul><ul><ul><li>The same sign convention holds as for the PV and FV formulas </li></ul></ul><ul><li>Click on the Excel icon for an example </li></ul>
    29. 29. Finding the Number of Payments – Example 5.6 <ul><li>Start with the equation and remember your logs. </li></ul><ul><ul><li>$ 1,000 = $ 20(1 – 1/1.015 t ) / .015 </li></ul></ul><ul><ul><li>.75 = 1 – 1 / 1.015 t </li></ul></ul><ul><ul><li>1 / 1.015 t = .25 </li></ul></ul><ul><ul><li>1 / .25 = 1.015 t </li></ul></ul><ul><ul><li>t = ln(1/.25) / ln(1.015) = 93.111 months = 7.75 years </li></ul></ul><ul><li>And this is only true if you don’t charge anything more on the card! </li></ul>
    30. 30. Finding the Number of Payments – Another Example <ul><li>Suppose you borrow $2,000 at 5% and you are going to make annual payments of $734.42. How long is it before you pay off the loan? </li></ul><ul><ul><li>$ 2,000 = $ 734.42(1 – 1/1.05 t ) / .05 </li></ul></ul><ul><ul><li>.136161869 = 1 – 1/1.05 t </li></ul></ul><ul><ul><li>1/1.05 t = .863838131 </li></ul></ul><ul><ul><li>1.157624287 = 1.05 t </li></ul></ul><ul><ul><li>t = ln(1.157624287) / ln(1.05) = 3 years </li></ul></ul>
    31. 31. Finding the Rate <ul><li>Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate? </li></ul><ul><ul><li>Sign convention matters!!! </li></ul></ul><ul><ul><li>60 N </li></ul></ul><ul><ul><li>10,000 PV </li></ul></ul><ul><ul><li>-207.58 PMT </li></ul></ul><ul><ul><li>CPT I/Y = .75% </li></ul></ul>
    32. 32. Annuity – Finding the Rate Without a Financial Calculator <ul><li>Trial and Error Process </li></ul><ul><ul><li>Choose an interest rate and compute the PV of the payments based on this rate </li></ul></ul><ul><ul><li>Compare the computed PV with the actual loan amount </li></ul></ul><ul><ul><li>If the computed PV > loan amount, then the interest rate is too low </li></ul></ul><ul><ul><li>If the computed PV < loan amount, then the interest rate is too high </li></ul></ul><ul><ul><li>Adjust the rate and repeat the process until the computed PV and the loan amount are equal </li></ul></ul>
    33. 33. Quick Quiz: Part 3 <ul><li>You want to receive $5,000 per month for the next 5 years. How much would you need to deposit today if you can earn .75% per month? </li></ul><ul><li>What monthly rate would you need to earn if you only have $200,000 to deposit? </li></ul><ul><li>Suppose you have $200,000 to deposit and can earn .75% per month. </li></ul><ul><ul><li>How many months could you receive the $5,000 payment? </li></ul></ul><ul><ul><li>How much could you receive every month for 5 years? </li></ul></ul>
    34. 34. Future Values for Annuities <ul><li>Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years? </li></ul><ul><ul><li>FV(Ordinary) = $ 2,000(1.075 40 – 1)/.075 = $ 454,513.04 </li></ul></ul><ul><ul><li>FV(Due) = $ 488,601.52 </li></ul></ul>
    35. 35. Annuity Due <ul><li>You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years? </li></ul><ul><ul><li>FV = $ 10,000[(1.08 3 – 1) / .08](1.08) = $ 35,061.12 </li></ul></ul>
    36. 36. Annuity Due Time Line 35,016.12 0 1 2 3 10000 10000 10000 32,464
    37. 37. Table 5.2
    38. 38. Example: Work the Web <ul><li>Another online financial calculator can be found at MoneyChimp </li></ul><ul><li>Click on the Web surfer and work the following example </li></ul><ul><ul><li>Choose calculator and then annuity </li></ul></ul><ul><ul><li>You just inherited $5 million. If you can earn 6% on your money, how much can you withdraw each year for the next 40 years? </li></ul></ul><ul><ul><li>Moneychimp assumes annuity due!!! </li></ul></ul><ul><ul><li>Payment = $313,497.81 </li></ul></ul>
    39. 39. Perpetuity – Example 5.7 <ul><li>Perpetuity formula: PV = C / r </li></ul><ul><li>Current required return: </li></ul><ul><ul><li>$ 40 = $ 1 / r </li></ul></ul><ul><ul><li>r = .025 or 2.5% per quarter </li></ul></ul><ul><li>Dividend for new preferred: </li></ul><ul><ul><li>$ 100 = C / .025 </li></ul></ul><ul><ul><li>C = $ 2.50 per quarter </li></ul></ul>
    40. 40. Quick Quiz: Part 4 <ul><li>You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month? </li></ul><ul><li>What if the first payment is made today? </li></ul><ul><li>You are considering preferred stock that pays a quarterly dividend of $1.50. If your desired return is 3% per quarter, how much would you be willing to pay? </li></ul>
    41. 41. Effective Annual Rate (EAR) <ul><li>This is the actual rate paid (or received) after accounting for compounding that occurs during the year </li></ul><ul><li>If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison. </li></ul>
    42. 42. Annual Percentage Rate <ul><li>This is the annual rate that is quoted by law </li></ul><ul><li>By definition APR = period rate times the number of periods per year </li></ul><ul><li>Consequently, to get the period rate we rearrange the APR equation: </li></ul><ul><ul><li>Period rate = APR / number of periods per year </li></ul></ul><ul><li>You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate </li></ul>
    43. 43. Computing APRs <ul><li>What is the APR if the monthly rate is .5%? </li></ul><ul><ul><li>.5%(12) = 6% </li></ul></ul><ul><li>What is the APR if the semiannual rate is .5%? </li></ul><ul><ul><li>.5%(2) = 1% </li></ul></ul><ul><li>What is the monthly rate if the APR is 12% with monthly compounding? </li></ul><ul><ul><li>12% / 12 = 1% </li></ul></ul><ul><ul><li>Can you divide the above APR by 2 to get the semiannual rate? NO!!! You need an APR based on semiannual compounding to find the semiannual rate. </li></ul></ul>
    44. 44. Things to Remember <ul><li>You ALWAYS need to make sure that the interest rate and the time period match. </li></ul><ul><ul><li>If you are looking at annual periods, you need an annual rate. </li></ul></ul><ul><ul><li>If you are looking at monthly periods, you need a monthly rate. </li></ul></ul><ul><li>If you have an APR based on monthly compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly </li></ul>
    45. 45. Computing EARs - Example <ul><li>Suppose you can earn 1% per month on $1 invested today. </li></ul><ul><ul><li>What is the APR? 1%(12) = 12% </li></ul></ul><ul><ul><li>How much are you effectively earning? </li></ul></ul><ul><ul><ul><li>FV = $ 1(1.01) 12 = $ 1.1268 </li></ul></ul></ul><ul><ul><ul><li>Rate = (1.1268 – 1) / 1 = .1268 = 12.68% </li></ul></ul></ul><ul><li>Suppose if you put it in another account, you earn 3% per quarter. </li></ul><ul><ul><li>What is the APR? 3%(4) = 12% </li></ul></ul><ul><ul><li>How much are you effectively earning? </li></ul></ul><ul><ul><ul><li>FV = $ 1(1.03) 4 = $ 1.1255 </li></ul></ul></ul><ul><ul><ul><li>Rate = (1.1255 – 1) / 1 = .1255 = 12.55% </li></ul></ul></ul>
    46. 46. EAR - Formula Remember that the APR is the quoted rate, and m is the number of compounds per year
    47. 47. Decisions, Decisions II <ul><li>You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use? </li></ul><ul><ul><li>First account: </li></ul></ul><ul><ul><ul><li>EAR = (1 + .0525/365) 365 – 1 = 5.39% </li></ul></ul></ul><ul><ul><li>Second account: </li></ul></ul><ul><ul><ul><li>EAR = (1 + .053/2) 2 – 1 = 5.37% </li></ul></ul></ul><ul><li>Which account should you choose and why? </li></ul>
    48. 48. Decisions, Decisions II Continued <ul><li>Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year? </li></ul><ul><ul><li>First Account: </li></ul></ul><ul><ul><ul><li>Daily rate = .0525 / 365 = .00014383562 </li></ul></ul></ul><ul><ul><ul><li>FV = $100(1.00014383562) 365 = $105.39 </li></ul></ul></ul><ul><ul><li>Second Account: </li></ul></ul><ul><ul><ul><li>Semiannual rate = .053 / 2 = .0265 </li></ul></ul></ul><ul><ul><ul><li>FV = $100(1.0265) 2 = $105.37 </li></ul></ul></ul><ul><li>You have more money in the first account. </li></ul>
    49. 49. Computing APRs from EARs <ul><li>If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get: </li></ul>
    50. 50. APR - Example <ul><li>Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay? </li></ul>
    51. 51. Computing Payments with APRs <ul><li>Suppose you want to buy a new computer system. The store will allow you to make monthly payments. The entire computer system costs $3,500. The loan period is for 2 years and the interest rate is 16.9% with monthly compounding. What is your monthly payment? </li></ul><ul><ul><li>Monthly rate = .169 / 12 = .01408333333 </li></ul></ul><ul><ul><li>Number of months = 2(12) = 24 </li></ul></ul><ul><ul><li>$ 3,500 = C[1 – 1 / (1.01408333333) 24 ] / .01408333333 </li></ul></ul><ul><ul><li>C = $ 172.88 </li></ul></ul>
    52. 52. Future Values with Monthly Compounding <ul><li>Suppose you deposit $50 per month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years? </li></ul><ul><ul><li>Monthly rate = .09 / 12 = .0075 </li></ul></ul><ul><ul><li>Number of months = 35(12) = 420 </li></ul></ul><ul><ul><li>FV = $50[1.0075 420 – 1] / .0075 = $147,089.22 </li></ul></ul>
    53. 53. Present Value with Daily Compounding <ul><li>You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit? </li></ul><ul><ul><li>Daily rate = .055 / 365 = .00015068493 </li></ul></ul><ul><ul><li>Number of days = 3(365) = 1,095 </li></ul></ul><ul><ul><li>FV = $15,000 / (1.00015068493) 1095 = $12,718.56 </li></ul></ul>
    54. 54. Quick Quiz: Part 5 <ul><li>What is the definition of an APR? </li></ul><ul><li>What is the effective annual rate? </li></ul><ul><li>Which rate should you use to compare alternative investments or loans? </li></ul><ul><li>Which rate do you need to use in the time value of money calculations? </li></ul>
    55. 55. Pure Discount Loans – Example 5.11 <ul><li>Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments. </li></ul><ul><li>If a T-bill promises to repay $10,000 in one year, and the market interest rate is 7 percent, how much will the bill sell for in the market? </li></ul><ul><ul><li>PV = $10,000 / 1.07 = $9,345.79 </li></ul></ul>
    56. 56. Interest-Only Loan - Example <ul><li>Consider a 5-year, interest-only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually. </li></ul><ul><ul><li>What would the stream of cash flows be? </li></ul></ul><ul><ul><ul><li>Years 1 – 4: Interest payments of .07( $ 10,000) = $ 700 </li></ul></ul></ul><ul><ul><ul><li>Year 5: Interest + principal = $ 10,700 </li></ul></ul></ul><ul><li>This cash flow stream is similar to the cash flows on corporate bonds and we will talk about them in greater detail later. </li></ul>
    57. 57. Amortized Loan with Fixed Payment - Example <ul><li>Each payment covers the interest expense; plus, it reduces principal </li></ul><ul><li>Consider a 4-year loan with annual payments. The interest rate is 8% and the principal amount is $5,000. </li></ul><ul><ul><li>What is the annual payment? </li></ul></ul><ul><ul><ul><li>$ 5,000 = C[1 – 1 / 1.08 4 ] / .08 </li></ul></ul></ul><ul><ul><ul><li>C = $ 1,509.60 </li></ul></ul></ul>
    58. 58. Amortization Table for Example 4,999.99 1,038.41 6,038.40 Totals .01 1,397.78 111.82 1,509.60 1,397.79 4 1,397.79 1,294.24 215.36 1,509.60 2,692.03 3 2,692.03 1,198.37 311.23 1,509.60 3,890.40 2 3,890.40 1,109.60 400.00 1,509.60 5,000.00 1 End. Balance Principal Paid Interest Paid Total Payment Beg. Balance Year
    59. 59. Example: Spreadsheet Strategies <ul><li>Each payment covers the interest expense; plus, it reduces principal </li></ul><ul><li>Consider a 4-year loan with annual payments. The interest rate is 8% and the principal amount is $5,000. </li></ul><ul><ul><li>What is the annual payment? </li></ul></ul><ul><ul><ul><li>4 N </li></ul></ul></ul><ul><ul><ul><li>8 I/Y </li></ul></ul></ul><ul><ul><ul><li>5,000 PV </li></ul></ul></ul><ul><ul><ul><li>CPT PMT = -1,509.60 </li></ul></ul></ul><ul><li>Click on the Excel icon to see the amortization table </li></ul>
    60. 60. Example: Work the Web <ul><li>Several Web sites have calculators that will prepare amortization tables quickly </li></ul><ul><li>One such site is LendingTree.com </li></ul><ul><li>Click on the Web surfer and follow knowledge center, mortgage calculators, “How much will your mortgage payments be” and enter the following information: </li></ul><ul><ul><li>Loan amount = $ 20,000 </li></ul></ul><ul><ul><li>Term = 10 years </li></ul></ul><ul><ul><li>Interest rate = 7.625% </li></ul></ul><ul><ul><li>What is the monthly payment? </li></ul></ul><ul><ul><li>Using the calculator you will get $238.71 </li></ul></ul><ul><ul><li>Clicking “View Report” will give you the amortization table </li></ul></ul>
    61. 61. Quick Quiz: Part 6 <ul><li>What is a pure discount loan? What is a good example of a pure discount loan? </li></ul><ul><li>What is an interest-only loan? What is a good example of an interest-only loan? </li></ul><ul><li>What is an amortized loan? What is a good example of an amortized loan? </li></ul>
    62. 62. Comprehensive Problem <ul><li>An investment will provide you with $100 at the end of each year for the next 10 years. What is the present value of that annuity if the discount rate is 8% annually? </li></ul><ul><li>What is the present value of the above if the payments are received at the beginning of each year? </li></ul><ul><li>If you deposit those payments into an account earning 8%, what will the future value be in 10 years? </li></ul><ul><li>What will the future value be if you open the account with $1,000 today, and then make the $100 deposits at the end of each year? </li></ul>

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