1. The problem involves calculating the Future Value (FV) of a series of cash flows (CF) compounded at an interest rate $r$ over $T$ periods, and then finding the Present Value (PV) of that future amount at time $t$. 2. The formula for Future Value of an annuity is: $$FV_T = CF \times \frac{(1+r)^T - 1}{r}$$ This formula calculates the accumulated value of equal cash flows made at the end of each period, compounded at rate $r$. 3. To find the Present Value of the future amount at time $t$, we use: $$PV = \frac{FV_t}{(1+r)^t}$$ This discounts the future value back to the present time. 4. Important rules: - $r$ is the interest rate per period. - $T$ is the total number of periods for the annuity. - $t$ is the time at which we want to find the present value. 5. Example intermediate step (if given values): Suppose $CF=100$, $r=0.05$, $T=10$, and $t=5$. Calculate $FV_T$: $$FV_T = 100 \times \frac{(1+0.05)^{10} - 1}{0.05}$$ Calculate $(1+0.05)^{10}$: $$1.05^{10} = 1.62889$$ So, $$FV_T = 100 \times \frac{1.62889 - 1}{0.05} = 100 \times \frac{0.62889}{0.05} = 100 \times 12.5778 = 1257.78$$ 6. Calculate Present Value at $t=5$: $$PV = \frac{1257.78}{(1+0.05)^5} = \frac{1257.78}{1.27628} = 985.61$$ This means the present value of the future cash flows at time 5 is approximately 985.61. This explanation covers the formulas and their application step-by-step.