2. slide no.: 1 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Finanzierung und Investition Schwerpunktthema 3 Grundlagen der Finanzmathematik Fachhochschule für Wirtschaft Berlin Berlin School of Economics Sommer 2007

3. slide no.: 2 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 1: One single future pay- ment date, r=10% Formula: Example: +16T t1t1 t2t2 t3t3 t4t4 t0t0 = 10,9 Basic Principals of Cash Flow Valuation - Present Values -

4. slide no.: 3 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 2: Two single cash flows at future payment dates, r=10% Formula: +16T t1t1 t2t2 t3t3 t4t4 t0t0 Example: = 18,4 +10T Basic Principles of Cash Flow Valuation – Present Values -

5. slide no.: 4 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 3: annuities over 4 years, no growth, r=10% Formula: t1t1 t2t2 t3t3 t4t4 t0t0 Example: = 31,7 +10T Basic Principles of Cash Flow Valuation - Present Values -

6. slide no.: 5 Prof. 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: Example: t1t1 t2t2 t3t3 t4t4 t0t0 = 31,7 +?T

7. slide no.: 6 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 6: Perpetual cash flows without growth, r=10% Formula: Example: t1t1 t2t2 t3t3 tt t0t0 = 100,0 +10T........ Basic Principles of Cash Flow Valuation - Present Values -

8. slide no.: 7 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 7: perpetual cash flows with constant growth, r=10%, growth=5% p.a. Formula: Example: t1t1 t2t2 t3t3 tt t0t0 = 200,0 + 11,02 T + 10,5 T + 10 T............. Basic Principles of Cash Flow Valuation - Present Values -

9. slide no.: 8 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 5: Annuities over 4 years, constant growth, r=10%, g=5% Formula: Example: t1t1 t2t2 t3t3 t4t4 t0t0 = 33,9 11,03 T 11,58 T 10,5 T 10 T Basic Principles of Cash Flow Valuation - Present Values -

10. slide no.: 9 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 1: One single investment today, one single future pay- ment date, r=10% Formula: Example: +16T t1t1 t2t2 t3t3 t4t4 t0t0 10,9 Basic Principals of Cash Flow Valuation - Future Values -

11. slide no.: 10 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 2: Two single payments now, one future payment date, r=10% Formula: Example: 16T t1t1 t2t2 t3t3 t4t4 t0t0 36,735 10T Basic Principles of Cash Flow Valuation – Future Values -

12. slide no.: 11 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 3: Annuities over 4 years, no growth, r=10% Formula: Example: t1t1 t2t2 t3t3 t4t4 t0t0 46,410 +10T Basic Principles of Cash Flow Valuation - Future Values -

13. slide no.: 12 Prof. 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: Example: Basic Principals of Cash Flow Valuation Future Value of n-Payments Per Year 41,525 Q1Q1 Q2Q2 Q3Q3 Q4Q4 Q0Q0 10,000

14. slide no.: 13 Prof. 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 t 1 : Formula: Example: Basic Principals of Cash Flow Valuation - Return of Investment (2 Payments) - +16T t1t1 t2t2 t3t3 t4t4 t0t0 10,9 r = 46,41 %

15. slide no.: 14 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 2: An investment of 10.928 now, offers in t 4 (after 4 years) a single cash flow of 16.000: Formula: Example: Basic Principals of Cash Flow Valuation - Return on Investment (2 Payments) - +16T t1t1 t2t2 t3t3 t4t4 t0t0 10,9 r = 12,47 %

16. slide no.: 15 Prof. 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: t1t1 t2t2 t3t3 t4t4 t0t0 - 10.928 + 5.000 + 5.000 + 5.000 + 5.000

17. slide no.: 16 Prof. Dr. Rainer Stachuletz – Berlin School of Economics Case 2: An investment of 10.928 now, offers in t 4 (after 4 years) a single cash flow of 16.000: Formula: Example: Basic Principals of Cash Flow Valuation - Return on Investment (2 Payments) - +16T t1t1 t2t2 t3t3 t4t4 t0t0 10,9 r = 12,47 %