slide no.: 1Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Finanzierung und Investition

Schwerpunktthema 3

Grundlagen der Finanzmathematik

Fachhochschule für Wirtschaft BerlinBerlin School of Economics

Sommer 2007

slide no.: 2Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 1: One single future pay-ment date, r=10%

Formula:

tt rCPV 1

Example:

92810

6830000016

1001

100016

4

.

..

..

PV

+16T

t1 t2 t3 t4t0

=10,9

Basic Principals of Cash Flow Valuation

- Present Values -

slide no.: 3Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 2: Two single cash flows at future payment dates, r=10%

Formula:

+16T

t1 t2 t3 t4t0

n

t

tt rCPV

1

1

Example:

44118

928105137

1001

00016

1001

0001043

.

..

.

.

.

.

PV

=18,4 +10T

Basic Principles of Cash Flow Valuation

– Present Values -

slide no.: 4Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 3: annuities over 4 years, no growth, r=10%

Formula:

t1 t2 t3 t4t0

n

n

trr

rCPV

1

11

Example:

69931

1699300010

1001100

1100100010

4

4

.

,.

.,

,.

PV

PV

PV

=31,7

+10T +10T+10T+10T

Basic Principles of Cash Flow Valuation

- Present Values -

slide no.: 5Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Basic Principles of Cash Flow Valuation

- NPV to Annuities - Case 4: 31,7 T€ are to be transformed into 4 equally high annual payments; r=10%

Formula:

11

10

n

n

r

rrPVA

Example:

000.10

3154,0700.31

110,01

10.0110,0700.31 4

4

A

A

At1 t2 t3 t4t0

=31,7

+?T + ?T+?T+?T

slide no.: 6Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 6: Perpetual cash flows without growth, r=10%

Formula:

r

CPV t

Example:

000100

100

00010

.

,

.

PV

PVt1 t2 t3 tt0

=100,0

+10T +10T+10T+10T ........

Basic Principles of Cash Flow Valuation

- Present Values -

slide no.: 7Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 7: perpetual cash flows with constant growth, r=10%, growth=5% p.a. Formula:

gr

CPV t

Example:

000200

050100

00010

.

,,

.

PV

PVt1 t2 t3 tt0

=200,0

+11,02

T

+10,5

T

+10T

.............

Basic Principles of Cash Flow Valuation

- Present Values -

slide no.: 8Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 5: Annuities over 4 years, constant growth, r=10%, g=5% Formula:

Example:

95933

3959300010

1001050100

0501100100010

4

44

.

,.

.,,

,,.

PV

PV

PVt1 t2 t3 t4t0

=33,9 11,03

T11,58

T10,5

T10T

n

nn

t

n

t

tt

rgr

grC

rCPV

1

11

11

Basic Principles of Cash Flow Valuation

- Present Values -

slide no.: 9Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 1: One single investment today, one single future pay-ment date, r=10%

Formula:

tt rCFV 1

Example:

000.16

4641.1928.10

10.01928.10 4

FV+16T

t1 t2 t3 t4t0

10,9

Basic Principals of Cash Flow Valuation

- Future Values -

slide no.: 10Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 2: Two single payments now, one future payment date, r=10%

Formula:

n

t

tt rCFV

1

1

Example:

60,735.36

310.1360,425.23

10,1000.1010,1000.16 34

FV

16T

t1 t2 t3 t4t0

36,735

10T

Basic Principles of Cash Flow Valuation

– Future Values -

slide no.: 11Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 3: Annuities over 4 years, no growth, r=10%

Formula:

r

rCFV

n

t

11

Example:

410.46

6410,4000.10

10,0

110,01000.10

4

FV

FV

FVt1 t2 t3 t4t0

46,410

+10T +10T+10T+10T

Basic Principles of Cash Flow Valuation

- Future Values -

slide no.: 12Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 4: You save 10,000 $ per quarter (end). At an annual per-centage rate of 10%, what will be the future value of your savings at the end of the year ?

Formula:

mr

1mr

1CFV

m

Example:

Basic Principals of Cash Flow Valuation

Future Value of n-Payments Per Year

$00.525,41

1525.4000,10410.0

1410.0

1000,10FV

4

41,525

Q1 Q2 Q3 Q4Q0

10,000 10,000 10,000 10,000

slide no.: 13Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 1: An investment of 10.928 now, leads to a later single cash flow of 16.000 in t1:

Formula:

1PV

Cr

Cr1PV

r1CPV

t t

tt

tt

Example:

%41,461928.10

000.16r

000.16r1928.10

r1

1000.16928.10

1

1

Basic Principals of Cash Flow Valuation- Return of Investment (2 Payments) -

+16T

t1 t2 t3 t4t0

10,9

r = 46,41 %

slide no.: 14Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 2: An investment of 10.928 now, offers in t4 (after 4 years) a single cash flow of 16.000:

Formula:

1PV

Cr

Cr1PV

r1CPV

t t

tt

tt

Example:

%47,121928.10

000.16r

000.16r1928.10

r1

1000.16928.10

4

4

4

Basic Principals of Cash Flow Valuation- Return on Investment (2 Payments) -

+16T

t1 t2 t3 t4t0

10,9

r = 12,47 %

slide no.: 15Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Basic Principles of Cash Flow Valuation - Return on Investment (5 Payments) -

Case 3: An investment of 10.928 now is expected to generate 5,000 p.a. over the next four years:

Formula:

Example:

t1 t2 t3 t4t0

-10.928 +

5.000+

5.000+

5.000+

5.000

0r1

CCNPV

n

1t

tt

0

12

12

11 NPVNPV

rrNPVrr

%49,29r

0149,028,0r

28,21646,74

28,030,028,21628,0r

slide no.: 16Prof. Dr. Rainer Stachuletz – Berlin School of Economics

Case 2: An investment of 10.928 now, offers in t4 (after 4 years) a single cash flow of 16.000:

Formula:

1PV

Cr

Cr1PV

r1CPV

t t

tt

tt

Example:

%47,121928.10

000.16r

000.16r1928.10

r1

1000.16928.10

4

4

4

Basic Principals of Cash Flow Valuation- Return on Investment (2 Payments) -

+16T

t1 t2 t3 t4t0

10,9

r = 12,47 %