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**