Answer: $73,601

## Explanation This is an **annuity due** problem because the first payment is made today (at time 0). **Key Information:** - Payment amount (PMT) = $10,000 - Number of payments (n) = 10 - Discount rate (r) = 6% = 0.06 - First payment today → Annuity due **Formula for Present Value of Annuity Due:** \[ PV_{\text{due}} = PMT \times \left[ \frac{1 - (1 + r)^{-n}}{r} \right] \times (1 + r) \] **Step-by-step calculation:** 1. **Calculate the present value of an ordinary annuity (payments at end of period):** \[ PV_{\text{ordinary}} = 10,000 \times \left[ \frac{1 - (1.06)^{-10}}{0.06} \right] \] \[ PV_{\text{ordinary}} = 10,000 \times \left[ \frac{1 - 0.558394}{0.06} \right] \] \[ PV_{\text{ordinary}} = 10,000 \times \left[ \frac{0.441606}{0.06} \right] \] \[ PV_{\text{ordinary}} = 10,000 \times 7.360087 \] \[ PV_{\text{ordinary}} = 73,600.87 \] 2. **Convert to annuity due by multiplying by (1 + r):** \[ PV_{\text{due}} = 73,600.87 \times 1.06 \] \[ PV_{\text{due}} = 78,016.92 \] **Alternative direct calculation:** \[ PV_{\text{due}} = 10,000 \times \left[ \frac{1 - (1.06)^{-10}}{0.06} \right] \times 1.06 \] \[ PV_{\text{due}} = 10,000 \times 7.360087 \times 1.06 \] \[ PV_{\text{due}} = 78,016.92 \] **Verification with financial calculator:** - Set calculator to BGN mode (for annuity due) - N = 10, I/Y = 6, PMT = 10,000, FV = 0 - Compute PV = **$78,016.92** **Comparing to options:** - A. $72,098 → This would be the present value of an ordinary annuity (payments at end of period) - B. $73,601 → This is the present value of an ordinary annuity (close to our $73,600.87) - C. $78,017 → This is the present value of the annuity due (our calculated $78,016.92) **Answer: C ($78,017)** is correct because: 1. Payments start today → Annuity due 2. Present value of annuity due = Present value of ordinary annuity × (1 + r) 3. The calculated value of $78,016.92 is closest to $78,017

Author: LeetQuiz .