1. Time Value of Money
Money Value of Time???

2. Interest Rates Why interest rates are positive?
People have ‘positive time preference’
Behavior of human beings
Current resources have productive uses
Technology and natural process

3. Simple vs. Compound Interest
Simple Interest
No interest is earned on interest money paid in the previous periods
Money grows at a slower rate
Compound Interest
Interest is earned on interest money paid in the previous periods
Money grows at a faster rate

4. Simple Interest Example
$100 at 8% simple annual interest for 2 years
First year interest
100 x (.08) = $8 Total = = $___
Second year interest
100 x (.08) = $8 Total = = $___
Total Interest after 2 years: = $__

5. Another example
You deposit $5000 into a savings account that earns 13% simple annual interest. What is the amount in the account after 6 years?
Answer:_________
What is the total amount of interest earned?

6. Compound Interest Example
Invest $100 at 8% compounded annually for 2 years
Total after first year:
100 x ( ) = $108
Total after second year
108 x ( ) = $_____
Total Interest = = $______

7. Compound Interest Example
Year Begin. Amount Interest Earned Ending Amount
1 $ $10.00 $110.00
Total interest $61.05
[What would be the total interest earned in simple interest case? Ans: $_______ ]

8. Future Value for a Lump Sum
Notice that
1. $ = $ ( )
2. $ = $ ( ) = $100 * 1.1 * 1.1 = $100 * 1.12
3. $ = $ ( ) = $100 * 1.1 * 1.1 * 1.1
= $ ________
In general, the future value, FVt, of $1 invested today at r% for t periods is
FVt = $1 * (1 + r)t
The expression (1 + r)t is called the future value factor.

9. FV on Calculator
What is the FV of $5000 invested at 12% per year for 4 years compounded annually?
Clear all memory: CLEAR ALL
Ensure # compounding periods is 1:
Enter amount invested today: -5000
Enter # of years: 4
Enter interest rate: 12
Find Future Value:
Answer: $___________
P/YR PV N
I/YR FV

10. Notice.. You entered $5000 as a negative amount
You got FV answer as a positive amount
Why the negative sign?
It turns out that the calculator follows ‘cash flow convention’
Cash outflow is negative (i.e. money going out)
Cash inflow is positive (i.e. money coming in)

11. Another example
Calculate the future value of $500 invested today at 9% per year for 35 years
Answer: ________

12. Present Values Here you simply reverse the question You are given
Future Value
Number of Periods
Interest Rate
and need to find the sum (PRESENT VALUE) needed today to achieve that FV

13. Present Value for a Lump Sum
Q. Suppose you need $20,000 in three years to pay tuition at SU. If you can earn 8% on your money, how much do you need today?
A. Here we know the future value is $20,000, the rate (8%), and the number of periods (3). What is the unknown present amount (called the present value)?
From before:
FVt = PV x (1 + r)t
$20,000 = PV __________
Rearranging:
PV = $20,000/(1.08)3
= $_____________

14. In general, the present value, PV, of a $1 to be received in t periods when the rate is r is
PV = FVt
(1+r)t
Present Value Factor = (1+r)t
‘r’ is also called the discount rate

15. PV on Calculator
Your friend promises to pay you $5,000 after 3 years. How much are you willing to pay her today? You can earn 8% compounded annually elsewhere.
Clear all memory: CLEAR ALL
Ensure # compounding periods is 1:
Enter amount future value:
Enter # of years: 3
Enter interest rate: 8
Find Present Value:
Answer: $___________
P/YR FV N
I/YR PV

16. Another PV example
Vincent van Gogh painted Portrait of Dr. Gachet in It sold in 1987 for $82.5 million. How much should he have sold it in 1889 if annual interest rate over the period was 9%?
Answer: _____________

17. Vincent Van Gogh
The Portrait of Dr Gachet

18. Present Value of $1 for Different Periods and Rates
1.00
.90
.80
.70
.60
.50
.40
.30
.20
.10
r = 0%
Present value of $1 ($)
r = 5%
r = 10%
r = 15%
r = 20%
Time (years)

19. Notice... As time increases, present value declines
As interest rate increases, present value declines
The rate of decline is not a straight line!

20. Notice Four Components
Present Value (PV)
Future Value at time t (FVt)
Interest rate per period (r)
Number of periods (t)
Given any three, the fourth can be found

21. Finding ‘r’
You need $8,000 after four years. You have $7,000 today. What annual interest rate must you earn to have that sum in the future? Answer: __________

22. Finding ‘t’
How many years does it take to double your $100,000 inheritance if you can invest the money earning 11% compounded annually? Answer: __________

23. Note:
When calculating future value what you are doing is compounding a sum
When calculating present value, what you are doing is discounting a sum

24. FV - Multiple Cash Flows
You deposit $100 in one year $200 in two years $300 in three years How much will you have in three years? r = 7% per year.
Answer: ____________
Draw a time line!!!

25. PV - Multiple Cash Flows
An investment pays: $200 in year 1 $600 in year $400 in year 2 $800 in year 4 You can earn 12% per year on similar investments. What is the most you are willing to pay now for this investment?
Answer: __________
Draw time line!!!

26. Important…
You can add cash flows ONLY if they are brought back (or taken forward) to the SAME point in time
Adding cash flows occurring at different points in time is like adding apples and oranges!

27. Level Multiple Cash Flows
Examples of constant level cash flows for more than one period
Annuities
Perpetuities
Most of the time we assume that the cash flow occurs at the END of the period

28. Examples of Annuities Car loan payments Mortgage on a house
Most other consumer loans
Contributions to a retirement plan
Retirement payments from a pension plan

29. Saving a Fixed Sum
You save $450 in a retirement fund every month for the next 30 years. The interest rate earned is 10%. What is the accumulated balance at the end of 30 years?
This is Future Value of an Annuity

30. Future Value Calculated
Save $2,000 every year for 5 years into an account that pays 10%. What is the accumulated balance after 5 years?
Future value calculated by compounding each cash flow separately 1 2 3 4 5
Time (years)
$2,000
$2,000
$2,000
$2,000
$2, , , , $12,210.20
x 1.1
x 1.12
x 1.13
x 1.14
Total future value

31. FV of Annuity

32. Important to understand inputs
‘r’ is the interest rate per period
‘t’ is the # of periods.
For example,
if ‘t’ is # of years, ‘r’ is annual rate
if ‘t’ is # of months, ‘r’ is the monthly rate

33. FV of Annuity Example
You will contribute $5,000 per year for the next 35 years into a retirement savings plan. If your money earns 10% interest per year, how much will you have accumulated at retirement?
Draw a time line!!!

34. Time Line Notice: Payment begins at the end of first year 1 2 34 351 2 34 35
-5000
-5000
-5000
-5000
Notice: Payment begins at the end of first year

35. FV of Annuity on Calculator
Clear all memory: CLEAR ALL
Ensure # compounding periods is 1:
Enter payments:
Enter # of payments:
Enter interest rate: 10
Find Future Value:
Answer: $___________
P/YR
PMT N
I/YR FV

36. FV Annuity - A Twist..
You estimate you will need $1 million to live comfortably in retirement in 30 years. How much must you save monthly if your money earns 12% interest per year?
Note: Payments are monthly, interest quoted is annual!!!

37. Two ways to adjust for compounding periods
Divide annual interest rate by 12 and enter interest rate per month into calculator as the interest rate and leave “P/YR” as 1
Set “P/YR” on calculator as 12: 12 and enter the annual interest rate OR
P/YR

38. ‘N’ on calculator You can either:
Enter # of periods directly (360 in the example)
If you have set 12 as the P/YR then you can also enter it as 30
(notice it appears as 360) OR N

39. FV Annuity on Calculator (2)
Clear all memory: CLEAR ALL
Monthly-> # compounding periods is 12: 12
Enter Future Value: 1,000,000
Enter # of payments:
Enter interest rate: 12
Find payments:
Answer: $___________
P/YR FV N
I/YR
PMT
Note the
difference!

40. Present Value of Annuities
Here we bring multiple, level cash flows back to the present (year 0)
Typical examples are consumer loans where the loan amount is the PV and the fixed payments are the cash flows

41. PV of Annuity Example Cash flow per period (CFt) = $500
Number of periods (t) = 4 years
Interest Rate (r) = 9% per year
What is the present value (PV) = ?
ALWAYS DRAW A TIME LINE!!!

42. PV of Annuity on Calculator
Clear all memory: CLEAR ALL
Ensure # compounding periods is 1:
Enter payments: 500
Enter # of payments: 4
Enter interest rate: 9
Find Present Value:
Answer: $___________
P/YR
PMT N
I/YR PV

43. PV of Annuity
Again: ‘r’ and ‘t’ must match – i.e. if t is # of months, r must be monthly rate

44. Car Loan Example Car costs $ 20,000 Interest rate per month = 1%
5-year loan ---> number of months = t = 60
What is the monthly payment?
Answer: ___________

45. Mortgage payments House cost $250,000 Mortgage Rate = 7.5% annually
Term of loan = 30 years
Payments made monthly
What are your payments?
Answer: _____________

46. To Reiterate... Be VERY careful about compounding periods
Problem can state annual interest rate, but the cash flows can be monthly, quarterly…
The convention is to state interest rate annually (Annual Percentage Rate)

47. Perpetuity
Annuity forever
Examples: Preferred Stock, Consols

48. Perpetuity
Note: C and r measured over same interval

49. Perpetuity Example
Preferred stock pays $1.00 dividend per quarter. The required return, r, is 2.5% per quarter.
What is the stock value?

50. Perpetuity Example
Steve Forbes’s flat-tax proposal was expected to save him $500,000 a year forever if passed. He spent $40,000,000 of his own money for campaign
Charge: He was running for presidency for personal gain
Did the charge make sense

51. Forbes continued... What should be ‘r’ in the example?
At what ‘r’ would Forbes have gained from being a president and steamrolling flat-tax proposal?

52. Compounding Periods Interest can be compounded
Annually Semiannually
Monthly Daily Continuously
Smaller the compounding period, faster is the growth of money
The same PV or FV formula can be used: BUT UNDERSTAND THE INPUTS!!

53. Compounding example Invest $5,000 in a 5-year CD
Quoted Annual Percentage Rate (APR) = 15%
Calculate FV5 for annual, semi-annual, monthly and daily compounding
Key: Adjust “P/YR” on calculator

54. Answers: Annual: $10,056.78 Semi-annual $10,305.16 Monthly: $10,535.91
Daily: $10,583.37
Continuous Compounding???

55. Continuous compouding
Compounded every instant “microsecond”
r = interest rate per period
t = number of periods
Previous example answer: $ 10,585.00

56. Continuous compounding example
Invest $4,500 in an account paying 9.5% compounded continuously
What is the balance after 4 years? Answer: _________

57. Quoted vs. Effective Interest Rates
Quoted Rate: Usually stated annually along with compounding period (APR)
e.g. 10% compounded quarterly
Effective Annual Rate (EAR): Interest rate actually earned IF the compounding period were one year

58. EAR
m = number of compounding periods in a year

59. EAR on Calculator
What is the EAR for quoted rate of 15% per year compounded quarterly?
Set number of periods per year: 4
Enter quoted annual rate: 15
Compute EAR:
Answer: _______
P/YR
I/YR
EFF%

60. EAR Example Compute EAR for 12% compounded
Annually
Quarterly
Monthly
Daily
Answers: ____ , ____ , ____ , ____

61. EAR for Continuous compounding
Example: Quoted rate is 10% compounded continuously
EAR = _____%

62. Complicatons to TVM When payments begin beyond year 1
PV and FV combined
When payments begin in year 0 (Annuities Due)

63. Payments beyond year 1
A car dealer offers ‘no payments for next 12 months’ deal on a $15,000 car. After that, you will pay monthly payments for the next 4 years. r = 10% APR. What are your monthly payments?
Answer: ___________

64. PV and FV combined
How much must you invest per year to have an amount in 20 years that will provide an annual income of $12,000 per year for 5 years? r = 8% annually.
Answer: ___________

65. PV and FV combined 2 You have 2 options:
Receive $100 for next 10 years only
Receive $100 forever beginning in year 11
If r = 10% which one would you prefer?
At what interest rate are you indifferent between the two options?

66. Annuities Due Payments begin in year 0 Trick:
Ex. Rent/Lease Payments
Trick:
Adjust BEG/END on calculator to BEG
Leave to END, but multiply (1+r) for both PV and FV OR

67. Annuity Due Example
Find PV of a 4-year (5 payment), $400 annuity due. r = 10%
Find FV in year 5 of the above annuity due
Answers:
PV = $1,667.95
FV5 = $2,686.24 1 2 3 4 5
Time (years)
$400
$400
$400
$400
$400 FV

68. Another Example
You start to contribute $500 every month to your IRA account beginning immediately. How much will you accumulate at the end of first year? The return on your investment is 20% per year.
Note: ‘Return’ here is just another term for the interest rate
Answer: $_______

69. Tricky but Legal...
Add-on Interest Called ‘add-on’ interest because interest is added on to the principal before the payments are calculated
Points on a Loan: Percentage of loan amount reduced up front
Used in home mortgages

70. Example: Add-on Interest
You are offered the opportunity to borrow $1,000 for 3 years at 12% ‘add-on’ interest. The lender calculates the payment as follows: Amt. owed in 3 years: $1000 x (1+.12)3 = 1,405 Monthly Payment = $1,405 / 36 = $39
What is the effective annual rate (EAR)?
Steps:
Calculate the APR interest (I/YR)
Use answer to calculate the EAR

71. Add-on Example (2)
Calcuate the EAR on a 6-year, $7,000 loan at 13% ‘add-on’ interest. The payments are monthly.
Answer: ________

72. Example: Points on a Loan
1-year loan of $100. r = 10% + 2 points [Note: 1 point = 1% of loan amount. Hence you pay upfront $2 to lender. Hence you are actually getting only $98, not $100]
What is the EAR?
$110 = $98 (1+r) r = 12.24%

73. Points on a loan (2)
Calculate the EAR on a 10-year, $110,000 mortgage when interest rate quoted is 7.75% + 1 point. The payments are monthly
Answer: _________

74. Balloon Payments
Amount on the loan outstanding after a certain number of payments have been made
Sometimes called ‘residual’ on a loan
e.g. when you want to pay off a loan early

75. Balloon Example
You borrowed $90,000 on a house for 30 years 10 years ago. The annual interest rate then was 17%. The payments are monthly. Since interest rate has fallen, you want to payoff the remaining amount on the loan and refinance it. What is the outstanding amount to be paid off? (Note: Payments are $1,283.11)
Answer: $__________

76. Two ways to calculate Balloons
First calculate payments
Take the present value of the remaining (unpaid) payments
Use amortization function on calculator
Enter the period : period
Enter , and then OR
INPUT
AMORT = = =

77. Another Example..
What is the outstanding balance on a 5 year $19,000 car loan at 11% interest after 2-1/2 years have passed? The payments are monthly.
Answer: $____________

78. TVM TIPS Draw time line! Check & set BEG/END on calculator
Check & set P/YR on calculator
Check & set # of decimal places to 4

79. TVM Tips Continued... Clear all previously stored #’s in memory
Especially true when same problem requires multiple TVM calculations
Make sure that for FV and PV calculation, you have correctly signed (+/-) the cash flows