1. TIME VALUE OF MONEY
CHAPTER 5

2. Overview Lump sum payments Annuities Perpetuities
Complex Cash Flow Streams
Compounding Frequency
EAR and APR

3. Lump Sum Payments
Calculating Present and Future Values of Lump Sum Payments

4. Future Value of a Single Amount
The future value of a single amount is the amount of money that a dollar will grow to at some point in the future.
Assume we will save $1,000 for three years and earn 6% interest compounded annually.
1st year: $1, × = $1,060.00
2nd year: $1, × = $1,123.60
3rd year: $1, × = $1,191.02
The future value of a single amount is the amount of money that a dollar will grow to at some point in the future.
Recall our previous example where we assume will save $1,000 for three years and earn 6% interest compounded annually.

5. Future Value of a Single Amount
Writing in a more efficient way, we can say
$1,000 × 1.06 × 1.06 × 1.06 = $1,191.02 or
$1,000 × [1.06]3 = $1,191.02
Writing in a more efficient way, we can say $1,000 times 1.06 times 1.06 times 1.06, or even more concise is $1,000 times 1.06 to the third.

6. Future Value of a Single Amount
$1,000 × [1.06]3 = $1,191.02
We can generalize this as . . .
Number
of Compounding Periods
FV = PV (1 + i)n
In fact, the future value of any invested amount can be determined using this concise formula.
Future
Value
Present Value
Interest
Rate

7. Present Value of a Single Amount
Instead of asking what is the future value of a current amount, we might want to know what amount we must invest today to accumulate a known future amount.
This is a present value question.
Present value of a single amount is today’s equivalent to a particular amount in the future.
Instead of asking what is the future value of a current amount, we might want to know what amount we must invest today to accumulate a known future amount.
This is a present value question.
Present value of a single amount is today’s equivalent to a particular amount in the future.

8. Present Value of a Single Amount
Remember our equation?
FV = PV (1 + i) n
We can solve for PV and get
Using our previous equation for future value, we can solve for present value by dividing the future value by 1 plus the interest rate raised to the number of periods. Another way to solve for the present value is to use the Present Value of $1 table in your textbook. FV
(1 + i)n
PV =

9. Present Value of a Single Amount
Assume you plan to buy a new car in 5 years and you think it will cost $20,000 at that time.
Example 5.1
What amount must you invest today in order to accumulate $20,000 in 5 years, if you can earn 8% interest compounded annually?
Solution of Example 5.1
i = .08, n = 5, FV = 20,000
PV = FV / (1 + i)n
PV = 20,000 / (1.08)5 = $13,611.60
If you deposit $13, now, at 8% annual interest, you will have $20,000 at the end of 5 years.
Assume you plan to buy a new car in 5 years and you think it will cost $20,000 at that time.
What amount must you invest today in order to accumulate $20,000 in 5 years, if you can earn 8% interest compounded annually?

10. Solving for Other Values
FV = PV (1 + i)n
Number
of Compounding Periods
Future
Value
Present
Value
Interest
Rate
There are four variables needed when determining the time value of money.
If you know any three of these, the fourth can be determined.
There are four variables needed when determining the time value of money. If you know any three of these, the fourth can be determined by using a little algebra.

11. Determining the Unknown Interest Rate
Example 5.2
Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to?
a. 3.5%
b. 4.0%
c. 4.5%
d. 5.0%
Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to?

12. Determining the Unknown Interest Rate
Example 5.2 Solution
Suppose a friend wants to borrow $1,000 today and promises to repay you $1,092 two years from now. What is the annual interest rate you would be agreeing to?
a. 3.5%
b. 4.0%
c. 4.5%
d. 5.0%

13. Annuities
Calculating Present and Future Values of Annuities

14. Ordinary Annuities
An annuity is a series of equal dollar payments that are made at the end of equidistant points in time such as monthly, quarterly, or annually over a finite period of time.
If payments are made at the end of each period, the annuity is referred to as ordinary annuity.
Example 5.3 How much money will you accumulate by the end of year 10 if you deposit $3,000 each for the next ten years in a savings account that earns 5% per year?

15. The Future Value of an Ordinary Annuity
The time line: i=5%
Time
Cashflow: …
FV [?]
We want to know the future value of the 10 cash flows.
We can compute the future value of each cash flow and sum them together:
3000(1.05) (1.05)8 + … = 37,733.68 1
2 … 10

16. The Future Value of an Ordinary Annuity
The earlier cash flows have higher future values because they have more years to earn interest.
Year 1 cash flow can earn 9 years of interest.
Year 10 cash flow does not earn any interest.

17. The Future Value of an Ordinary Annuity
Since the annuity cash flow has a strong pattern, we can also compute the future value of the annuity using a simple formula:
FVn = FV of annuity at the end of nth period.
PMT = annuity payment deposited or received at the end of each period.
i = interest rate per period
n = number of periods for which annuity will last.

18. Example 5.3 (Continued) = $3,000 { [0.63] ÷ (.05) } = $3,000 {12.58}
$3,000 for 10 years at 5% rate. Use the formula
FV = $3000 {[ (1+.05) ] ÷ (.05)}
= $3,000 { [0.63] ÷ (.05) }
= $3,000 {12.58}
= $37,733.68

19. Solving for PMT in an Ordinary Annuity
Instead of figuring out how much money you will accumulate (i.e. FV), you may like to know how much you need to save each period (i.e. PMT) in order to accumulate a certain amount at the end of n years.
In this case, we know the values of n, i, and FVn in the formula FVn=PMT [((1+i)n-1)/i], and we need to determine the value of PMT.
PMT=FVn/[((1+i)n-1)/i].

20. Examples
5-4 You would like to have $25,000 saved 6 years from now to pay towards your down payment on a new house. If you are going to make equal annual end-of-year payments to an investment account that pays 7%, how big do these annual payments need to be?
5-5 How much must you deposit in a savings account each year, earning 8% interest in order to accumulate $5,000 at the end of 10 years?
5-6 If you can earn 12% on your investments, and you would like to accumulate $100,000 for your child’s education at the end of 18 years, how much must you invest annually to reach your goal?
Verify the answers: ; ;

21. The Present Value of an Ordinary Annuity
The present value of an ordinary annuity measures the value today of a stream of cash flows occurring in the future.
Example 5-7: What is the value today or lump sum equivalent of receiving $3,000 every year for the next 30 years if the interest rate is 5%?
If I know its future value, I can compute its present value.
PV= FVn/(1+i)n, where
= PMT[ ((1-(1+i)-n)/i]
For the example, FV=199, PV=46,

22. One can also compute the PV of each cash flow
and sum them up.

23. Formulas for the Present and Future Values of an Ordinary Annuity

24. Checkpoint 5.1: Check Yourself
The Present Value of an Ordinary Annuity
Your grandmother has offered to give you $1,000 per year for the next 10 years. What is the present value of this 10-year, $1,000 annuity discounted back to the present at 5%?

25. Checkpoint 5.1
Verify the answer: ;

26. Checkpoint 5.2: Check Yourself
What is the present value of an annuity of $10,000 to be received at the end of each year for 10 years given a 10 percent discount rate?
Verify the Answer: 61,445.67

27. Amortized Loans
An amortized loan is a loan paid off in equal payments – consequently, the loan payments are an annuity.
Examples: Home mortgage loans, Auto loans
In an amortized loan, the present value can be thought of as the amount borrowed, n is the number of periods the loan lasts for, i is the interest rate per period, and payment is the loan payment that is made.

28. Example
Example 5.8 Suppose you plan to get a $9,000 loan from a furniture dealer at 18% annual interest with annual payments that you will pay off in over five years. What will your annual payments be on this loan?
PMT=PV/[(1-(1+i)n)/i] =2,

29. The Loan Amortization Schedule: How interest and principal are accounted for?
Year
Amount Owed on Principal at the Beginning of the Year (1)
Annuity Payment (2)
Interest Portion of the Annuity (3) = (1) × 18%
Repayment of the Principal Portion of the Annuity (4) =
(2) –(3)
Outstanding Loan Balance at Year end, After the Annuity Payment (5)
=(1) – (4) 1
$9,000
$2,878
$1,620.00
$1,258.00
$7,742.00 2
$7,742
$1,393.56
$1,484.44
$6,257.56 3 $
$1,126.36
$1,751.64
$4,505.92 4
$811.07
$2,066.93
$2,438.98 5
$439.02
$0.00

30. The Loan Amortization Schedule How interest and principal are accounted for?
We can observe the following from the table:
Size of each payment remains the same.
However, Interest payment declines each year as the amount owed declines and more of the principal is repaid.

31. Amortized Loans with Monthly Payments
Many loans such as auto and home loans require monthly payments. This requires converting n to number of months and computing the monthly interest rate.
Example 5-9 You have just found the perfect home. However, in order to buy it, you will need to take out a $300,000, 30- year mortgage at an annual rate of 6 percent. What will your monthly mortgage payments be?
n=30*12=360. i=6%/12=0.5%.
PMT=300000/[(1-(1/ )/0.005] = $

32. Annuities Due
Annuity due is an annuity in which all the cash flows occur at the beginning of the period. For example, rent payments on apartments are typically annuity due as rent is paid at the beginning of the month.
Computation of future/present value of an annuity due requires compounding the cash flows for one additional period, beyond an ordinary annuity.
FV or PV (annuity due) = (FV or PV (ordinary annuity)x(1+i)

33. Examples
Example 5.10 where we calculated the future value of 10-year ordinary annuity of $3,000 earning 5% to be $37,734. What will be the future value if the deposits of $3,000 were made at the beginning of the year i.e. the cash flows were annuity due?
Just compound the future value for the ordinary annuity for one more period: FV=37734 x 1.05=39,620.7
Checkpoint 5.2 where we computed the PV of 10-year ordinary annuity of $10,000 at a 10% discount rate to be equal to $61,446. What will be the present value if $10,000 is received at the beginning of each year i.e. the cash flows were annuity due?
Just compound the PV of the ordinary annuity for one more period: PV=61446x1.1=67,590.6

34. Perpetuities

35. Perpetuities
A perpetuity is an annuity that continues forever or has no maturity. For example, a dividend stream on a share of preferred stock.

36. Present Value of Perpetuity
with n=infinity
Present Value of a perpetuity= PMT/ i
PMT = level (constant) payment per period.
I = rate per period.

37. Examples
Example 5.11 What is the present value of $600 perpetuity at 7% discount rate?
PV=600/0.07=
If you decide to rent an apartment with a fixed rent of $2,000 per month and live there forever (subletting it to your children after you die), how much is this apartment worth if the mortgage rate is 6% per year.
The present value of paying $2000 per month forever at 6% rate per year is: PV=2000/(0.06/12)=400,000.
PV = $600 ÷ .07 = $8,571.43

38. Checkpoint 5.3: Check Yourself
The Present Value of a Perpetuity
What is the present value of a perpetuity of $500 paid annually discounted back to the present at 8 percent?
What is the present value of stream of payments equal to $90,000 paid annually forever and discounted back to the present at 9 percent?
Verify the Answer: 6250; 1,000,000

39. Complex Cash Flow Streams

40. Complex Cash Flow Streams
The cash flows streams in the business world may not always involve one type of cash flows. The cash flows may have a mixed pattern. For example, different cash flow amounts mixed in with annuities.
For example, figure 5-4 summarizes the cash flows for Marriott.

41. Complex Cash Flow Streams (cont.)
Fig. 5-4

42. Complex Cash Flow Streams
In this case, we can find the present value of the project by summing up all the individual cash flows by proceeding in four steps:
Find the present value of individual cash flows in years 1, 2, and 3.
Find the present value of ordinary annuity cash flow stream from years 4 through 10.
Discount the present value of ordinary annuity (step 2) back three years to the present.
Add present values from step 1 and step 3.

43. Checkpoint 5.4: Check Yourself
The Present Value of a Complex Cash Flow Stream What is the present value of cash flows of $500 at the end of years 1 through 3, a cash flow of a negative $800 at the end of year 4, and cash flows of $800 at the end of years 5 through 10 if the appropriate discount rate is 5%?

44. Checkpoint 6.6 PV of 3x5000=500*[(1-1.05-3)/.05]=1361.62
Year 4 value of 6x800= 800*[( )/.05]=
PV= /1.054=
Total PV= =

45. Checkpoint 5.5: Check Yourself
What is the present value of cash flows of $300 at the end of years 1 through 5, a cash flow of negative $600 at the end of year 6, and cash flows of $800 at the end of years 7-10 if the appropriate discount rate is 10%?

46. Steps Group the cash flow in to three types, all with i=10%
$300 from year 1 to 5
-$600 at year 6
$800 from year 7-10
Find PV for each group:
PV=300[( )/0.1]=
PV=-600/1.16=
PV={800[( )/0.1]}/1.16= (two steps here)
Total PV=

47. Compounding Frequency

48. Frequency of Compounding
General Formula:
FVn = PV0(1 + [i/m])mn
n: Number of Years
m: Compounding Periods per Year
i: Annual Interest Rate
FVn,m: FV at the end of Year n
PV0: PV of the Cash Flow today

49. Impact of Frequency
Julie Miller has $1,000 to invest for 2 Years at an annual interest rate of 12%.
Annual FV2 = 1,000(1+ [.12/1])(1)(2) = 1,254.40
Semi FV2 = 1,000(1+ [.12/2])(2)(2)= 1,262.48
Qrtly FV2= 1,000(1+ [.12/4])(4)(2) = 1,266.77
Monthly FV2= 1,000(1+ [.12/12])(12)(2) = 1,269.73
Daily FV2= 1,000(1+[.12/365])(365)(2) = 1,271.20
The result indicates that a $1,000 investment that earns a 12% annual rate compounded semi-annually for 2 years will earn a future value of $1,262.48
The result indicates that a $1,000 investment that earns a 12% annual rate compounded quarterly for 2 years will earn a future value of $1,
The result indicates that a $1,000 investment that earns a 12% annual rate compounded monthly for 2 years will earn a future value of $1,
The result indicates that a $1,000 investment that earns a 12% annual rate compounded daily for 2 years will earn a future value of $1,

50. EAR and APR

51. Effective Annual Rate (EAR)
This is the actual rate paid (or received) after accounting for compounding that occurs during the year.
If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.
Where m is the number of compounding periods per year

52. Annual Percentage Rate
This is the annual rate that is quoted by law
By definition APR = period rate times the number of periods per year
Consequently, to get the period rate we rearrange the APR equation:
Periodic rate = APR / number of periods per year
You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the periodic rate

53. Computing APRs What is the APR if the monthly rate is .5%?
.5(12) = 6%
What is the APR if the semiannual rate is .5%?
.5(2) = 1%
What is the monthly rate if the APR is 12% with monthly compounding?
12 / 12 = 1%

54. Things to Remember
You ALWAYS need to make sure that the interest rate and the time period match.
If you are looking at annual periods, you need an annual rate.
If you are looking at monthly periods, you need a monthly rate.
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.

55. EAR - Formula Remember that the APR is the quoted rate
m is the number of compounding periods per year

56. Decisions, Decisions!
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 and why?
Rates are quoted on an annual basis. The given numbers are APRs, not daily or semiannual rates.
First account:
EAR = ( /365)365 – 1 = 5.39%
Second account:
EAR = ( /2)2 – 1 = 5.37%
Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year?
First Account:
Daily rate = / 365 =
FV = 100( )365 =
Second Account:
Semiannual rate = / 2 = .0265
FV = 100(1.0265)2 =
You will have more money if you put money in the first account.

57. Computing APRs from EARs
If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get:

58. APR - Example
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?

59. Formulas

61. 6. For Future value of Annuity due
Future value of ordinary annuity * (1+i)
7. For Present value of Annuity due
Present value of ordinary annuity * (1+i)
8. Present Value of a Perpetuity =