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Suppose I want to buy a \$200000 home, and take a 30 year mortgage on it. My interest rate is 7%. I have saved \$20000 to make a down payment.

Question: What will my monthly payment be? What will be the total amount that I pay the bank? Finally, if I double my first payment, then how long will it take before the loan is paid off?

Work: For the first question I have \$1197.55 thanks to the down payment which means I pay the bank \$431,118.00, but I cannot figure out the last question. I know the answer is allegedly 29 years and 3 months, but I cannot figure out at all how this is obtained.

Any ideas?

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1 Answer 1

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$$180000 = K a_{\overline{360} \rceil j} = K \frac{1 - (1 + 0.07/12)^{-360}}{(0.07/12)} = 150.308K,$$ hence $K = 1197.5444913$ is the level payment; rounding up ensures the payment is in full, since choosing $K = 1197.54$ means that the final payment is about $0.675$ short.

If the first payment is doubled, then the present value of the cash flow at $m$ months is $$K a_{\overline{m} \rceil j} + Kv,$$ thus we seek the smallest $m$ such that this expression equals $180000$; i.e., $$m \ge -\frac{\log \left(\frac{1448449}{1440000}-\frac{1050}{K}\right)}{\log \left(\frac{1207}{1200}\right)} \approx 352.0013.$$ Thus the loan is paid off upon the $352^{\rm th}$ payment. If we choose $K = 1197.54$ (the rounded down value) then the final payment will be short about $0.883$; if we choose the precise value of $K$, then the final payment will be short by about $0.208$. If we choose the rounded up value of $K$ the final payment will result in overpayment by $0.62$.

Another way to reason about the second part--i.e., doubling the initial payment--is to note that doing so effectively increases the down payment made by the present value of the excess of the first installment. That is to say, if you are supposed to pay $1197.55$ each month, then the present value of that first month's excess payment at the time the loan is taken out, is $1197.55(1 + 0.07/12)^{-1} = 1190.60$. Thus you are in effect looking at a loan of $200000 - 20000 - 1190.60 = 178809.40$ with level payments of $1197.55$ per month at a nominal rate of $7\%$ convertible monthly, and you need to find the term of such a loan.

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  • $\begingroup$ Hm interesting. I came up with $352.086386188$. Guess there are some rounding issues. Regardless, it looks like the answer should be 29 years and 4 months as opposed to 29 years and 3 months no? $\endgroup$
    – user407200
    May 9, 2017 at 6:14
  • $\begingroup$ @rudy Yes. Another way to put it is that by doubling the initial payment, the loan is paid off 8 months early, not 9. $\endgroup$
    – heropup
    May 9, 2017 at 6:21
  • $\begingroup$ Quick aside on what I did (maybe you can point out where some imprecision was introduced on my part perhaps): I basically let the amount of the annuity be $A_f=R+R(1+i)+R(1+i)^2+\cdots+2R(1+i)^{n-1}$ where $R$ is the periodic payment, and then I went from there. Anything wrong with that (the last term of the sum being $2R(1+i)^{n-1}$ because it represents twice the monthly payment $R$ being paid out at the beginning). If this isn't correct, then I am curious as to how our end results are so close. With such large numbers I would expect the error to be larger. $\endgroup$
    – user407200
    May 9, 2017 at 6:23

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