Mooncorp Insurance has quoted you an annual premium to insure your car of $3100. You are offered a 15% discount if you pay the lump sum immediately. They also offer an alternative payment method. You can pay the account in full by making 12 equal beginning of the month payments of $270, rather than the lump sum. What is the effective annual opportunity cost of paying monthly?

provide an explanation of this opportunity cost.

I figured out the IRR but not sure about the meaning of opportunity annual cost in this case. Can anyone help?

cash price = .85(3100) = 2635

So 2635 becomes the present value of the debt.

are we doing this by simple interest or compound interest?

by compound interest
270( 1 - (1+i)^-12)/i = 2635
this equation is very difficult to solve, and you need some special procedures.
I will just send it through Wolfram (I changed the i to x to avoid confusion with compex numbers)
http://www.wolframalpha.com/input/?i=solve+270%28+1+-+%281%2Bx%29%5E-12%29%2Fx+%3D+2635
i = .0333256 ---> monthly interest
so the effective annual rate compounded monthly is
.3999 or 39.99% , lets say 40%

simple interest:
total paid = 12(270) = 3240
interest paid = 3240 - 2635 = 605

I = PRT
605 = 2635(R)(1)
R = .2296 or 22.96%

Of course the company will state it as 22.96% interest rate without mentioning that it would be simple interest.
In reality, the 40% compound rate would be the correct rate

The effective annual opportunity cost of paying monthly refers to the cost of forgoing the discount offered for paying the full annual premium upfront. In this case, Mooncorp Insurance is offering a 15% discount if you pay the lump sum immediately. The opportunity cost is the amount of money you would save by paying the lump sum upfront instead of using the alternative payment method, which is paying in 12 equal monthly installments.

To calculate the effective annual opportunity cost, we need to compare the total amount paid under each payment option.

First, let's calculate the total amount paid if you choose to pay monthly:
Total monthly payment = Monthly payment * Number of payments
Total monthly payment = $270 * 12 = $3240

Now, let's calculate the discount you would receive if you pay upfront:
Discount = Annual premium * Discount rate
Discount = $3100 * 0.15 = $465

The effective annual opportunity cost can be calculated by subtracting the discount from the total monthly payment:
Opportunity cost = Total monthly payment - Discount
Opportunity cost = $3240 - $465 = $2775

Therefore, the effective annual opportunity cost of paying monthly is $2775. This means that by choosing the monthly payment option, you would end up paying an additional $2775 over the course of the year compared to paying upfront.