1. Let's start by stating the problem: You want to understand the exact formula for a deferred annuity. 2. A deferred annuity is a series of payments that begin after a certain delay period. The formula calculates the present value of these future payments. 3. The exact formula for the present value $PV$ of a deferred annuity with payment $P$, interest rate per period $i$, number of payments $n$, and deferral period $d$ is: $$PV = P \times \frac{1 - (1+i)^{-n}}{i} \times (1+i)^{-d}$$ 4. Explanation: - $\frac{1 - (1+i)^{-n}}{i}$ is the present value of an annuity immediate for $n$ payments. - Multiplying by $(1+i)^{-d}$ discounts this value back $d$ periods to account for the deferral. 5. Important rules: - Interest rate $i$ must be per payment period. - Number of payments $n$ and deferral $d$ are in the same time units. 6. Example: Suppose $P=100$, $i=0.05$, $n=10$, and $d=5$. Calculate the annuity factor: $$\frac{1 - (1+0.05)^{-10}}{0.05} = \frac{1 - (1.05)^{-10}}{0.05} \approx \frac{1 - 0.6139}{0.05} = \frac{0.3861}{0.05} = 7.722$$ Discount for deferral: $$(1+0.05)^{-5} = (1.05)^{-5} \approx 0.7835$$ Present value: $$PV = 100 \times 7.722 \times 0.7835 = 100 \times 6.048 = 604.8$$ 7. So, the present value of the deferred annuity is approximately 604.8. This formula ensures you correctly account for both the annuity payments and the deferral period.