FINANCIAL MANAGEMENT PART 5

1. Time Value of Money Chapter 5

8. Table 5-1: The Future Value Factor for k and n FVF k,n = (1+k) n 5 6% 1.3382

11. Financial Calculators N I/Y PMT FV PV Number of time periods Interest rate (%) Future Value Present Value Payment Basic Calculator Keys

12. Financial Calculators N I/Y PMT PV FV 1 6 0 5000 4,716.98 Answer Q: What is the present value of $5,000 received in one year if interest rates are 6%? A: Input the following values on the calculator and compute the PV: Example

14. The Expression for the Present Value of an Amount—Example N PV PMT FV 3 -850 0 983.96 5.0 I/Y Answer Or you could look up the interest factor of 0.8639 [850  983.96] in the proper table with n=3. Example Q: What interest rate will grow $850 into $983.96 in three years? A: Using the FV equation of FV = PV (1 + i) n , the answer is:

15. The Expression for the Present Value of an Amount—Example Q: How long does it take money invested at 14% to double? A: The future value must be twice the present value, so make up some values for present and future values that meet that criterion, such as $1 doubling to $2. N PV PMT FV 5.29 -1 0 2 14 I/Y Answer Or you could look up the interest factor of 2.00 [2  1] in the proper table with k=14%. Example

17. Figure 5.1: Ordinary Annuity

18. Figure 5.2: Annuity Due

20. Figure 5.4: FV of a Three-Year Ordinary Annuity

23. The Future Value of an Annuity—Solving Problems—Example Q: The Brock Corporation owns the patent to an industrial process and receives license fees of $100,000 a year on a 10-year contract for its use. Management plans to invest each payment until the end of the contract to provide funds for development of a new process at that time. If the invested money is expected to earn 7%, how much will Brock have after the last payment is received? A: Using the FVA n = PMT[FVFA k,n ] equation and looking up the interest factor of the annuity at an n of 10 and a k of 7, we obtain an interest factor of 13.8164. This gives up a FVA 10 of $100,000[13.8164] = $1,381,640. FV N PMT I/Y 1,381,645 10 100000 7 0 PV Answer Example

25. The Sinking Fund Problem –Example Q: The Greenville Company issued bonds totaling $15 million for 30 years. The bond agreement specifies that a sinking fund must be maintained after 10 years, which will retire the bonds at maturity. Although no one can accurately predict interest rates, Greenville’s bank has estimated that a yield of 6% on deposited funds is realistic for long-term planning. How much should Greenville plan to deposit each year to be able to retire the bonds with the money put aside? A: The time period of the annuity is the last 20 years of the bond issue’s life. Input the following keystrokes into your calculator. PMT N FV I/Y 407,768.35 20 15,000,000 6 0 PV Answer Example

27. Figure 5.5: The Effect of Compound Interest

29. The Effective Annual Rate—Example Example Q: If 12% is compounded monthly, what annually compounded interest rate will get a depositor the same interest? A: If your initial deposit were $100, you would have $112.68 after one year of 12% interest compounded monthly. Thus, an annually compounded rate of 12.68% [($112.68  $100) – 1] would have to be earned.

31. Table 5.3

33. The Effective Annual Rate – Example Example PMT N FV I/Y 431.22 30 15,000 1 0 PV Answer Q: You want to buy a car costing $15,000 in 2½ years. You plan to save the money by making equal monthly deposits in your bank account, which pays 12% compounded monthly. How much must you deposit each month? A: This is a future value of an annuity problem with a 1% monthly interest rate and a 30-month time period. Input the following keystrokes into your calculator.

36. The Present Value of an Annuity—Solving Problems–Example Q: The Shipson Company has just sold a large machine to Baltimore Inc. on an installment contract. The contract calls for Baltimore to make payments of $5,000 every six months (semiannually) for 10 years. Shipson would like its cash now and asks its bank to discount the contract and pay it the present (discounted) value. Baltimore is a good credit risk, so the bank is willing to discount the contract at 14% compounded semiannually. How much should Shipson receive? A: The contract represents an annuity with payments of $5,000. Adjust the interest rate and number of periods for semiannual compounding and solve for the present value of the annuity. Example PV N FV I/Y 52,970.07 20 0 7 5000 PMT Answer This can also be calculated using the PVA formula of PVA = PMT[PVFA k, n ] with an n of 20 and a k of 7%, resulting in PVA = $5,000[10.594] = $52,970.

40. Amortized Loans—Example Example PMT N PV I/Y 293.75 48 10,000 1.5 0 FV Answer This can also be calculated using the PVA formula of PVA = PMT[PVFA k, n ] with an n of 48 and a k of 1.5%, resulting in $10,000 = PMT[34.0426] = $293.75. Q: Suppose you borrow $10,000 over four years at 18% compounded monthly repayable in monthly installments. How much is your loan payment? A: Adjust your interest rate and number of periods for monthly compounding and input the following keystrokes into your calculator.

41. Amortized Loans—Example PV N FV I/Y 15,053.75 36 0 1 500 PMT Answer Example This can also be calculated using the PVA formula of PVA = PMT[PVFA k, n ] with an n of 36 and a k of 1%, resulting in PVA = $500[30.1075] = $15,053.75. Q: Suppose you want to buy a car and can afford to make payments of $500 a month. The bank makes three-year car loans at 12% compounded monthly. How much can you borrow toward a new car? A: Adjust your k and n for monthly compounding and input the following calculator keystrokes.

43. Loan Amortization Schedules—Example Example Q: Develop an amortization schedule for the loan demonstrated in Example 5.12. Note that the Interest portion of the payment is decreasing while the Principal portion is increasing.

46. Mortgage Loans—Example Example PMT N FV I/Y 1,015.65 360 0 .5979 150,000 PV Answer $150,944 Net interest cost $64,690 Tax Savings @ 30% $215,634 Total Interest $150,000 - Original Loan $365,634 Total payments 360 X number of payments $1,015.65 Monthly payment Q: Calculate the monthly payment for a 30-year 7.175% mortgage of $150,000. Also calculate the total interest paid over the life of the loan. A: Adjust the n and k for monthly compounding and input the following calculator keystrokes.

48. The Annuity Due—Example Example FV N PMT I/Y 3,020,099 x 1.02 = 3,080,501 40 50,000 2 0 PV Answer NOTE: Advanced calculators allow you to switch from END (ordinary annuity) to BEGIN (annuity due) mode. Q: The Baxter Corporation started making sinking fund deposits of $50,000 per quarter today. Baxter’s bank pays 8% compounded quarterly, and the payments will be made for 10 years. What will the fund be worth at the end of that time? A: Adjust the k and n for quarterly compounding and input the following calculator keystrokes.

51. Perpetuities—Example Example You may also work this by inputting a large n into your calculator (to simulate infinity), as shown below. PV N PMT I/Y 250 999 5 2 0 FV Answer Q: The Longhorn Corporation issues a security that promises to pay its holder $5 per quarter indefinitely. Money markets are such that investors can earn about 8% compounded quarterly on their money. How much can Longhorn sell this special security for? A: Convert the k to a quarterly k and plug the values into the equation.

53. Continuous Compounding—Example A: To determine the EAR of 12% compounded continuously, find the future value of $100 compounded continuously in one year, then calculate the annual return. Example NOTE: Some advanced calculators have a function to solve for the EAR with continuous compounding. Q: The First National Bank of Charleston is offering continuously compounded interest on savings deposits. Such an offering is generally more of a promotional feature than anything else. If you deposit $5,000 at 6½% compounded continuously and leave it in the bank for 3½ years, how much will you have? Also, what is the equivalent annual rate (EAR) of 12% compounded continuously? A: To determine the future value of $5,000, plug the appropriate values into the equation.

54. Table 5.5: Time Value Formulas

56. Multipart Problems—Example Example Q: Exeter Inc. has $75,000 invested in securities that earn a return of 16% compounded quarterly. The company is developing a new product that it plans to launch in two years at a cost of $500,000. Exeter’s cash flow is good now but may not be later, so management would like to bank money from now until the launch to be sure of having the $500,000 in hand at that time. The money currently invested in securities can be used to provide part of the launch fund. Exeter’s bank has offered an account that will pay 12% compounded monthly. How much should Exeter deposit with the bank each month to have enough reserved for the product launch?

58. Multipart Problems—Example Example To find the future value of the $75,000… PV N PMT I/Y 102,643 8 0 4 75,000 FV Answer To find the savings annuity value PMT N PV I/Y 14,731 24 0 1 397,355 FV Answer $500,000 - $102,645 = $397,355

60. Uneven Streams and Imbedded Annuities $100 $200 $300 Example A: We’ll start with a guess of 12% and discount each amount separately at that rate. This value is too low, so we need to select a lower interest rate. Using 11% gives us $471.77. The answer is between 8% and 9%. Q: Calculate the interest rate at which the present value of the stream of payments shown below is $500.