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Chapter 4

Chapter 4. The Time Value of Money (Part 2). Learning Objectives. Compute the future value of multiple cash flows. Determine the future value of an annuity. Determine the present value of an annuity.

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Chapter 4

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  1. Chapter 4 The Time Value of Money (Part 2)

  2. Learning Objectives Compute the future value of multiple cash flows. Determine the future value of an annuity. Determine the present value of an annuity. Adjust the annuity formula for present value and future value for an annuity due, and understand the concept of a perpetuity. Distinguish between the different types of loan repayments: discount loans, interest-only loans, and amortized loans. Build and analyze amortization schedules. Calculate waiting time and interest rates for an annuity.

  3. 4.1 Future Value of Multiple Payment Streams With unequal periodic cash flows, treat each of the cash flows as a lump sum and calculate its future value over the relevant number of periods. Sum up the individual future values to get the future value of the multiple payment streams.

  4. FIGURE 4.1 The time line of a nest egg

  5. 4.1 Future Value of Multiple Payment Streams (continued) Example 1: Future Value of an Uneven Cash Flow Stream Jim deposits $3,000 today into an account that pays 10% per year, and follows it up with 3 more deposits at the end of each of the next three years. Each subsequent deposit is $2,000 higher than the previous one. How much money will Jim have accumulated in his account by the end of three years?

  6. 4.1 Future Value of Multiple Payment Streams (Example 1 Answer) FV = PV x (1+r)n FV of Cash Flow at T0 = $3,000 x (1.10)3 = $3,000 x 1.331= $3,993.00 FV of Cash Flow at T1 = $5,000 x (1.10)2 = $5,000 x 1.210 = $6,050.00 FV of Cash Flow at T2 = $7,000 x (1.10)1 = $7,000 x 1.100 = $7,700.00 FV of Cash Flow at T3 = $9,000 x (1.10)0 = $9,000 x 1.000 = $9,000.00 Total = $26,743.00

  7. 4.2 Future Value of an Annuity Stream Annuities are equal, periodic outflows/inflows., e.g. rent, lease, mortgage, car loan, and retirement annuity payments. An annuity stream can be: - annuity due: at the start of each period as is true of rent and insurance payments, or - ordinary annuity: at the end of each period as in the case of mortgage and loan payments.

  8. 4.2 Future Value of an Annuity Stream - Continued • The formula for calculating the future value of an ordinary annuity is as follows: FV = PMT * (1+r)n -1 r • where PMT is the term used for the equal periodic cash flow, r is the rate of interest, and n is the number of periods involved. • The last portion of the equation is the Future Value Interest Factor of an Annuity (FVIFA).

  9. 4.2 Future Value of an Annuity Stream (continued) Example 2: Future Value of an Ordinary Annuity Stream Jill has been faithfully depositing $2,000 at the end of each year for the past 10 years into an account that pays 8% per year. How much money will she have accumulated in the account?

  10. Example 2 Answer Future Value of Payment One = $2,000 x 1.089 = $3,998.01 Future Value of Payment Two = $2,000 x 1.088 = $3,701.86 Future Value of Payment Three = $2,000 x 1.087 = $3,427.65 Future Value of Payment Four = $2,000 x 1.086 = $3,173.75 Future Value of Payment Five = $2,000 x 1.085 = $2,938.66 Future Value of Payment Six = $2,000 x 1.084 = $2,720.98 Future Value of Payment Seven = $2,000 x 1.083 = $2,519.42 Future Value of Payment Eight = $2,000 x 1.082 = $2,332.80 Future Value of Payment Nine = $2,000 x 1.081 = $2,160.00 Future Value of Payment Ten = $2,000 x 1.080 = $2,000.00 Total Value of Account at the end of 10 years $28,973.13 4.2 Future Value of an Annuity Stream (continued)

  11. 4.2 Future Value of an Annuity Stream (continued) Example 2 (Answer) FORMULA METHOD FV = PMT * (1+r)n -1 r where, PMT = $2,000; r = 8%; and n=10. FVIFA [((1.08)10 - 1)/.08] = 14.486562, FV = $2000*14.486562  $28,973.13 USING A FINANCIAL CALCULATOR N= 10; PMT = -2,000; I = 8; PV=0; CPT FV = 28,973.13

  12. 4.2 Future Value of an Annuity Stream (continued) USING FVIFA TABLE (Appendix 3 in text) Find the FVIFA in the 8% column and the 10 period row: FVIFA = 14.4865 FV = 2000*14.4865 = $28.973.13

  13. 4.3 Present Value of an Annuity To calculate the value of a series of equal periodic cash flows at the current point in time, we can use the following simplified formula: The last portion of the equation is the Present Value Interest Factor of an Annuity (PVIFA). Practical applications include figuring out the nest egg needed prior to retirement or the lump sum needed for college expenses.

  14. FIGURE 4.4 Time line of present value of annuity stream.

  15. 4.3 Present Value of an Annuity (continued) Example 3: Present Value of an Annuity. John wants to make sure that he has saved up enough money prior to the year in which his daughter begins college. Based on current estimates, he figures that college expenses will amount to $40,000 per year for 4 years (ignoring any inflation or tuition increases during the 4 years of college). How much money will John need to have accumulated in an account that earns 7% per year, just prior to the year that his daughter starts college?

  16. 4.3 Present Value of an Annuity (continued) Example 3 (Answer) USING THE EQUATION Calculate the PVIFA value for n=4 and r=7%3.387211. Then, multiply the annuity payment by this factor to get the PV: PV = $40,000 x 3.387211 = $135,488.45

  17. 4.3 Present Value of an Annuity (continued) Example 3 (Answer—continued) USING A FINANCIAL CALCULATOR: Set the calculator for an ordinary annuity (END mode) and then enter: N= 4; PMT = 40,000; I = 7; FV=0; CPT PV = 135,488.45

  18. 4.3 Present Value of an Annuity (continued) Example 3 (Answer—continued) USING A PVIFA TABLE (APPENDIX 4) For r =7% and n = 4, PVIFA =3.3872 PVA = PMT*PVIFA = 40,000*3.3872 = $135,488 (Notice the slight rounding error!)

  19. 4.4 Annuity Due and Perpetuity A cash flow stream such as rent, lease, and insurance payments, which involves equal periodic cash flows that begin right away or at the beginning of each time interval, is known as an annuity due. Figure 4.5

  20. 4.4 Annuity Due and Perpetuity PV annuity due = PV ordinary annuity x (1+r) FV annuity due = FV ordinary annuity x (1+r) PV annuity due > PV ordinary annuity FV annuity due > FV ordinary annuity Can you see why?

  21. 4.4 Annuity Due and Perpetuity (continued) Example 4: Annuity Due versus Ordinary Annuity Let’s say that you are saving up for retirement and decide to deposit $3,000 each year for the next 20 years into an account that pays a rate of interest of 8% per year. By how much will your accumulated nest egg vary if you make each of the 20 deposits at the beginning of the year, starting right away, rather than at the end of each of the next twenty years?

  22. 4.4 Annuity Due and Perpetuity (continued) Example 4 (Answer) Given information: PMT = -$3,000; n=20; i= 8%; PV=0; FV ordinary annuity = $3,000 * [((1.08)20 - 1)/.08] = $3,000 * 45.76196 = $137,285.89 FV of annuity due = FV of ordinary annuity * (1+r) FV of annuity due = $137,285.89*(1.08) = $148,268.76

  23. 4.4 Annuity Due and Perpetuity (continued) Perpetuity A perpetuity is an equal periodic cash flow stream that will never cease. The PV of a perpetuity is calculated by using the following equation:

  24. 4.4 Annuity Due and Perpetuity (continued) Example 5: PV of a Perpetuity If you are considering the purchase of a consol that pays $60 per year forever, and the rate of interest you want to earn is 10% per year, how much money should you pay for the consol?  Answer: r=10%, PMT = $60, and PV = ($60/.1) = $600. So $600 is the most you should pay for the consol.

  25. 4.5 Three Payment Methods Loan payments can be structured in one of 3 ways: Discount loan Principal and interest is paid in lump sum at end Interest-only loan Periodic interest-only payments, with principal due at end Amortized loan Equal periodic payments of principal and interest

  26. 4.5 Three Payment Methods (continued) Example 6: Discount loan versus Interest-only loan versus Amortized loans Roseanne wants to borrow $40,000 for a period of 5 years. The lender offers her a choice of three payment structures: Pay all of the interest (10% per year) and principal in one lump sum at the end of 5 years. Pay interest at the rate of 10% per year for 4 years and then a final payment of interest and principal at the end of the 5th year. Pay 5 equal payments at the end of each year inclusive of interest and part of the principal. Under which of the three options will Roseanne pay the least interest and why? Calculate the total amount of the payments and the amount of interest paid under each alternative.

  27. 4.5 Three Payment Methods (continued) Option 1: Discount Loan Since all the interest and the principal is paid at the end of 5 years, we can use the FV of a lump sum equation to calculate the payment required: FV = PV x (1 + r)n FV5 = $40,000 x (1+0.10)5 = $40,000 x 1.61051 = $64, 420.40 Interest paid = Total payment - Loan amount Interest paid = $64,420.40 - $40,000 = $24,420.40

  28. 4.5 Three Payment Methods (continued) Option 2: Interest-Only Loan Annual Interest Payment (Years 1-4) = $40,000 x 0.10 = $4,000 Year 5 payment = Annual interest payment + Principal payment = $4,000 + $40,000 = $44,000 Total payment = $16,000 + $44,000 = $60,000 Interest paid = $60,000 - $40,000 = $20,000

  29. 4.5 Three Payment Methods (continued) Option 3: Amortized Loan By using financial calculator: n = 5; I = 10%; PV=$40,000; FV = 0;CPT PMT=$10,551.86 Or we can solve for PMT, PMT=PV/1-[(1/(1+r) n]/r = 40,000/1-[1/(1.1) 5]/0.1 = $10,551.86 Total payments = 5*$10,551.8 = $52,759.31 Interest paid = Total Payments - Loan Amount = $52,759.31-$40,000 Interest paid = $12,759.31 Loan Type Total Payment Interest Paid Discount Loan $64,420.40 $24,420.40 Interest-only Loan $60,000.00 $20,000.00 Amortized Loan $52,759.31 $12,759.31

  30. 4.6 Amortization Schedules An amortization schedule is a tabular listing of the allocation of each loan payment towards interest and principal reduction. It can help borrowers and lenders figure out the payoff balance on an outstanding loan. Procedure Compute the amount of each equal periodic payment (PMT). Calculate interest on unpaid balance at the end of each period, subtract it from the PMT, and reduce the loan balance by the remaining amount. Continue the process for each payment period, until you get a zero loan balance.

  31. 4.6 Amortization Schedules (continued) Example 7: Loan Amortization Schedule. Prepare a loan amortization schedule for the amortized loan option given in Example 6. What is the loan payoff amount at the end of 2 years? PV = $40,000; n=5; i=10%; FV=0; CPT PMT = $10,551.89

  32. 4.6 Amortization Schedules (continued) The loan payoff amount at the end of 2 years is $26,241.03

  33. 4.7 Waiting Time and Interest Rates for Annuities Problems involving annuities typically have 4 variables: i.e. PVorFV, PMT, r, n. If any 3 of the 4 variables are given, we can easily solve for the fourth one. This section deals with the procedure of solving problems where either n or r is not given. For example: Finding out how many deposits (n) it would take to reach aretirement or investment goal Figuring out the rate of return (r) required to reach a retirement goal, given fixed monthly deposits

  34. 4.7 Waiting Time and Interest Rates for Annuities (continued) Example 8: Solving for the Number of Annuities Involved Martha wants to save up $100,000 as soon as possible so that she can use it as a down payment on her dream house. She figures that she can easily set aside $8,000 per year and earn 8% annually on her deposits. How many years will Martha have to wait before she can buy that dream house?

  35. 4.7 Waiting Time and Interest Rates for Annuities (continued) Using the formula, solve for n FV= PMT * [(1+r) n -1]/r 100,000 = 8000 * [(1.08) n -1]/0.08 100,000/8000 = [(1.08) n -1]/0.08 1=(1.08) n -1 (1.08) n =2 n= ln (2)/ln (1.08)= 9.00647 USING A FINANCIAL CALCULATOR INPUT ? 8.0 0 -8000 100000 TVM KEYS N I/Y PV PMT FV Compute 9.00647

  36. 4.8 Solving a Lottery Problem In the case of lottery winnings, there are two choices: Annual lottery payment for fixed number of years Lump sum payout How do we make an informed judgment? We need to figure out the implied rate of return of both options using TVM functions.

  37. 4.8 Solving a Lottery Problem (continued) Example 9: Calculating an Implied Rate of Return, Given an Annuity Let’s say that you have just won the state lottery. The authorities have given you a choice of either taking a lump sum of $26,000,000 or a 30-year annuity of $1,625,000. Both payments are assumed to be after-tax. What will you do?

  38. 4.8 Solving a Lottery Problem (continued) Using the TVM keys of a financial calculator, enter: PV=26,000,000; FV=0; N=30; PMT = -$1,625,000; CPT I = 4.65283% 4.65283% = rate of interest used to determine the 30-year annuity of $1,625,000 versus the $26,000,000 lump sum payout. Choice: If you can earn an annual after-tax rate of return higher than 4.65% over the next 30 years, go with the lump sum. Otherwise, take the annuity option.

  39. 4.8 Solving a Lottery Problem (continued) Using the financial table: Then, go to PV of annuity table and go to the 30 period row and try to find 16 and see it under which column.

  40. 4.9 Ten Important Points about the TVM Equation Amounts of money can be added or subtracted only if they are at the same point in time. The timing and the amount of the cash flow are what matters. It is very helpful to lay out the timing and amount of the cash flow with a time line. Present value calculations discount all future cash flow back to current time. Future value calculations value cash flows at a single point in time in the future.

  41. 4.9 Ten Important Points about the TVM Equation (continued) An annuity is a series of equal cash payments at regular intervals across time. The time value of money equation has four variables but only one basic equation, and so you must know three of the four variables before you can solve for the missing or unknown variable. There are three basic methods to solve for an unknown time value of money variable: (1) Using equations and calculating the answer; (2) Using the TVM keys on a calculator; (3) Using financial functions from a spreadsheet.

  42. 4.9 Ten Important Points about the TVM Equation (continued) There are 3 basic ways to repay a loan: Discount loans Interest-only loans Amortized loans Despite the seemingly accurate answers from the time value of money equation, in many situations not all the important data can be classified into the variables of present value, time, interest rate, payment, or future value.

  43. ADDITIONAL PROBLEMS WITH ANSWERSProblem 1 Future Value ofUneven Cash Flows. If Mary deposits $4000 a year for three years, starting a year from today, followed by 3 annual deposits of $5000 into an account that earns 8% per year, how much money will she have accumulated in her account at the end of 10 years?

  44. ADDITIONAL PROBLEMS WITH ANSWERSProblem 1 (ANSWER) Future Value in Year 10 = $4000*(1.08)9 + $4000*(1.08)8 + $4000*(1.08)7 + $5000*(1.08)6 + $5000*(1.08)5 + $5000*(1.08)4 =$4000*1.999+$4000*1.8509+ $4000*1.7138+$5000*1.5868+ $5000*1.4693+$5000*1.3605 =$7,996+$7,403.6+$6,855.2+ $7,934+ $7,346.5+6,802.5 =$44,337.8

  45. ADDITIONAL PROBLEMS WITH ANSWERSProblem 2 Present Value of Uneven Cash Flows: Jane Bryant has just purchased some equipment for her beauty salon. She plans to pay the following amounts at the end of the next five years: $8,250; $8,500; $8,750; $9,000; and $10,500. If she uses a discount rate of 10 percent, what is the cost of the equipment that she purchased today?

  46. ADDITIONAL PROBLEMS WITH ANSWERSProblem 2 (ANSWER)

  47. ADDITIONAL PROBLEMS WITH ANSWERSProblem 3 Computing an Annuity Payment The Corner Bar & Grill is in the process of taking a five-year loan of $50,000 with First Community Bank. The bank offers the restaurant owner his choice of three payment options: Pay all of the interest (8% per year) and principal in one lump sum at the end of 5 years; Pay interest at the rate of 8% per year for 4 years and then a final payment of interest and principal at the end of the 5th year; Pay 5 equal payments at the end of each year inclusive of interest and part of the principal. Under which of the three options will the owner pay the least interest and why?

  48. ADDITIONAL PROBLEMS WITH ANSWERSProblem 3 (ANSWER) Under Option 1: Principal and Interest Due at end Payment at the end of year 5 = FVn = PV x (1 + r)n FV5 = $50,000 x (1+0.08)5 = $50,000 x 1.46933 = $73,466.5 Interest paid = Total payment - Loan amount Interest paid = $73,466.5 - $50,000 = $23,466.50

  49. ADDITIONAL PROBLEMS WITH ANSWERSProblem 3 (ANSWER continued) Under Option 2: Interest-only Loan Annual Interest Payment (Years 1-4) = $50,000 x 0.08 = $4,000 Year 5 payment = Annual interest payment + Principal payment = $4,000 + $50,000 = $54,000 Total payment = $16,000 + $54,000 = $70,000 Interest paid = $20,000

  50. ADDITIONAL PROBLEMS WITH ANSWERSProblem 3 (ANSWER continued) Under Option 3: Amortized Loan To calculate the annual payment of principal and interest, we can use the PV of an ordinary annuity equation and solve for the PMT value using n = 5; I = 8%; PV=$50,000; and FV = 0. PMT $12,522.82 Total payments = 5*$12,522.82 = $62,614.11 Interest paid = Total Payments - Loan Amount = $62,614.11-$50,000 Interest paid = $12,614.11

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