510chapter6fall2009afterclass10222009

510chapter6fall2009afterclass10222009 - Chapter 6:...

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Unformatted text preview: Chapter 6: Accounting and the Time Value of Money Single-Sum Problems Present Value $15,000 Future Value? 0 1 2 3 4 5 6 BE6­1 Chris Spear invested $15,000 today in a fund that earns 8% compounded semiannually. To what amount will the investment grow in 3 years? What table do we use? Present Value of a Single Sum BE6­2: Tony Bautista needs $25,000 in 4 years. What amount must he invest today if his investment earns 12% compounded quarterly? Present Value? Future Value $25,000 0 1 2 3 4 5 6 What table do we use? Present Value of a Single Sum i=3% n=16 $25,000 $25,000 Future Value x .62317 Factor = $15,579 Present Value Single­Sum Problems Single­Sum Problems Solving for Other Unknowns Example—Computation of the Number of Periods The Village of Somonauk wants to accumulate $70,000 for the construction of a veterans monument in the town square. At the beginning of the current year, the Village deposited $47,811 in a memorial fund that earns 10% interest compounded annually. How many years will it take to accumulate $70,000 in the memorial fund? Single­Sum Problems Single­Sum Problems Example—Computation of the Number of Periods Illustration 6­14 Using the future value factor of 1.46410, refer to Table 6­1 and read down the 10% column to find that factor in the 4­period row. Single­Sum Problems Single­Sum Problems Example—Computation of the Number of Periods Illustration 6­14 Using the present value factor of .68301, refer to Table 6­2 and read down the 10% column to find that factor in the 4­period row. Single­Sum Problems Single­Sum Problems Solving for Other Unknowns Example—Computation of the Interest Rate Single­Sum Problems Single­Sum Problems Example—Computation of the Interest Rate Illustration 6­16 Using the future value factor of 1.76234, refer to Table 6­1 and read across the 5­period row to find the factor. Single­Sum Problems Single­Sum Problems Example—Computation of the Interest Rate Illustration 6­16 Using the present value factor of .56743, refer to Table 6­2 and read across the 5­period row to find the factor. BE6-3 C a nd ic e Willis will inve s t $ 3 0 ,0 0 0 to d a y . S h e ne e d s $ 1 5 0 ,0 0 0 in 2 1 ye a rs . Wh a t a nnua l inte re s t ra te m us t s h e e a rn? BE6­3 PV=$30,000 FV=150,000 0 1 2 3 21 150,000=30,000FVp(21,i) 5 =FVp(21,i) (p.308) i=8% BE6-4 Bo Newman will invest $10,000 today in a fund that earns 5% annual interest. How many years will it take for the fund to grow to $17,100? BE6­4 PV=$10,000 FV=$17,100 0 1 2 3 n 17,100=10,000FVp(5%, n) 1.71 =FVp(5%,n) (p. 308) n=11 BE6-7 John Fillmore’s lifelong dream is to own his own fishing boat to use in his retirement. Jack has recently come into an inheritance of $400,000. He estimates that the boat he wants will cost $300,000 when he retires in 5 years. How much of his inheritance must he invest at an annual rate of 12% (compounded annually) to buy the boat at retirement? BE6­7 PV=400,000 FV=$300,000 0 1 2 3 4 5 PV=300,000*PVf(12%,5) PV=300,000*.56743 PV=170,229 (p.310) BE6-8 Refer to the data in BE6-7. Assuming quarterly compounding of amounts invested at 12%, how much of Jack Thompson’s inheritance must be invested to have enough at retirement to buy the boat? BE6-8 FV=300,000 0 1 2 3 20 PV=300,000*PVf(3%,20) PV=300,000*.55368=$166,104 (p. 310) Annuities Annuity requires the following: (1) (2) (3) Periodic payments or receipts (called rents) of the same amount, The same­length interval between such rents, and Compounding of interest once each interval. Two Types Ordinary annuity ­ rents occur at the end of each period. Annuity Due ­ rents occur at the beginning of each period. Ordinary v. Annuity Due Note: The difference between an ordinary annuity and an annuity due is that: – Each rent or payment compounds (interest added) one more period in a annuity due, future value situation. – Each rent or payment is discounted (interest removed) one less period under the annuity due situation. – Given the same i, n and periodic rent, the annuity due will always yield a greater present value (less interest removed) and a greater future value (more interest added). 14 14 Annuities Future Value of an Ordinary Annuity Rents occur at the end of each period. No interest during 1st period. Present Value $20,000 20,000 20,000 20,000 20,000 20,000 Future Value 20,000 20,000 0 1 2 3 4 5 6 7 8 Future Value of an Ordinary Annuity Illustration: Assume that $1 is deposited at the end of each of 5 years (an Illustration: ordinary annuity) and earns 12% interest compounded annually. Following is the computation of the future value, using the “future value of 1” table (Table 6­1) for each of the five $1 rents. Future Value of an Ordinary Annuity A formula provides a more efficient way of expressing the future value of an ordinary annuity of 1. Where: R = periodic rent FVF­OA = future value factor of an ordinary annuity n,i i = rate of interest per period n = number of compounding periods Future Value of an Ordinary Annuity Illustration: What is the future value of five $5,000 deposits made at the end of each of the next 5 years, earning interest of 12%? Illustration 6­19 = $31,764.25 Future Value of an Ordinary Alternate Calculation Annuity Illustration: What is the future value of five $5,000 deposits made at the end of each of the next 5 years, earning interest of 12%? What table do we use? Future Value of an Ordinary Annuity i=12% n=5 What factor? $5,000 $5,000 Deposits x 6.35285 Factor = $31,764 Present Value Future Value of an Ordinary Annuity Present Value $30,000 30,000 30,000 30,000 30,000 30,000 Future Value 30,000 30,000 0 1 2 3 4 5 6 7 8 BE6­13 Adams Inc. will deposit $30,000 in a 12% fund at the end of each year for 8 years beginning December 31, Year 1. What amount will be in the fund immediately after the last deposit? What table do we use? Future Value of an Ordinary Annuity i=12% n=8 $30,000 Deposit x 12.29969 Factor = $368,991 Future Value Annuities Future Value of an Annuity Due Rents occur at the beginning of each period. Interest will accumulate during 1st period. Annuity Due has one more interest period than Ordinary Annuity. Factor = multiply future value of an ordinary annuity factor by 1 plus the interest rate. $20,000 20,000 20,000 20,000 20,000 20,000 20,000 20,000 Future Value 0 1 2 3 4 5 6 7 8 Comparison of Ordinary Annuity with an Annuity Due Future Value of an Annuity Due Future Value of an Annuity Due Computation of Rent Illustration: Assume that you plan to accumulate $14,000 for a down Illustration: payment on a condominium apartment 5 years from now. For the next 5 years, you earn an annual return of 8% compounded semiannually. How much should you deposit at the end of each 6semiannually. month period? R = $1,166.07 Future Value of an Annuity Due Computation of Rent $14,000 12.00611 Alternate Calculation = $ $1,166.07 Future Value of an Annuity Due Computation of Number of Periodic Rents Illustration: Suppose that a company’s goal is to accumulate $117,332 by making periodic deposits of $20,000 at the end of each year, which will earn 8% compounded annually while accumulating. How many deposits must it make? How 5.86660 Future Value of an Annuity Due Computation of Future Value Illustration: Mr. Goodwrench deposits $2,500 today in a savings account that earns 9% interest. He plans to deposit $2,500 every year for a total of 30 years. How much cash will Mr. Goodwrench accumulate in his retirement savings account, when he retires in 30 years? years? Future Value of an Annuity Due Present Value 30,000 $30,000 30,000 30,000 30,000 30,000 30,000 Future Value 30,000 0 1 2 3 4 5 6 7 8 Adams Inc. will deposit $30,000 in a 12% fund at the beginning of each year for 8 years beginning January 1, Year 1. What amount will be in the fund at the end of Year 8? What table do we use? Future Value of an Annuity Due i=12% n=8 12.29969 x 1.12 = 13.775652 $30,000 Deposit Deposit x 13.775652 Factor = $413,270 Future Value Present Value of an Ordinary Annuity Present Value of an Ordinary Annuity Present value of a series of equal amounts to be withdrawn or received at equal intervals. Periodic rents occur at the end of the period. Present Value $100,000 100,000 100,000 100,000 ..... 100,000 100,000 0 1 2 3 4 19 20 Present Value of an Ordinary Annuity Illustration: Assume that $1 is to be received at the end of each of 5 Illustration: periods, as separate amounts, and earns 12% interest compounded annually. Present Value of an Ordinary Annuity A formula provides a more efficient way of expressing the present value of an ordinary annuity of 1. Where: Present Value of an Ordinary Annuity Illustration: What is the present value of rental receipts of $6,000 each, to be received at the end of each of the next 5 years when discounted at 12%? Present Value of an Ordinary Annuity Present Value $100,000 100,000 100,000 100,000 ..... 100,000 100,000 0 1 2 3 4 19 20 Jaime Yuen wins $2,000,000 in the state lottery. She will be paid $100,000 at the end of each year for the next 20 years. How much has she actually won? Assume an appropriate interest rate of 8%. What table do we use? Present Value of an Ordinary Annuity i=5% n=20 $100,000 Receipts x 9.81815 Factor = $981,815 Present Value Present Value of an Annuity Due Present Value of an Annuity Due Present value of a series of equal amounts to be withdrawn or received at equal intervals. Periodic rents occur at the beginning of the period. Present Value $100,000 100,000 100,000 100,000 100,000 100,000 ..... 0 1 2 3 4 19 20 Present Value of an Annuity Due Comparison of Ordinary Annuity with an Annuity Due Illustration 6­31 Present Value of an Annuity Due Illustration: Space Odyssey, Inc., rents a communications satellite for 4 years Illustration with annual rental payments of $4.8 million to be made at the beginning of each year. If the relevant annual interest rate is 11%, what is the present value of the rental obligations? Present Value of an Annuity Due Present Value $100,000 100,000 100,000 100,000 100,000 100,000 ..... 0 1 2 3 4 19 20 Jaime Yuen wins $2,000,000 in the state lottery. She will be paid $100,000 at the beginning of each year for the next 20 years. How much has she actually won? Assume an appropriate interest rate of 8%. What table do we use? Present Value of an Annuity Due i=8% n=20 $100,000 $100,000 Receipts x 10.60360 Factor = $1,060,360 Present Value BE6-5 Sally Medavoy will invest $8,000 a year for 20 years in a fund that will earn 12% annual interest. If the first payment occurs at year end, what amount will be in the fund in 20 years? If the first payment into the fund occurs today, what amount will be in the fund in 20 years? BE6­5 $5,000 $5,000 0 1 2 3 20 Ordinary Annuity : FVa=8,000FVa-oa(12%,20) =8,000*72.05244=$576,420 (p.312) Annuity Due: FVa=8,000FVa(12%,20) =8,000(72.05244)*1.12=$645,590 BE6-6 Steve Madison needs $250,000 in 10 years. How much must he invest at the end of each year at 11% interest, to meet his needs. BE6­6 FV=$250,000 0 1 2 3 10 250,000=PMT*FVa-oa(10,11%) 250,000=PMT*16.72201 14,950=PMT (p.312) BE6-9 Morgan Freeman is investing $16,380 at the end of each year in a fund that earns 10% interest. In how many years will the fund be at $100,000 BE6­9 16,380 16,380 FV=$100,000 0 1 2 3 n 100,000=16,380FVa-oa(n, 10%) 6.105=FVa-oa(n, 10%) (p. 312) n=5 BE6-10 Henry Quincy wants to withdraw $30,000 each year for 10 years from a fund that earns 8% interest. How much must he invest today if the first withdrawal is at year end? How much must he invest today if the first withdrawal takes place immediately? BE6­10 30,000 30,000 0 1 2 3 10 PV=30,000*PVa-oa(8%,10)=30,000*6.71008 (p. 314) $201,302 PV=30,000*PVa-ad(8%,10)=30,000*7.24689 (p.316) $217,407 ...
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This note was uploaded on 12/29/2009 for the course ACC 5100 taught by Professor Andrews during the Fall '09 term at Wayne State University.

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