A security company offers to provide CCTV coverage for a parking garage for ten years for an initial payment of $50,000 and additional payments of $30,000 per year. What is the equivalent annual annuity of this deal, given a cost of capital of 5%?

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BlackTom AI · Accepted Answer

Question restatement: A security company offers to provide CCTV coverage for a parking garage for ten years for an initial payment of $50,000 and additional payments of $30,000 per year. What is the equivalent annual annuity of this deal, given a cost of capital of 5%?

Options: 
- $22,194 (approximate) 
- $36,475 (approximate) 
- $31,706 (approximate) 
- $25,365 (approximate)

To compare this project on an annual basis, we convert the net present value (NPV) of the cash flows into an equivalent annual annuity (EAA) using the given discount rate, i = 5% and n = 10 years.

Step 1: Compute the present value of all costs.
- There is $50,000 paid at time 0, which has PV = $50,000.
- There are $30,000 payments at the end of each year for 10 years. The PV of an annuity-immediate of $30,000 for 10 years at 5% is 30,000 × a-angle-5%,10.
- The annuity factor a-angle-5%,10 = (1 − (1 + i)^(-n)) / i = (1 − (1.05)^(-10)) / 0.05.
- (1.05)^(-10) ≈ 0.6139, so a-angle ≈ (1 − 0.6139) / 0.05 ≈ 0.3861 / 0.05 ≈ 7.7217.
- PV of the annual payments ≈ 30,000 × 7.7217 ≈ 231,651.
- Total PV of costs ≈ 50,000 + 231,651 ≈ 281,651.

Step 2: Convert the PV of costs to an equivalent annual annuity (EAA).
- The EAA factor for i = 5% and n = 10 is the capital recovery factor: [i × (1 + i)^n] / [(1 + i)^n − 1].
- (1.05)^10 ≈ 1.6289; numerator ≈ 0.05 × 1.6289 ≈ 0.08144; denominator ≈ 1.6289 − 1 = 0.6289; thus EAA factor ≈ 0.08144 / 0.6289 ≈ 0.1295.
- EAA ≈ PV of costs × 0.1295 ≈ 281,651 × 0.1295 ≈ 36,475.

Step 3: Apply the sign convention.
- Since these are costs (outflows), the equivalent annual amount is negative.
- Therefore, EAA ≈ −$36,475 per year.

Option-by-option analysis:
- Option A: −$22,194 would imply a much smaller annual equivalent than the calculated EAA; this would misstate the cost recovery rate and understate the annual burden relative to the actual PV of cash outlays.
- Option B: −$36,475 aligns closely with the computed capital-recovery-based EAA for the given numbers; it matches the standard method of converting PV of costs to an annual payment over 10 years at 5%.
- Option C: −$31,706 is too high relative to the PV of costs and the 5% capital recovery factor; it would correspond to a lower PV or a higher discount rate or a shorter horizon, none of which match the problem data.
- Option D: −$25,365 is also not consistent with the derived EAA; it lies between the PV-based calculation and the other alternatives but does not reflect the precise capital recovery calculation for i = 5% and n = 10.

Why the correct choice is correct (conceptual takeaway): the equivalent annual annuity represents the fixed annual cost that would have the same present value as paying $50,000 now plus $30,000 each year for 10 years, when discounted at 5%. Using the present value of costs and the capital recovery factor yields an EAA of about −$36,475 per year, which correctly translates the 10-year, front-loaded-cost structure into a single annualized cash flow.

In summary, the computation steps—PV of costs (50,000 immediate plus 30,000 × a-angle-5%,10) and then conversion to EAA via the 5% capital recovery factor—lead to an equivalent annual outflow of approximately −$36,475, corresponding to the second option.

A security company offers to provide CCTV coverage for a parking garage for ten years for an initial payment of $50,000 and additional payments of $30,000 per year. What is the equivalent annual annuity of this deal, given a cost of capital of 5%?

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