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 $20,000 per year.What is the equivalent annual annuity of this deal,given a cost of capital of 6%?
A)-$14,720
B)-$19,720
C)-$20,000
D)-$26,793
E)-$31,011

Question

Quizplus · Accepted Answer

To find the equivalent annual annuity, we need to find the level payment that would make the present value of the deal equal to the present value of a level payment annuity at a cost of capital of 6%.

The present value of the initial payment of $50,000 is simply $50,000.

The present value of the annual payments of $20,000 per year for ten years can be calculated using the formula for the present value of an annuity:

PV = Pmt x [1 - (1 + r)^-n] / r

where PV is the present value, Pmt is the payment amount, r is the cost of capital, and n is the number of payment periods.

Plugging in the numbers we get:

PV = $20,000 x [1 - (1 + 0.06)^-10] / 0.06
PV = $20,000 x 6.7101
PV = $134,202

So the total present value of the deal is:

$50,000 + $134,202 = $184,202

To find the equivalent annual annuity, we need to solve for the level payment that would make the present value of a level payment annuity equal to $184,202 at a cost of capital of 6%. We can use the formula for the present value of a level payment annuity:

PV = Pmt x [1 - (1 + r)^-n] / r

where PV is the present value, Pmt is the payment amount, r is the cost of capital, and n is the number of payment periods.

In this case, PV = $184,202, r = 0.06, and n = 10, so we can solve for Pmt:

$184,202 = Pmt x [1 - (1 + 0.06)^-10] / 0.06
$184,202 x 0.06 = Pmt x [1 - (1 + 0.06)^-10]
$11,052.12 = Pmt x 6.7101
Pmt = $11,052.12 / 6.7101
Pmt = $1,645.83

So the equivalent annual annuity is $1,645.83.

Therefore, the best choice is D.

A Security Company Offers to Provide CCTV Coverage for a Parking