# What is the present value of following cash flows at the interest rate of 8 % per year? (a) Rs. 40,000 received 5 years from now (b) Rs. 40,000 received 50 years from now (c) Rs. 20,000 received every year, beginning one year from now and ending 10 y

To calculate the present value (PV) of the future cash flows at an interest rate of 8% per year, we can use the present value formula for single cash flows and annuities.
The present value formula for a single future cash flow is:
\[
PV = \frac{FV}{(1 + r)^n}
\]
Where:
- \( PV \) = Present value
- \( FV \) = Future value (cash flow received in the future)
- \( r \) = Interest rate (as a decimal)
- \( n \) = Number of years until payment is received
For multiple cash flows (annuities), the present value formula is:
\[
PV = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
Where:
- \( PV \) = Present value of annuity
- \( C \) = Cash flow per period
- \( r \) = Interest rate (as a decimal)
- \( n \) = Number of periods
### Calculations:
#### (a) Rs. 40,000 received 5 years from now
\[
PV = \frac{40,000}{(1 + 0.08)^5} = \frac{40,000}{1.4693} \approx 27,249.45
\]
#### (b) Rs. 40,000 received 50 years from now
\[
PV = \frac{40,000}{(1 + 0.08)^{50}} = \frac{40,000}{46.9026} \approx 853.79
\]
#### (c) Rs. 20,000 received every year, beginning one year from now and ending 10 years from now
\[
PV = 20,000 \times \left( \frac{1 - (1 + 0.08)^{-10}}{0.08} \right) = 20,000 \times 5.8681 \approx 117,362.02
\]
#### (d) Rs. 20,000 received each year for 10 years beginning now
This is an annuity due, so we adjust the PV formula to account for the immediate payment:
\[
PV = 20,000 \times \left( \frac{1 - (1 + 0.08)^{-10}}{0.08} \right) \times (1 + 0.08) = 20,000 \times 5.8681 \times 1.08 \approx 127,370.79
\]
#### (e) Rs. 20,000 each year beginning one year from now and continuing forever
This is a perpetuity, and the present value formula is:
\[
PV = \frac{C}{r} = \frac{20,000}{0.08} = 250,000
\]
### Summary of Present Values:
- (a) Rs. 40,000 in 5 years: **Rs. 27,249.45**
- (b) Rs. 40,000 in 50 years: **Rs. 853.79**
- (c) Rs. 20,000 for 10 years (starting next year): **Rs. 117,362.02**
- (d) Rs. 20,000 for 10 years (starting now): **Rs. 127,370.79**
- (e) Rs. 20,000 per year forever: **Rs. 250,000**