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