SOLUTION: Suppose you can afford to pay at most $2650 per month for a mortgage payment. If the maximum amortization period you can get is 20 years, and you must pay 6% interest per year com

Question 1206600: Suppose you can afford to pay at most $2650 per month for a mortgage payment. If the maximum amortization period you can get is 20 years, and you must pay 6% interest per year compounded annually, what is the most expensive house you can buy? How much interest will you have paid to the lender at the end of the loan?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
you can afford to pay at most 2650 per month.
maximum amortization period is 20 years.
payment is 6% per year compounded annually.

i used the texas instruments business analyst 2 calculator.

inputs are:
present value = 0
future value = 0
payment at the end of each time period = 2650
number of time periods = 20 years * 12 months per year = 240 months.
interest rate per time period = 1.06 ^ (1/12) = 1.004867551 per month, minus 1 = .004867551 per month, * 100 = .4867551% per month.

calculator says that the present value of the loan is equal to 374668.4188.

that's the most expensive house you can guy with the amount of money that you have available.

the total interest that will be paid to the owner by the end of the loan period is equal to 240 * 2650 = 636000 minus 374668.4188 = 261331.5812.

that should be your answer.

note that the 6% per year is compounded annually.
this is different than if it was compounded monthly.

if it was compounded monthly, then the monthly interest rate would be 6% / 12 = .5% per month.

the effective annual interest rate would then be 1.005^12 = 1.061677812, minus 1 = .061677812, * 100 = 6.1677812% per year.

since it is compounded annually, then the monthly interest rate would 1.06 ^ 1/12 = 1.004867551, minus 1 = .004867551, * 100 = .4867551% per month.

the effective annual interest rate would then be 1.004867551 ^ 12 = 1.06, minus 1 = .06, * 100 = 6%.

that makes a difference, so please check to make sure they wanted the 6% to be compounded annually, rather than monthly.

if they wanted it compounded monthly, then this answer is incorrect.

i verified with excel that the answer provided here is correct if annual compounding of the interest rate is assumed.

Answer by ikleyn(52776) (Show Source): You can put this solution on YOUR website!
.
Suppose you can afford to pay at most $2650 per month for a mortgage payment. If the maximum amortization period
you can get is 20 years, and you must pay 6% interest per year compounded annually, what is the most expensive
house you can buy? How much interest will you have paid to the lender at the end of the loan?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

This problem is tricky, since the payments are monthly, while compounding are annually.
So, the payments are desynchronized with compounding.
It means that monthly payments lie in the bank with no move and wait for the end of a year -
only then they are compounded, according to the problem.

Classic formulas for loan/mortgage are applicable for synchronized payments/compounding.

But we can modify the situation EQUIVALENTLY to get payments/compounding synchronized.

Indeed, we actually have annual payments of 12*2650 = 31800 dollars each, compounded annually.

Thus, it works as a classic loan for 20 years with annual payments of $31800
at the end of each year, compounded annually at the annual rate of 6%.

Now apply a standard loan formula PMT = where PMT is the annual payment ($31800); L is the loaned amount; r = 0.06 is the percentage rate of rounding and n = 20 years. Then the formula becomes 31800 = . From it, we get L = = 364743.50. It means that the most expensive house you can buy under given conditions is for $364,743.50. <<<---=== ANSWER You will pay for the loan 20*12*2650 = 636000 dollars. It means that interest you have paid to the lender for the loan is 636,000 - 364,743.50 = 271256.50 dollars. <<<---=== ANSWER

At this point, the problem is solved completely.

//////////////////////////////////////////

The solution in the post by @Theo is inadequate to the problem.

It is because Theo introduces, considers and treats monthly compounding; but the bank does not perform
monthly compounding. According to the problem, the bank makes annual compounding, ONLY.

Theo introduces equivalent monthly rate; but it works as an equivalent scheme only under condition
when there are no intermediate compounding inside a year. When there are intermediate monthly compounding,
it immediately destroys equivalency.

Had the problem admit monthly compounding, the solution by @Theo would be correct.
But under the conditions, described in the post, @Theo' solution is inadequate.

It is why I called this problem "tricky".

It has a hidden underwater stone as a trap, and, therefore, should be treated carefully.

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