Time Value of Money

1
Time Value of Money
2
Cash Flow Time Lines

• CF Time Lines are a graphical representation of
  cash flows associated with a particular financial
  option.

NOTE Each tick mark denotes the end of one
period.
3
Cash Flow Time Lines

• Outflow A payment or disbursement of cash, such
  as for investment, or expenses.
• Inflow A receipt of cash, can be in the form of
  dividends, principal, annuity payments, etc.

4
Future Value (FV)
Present Value (PV)
FV

• Future Value (FV) The ending value of a cash
  flow (or series of cash flows) over a given
  period of time, when compounded for a specified
  interest rate.
• Compounding The process of calculating the
  amount of interest earned on interest.

5
FV Calculations

• Given
• PV 100
• i 5
• n 1
• INT (PV x i)
• Solution
• FVn PVINT
• PV (PV x i)
• PV(1i)

Solution FVn 100INT 100 (100 x 5)
100(1 0.05) 105
6
FV Calculations
5
0
1
2
3
?
-100
INT2
INT1
INT3
5.00
5.25
5.51
S15.76
Value at end of Period
110.25
115.76
105.00

• FV1 PV(1i)
• FV2 FV1(1i) PV(1i)(1i)
• FV3 FV2(1i) PV(1i)(1i)(1i)
• FVn PV(1i)n

7
FV Calculations

• Three ways to calculate Time Value of Money (TVM)
  solutions
• Numerical SolutionCalculate solution with
  formula
• Tabular SolutionsUse Interest Factor tables to
  calculate
• Financial Calculator SolutionsUse calculator

8
Numerical Solution

• Future Value Interest Factor for i and n
  (FVIFi,n) is the factor by which the principal
  grows over a specified time period (n) and rate
  (i).
• FVIFi,n (1i)n
• Given Solution
• PV 1 FVn PV(1i)n PV(FVIFi.n)
• i 5 FV5 1(1.05)5
• n 5 FV5 1.2763

9
Tabular Solution
Given i 5 n 5
FVn PV(1i)n PV(FVIFi.n)
FVIFi,n (1 i)n
10
Financial Calculator

• Points to remember when using your Financial
  Calculator
• Check your settings
• END / BGN
• P/Y
• Clear TVM memory
• Five Variables (N, I/Y, PV, PMT, FV) - with any 4
  the 5th can be calculated
• Given
• N 5
• I/Y 5
• PV 1
• PMT0
• FV ?

11
Present Value (PV)
5
0
1
2
3
?
105
PV
FV

• Present Value (PV) The current value of a future
  cash flow (or series of cash flows), when
  discounted for a specified period of time an
  rate.
• Discounting The process of calculating the
  present value of a future cash flow or series of
  cash flows.

12
PV Calculations
5
0
1
2
3
?
105

• Given
• FV 105
• i 5
• n 1
• Solution
• FVn PV(1i)n Solve for PV
• PVn FVn / (1i)n FVn1/(1i)n
• FVn(PVIFi,n)

Solution PVn 105/(10.05)1 100
13
PV Calculations
Given FV
5
0
1
2
3
110.25
105.00
115.7625
?
Value at end of Period

• 1.05

• 1.05

• 1.05

• FV2 FV3/(1i)
• FV1 FV2/(1i) FV3/(1i)/(1i)
• PV FV1/(1i) FV1/(1i)/(1i)/(1i)
• PVn FVn

14
Numerical Solution

• Present Value Interest Factor for i and n
  (PVIFi,n) is the discount factor applied to the
  FV in order to calculate the present value for a
  specific time period (n) and rate (i).
• PVIFi,n 1/(1i)n
• Given Solution
• FV 1 PVn FV1/(1i)n FV(PVIFi.n)
• i 5 PV5 1 1/(1.05)5
• n 5 PV5 0.7835

15
Tabular Solution
Given i 5 n 5
PVn FV1/(1i)n FV(PVIFi.n)
PVIFi,n (1 i)n
16
Financial Calculator

• Given
• N 5
• I/Y 5
• PV ?
• PMT0
• FV -1

Input
-1
5
5
0
N
I/Y
PV
PMT
FV
Output
.7835
17
Annuities

• Annuity a series of equal payments made at
  specific intervals for a specified period.
• Types of Annuities
• Ordinary (Deferred) Annuity - is an annuity in
  which the payments occur at the end of each
  period.
• Annuity Due - is an annuity in which the payments
  occur at the beginning of each period.

18
FV Ordinary Annuities

• Example You decide that starting a year from
  now you will deposit 1,000 each year in a
  savings account earning 8 interest per year.
  How much will you have after 4 years?

FVnPV(1i)n
FVAn PMT(1i)0 PMT(1i)1 PMT(1i)2 . . .
PMT(1i)n-1
19
FV Ordinary Annuities

• FVAn represents the future value of an ordinary
  annuity over n periods.
• FVAn PMT(1i)0 PMT(1i)1 PMT(1i)2 . . .
  PMT(1i)n-1
• PMT (1i)n-t PMT (1i)t
• PMT (1i)n-1 PMT

20
FV Ordinary Annuities

• Future Value Interest Factor for an Annuity
  (FVIFAi,n) is the future value interest factor
  for an annuity (even series of cash flows) of n
  periods compounded at i percent.

21
Numerical Solution

• Given
• PMT 1,000
• I 8
• N 4

Solution FVAn PMT (1i)n 1/i 1,000
(10.08)4 1/0.08 1,000 4.5061
4,506.11
22
Tabular Solution

• Given
• PMT 1,000
• I 8
• N 4
• FVAn PMT(FVIFAi,n)

PVIFAi,n
23
Financial Calculator

• Given
• N 4
• I/Y 8
• PV 0
• PMT1,000
• FV ?

Input
0
4
8
1,000
N
I/Y
PV
PMT
FV
Output
4,506.11
24
FV Annuity Due

• Example You decide that starting today you will
  deposit 1,000 each year in a savings account
  earning 8 interest per year. How much will you
  have after 4 years?

8
1,000
1,000
1,000
1,080.00
1,000
FVnPV(1i)n
1,166.40
1,259.71
1,360.49
4,866.60
S
25
FV Annuity Due

• FVA(Due)n represents the future value of an
  annuity due over n periods.

FVA(Due)n PMT (1i)t
26
FV Annuity Due

• Future Value Interest Factor for an Annuity Due
  (FVIFA(Due)i,n) is the future value interest
  factor for an annuity due of n periods compounded
  at i percent.

27
Numerical Solution
8
1,000
1,000
1,000
1,000

• Given
• PMT 1,000 - BGN
• I 8
• N 4

Solution FVA(Due)n PMT ((1i)n 1)/ix
(1i) 1,000 ((10.08)4 1)/0.08x
(10.08) 1,000 4.8666 4,866.60
28
Tabular Solution

• Given
• PMT 1,000 - BGN
• I 8
• N 4
• FVA(Due)n PMT(FVIFAi,n)(1i)

PVIFAi,n
29
Financial Calculator

• Given
• N 4
• I/Y 8
• PV 0
• PMT1,000 - BGN
• FV ?

BGN
Input
0
4
8
1,000
N
I/Y
PV
PMT
FV
Output
-4,866.60
30
PV Ordinary Annuities

• Example You decide that starting a year from
  now you will withdraw 1,000 each year for the
  next 4 years from a savings account which earns
  8 interest per year. How much do you need to
  deposit today?

The present value of an annuity is calculated by
adding the PV of the individually
discounted/compounded cash flows.
PVAn PMT1/(1i)1 PMT1/(1i)2 . . .
PMT1/(1i)n
31
PV Ordinary Annuities

• PVAn represents the present value of an ordinary
  annuity over n periods.
• PVAn PMT1/(1i)1 PMT1/(1i)2 . . .
  PMT1/(1i)n
• PMT (1i)t
• PMT

32
PV Ordinary Annuities

• Present Value Interest Factor for an Annuity
  (PVIFAi,n) is the present value interest factor
  for an annuity (even series of cash flows) of n
  periods compounded at i percent.

33
Numerical Solution

• Given
• PMT 1,000
• I 8
• N 4

Solution PVAn PMT 1-1/(1i)n/i 1,000
1-1/(10.08)4/0,08 1,000 3.3121
3,312.13
34
Tabular Solution

• Given
• PMT 1,000
• I 8
• N 4
• PVAn PMT(PVIFAi,n)

PVIFAi,n
Periods 7 8 9
1 0.9346 0.9259 0.9174
2 1.8080 1.7833 1.7591
3 2.6243 2.5771 2.5313
4 3.3872 3.3121 3.2397
5 4.1002 3.9927 3.8897
35
Financial Calculator

• Given
• N 4
• I/Y 8
• PV ?
• PMT1,000
• FV 0

Input
0
4
8
1,000
N
I/Y
PV
PMT
FV
Output
-3,3121.13
36
PV Annuity Due

• Example You decide that starting today you will
  withdraw 1,000 each year for the next four years
  from a savings account earning 8 interest per
  year. How much do you need today?

37
PV Annuity Due

• PVA(Due)n represents the future value of an
  annuity due over n periods.

38
PV Annuity Due

• Present Value Interest Factor for an Annuity Due
  (PVIFA(Due)i,n) is the present value interest
  factor for an annuity due of n periods compounded
  at i percent.

39
Numerical Solution
8
1,000
1,000
1,000
1,000

• Given
• PMT 1,000 - BGN
• I 8
• N 4

Solution PVA(Due)n PMT (1-1/(1i)n)/ix
(1i) 1,000 (1-1/(10.08)4)/0.08x
(10.08) 1,000 3.5771 3,577.10
40
Tabular Solution

• Given
• PMT 1,000 - BGN
• I 8
• N 4
• PVA(Due)n PMT(PVIFAi,n)(1i)

PVIFAi,n
Periods 7 8 9
1 0.9346 0.9259 0.9174
2 1.8080 1.7833 1.7591
3 2.6243 2.5771 2.5313
4 3.3872 3.3121 3.2397
5 4.1002 3.9927 3.8897
41
Financial Calculator

• Given
• N 4
• I/Y 8
• PV ?
• PMT1,000 - BGN
• FV 0

BGN
Input
0
4
8
1,000
N
I/Y
PV
PMT
FV
Output
-3,577.10
42
Solving for Interest Rates with Annuities

• PVAnPMT(PVIFAi,n)
• -3,239.72 1,000(PVIFAi,n)
• -3.2397 PVIFAi,n
• Numerical Solution
• Trial Error ? Solve for PVIFA

43
Solving for Interest Rates with Annuities

• Tabular Solution
• -3.2397 PVIFAi,n

PVIFAi,n
Periods 7 8 9
1 0.9346 0.9259 0.9174
2 1.8080 1.7833 1.7591
3 2.6243 2.5771 2.5313
4 3.3872 3.3121 3.2397
5 4.1002 3.9927 3.8897
44
Solving for Interest Rates with Annuities

• Financial Calculator
• N 4
• I/Y ?
• PV -3,239.72
• PMT1,000
• FV 0

Input
0
4
-3, 239.72
1,000
N
I/Y
PV
PMT
FV
Output
9
45
Perpetuities

• Perpetuity A perpetual annuity, an annuity
  which continues forever.
• Consol A perpetual bond issued by the British
  government where the proceeds were used to
  consolidate past debts.

PVP PMT / i
46
Perpetuities
PVA5,100 19,848
47
Uneven Cash Flow Streams
8
1,000
750
750
250
(231.48)
(643.00)
(595.37)
(735.03)
PVn FV1/(1i)n FV(PVIFi.n)
(2,204.88)
48
Semiannual and Other Compounding Periods

• Simple Interest Rate The interest rate used to
  compute the interest rate per period the quoted
  interest rate is always in annual terms.
• Effective Annual Rate (EAR) The actual interest
  rate being earned during a year when compounded
  interest is considered.

49
Semiannual and Other Compounding Periods

• Types of Compounding
• Annual Compounding
• Semiannual Compounding (Bonds)
• Quarterly (Stock Dividends)
• Daily (Bank Accounts/Credit Cards)
• EAR Formula

isimple
m
EAR 1 -1
m
50
Semiannual and Other Compounding Periods

• Annual Percentage Rate (APR) the periodic rate
  multiplied by the number of period per year.

51
Fractional Time Periods

• Use current formulas and convert time (n) into a
  fraction.

52
Amortized Loans

• Amortized Loan a loan that is repaid in equal
  payments (an annuity) over the life of the loan.
• Amortization Schedule A financial schedule
  illustrating each payment in the loan, and
  further breaking that down between principal and
  interest.