Solution:
Cash Flow(F )= $ 50,000
Rate = 10 %
Discounted Cash Flow of amount $50,000 (D)= [tex]F \times (\frac{1}{1+\frac{10}{100}})^1+ F \times (\frac{1}{1+\frac{10}{100}})^2+F \times (\frac{1}{1+\frac{10}{100}})^3+F \times (\frac{1}{1+\frac{10}{100}})^4+F \times (\frac{1}{1+\frac{10}{100}})^5+..........\\\\= F \times \frac{10}{11} + F \times [\frac{10}{11}]^2+ F [\times \frac{10}{11}]^3+..................................+ F [\times \frac{10}{11}]^{20}[/tex]
As, this is a geometric Progression.
Formula for Sum of n terms of geometric Progression having common ratio r,
[tex]S_{n}=\frac{ a \times (1- r^n)}{1-r}[/tex]
For, r < 1 and
for , r>1
[tex]S_{n}=\frac{ a \times (r^n-1)}{r-1}[/tex]
[tex]S_{20}=50,000 \times\frac{1 \times ( [\frac{11}{10}]^{20}-1)}{\frac{11}{10}-1}\\\\ S_{20}=50,000 \times \frac{6.7274-1}{1.1-1}\\\\ S_{20}=50,000 \times 57.27=2863500[/tex]
If cash flow(K) = $ 25,000
Then at the rate of 10 % ,
Value after 20 years is given by:
Discounted Cash Flow of amount $25,000 (D)= [tex]K \times (\frac{1}{1+\frac{10}{100}})^1+ K \times (\frac{1}{1+\frac{10}{100}})^2+K \times (\frac{1}{1+\frac{10}{100}})^3+K \times (\frac{1}{1+\frac{10}{100}})^4+K \times (\frac{1}{1+\frac{10}{100}})^5+..........\\\\ =K \times \frac{10}{11} + K \times [\frac{10}{11}]^2+ K [\times \frac{10}{11}]^3+..................................+ K [\times \frac{10}{11}]^{20}[/tex]
for , r>1, formula for sum of n terms of geometric series
[tex]S_{n}=\frac{ a \times (r^n-1)}{r-1}[/tex]
[tex]K_{20}=25,000 \times\frac{1 \times ( [\frac{11}{10}]^{20}-1)}{\frac{11}{10}-1}\\\\ K_{20}=25,000 \times \frac{6.7274-1}{1.1-1}\\\\ K_{20}=25,000 \times 57.27=1431750[/tex]
