Mortgage Question

Steven G

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Suppose you have a 20 year mortgage. When you get your mortgage statement there is an interest charge and a principle charge. If each month you pay twice the principle payment and the interest charge will you finish paying off the mortgage in half the time?
 
Suppose you have a 20 year mortgage. When you get your mortgage statement there is an interest charge and a principle charge. If each month you pay twice the principle payment and the interest charge will you finish paying off the mortgage in half the time?
Initial Impression: The principal portion is monotonic increasing, You are doubling the least of the possible values. Thus, one would have to conclude that "half the time" is a bit optimistic.

2nd Impression: Individual contracts may have unusual early payoff rules. Read your contract carefully.

3rd Impression: There is not enough information to calculate this exactly. One must know the actual payment splitting methodology. See "Rule of 78's", for example.

4th Impression, surely this is related to the interest rate, itself.

5% 240 is shortened to 134 (55.8% > 1/2)
6% 240 is shortened to 136 (56.7%)
8% 240 is shortened to 141 (58.8%)
 
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Ya been drinking Jomo?!
Whadda heck is a "principle charge"? D'you mean "principal"?
Anyway, a mortgage payment works this way:
it is first applied to pay interest owing, and anything left
over is applied to principal.
Smoking not drinking.
When you receive a mortgage bill say for $2000 it says that the interest for that period is $1600 and $400 goes to the principal. Suppose for this month you pay $2400 = $1600 + 2*$400. Capisce??

And then you do the same for every payment.

So your statement above that you first pay off the interest is not true. You build up equity all the time.

I am a bit nervous saying this to you as you were into finance but I see it right in my mortgage bill each month.
 
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Initial Impression: The principal portion is monotonic increasing agreed, You are doubling the least of the possible values not true, since towards the end of the 20 years the interest will be less than the principal. Thus, one would have to conclude that "half the time" is a bit optimistic. I wonder

2nd Impression: Individual contracts may have unusual early payoff rules. Read your contract carefully. Yeah, but this is in theory so assume no payoff rules.

3rd Impression: There is not enough information to calculate this exactly. One must know the actual payment splitting methodology. See "Rule of 78's", for example.

4th Impression, surely this is related to the interest rate, itself. I wonder exactly how much the interest rate has to do with it as you are always paying the interest each month.

5% 240 is shortened to 134 (55.8% > 1/2)
6% 240 is shortened to 136 (56.7%)
8% 240 is shortened to 141 (58.8%)
Please my comments in red above. I am assuming the last three lines you wrote are actual computations to my questions? If yes, thanks! and note that the interest rate did not have the biggest effect on the outcome. It now makes sense to me that as interest rate goes to 0 the payoff time will approach 1/2.

What assumptions did you make to get those numbers?
 
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A constant mortgage payment (P) to pay-off a loan amount (A) is calculated according to the following equation:

\(\displaystyle \displaystyle{P = \dfrac{A * r * (1+r)^n}{(1+r)^n - 1} \ = \ \dfrac{A * r}{1 - (1+r)^{-n}} }\) where 'n' is the number of payments and 'r' is the interest rate for the period. This equation is non-linear in 'n'.

Jomo, in your scheme though the payment 'P' is not fixed over the loan period. So it will be hard to calculate analytically.

However, I would use a spreadsheet to observe the trend and come to conclusion.
 
I am sure that you could create a closed form solution to this problem, but it could only be solved by numerical methods. And it will certainly not be linear, both because it will be an exponential and because the payment will be increasing over time. Denis is right; solve it by computer.
 
Please my comments in red above. I am assuming the last three lines you wrote are actual computations to my questions? If yes, thanks! and note that the interest rate did not have the biggest effect on the outcome. It now makes sense to me that as interest rate goes to 0 the payoff time will approach 1/2.

What assumptions did you make to get those numbers?

Initial Impression: The principal portion is monotonic increasing agreed, You are doubling the least of the possible values not true, since towards the end of the 20 years the interest will be less than the principal. Thus, one would have to conclude that "half the time" is a bit optimistic. I wonder

"not true" - Yes, it is true. Think it over again. We're talking about the regular amortization - prior to any advance payment of principal. If you agree that it is monotonic increasing, there is no other conclusion. The relationship of the principal to the interest is inconsequential

"I wonder" - No need, since it is true.

The interest rate always makes a difference. If it didn't, why would anyone even bother to state an interest rate? Loan Sharking could be rampant, but no one would care.

I used a standard amortization with monthly compounding to accomplish the suggested annual effective rate. I calculated the entire amortization schedule under normal circumstances, then followed with a second version with no change other than doubling the principal amount of each payment.

I found no reason to revise any of my previous comments.
 
From what I can decipher out of your smoky question,
you are trying to calculate the effect of paying an extra
amount each month equal to the portion of the payment
that was applied to principal, as example $400 using the
above example, thereby resulting in a balance of
$159600.00 *** - 400 = 159200 at end of 1st month...YES??
Yes
 
A constant mortgage payment (P) to pay-off a loan amount (A) is calculated according to the following equation:

\(\displaystyle \displaystyle{P = \dfrac{A * r * (1+r)^n}{(1+r)^n - 1} \ = \ \dfrac{A * r}{1 - (1+r)^{-n}} }\) where 'n' is the number of payments and 'r' is the interest rate for the period. This equation is non-linear in 'n'.

Jomo, in your scheme though the payment 'P' is not fixed over the loan period. So it will be hard to calculate analytically.

However, I would use a spreadsheet to observe the trend and come to conclusion.
Yes, the payment will not be the same. It actually will be increasing since more and more of the monthly payment will be going toward the principal. I would appreciate it if you see the trend. Thanks.
 
Initial Impression: The principal portion is monotonic increasing agreed, You are doubling the least of the possible values not true, since towards the end of the 20 years the interest will be less than the principal. Thus, one would have to conclude that "half the time" is a bit optimistic. I wonder

"not true" - Yes, it is true. Think it over again. We're talking about the regular amortization - prior to any advance payment of principal. If you agree that it is monotonic increasing, there is no other conclusion. The relationship of the principal to the interest is inconsequential

The principal portion is monotonic increasing since after each monthly payment you lowered the remaining balance by making a payment that includes ALL the interest owed for that month plus some for the principal. Now next month the interest for the month is a bit less since you are being charged interest on the (lower) remaining balance. Now since you always make the same payment each month and less is going towards interest then more money goes towards paying off the principal
"I wonder" - No need, since it is true.
Your own numbers show that the new payoff date is about 1/2

The interest rate always makes a difference. If it didn't, why would anyone even bother to state an interest rate? Loan Sharking could be rampant, but no one would care.
It really depends on what you mean when you say the interest always makes a difference. If you mean if the interest rate is higher than your monthly payment will be significantly more than of course you are right. If you mean that the payoff time will be significantly different if you double your principal payment I do not think it matters that much according to the data you provided. Of course significantly different and I do not think it matters that much is not very precise.

I used a standard amortization with monthly compounding to accomplish the suggested annual effective rate. I calculated the entire amortization schedule under normal circumstances, then followed with a second version with no change other than doubling the principal amount of each payment.

I found no reason to revise any of my previous comments.
My comments are above
 
J: since you always make the same payment each month

T: There's your problem, right there. If you double the principle from each payment in the original amortization, you will not be paying the same each month. It seems we have not sufficiently defined the problem.

J: Your own numbers show that the new payoff date is about 1/2

T: "About half", but a little more. Exactly what I theorized and then demonstrated.
J: I do not think it matters that much ...

T: Again, here is the problem. Do you want a precise calculation or "matters much". Please define the problem instead of discounting precision with vague declarations.
 
Suppose you have a 20 year mortgage. When you get your mortgage statement there is an interest charge and a principle charge. If each month you pay twice the principle payment and the interest charge will you finish paying off the mortgage in half the time?

Okay, I devised a new scheme.

1) You must choose the 1st payment. Standard 20 year monthly. 5% annual effective gives $1,307.67
2) Pay twice the Principal when you get the statement - assume this happens simultaneously with the payment on the statement. Thus, we paid $814.82 interest and $985.70 principal.
3) Here's the problem. Just exactly what is the 2nd payment? I recalculated it with 239 months at the same interest rate and the new principal. 5% annual effective gives $1,304.44. If you do not recalculate, and retain the $1,307.67, you get the results already reported.

Effects:

Change in Duration of Loan: None. Entirely insensitive to interest rate, term, or original amount of loan.
Principal Portion: Increases through 70 payments, then begins to shrink.
Interest Portion: This is a little harder to describe:

Years 1-5: 40% vs 51% -- Without the extra principal payment, 40% of the eventual total interest is paid in the first 5 years. With the extra, 51%.
Years 6-10: 31% vs 32%
Years 11-15: 21% vs 15%
Years 16-20: 8% vs 3%

Total Interest at 5% Annual Effective $114K vs $82K.

We significantly decreased the total interest, but packed more of it up front.

Note: It's still not the same payment every month.
 
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