Basic Principles of Valuation - Uni Konstanz · Basic Principles of Valuation C=f(S,t) u d 6 Net Present Value and the Creation of Value I The ‘great deal’ is worth as much as

Basic Principles of Valuation

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Jens Carsten Jackwerth

University of Konstanz

[email protected]

http://www.wiwi.uni-konstanz.de/jackwerth/


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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


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


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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


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



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



-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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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?


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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

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Basic Principles of Valuation - Uni Konstanz · Basic Principles of Valuation C=f(S,t) u d 6 Net Present Value and the Creation of Value I The ‘great deal’ is worth as much as