Answer: $7,221.

## Explanation This is a **delayed perpetuity** problem. The investment pays $1,000 per year forever, but the first payment occurs **6 years from today**. ### Step 1: Value of perpetuity at time t=5 A perpetuity paying $1,000 per year starting at the end of year 1 has a present value formula: \[ PV = \frac{PMT}{r} \] Where: - PMT = $1,000 - r = 9% = 0.09 If the perpetuity started at t=1, its value at t=0 would be: \[ \frac{1000}{0.09} = \$11,111.11 \] However, our perpetuity starts at **t=6**, which means at **t=5** (one period before the first payment), the perpetuity value would be: \[ PV_{t=5} = \frac{1000}{0.09} = \$11,111.11 \] ### Step 2: Discount back to today (t=0) We need to discount this value from t=5 back to t=0: \[ PV_{t=0} = \frac{PV_{t=5}}{(1+r)^5} = \frac{11,111.11}{(1.09)^5} \] ### Step 3: Calculate \[ (1.09)^5 = 1.538624 \] \[ PV_{t=0} = \frac{11,111.11}{1.538624} = \$7,221.02 \] ### Step 4: Verify with options $7,221.02 is closest to **$7,221** (Option C). **Alternative approach**: The present value of a perpetuity starting at time n is: \[ PV = \frac{PMT}{r} \times \frac{1}{(1+r)^{n-1}} \] Where n is the period when the first payment occurs. Here n=6: \[ PV = \frac{1000}{0.09} \times \frac{1}{(1.09)^{5}} = 11,111.11 \times 0.649931 = \$7,221.02 \] **Why not the other options?** - **$4,486**: This would be if you incorrectly discounted for 6 periods instead of 5 - **$6,625**: This might be if you used the wrong discount factor or miscalculated the perpetuity value

Author: LeetQuiz .