gsb711-lecture-note-03-the-time-value-of-money
TRANSCRIPT
GSB711 Managerial Finance – Topic 01 Page No. 1 GSB711 Managerial Finance – Topic 03 Page No. 1
The Time Value of Money
Topic 03GSB711 – Managerial Finance
Reading:Chapter: The Time Value of Money (Pages 108 – 152)
Questions: 3, 4, 5, 6, 10, 11, 12 and 18 Problems: 22, 25, 27, 34, 35, 36, 40, 44, 46, 48, 50, 55, 59, 63, 68,
76, and 80.
GSB711 Managerial Finance – Topic 01 Page No. 2 GSB711 Managerial Finance – Topic 03 Page No. 2
Topics Covered
• Future Values and Compound Interest• Present Values• Multiple Cash Flows• Level Cash Flows
– Perpetuities and Annuities• Effective Annual Interest Rates• Inflation & Time Value
GSB711 Managerial Finance – Topic 01 Page No. 3 GSB711 Managerial Finance – Topic 03 Page No. 3
Future Values
Future Value - Amount to which an investment will grow after earning interest.
Compound Interest - Interest earned on interest.
Simple Interest - Interest earned only on the original investment.
GSB711 Managerial Finance – Topic 01 Page No. 4 GSB711 Managerial Finance – Topic 03 Page No. 4
Future Values
Example - Simple InterestInterest earned at a rate of 6% for five years on a principal balance of $100.
Interest Earned Per Year = 100 x .06 = $ 6
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Example - Simple InterestInterest earned at a rate of 6% for five years on a principal balance of $100.
Today Future Years 1 2 3 4 5
Interest Earned
Value 100
Future Values
6106
6112
6118
6124
6130
Value at the end of Year 5 = $130
GSB711 Managerial Finance – Topic 01 Page No. 6 GSB711 Managerial Finance – Topic 03 Page No. 6
Future Values
Example - Compound InterestInterest earned at a rate of 6% for five years on the previous year’s balance.
Interest Earned Per Year =Prior Year Balance x .06
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Example - Compound InterestInterest earned at a rate of 6% for five years on the previous year’s balance.
Today Future Years 1 2 3 4 5
Interest EarnedValue 100
Future Values
6106
6.36112.36
6.74119.10
7.15126.25
7.57133.82
Value at the end of Year 5 = $133.82
GSB711 Managerial Finance – Topic 01 Page No. 8 GSB711 Managerial Finance – Topic 03 Page No. 8
Future Values
Future Value of $100 = FV
FV r t $100 ( )1
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Future Values
FV r t $100 ( )1
Example - FV
What is the future value of $100 if interest is compounded annually at a rate of 6% for five years?
82.133$)06.1(100$ 5 FV
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0
200
400
600
800
1000
1200
1400
1600
1800
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of Years
FV
of
$100
0%
5%
10%
15%
Future Values with Compounding
Interest Rates
GSB711 Managerial Finance – Topic 01 Page No. 11 GSB711 Managerial Finance – Topic 03 Page No. 11
Manhattan Island SalePeter Minuit bought Manhattan Island for $24 in 1626. Was this a good deal?
trillion
FV
63.140$
)08.1(24$ 382
To answer, determine $24 is worth in the year 2008, compounded at 8%.
FYI - The value of Manhattan Island land is well below this figure.
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Present ValuesPresent Value
Value today of a future cash
flow.
Discount Rate
Interest rate used to compute
present values of future cash flows.
Discount Factor
Present value of a $1 future payment.
GSB711 Managerial Finance – Topic 01 Page No. 13 GSB711 Managerial Finance – Topic 03 Page No. 13
Present Values
Present Value = PV
PV = Future Value after t periods
(1+r) t
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Present Values
ExampleYou just bought a new computer for $3,000. The payment terms are 2 years same as cash. If you can earn 8% on your money, how much money should you set aside today in order to make the payment when due in two years?
572,2$2)08.1(3000 PV
GSB711 Managerial Finance – Topic 01 Page No. 15 GSB711 Managerial Finance – Topic 03 Page No. 15
Present Values
Discount Factor = DF = PV of $1
• Discount Factors can be used to compute the present value of any cash flow.
DFr t
1
1( )
GSB711 Managerial Finance – Topic 01 Page No. 16 GSB711 Managerial Finance – Topic 03 Page No. 16
• The PV formula has many applications. Given any variables in the equation, you can solve for the remaining variable.
PV FVr t
1
1( )
Time Value of Money(applications)
GSB711 Managerial Finance – Topic 01 Page No. 17 GSB711 Managerial Finance – Topic 03 Page No. 17
0
20
40
60
80
100
120
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Number of Years
PV
of
$100
0%
5%
10%
15%
Present Values with Compounding
Interest Rates
GSB711 Managerial Finance – Topic 01 Page No. 18 GSB711 Managerial Finance – Topic 03 Page No. 18
• Value of Free Credit• Implied Interest Rates• Internal Rate of Return• Time necessary to accumulate funds
Time Value of Money(applications)
GSB711 Managerial Finance – Topic 01 Page No. 19 GSB711 Managerial Finance – Topic 03 Page No. 19
PV of Multiple Cash FlowsExample
Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer?
$15,133.06 PVTotal
36.429,3
70.703,3
8,000.00
2
1
)08.1(
000,42
)08.1(
000,41
payment Immediate
PV
PV
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Present Values
Present Value
Year 0
4000/1.08
4000/1.082
Total
= $3,703.70
= $3,429.36
= $15,133.06
$4,000
$8,000
Year0 1 2
$ 4,000
$8,000
GSB711 Managerial Finance – Topic 01 Page No. 21 GSB711 Managerial Finance – Topic 03 Page No. 21
Perpetuities & Annuities
Finding the present value of multiple cash flows by using a spreadsheet
Time until CF Cash flow Present value Formula in Column C0 8000 $8,000.00 =PV($B$11,A4,0,-B4)1 4000 $3,703.70 =PV($B$11,A5,0,-B5)2 4000 $3,429.36 =PV($B$11,A6,0,-B6)
SUM: $15,133.06 =SUM(C4:C6)
Discount rate: 0.08
GSB711 Managerial Finance – Topic 01 Page No. 22 GSB711 Managerial Finance – Topic 03 Page No. 22
PV of Multiple Cash Flows
• PVs can be added together to evaluate multiple cash flows.
PV C
r
C
r
1
12
21 1( ) ( )....
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Perpetuities & Annuities
Perpetuity A stream of level cash payments that
never ends.
Annuity Equally spaced level stream of cash
flows for a limited period of time.
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Perpetuities & Annuities
PV of Perpetuity Formula
C = cash payment r = interest rate
PV Cr
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Perpetuities & Annuities
Example - PerpetuityIn order to create an endowment, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?
PV 100 00010 000 000,. $1, ,
GSB711 Managerial Finance – Topic 01 Page No. 26 GSB711 Managerial Finance – Topic 03 Page No. 26
Perpetuities & Annuities
Example - continuedIf the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?
PV
1 000 000
1 10 3 315, ,
( . )$751,
GSB711 Managerial Finance – Topic 01 Page No. 27 GSB711 Managerial Finance – Topic 03 Page No. 27
Perpetuities & Annuities
PV of Annuity Formula
C = cash payment r = interest rate t = Number of years cash payment is received
PV C r r r t
1 11( )
GSB711 Managerial Finance – Topic 01 Page No. 28 GSB711 Managerial Finance – Topic 03 Page No. 28
Perpetuities & Annuities
PV Annuity Factor (PVAF) - The present value of $1 a year for each of t years.
PVAF r r r t
1 11( )
GSB711 Managerial Finance – Topic 01 Page No. 29 GSB711 Managerial Finance – Topic 03 Page No. 29
Perpetuities & Annuities
Example - AnnuityYou are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)? PV
PV
4 000
947 41
110
110 1 10 3,
$9, .
. . ( . )
GSB711 Managerial Finance – Topic 01 Page No. 30 GSB711 Managerial Finance – Topic 03 Page No. 30
Perpetuities & Annuities
Applications• Value of payments• Implied interest rate for an annuity• Calculation of periodic payments
– Mortgage payment– Annual income from an investment payout– Future Value of annual payments
FV C PVAF r t ( )1
GSB711 Managerial Finance – Topic 01 Page No. 31 GSB711 Managerial Finance – Topic 03 Page No. 31
Perpetuities & Annuities
Example - Future Value of annual paymentsYou plan to save $4,000 every year for 20 years and then retire. Given a 10% rate of interest, what will be the FV of your retirement account?
FV
FV
4 000 1 10
100
110
110 1 10
2020, ( . )
$229,
. . ( . )
GSB711 Managerial Finance – Topic 01 Page No. 32 GSB711 Managerial Finance – Topic 03 Page No. 32
Effective Interest Rates
Annual Percentage Rate - Interest rate that is annualized using simple interest.
Effective Annual Interest Rate - Interest rate that is annualized using compound interest.
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Effective Interest Rates
exampleGiven a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?
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Effective Interest Rates
exampleGiven a monthly rate of 1%, what is the Effective Annual Rate(EAR)? What is the Annual Percentage Rate (APR)?
12.00%or .12=12 x .01=APR
12.68%or .1268=1-.01)+(1=EAR
r=1-.01)+(1=EAR12
12
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Inflation
Inflation - Rate at which prices as a whole are increasing.
Nominal Interest Rate - Rate at which money invested grows.
Real Interest Rate - Rate at which the purchasing power of an investment increases.
GSB711 Managerial Finance – Topic 01 Page No. 36 GSB711 Managerial Finance – Topic 03 Page No. 36
InflationA
nnua
l Inf
latio
n, %
Annual U.S. Inflation Rates from 1900 - 2007
GSB711 Managerial Finance – Topic 01 Page No. 37 GSB711 Managerial Finance – Topic 03 Page No. 37
Inflation
1 real interest rate = 1+nominal interest rate1+inflation rate
approximation formula
Real int. rate nominal int. rate - inflation rate
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InflationExample
If the interest rate on one year govt. bonds is 6.0% and the inflation rate is 2.0%, what is the real interest rate?
4.0%or .04=.02-.06=ionApproximat
3.9%or .039 = rateinterest real
1.039 =rateinterest real1
=rateinterest real1 +.021+.061
GSB711 Managerial Finance – Topic 01 Page No. 39 GSB711 Managerial Finance – Topic 03 Page No. 39
Inflation
• Remember: Current dollar cash flows must be discounted by the nominal interest rate; real cash flows must be discounted by the real interest rate.
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Summary
• Time has value• Future Value – Compounding• Present Value – Discounting• Multiple Cash Flows
– Perpetuities– Annuities
• Effective Annual Rate and Annual Percentage Rate
• Inflation and time value - Consistency