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