1. Chapter 6 - Time Value of Money

2. Time Value of Money
A sum of money in hand today is worth more than the same sum promised with certainty in the future.
Think in terms of money in the bank
The value today of a sum promised in a year is the amount you'd have to put in the bank today to have that sum in a year.
Example: Future Value (FV) = $1,000
k = 5%
Then Present Value (PV) = $ because
$ x .05 = $47.62
and $ $47.62 = $1,000.00

3. Time Value of Money Present Value Terminology
The amount that must be deposited today to have a future sum at a certain interest rate
Terminology
The discounted value of a future sum is its present value

4. Four different types of problem
Outline of Approach
Four different types of problem
Amounts
Present value
Future value
Annuities
Present value
Future value

5. Develop an equation for each Time lines - Graphic portrayals
Outline of Approach
Develop an equation for each
Time lines - Graphic portrayals
Place information on the time line

6. The Future Value of an Amount
How much will a sum deposited at interest rate k grow into over some period of time
If the time period is one year:
FV1 = PV(1 + k)
If leave in bank for a second year:
FV2 = PV(1 + k)(1 ─ k)
FV2 = PV(1 + k)2
Generalized:
FVn = PV(1 + k)n

7. The Future Value of an Amount
(1 + k)n depends only on k and n Define Future Value Factor for k,n as: FVFk,n = (1 + k)n Substitute for: FVn = PV[FVFk,n]

8. The Future Value of an Amount
Problem-Solving Techniques
All time value equations contain four variables
In this case PV, FVn, k, and n
Every problem will give you three and ask for the fourth.

9. Concept Connection Example 6-1 Future Value of an Amount
How much will $850 be worth in three years at 5% interest?
Write Equation 6.4 and substitute the amounts given.
FVn = PV [FVFk,n ]
FV3 = $850 [FVF5,3]

10. Concept Connection Example 6-1 Future Value of an Amount
Look up FVF5,3 in the three-year row under the 5% column of Table 6-1, getting

11. Concept Connection Example 6-1 Future Value of an Amount
Substitute the future value factor of for FVF5,3
FV3 = $850 [FVF5,3]
FV3 = $850 [1.1576]
= $983.96

12. Financial Calculators
Work directly with equations
How to use a typical financial calculator
Five time value keys
Use either four or five keys
Some calculators require inflows and outflows to be of different signs
If PV is entered as positive the computed FV is negative

13. Financial Calculators
Basic Calculator functions

14. Financial Calculators
What is the present value of $5,000 to be received in one year if the interest rate is 6%?
Input the following values on the calculator and compute the PV: N
I/Y
PMT PV FV 1 6
5000
4,716.98
Answer

15. The Present Value of an Amount
PV= FVn [PVFk,n ]
Future and present value factors are reciprocals
Use either equation to solve any amount problems

16. Concept Connection Example 6-3 Finding the Interest Rate
what interest rate will grow $850 into $ in three years. Here we have FV3, PV, and n, but not k.
Use Equation 6.7
PV= FVn [PVFk,n ]

17. Concept Connector Example 6-3
PV= FVn [PVFk,n ]
Substitute for what’s known
$850= $ [PVFk,n ]
Solve for [PVFk,n ]
[PVFk,n ] = $850/ $983.96
[PVFk,n ] = .8639
Find in Appendix A (Table A-2). Since n=3 search only row 3, and find the answer to the problem is (5% ) at top of column.

18. Concept Connection Example 6-3 Finding the Interest Rate

19. Annuity Problems Annuities
A finite series of equal payments separated by equal time intervals
Ordinary annuities
Annuities due

20. Figure 6-1 Future Value: Ordinary Annuity

21. Figure 6-2 Future Value: Annuity Due

22. The Future Value of an Annuity—Developing a Formula
The sum, at its end, of all payments and all interest if each payment is deposited when received
Figure 6-3 Time Line Portrayal of an Ordinary Annuity

23. Figure 6-4 Future Value of a Three-Year Ordinary Annuity

24. For a 3-year annuity, the formula is:
FVFAk,n

25. The Future Value of an Annuity—Solving Problems
Four variables in the future value of an annuity equation
FVAn future value of the annuity
PMT payment
k interest rate
n number of periods
Helps to draw a time line

26. Concept Connection Example 6-5 The Future Value of an Annuity
Brock Corp. will receive $100K per year for 10 years and will invest each payment at 7% until the end of the last year.
How much will Brock have after the last payment is received?

27. Concept Connection Example 6-5 The Future Value of an Annuity
FVAn = PMT[FVFAk,n]
FVFA 7,10 =
FVA10 = $100,000[ ] = $1,381,640

28. The Sinking Fund Problem
Companies borrow money by issuing bonds
No repayment of principal is made during the bond’s life
Principal is repaid at maturity in a lump sum
A sinking fund provides cash to pay off principal at maturity
See Concept Connection Example 6-6

29. Compound Interest and Non-Annual Compounding
Earning interest on interest
Compounding periods
Interest is usually compounded annually, semiannually, quarterly or monthly

30. Figure 6-5 The Effect of Compound Interest

31. The Effective Annual Rate
Effective annual rate (EAR)
The annually compounded rate that pays the same interest as a lower rate compounded more frequently

32. Year-end Balances at Various Compounding Periods for $100 Initial Deposit and knom = 12%
Table 6.2

33. The Effective Annual Rate
EAR can be calculated for any compounding period using the formula
m is number of compounding periods per year
Effect of more frequent compounding is greater at higher interest rates

34. The APR and the EAR
The annual percentage rate (APR) associated with credit cards is actually the nominal rate and is less than the EAR

35. Compounding Periods and the Time Value Formulas
n must be compounding periods
k must be the rate for a single
E.g. for quarterly compounding
k = knom divided by 4, and
n = years multiplied by 4

36. Save up to buy a $15,000 car in 2½ years.
Concept Connection Example 6-7 Compounding periods and Time Value Formulas
Save up to buy a $15,000 car in 2½ years.
Make equal monthly deposits in a bank account which pays 12% compounded monthly
How much must be deposited each month?
A “Save Up” problem
Payments plus interest accumulates to a known amount Save ups are always FVA problems

37. Concept Connection Example 6-7 Compounding periods and Time Value Formulas
Calculate k and n for monthly compounding,
and
n = 2.5 years x 12 months/year = 30 months.

38. Concept Connection Example 6-7 Compounding periods and Time Value Formulas
Write the future value of an annuity expression and substitute.
FVAN = PMT [FVFAk,n ]
$15,000 = PMT [FVFA1,30 ]
From Appendix A (Table A-3) FVFA1,30 = substituting
$15,000 = PMT [ ]
Solve for PMT
PMT = $431.22

39. Concept Connection Example 6-7 Compounding periods and Time Value Formulas

40. Figure 6-6 Present Value of a Three-period Ordinary Annuity

41. The Present Value of an Annuity Developing a Formula
Sum of the present values of all of the annuity’s payments
PVFAk,n

42. The Present Value of an Annuity—Solving Problems
There are four variables
PVA present value of the annuity
PMT payment
k interest rate
n number of periods
Problems give 3 and ask for the fourth

43. Concept Connection Example 6.9 PVA - Discounting a Note
Shipson Co. will receive $5,000 every six months (semiannually) for 10 years. The firm needs cash now and asks its bank to discount the contract and pay Shipson the present value of the expected annuity. This is a common banking service called discounting.
If the payer has good credit, the bank will discount the contract at the current rate of interest, 14% compounded semiannually and pay Shipson the present value of the annuity of the expected payments. How much should Shipson receive?
Solution:

44. Amortized Loans
An amortized loan’s principal is paid off over its life along with interest
Constant Payments are made up of a varying mix of principal and interest
The loan amount is the present value of the annuity of the payments

45. Concept Connection Example 6-11 Amortized Loan – Finding PMT
What are the monthly payments on a $10,000, four year loan at 18% compounded monthly?
Solution:
k= kNom /12 = 18%/12 =1.5%
n = 4 years x 12 months/ year = 48 months
PVA = PMT [PVFAk,n ]
$10,000 = PMT [PVFAk,n ]
PVFA15,48 =
$10,000 = PMT [ ]
PMT = $293.75

46. n=3 years x 12 months/year = 36 months
Concept Connection Example Amortized Loan – Finding Amount Borrowed
A car buyer and can make monthly payments of $500. How much can she borrow with a three-year loan at 12% compounded monthly.
Solution:
k= kNom /12 = 12%/12 =1%
n=3 years x 12 months/year = 36 months
PVA= PMT [PVFAk,n ]
PVA = PMT [PVFA1,36 ]
Appendix A (Table A-4) gives PVFA1,36 =
PVA= 500 ( )
PVA =$15,053.75

47. Loan Amortization Schedules
Shows interest and principal in each loan payment
Also shows beginning and ending balances of unpaid principal for each period
To construct we need to know
Loan amount (PVA)
Payment (PMT)
Periodic interest rate (k)

48. Table 6-4 Partial Amortization Schedule
Develop an amortization schedule for the loan in Example 6 -12
Note that the Interest portion of the payment is decreasing while the Principal portion is increasing.

49. Often the largest financial transaction in a person’s life
Mortgage Loans
Used to buy real estate
Often the largest financial transaction in a person’s life
Mortgages are typically amortized loans, compounded monthly over 30 years
Early years most of payment is interest
Later on principal is reduced quickly

50. Concept Connection 6-13 Interest Content of Early Loan Payment
Calculate interest in the first payment on a 30-year,
$100,000 mortgage at 6%, compounded monthly.
Solution:
n= 30 years x 12 months/year = 360
k=6%/12 months/year = .5%
PVA= PMT [PVFAk,n ]
$100,000 = PMT [PVFA.5,360 ]
$100,000 = PMT [ ]
PMT = $599.55
First month’s interest = $100,000 x .005 = $500
leaving $99.55 to reduce principal.
First payment is 83.4%

51. Concept Connection 6-13 Interest Content of Early Loan Payment
Next, solve for the monthly payment
PVA= PMT [PVFAk,n ]
$100,000 = PMT [PVFA.5,360 ]
$100,000 = PMT [ ]
PMT = $599.55
The first month’s interest is .5% of $100,000
$100,000 x .005 = $500
The $500 of the first payment goes to interest, leaving $99.55 to reduce principal. The first payment is 83.4%

52. Concept Connection 6-13 Interest Content of Early Loan Payment

53. Mortgage Loans Implications of mortgage payment pattern
Early mortgage payments provide a large tax savings, reducing the effective cost of borrowing
Halfway through a mortgage’s life, half of the loan is not yet paid off
Long-term loans result in large total interest amounts over the life of the loan
Adjustable rate mortgage (ARM)

54. Payments occur at beginning of periods
The Annuity Due
Payments occur at beginning of periods
The future value of an annuity due
Each PMT earns interest one period longer
Formulas adjusted by multiplying by(1+k)
FVAdn = PMT [FVFAk,n](1+k)
PVAdn = PMT [PVFAk,n](1+k)

55. Figure 6-7 Future Value of a Three-Period Annuity Due

56. Concept Connection Example 6-17 Annuity Due
Baxter Corp started 10 years of $50,000 quarterly sinking fund deposits today at 8% compounded quarterly. What will the fund be worth in 10 years?
Solution:
k = 8%/4 = 2%
n = 10 years x 4 quarters/year x 40 quarters
FVAdn = PMT [FVFAk,n](1+k)
FVAd40 = $50,000[FVFA2,40](1+.02)
FVAd40 = from Appendix A (Table A-3).
FVAd40 = $50,000[ ](1.02)
=$3,080,502

57. Recognizing Types of Annuity Problems
Annuity problems always involve a stream of equal payments with a transaction at either the end or the beginning
End — future value of an annuity
Beginning — present value of an annuity

58. Perpetuities A stream of regular payments goes on forever
An infinite annuity
Future value of a perpetuity
Makes no sense because there is no end point
Present value of a perpetuity
The present value of payments is a diminishing series
Results in a very simple formula

59. Example 6-18 Perpetuities – Preferred Stock
Longhorn Corp issues a security that pays $5 per quarter indefinitely. Similar issues earn 8% compounded. How much can Longhorn sell this security for?
Solution: Longhorn’s security pays a quarterly perpetuity. It is worth the perpetuity’s present value calculated using the current quarterly interest rate.
k = .08 / 4 = .02
PVP = PMT / k = $5.00/.02 = $250

60. Continuous Compounding
Compounding periods can be any length
As the time periods become infinitesimally short, interest is compounded continuously
To determine the future value of a continuously compounded value:

61. Example 6-20 Continuous Compounding
First Bank is offering 6½% compounded continuously on savings deposits. If $5,000 is deposited and left for 3½ years, how much will it grow into?
Solution:

62. Multipart Problems
Time value problems are often combined due to the complexity of real situations
A time line portrayal can be critical to keeping things straight

63. Concept Connection Example 6-21 Simple Multipart
Exeter Inc. has $75,000 in securities earning 16% compounded quarterly. The company needs $500, in two years.
Management will deposit money monthly at 12% compounded monthly to be sure of having the cash.
How much should Exeter deposit each month.
Solution: Calculate the future value of the $75,000 and subtract it from $500,000 to get the contribution required from the deposit annuity.
Then solve a save up problem (future value of an annuity) for the payment required to get that amount.

64. Concept Connection Example 6-21 Simple Multipart

65. Concept Connection Example 6-21 Simple Multipart
Find the future value of $75,000 with Equation 6.4.
FVn = PV [PVFk,n ]
FV8 = $75,000 [FVF4,8]
= $75,000 [1.3686]
= $102,645
Then the savings annuity must provide:
$500,000 - $102,645 = $397,355

66. Concept Connection Example 6-21 Simple Multipart
Use Equation 6.13 to solve for the required payment.
FVAn = PMT [FVFA k,n ] $397,355 = PMT [FVFA1,24]
$397,355 = PMT [ ]
PMT = $14,731

67. Uneven Streams and Imbedded Annuities
Many real problems have uneven cash flows
These are NOT annuities
For example, determine the present value of the following stream of cash flows
Must discount each cash flow individually

68. Example 6-23 Present Value of an Uneven Stream of Payments
Calculate the interest rate at which the present value of the stream of payments shown below is $500.
$100
$200
$300
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 $ The answer is between 8% and 9%.

69. Sometimes uneven streams cash have annuities embedded within them
Imbedded Annuities
Sometimes uneven streams cash have annuities embedded within them
Use the annuity formula to calculate the present or future value of that portion of the problem

70. Present Value of an Uneven Stream