Present value, annuity, perpetuity Financial Economics 2012 höst.
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Transcript of Present value, annuity, perpetuity Financial Economics 2012 höst.
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Present value, annuity, perpetuity
Financial Economics2012 höst
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How to Calculate Present Values
LEARNING OBJECTIVES • Time value of money• how to calculate present value of future cash flows.• To calculate the present value of perpetuities, growing
perpetuities, annuities and growing annuities.• Compound interest and simple interest• Nominal and effective interest rates.• To understand value additive property and the concept of
arbitrage.• the net present value rule and the rate of return rule.
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Time value of money• The value of 1 € today is not the same as 1 € in a
year’s time• Suppose the interest rate on savings is 2% per year. • At the end of year 1: 1 € × (1+0,02)=1,02• At the end of year 2: 1,02€ × (1+0,02) =1,0404 year 3: 1,0404 × (1+0,02)=1,0612 i.e. FV = 1€× (1 + 0,02)3 year FV
1 1
2 1,02
3 1,0404
4 1,061208
5 1,082432
6 1,104081
7 1,126162
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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
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Calculating future value of 1$
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 370
0.5
1
1.5
2
2.5
Value of 1 $ investment in t year’s timecompound interest rate at 2%
FV
Year
Note that the 1 $ doubled in about 37 year’s time, given interest rate 2%.
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future value and present value
• FV = PV× (1 + r)t
where FV = Future value PV = Present value r = interest rate t = number of years (Periods)
It is readily seen, PV = FV/ (1 + r)t
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Manhattan Island SaleThe Power of Compounding!
Peter 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. Obs! 8% is way to high as historical return! Test 5% 3% or 2 % instead. The value of the purchase then becomes catastrophically exponentially lower!
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Future Values
Compound Interest - Interest earned on interest.
Simple Interest - Interest earned only on the original investment.
Future Value - Amount to which an investment will grow after earning interest.
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Future ValuesExample - Simple Interest
Interest earned at a rate of 6% for five years on a principal balance of $100.
Interest Earned Per Year = 100 × .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 EarnedValue 100
Future Values
6106
6112
6118
6124
6130
Value at the end of Year 5 = $130
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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
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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
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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
Equals to Present value of
a $1 future payment.
DFr t
1
1( )
r
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Present Values
Present Value = PV
PV = Future Value after t periods
(1+r) t
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Present Values
Example 5.2You just bought a new computer for $3,000. The payment due in 2 years. That means you need to pay 3000$ after 2 years. 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
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Present Values
Discount Factor = DF Note that discount factor is the same as PV of $1 in t
year´s time
• Discount Factors can be used to compute the present value of any cash flow.
• PV=DF*FV
DFr t
1
1( )
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• 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)
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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
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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, ,
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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,
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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( )
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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( )
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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, .
. . ( . )
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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
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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,
. . ( . )
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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.
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InflationA
nnua
l Inf
lati
on, %
Annual U.S. Inflation Rates from 1900 - 2007
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Inflation
1 real interest rate = 1+nominal interest rate1+inflation rate
approximation formula
rateinflation -rateinterest nominalrateinterest Real
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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
Savings
Bond
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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.