Time Value of Money

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 100 8 ___
• Second year interest
• 100 x (.08) 8 Total 100 8 8 ___
• Total Interest after 2 years 8 8 __

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?
• Answer_________

6
Compound Interest Example

• Invest 100 at 8 compounded annually for 2
  years
• Total after first year
• 100 x (1 .08) 108
• Total after second year
• 108 x (1 .08) _____
• Total Interest 116.64 - 100 ______

7
Compound Interest Example

• Year Begin. Amount Interest Earned
  Ending Amount
• 1 100.00 10.00 110.00
• 2 110.00 11.00 121.00
• 3 121.00 12.10 133.10
• 4 133.10 13.31 146.41
• 5 146.41 14.64 161.05
• 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. 110 100 (1 .10)
• 2. 121 110 (1 .10) 100 1.1
  1.1 100 1.12
• 3. 133.10 121 (1 .10) 100 1.1
  1.1 1.1
• 100 ________
• 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 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
• (1r)t
• Present Value Factor 1 (1r)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 1
  
• Enter amount future value 5000
• 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 1889. 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

• Presentvalueof 1 ()

r 5
r 10
r 15
r 20
Time(years)
1 2 3 4 5 6
7 8 9 10
19
Notice...

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

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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 yearsHow 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 3 400 in year 2 800 in year 4You
  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

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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 bycompounding each cash
flow separately
Time(years)
2,000
2,000
2,000
2,000
2,000.02,200.02,420.02,662.02.928.212,210.2
0
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
0
35
34
2
1
-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 1
  
• Enter payments -5000
• Enter of payments 35
• 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 12and 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-gt compounding periods is 12 12
  
• Enter Future Value 1,000,000
• Enter of payments 30
• 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 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 ---gt 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 Forbess 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
• Ex. Rent/Lease Payments
• Trick
• Adjust BEG/END on calculator to BEG
• Leave to END, but multiply (1r) 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

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 InterestCalled 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

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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 followsAmt. owed in 3
  years 1000 x (1.12)3 1,405Monthly 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

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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 (1r)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