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Basic Principles of Valuation
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Basic Principles of Valuation
Jens Carsten Jackwerth
University of Konstanz
http://www.wiwi.uni-konstanz.de/jackwerth/
Basic Principles of Valuation
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Outline
Motivation
Definition of the Net Present Value
The Net Present Value and the creation of value
Discounting
Perpetuities and annuities
Compounding
Assumptions and computation
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Motivation
A ‘great deal’:
year 0 1 2 3
cash flow -100 -50 30 200
Net cash flow is
Decisions based on net cash flows do not take
opportunity costs into account
0802003050100
CCCC 3210
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Net Present Value I
Interest rate is r
Net Present Value = sum of discounted net cash flows
Net Present Value of the ‘great deal’ at r = 10% is
029.6175.1145.5
0.1)(1
200
0.1)(1
30
0.11
50--100NPV
32
T
T
3
3
2
210
r)(1
C...
r)(1
C
r)(1
C
r1
CCNPV
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Net Present Value II
Discount rate
corresponds to the respective opportunity cost of capital, i.e.
riskless cash flows are discounted with the risk-free rate
risky cash flows need to be discounted with an interest rate with
the same level of risk
Net Present Value
positive net present value creates wealth
negative net present value destroys wealth
Present Value (more general than NPV)
present value = sum of discounted net cash flow
(often excludes investments)
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Net Present Value and the Creation of
Value I
The ‘great deal’ is worth as much as having €29.60 in the bank, as
one can replicate this cash flow as follows
year 0: borrow €29.60 and consume
borrow €100.00 and invest in the project
year 1: bank balance is (€29.60 + €100.00) 1.1 = €142.56
borrow €50.00 and invest in the project
year 2: bank balance is (€50.00 + €142.56) 1.1 = €211.82
receive €30.00 from the project and pay back loan
year 3: bank balance is (-€30.00 + €211.82) 1.1 = €200.00
receive €200.00 from the project and pay back loan
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Net Present Value and the Creation of
Value II
Assumptions
firm is free to borrow and lend
counterexample: banks have to hold reserves
credit interest = debit interest
counterexample: personally, you can borrow on a credit card at
25% but you lend on your account at 0%
shareholders’ interests in the firm are solely financial
counterexample: Frieda Springer would not sell her shares in
order keep control of the publisher „Springer Verlag“
shareholders are free to buy or sell shares
counterexample: In the UK, the state owns a „golden share“ and
thus has certain rights, here for instance the right to veto a sale of
defense companies
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Future Values
How much is €1 worth tomorrow?
Borrowing and lending is possible at r = 10%
€1.00 today = €1 ∙ (1+0.1) in 1 year = €1.10 in 1 year
= €1.1 ∙ (1+0.1) in 2 years
= €1 ∙ (1+0.1)2 in 2 years = €1.21 in 2 years
= €1 ∙ (1+0.1)3 in 3 years = €1.331 in 3 years
€a today = €a ∙ (1+ r )T in T years = €A in T years
€A is the future value in T years of €a invested now
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Present Values
How much is €1, received in one year, worth today?
€a is the Present Value of €A, received in T years
The discount factor is
Example: €1, received in 3 years, is worth today at r = 10%
T)r1(
1
751.0€)1.01(
1€3
€0.910.11
€1€a€10.1)(1 €a
T
T
r)(1
€A€a€Ar)(1€a
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Perpetuity
Constant cash flow in perpetuity
cash flow is P per year, starting in year one
discount rate is r
present value is obtained using the geometric series
year 0 1 2 … T … PV
cash flow 0 P P P P P P / r
Example
endowing a chair in finance at the university
interest rate is 10%
desired contribution is €50,000 per year in perpetuity
The required endowment today is €50,000 / 0.1 = €500,000
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Growing Perpetuity
Growing cash flow in perpetuity
cash flow is P in year one
growth rate of cash flows per year is g, starting in year two
discount rate is r
year 0 1 2 … T … PV
cash flow 0 P P(1+g) … P(1+g)T-1 … P / (r - g)
Example
endowing a chair in finance at the university
contributions are € 50,000 in year one
the contributions have to increase by 5% per year, r is 10%
The required endowment today is
€50,000 / (0.1 - 0.05) = €1,000,000
Basic Principles of Valuation
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Annuity I
Derivation of the PV for constant cash flows over T years
cash flow is P per year, starting in year 1, lasting T years
year 0 1 2 … T … PV
cash flow 0 P P P P 0 ??
replicate cash flows using two perpetuities
perpetuity 1 0 P P P P P P / r
perpetuity 2 0 0 0 0 0 P (P / r) / (1 + r)T
PV(Annuity) = PV(Perpetuity 1) - PV(Perpetuity 2) =
TT )r1(
11
r
P
)r1(r
P
r
P
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Annuity II
The T-year annuity factor with interest rate r is
Example
endowing a chair in finance at the university
discount rate is r = 10%
contributions are €50,000 per year for 5 years
The required endowment is
T)r1(
11
r
1
€189,5390.1)(1
11
0.1
€50,0005
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Compound Interest I
Compound interest is the interest on the reinvestment of
the interest paid during the investment period
Invest €100 for one year at 8%
with annual compounding one obtains
with semi-annual compounding one obtains
with quarterly compounding one obtains
with n-times compounding per year one obtains
1080.08)(1100
108.16(1.04)1002
0.081
2
0.081100 2
108.24 (1.02) 100 4
0.08 1 100 4
4
n
0.08 1 100
n
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Compound Interest II
Limit as n goes to infinity
As the time between payments of interest (and its
reinvestment) becomes smaller, money in the account
grows and finally converges
Relation between annually compounding and n-times per
year compounding
r1
n
r1
n
n
1r)(1nr n
1
n
108.33e100n
0.081100 0.08
n
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Discount Factors
annual compounding at interest rate r
n-times compounding at interest rate rn
continuous compounding at interest rate R
T)r1(
1
nT
n
n
r1
1
RT
RTe
e
1
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Methods for Computing the
Net Present Value
Example of a discount table at r = 10%
year 0 1 2 3 total
cash flow -100 -50 30 200
discount factor 1 0.909 0.826 0.751
present values -100 -45.5 24.8 150.3 29.6
Calculator
divide cash flow in each year t by (1 + r)t
then sum over all periods
Spreadsheet in Excel computes PV
NPV(r, A2:C2) - 100
note: C1 is in cell A2; C0 = -100 needs to be added
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BMA, 9th edition: Chapter 3 Question 14
BMA, 10th edition: Chapter 2 Question 14
BMA, 11th edition: Chapter 2 Question 14
A factory costs €800,000. You expect that it will produce
an inflow after operating costs of €170,000 a year for 10
years. If the opportunity cost of capital is 14%, what is the
net present value of the factory? What will the factory be
worth at the end of five years?
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BMA, 9th edition: n/a
BMA, 10th edition: n/a
BMA, 11th edition: n/a
-year annuity factor at 14%
PV at end of year five 170,00∙ 5-year annuity factor at
14%
10170,000PV
86,720investmentinitialPVNPV
886,7205.216170,000
1.14
11
0.14
1170,000
10
583,6103.433170,000
1.14
11
0.14
1170,000
5
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BMA, 9th edition: Chapter 3 Question 22
BMA, 10th edition: Chapter 2 Question 22
BMA, 11th edition: Chapter 2 Question 22
In order to finance a car with a list price of €10,000,
Kangaroo Autos makes the following offer: You pay
€1,000 down and then €300 a month for the next 30
months.
Turtle Motors next door does not offer free credit but
will give you €1,000 off the list price. If the rate of
interest is 10% a year, which company is offering the
better deal?
Basic Principles of Valuation
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BMA, 9th edition: n/a
BMA, 10th edition: n/a
BMA, 11th edition: n/a
PV Kangaroo annuity factor for 30 periods at
0.79% per period
PV Turtle
3001,000
112
monthlyr (1.1) 1 0.0079
8,986 26.623001,000
1.0079
11
0.0079
13001,000
30
9,0001,00010,000