The Time Value of Money

1
The Time Value of Money
Chapter 9
2
The Time Value of Money

• Which would you rather have ?
• 100 today - or
• 100 one year from today
• Sooner is better !

3
The Time Value of Money

• How about 100 today or 105 one year from today?
• We revalue current dollars and future dollars
  using the time value of money
• Cash flow time line graphically shows the timing
  of cash flows

4
Cash Flow Time Lines

• Time 0 is today Time 1 is one period from today

Interest rate
Time
5
Cash Flows
-100
105
Outflow
Inflow
5
Future Value

• Compounding
• the process of determining the value of a cash
  flow or series of cash flows some time in the
  future when compound interest is applied

6
Future Value

• PV present value or starting amount, say, 100
• i interest rate, say, 5 per year would be
  shown as 0.05
• INT dollars of interest you earn during the
  year 100 ? 0.05 5
• FVn future value after n periods or 100 5
  105 after one year
• 100 (1 0.05) 100(1.05) 105

7
Future Value
8
Future Value

• The amount to which a cash flow or series of cash
  flows will grow over a given period of time when
  compounded at a given interest rate

9
Compounded Interest

• Interest earned on interest

10
Cash Flow Time Lines
5
Time
-100
Interest
5.00
5.25
5.51
5.79
6.08
Total Value 105.00
110.25
115.76
121.55
127.63
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Future Value Interest Factor for i and n (FVIFi,n)

• The future value of 1 left on deposit for n
  periods at a rate of i percent per period
• The multiple by which an initial investment grows
  because of the interest earned

12
Future Value Interest Factor for i and n (FVIFi,n)

• FVn PV(1 i)n PV(FVIFi,n)

For 100 at i 5 and n 5 periods
13
Future Value Interest Factor for i and n (FVIFi,n)

• FVn PV(1 i)n PV(FVIFi,n)

For 100 at i 5 and n 5 periods
100 (1.2763) 127.63
14
Financial Calculator Solution

• Five keys for variable input
• N the number of periods
• I interest rate per periodmay be I, INT, or
  I/Y
• PV present value
• PMT annuity payment
• FV future value

15
Two Solutions

• Find the future value of 100 at 5 interest per
  year for five years
• 1. Numerical Solution

5.25
5.51
5.79
6.08
110.25
115.76
121.55
127.63
FV5 100(1.05)5 100(1.2763) 127.63
16
Two Solutions

• 2. Financial Calculator Solution

Inputs N 5 I 5 PV -100 PMT 0 FV
? Output 127.63
17
Graphic View of the Compounding Process Growth

• Relationship among Future Value, Growth or
  Interest Rates, and Time

Future Value of 1
i 15
i 10
i 5
i 0
2
4
6
8
10
0
Periods
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Present Value

• Opportunity cost
• the rate of return on the best available
  alternative investment of equal risk
• If you can have 100 today or 127.63 at the end
  of five years, your choice will depend on your
  opportunity cost

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Present Value

• The present value is the value today of a future
  cash flow or series of cash flows
• The process of finding the present value is
  discounting, and is the reverse of compounding
• Opportunity cost becomes a factor in discounting

20
Cash Flow Time Lines
5
PV ?
127.63
21
Present Value

• Start with future value
• FVn PV(1 i)n

22
Two Solutions

• Find the present value of 127.63 in five years
  when the opportunity cost rate is 5
• 1. Numerical Solution

1.05
1.05
1.05
1.05
110.25
115.76
121.55
105.00
-100.00
23
Two Solutions

• Find the present value of 127.63 in five years
  when the opportunity cost rate is 5
• 2. Financial Calculator Solution

Inputs N 5 I 5 PMT 0 FV 127.63 PV
? Output -100
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Graphic View of the Discounting Process
Present Value of 1

• Relationship among Present Value, Interest Rates,
  and Time

i 0
i 5
i 10
i 15
2
4
6
8
10
12
14
16
18
20
Periods
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Solving for Time and Interest Rates

• Compounding and discounting are reciprocals
• FVn PV(1 i)n

Four variables PV, FV, i and n If you know any
three, you can solve for the fourth
26
Solving for i

• For 78.35 you can buy a security that will pay
  you 100 after five years
• We know PV, FV, and n, but we do not know i

FVn PV(1 i)n 100 78.35(1 i)5 Solve
for i
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Numerical Solution

• FVn PV(1 i)n
• 100 78.35(1 i)5

28
Financial Calculator Solution

• Inputs N 5 PV -78.35 PMT 0 FV
  100 I ?
• Output 5
• This procedure can be used for any rate or value
  of n, including fractions

29
Solving for n

• Suppose you know that the security will provide a
  return of 10 percent per year, that it will cost
  68.30, and that you will receive 100 at
  maturity, but you do not know when the security
  matures. You know PV, FV, and i, but you do not
  know n - the number of periods.

30
Solving for n

• FVn PV(1 i)n
• 100 68.30(1.10)n
• By trial and error you could substitute for n and
  find that n 4

0
1
2
n-1
n?
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Financial Calculator Solution

• Inputs I 10 PV -68.30 PMT 0 FV
  100 N ?
• Output 4.0

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Annuity

• An annuity is a series of payments of an equal
  amount at fixed intervals for a specified number
  of periods
• Ordinary (deferred) annuity has payments at the
  end of each period
• Annuity due has payments at the beginning of each
  period
• FVAn is the future value of an annuity over n
  periods

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Future Value of an Annuity

• The future value of an annuity is the amount
  received over time plus the interest earned on
  the payments from the time received until the
  future date being valued
• The future value of each payment can be
  calculated separately and then the total summed

34
Future Value of an Annuity

• If you deposit 100 at the end of each year for
  three years in a savings account that pays 5
  interest per year, how much will you have at the
  end of three years?

100
100.00
100 (1.05)0
105.00 100 (1.05)1
110.25 100 (1.05)2
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Future Value of an Annuity
36
Future Value of an Annuity

• Financial calculator solution
• Inputs N 3 I 5 PV 0 PMT -100 FV
  ?
• Output 315.25
• To solve the same problem, but for the present
  value instead of the future value, change the
  final input from FV to PV

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Annuities Due

• If the three 100 payments had been made at the
  beginning of each year, the annuity would have
  been an annuity due.
• Each payment would shift to the left one year and
  each payment would earn interest for an
  additional year (period).

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Future Value of an Annuity

• 100 at the end of each year

100
100.00
100 (1.05)0
105.00 100 (1.05)1
110.25 100 (1.05)2
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Future Value of an Annuity Due

• 100 at the start of each year

40
Future Value of an Annuity Due

• Numerical solution

41
Future Value of an Annuity Due

• Numerical solution

42
Future Value of an Annuity Due

• Financial calculator solution
• Inputs N 3 I 5 PV 0 PMT -100 FV
  ?
• Output 331.0125

43
Present Value of an Annuity

• If you were offered a three-year annuity with
  payments of 100 at the end of each year
• Or a lump sum payment today that you could put in
  a savings account paying 5 interest per year
• How large must the lump sum payment be to make it
  equivalent to the annuity?

44
Present Value of an Annuity
45
Present Value of an Annuity

• Numerical solution

46
Present Value of an Annuity
47
Present Value of an Annuity

• Financial calculator solution
• Inputs N 3 I 5 PMT -100 FV 0
  PV ?
• Output 272.325

48
Present Value of an Annuity Due

• Payments at the beginning of each year
• Payments all come one year sooner
• Each payment would be discounted for one less
  year
• Present value of annuity due will exceed the
  value of the ordinary annuity by one years
  interest on the present value of the ordinary
  annuity

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Present Value of an Annuity Due
0
1
2
3
5
100
100
50
Present Value of an Annuity Due

• Numerical solution

51
Present Value of an Annuity Due
52
Present Value of an Annuity Due

• Financial calculator solution
• Switch to the beginning-of-period mode, then
  enter
• Inputs N 3 I 5 PMT -100 FV 0
  PV ?
• Output 285.94
• Then switch back to the END mode

53
Solving for Interest Rates with Annuities

• Suppose you pay 846.80 for an investment that
  promises to pay you 250 per year for the next
  four years, with payments made at the end of each
  year

54
Solving for Interest Rates with Annuities

• Numerical solution
• Trial and error using different values for i
  using until you find i where the present value of
  the four-year, 250 annuity equals 846.80. The
  solution is 7.

55
Solving for Interest Rates with Annuities

• Financial calculator solution
• Inputs N 4 PV -846.8 PMT 250 FV 0
  I ?
• Output 7.0

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Perpetuities

• Perpetuity - a stream of equal payments expected
  to continue forever
• Consol - a perpetual bond issued by the British
  government to consolidate past debts in general,
  and perpetual bond

57
Uneven Cash Flow Streams

• Uneven cash flow stream is a series of cash flows
  in which the amount varies from one period to the
  next
• Payment (PMT) designates constant cash flows
• Cash Flow (CF) designates cash flows in general,
  including uneven cash flows

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Present Value of Uneven Cash Flow Streams

• PV of uneven cash flow stream is the sum of the
  PVs of the individual cash flows of the stream

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Future Value of Uneven Cash Flow Streams

• Terminal value is the future value of an uneven
  cash flow stream

60
Solving for i with Uneven Cash Flow Streams

• Using a financial calculator, input the CF values
  into the cash flow register and then press the
  IRR key for the Internal Rate of Return, which is
  the return on the investment.

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Compounding Periods

• Annual compounding
• interest is added once a year
• Semiannual compounding
• interest is added twice a year
• 10 annual interest compounded semiannually would
  pay 5 every six months
• adjust the periodic rate and number of periods
  before calculating

62
Interest Rates

• Simple (Quoted) Interest Rate
• rate used to compute the interest payment paid
  per period
• Effective Annual Rate (EAR)
• annual rate of interest actually being earned,
  considering the compounding of interest

63
Interest Rates

• Annual Percentage Rate (APR)
• the periodic rate multiplied by the number of
  periods per year
• this is not adjusted for compounding
• More frequent compounding

64
Amortized Loans

• Loans that are repaid in equal payments over its
  life
• Borrow 15,000 to repay in three equal payments
  at the end of the next three years, with 8
  interest due on the outstanding loan balance at
  the beginning of each year

65
Amortized Loans
0
1
2
3
8
PMT
PMT
PMT
15,000
66
Amortized Loans

• Numerical Solution

67
Amortized Loans

• Financial calculator solution
• Inputs N 3 I 8 PV 15000 FV 0
  PMT ?
• Output -5820.5

68
Amortized Loans

• Amortization Schedule shows how a loan will be
  repaid with a breakdown of interest and principle
  on each payment date

aInterest is calculated by multiplying the loan
balance at the beginning of the year by the
interest rate. Therefor, interest in Year 1 is
15,000(0.08) 1,200 in Year 2, it is
10,379.50(0.08)830.36 and in Year 3, it is
5,389.36(0.08) 431.15 (rounded). bRepayment
of principal is equal to the payment of 5,820.50
minus the interest charge for each year. cThe
0.01 remaining balance at the end of Year 3
results from rounding differences.
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Comparing Interest Rates

• 1. Simple, or quoted, rate, (isimple)
• rates compare only if instruments have the same
  number of compounding periods per year
• 2. Periodic rate (iPER)
• APR represents the periodic rate on an annual
  basis without considering interest compounding
• APR is never used in actual calculations

70
Comparing Interest Rates

• 3. Effective annual rate, EAR
• the rate that with annual compounding (m1) would
  obtain the same results as if we had used the
  periodic rate with m compounding periods per year

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End of Chapter 9
The Time Value of Money