Does a home mortgage use Simple or Compund Interest?

[JamesSFO]

Okay suppose you have a savings account that gives 0.5% interest per month. You start with $100,000 and withdraw $1000 each month. Clearly the savings account has compound interest (compounding monthly), and surely no-one would dispute this. Every month the balance in the savings account is identical to the balance on the mortgage. The math is identical. Both the savings account and the mortgage, use exactly the same math, and so obviously, the mortgage uses compound interest.

I just don't see why people aren't getting this basic math. It's just not that complicated.

Compound is referring to whether interest is paid on interest. Imagine (a non-existent) banking product where interest payments were put into a separate non-interest bearing account. Thus the 0.5% interest each month goes into the separate account. So when you go to calculate the interest for the next month the prior interest paid is not earning more money. That's how mortgages work.

EDIT TO ADD: Also, I think your example is wrong because the monthly compound interest on the savings account would typically be calculated on the average daily balance while the mortgage would be calculated on the principal balance at the start of the period.

[555]

The general prinicipal [ed: principle?] is that "adding" interest is actually multiplying, and that's why mortgages use compound interest.

{{later post}}

No. The first reply was totally incorrect which is why it needed to be corrected.

Except that most US mortgages do NOT charge interest on interest, and thus are not considered to be compound interest loans. Check out the mortgage professor website (first response). The confusion is caused by the daily interest accrual approach on many (most?) loans which LOOKS a lot like compound interest.

The first reply was not incorrect nor are many of the websites that discuss this very issue...

Firstly the issue of daily interest accrual has little relevance to this discussion. We're just talking about what happens each month.

Also US mortgages most definitely DO charge interest on interest. Just look at Kosmo's example.
http://www.bogleheads.org/forum/viewtop ... 9#p1768709
$2.50 of the interest in the 2nd month is interest on the $500 interest from the first month. And see my comparison to savings accounts above. Savings accounts and mortgages compound in exactly the same way.

See here
http://en.wikipedia.org/wiki/Compound_i ... e_payments
for an explanation of the math of how typical mortgages have interest that is compounded monthly.

[555]

Okay suppose you have a savings account that gives 0.5% interest per month. You start with $100,000 and withdraw $1000 each month. Clearly the savings account has compound interest (compounding monthly), and surely no-one would dispute this. Every month the balance in the savings account is identical to the balance on the mortgage. The math is identical. Both the savings account and the mortgage, use exactly the same math, and so obviously, the mortgage uses compound interest.

I just don't see why people aren't getting this basic math. It's just not that complicated.

Compound is referring to whether interest is paid on interest. Imagine (a non-existent) banking product where interest payments were put into a separate non-interest bearing account. Thus the 0.5% interest each month goes into the separate account. So when you go to calculate the interest for the next month the prior interest paid is not earning more money. That's how mortgages work.

EDIT TO ADD: Also, I think your example is wrong because the monthly compound interest on the savings account would typically be calculated on the average daily balance while the mortgage would be calculated on the principal balance at the start of the period.

No you're completely wrong about how mortgages work. Interest is "added" (again "adding" interest is really multipying) each month and becomes part of the balance on which future interest is calculated.

Also the daily vs monthly issue is a red herring here. I was hypothesizing a savings account in which everything happened monthly (just to synchronize it with a mortgage). But the standard feature is compounding, and I made it clear that a savings account and a mortgage are identical in the way that interest is compounded. Just look at the math. That's what it comes down to.

[Jack]

No you're completely wrong about how mortgages work. Interest is "added" (again "adding" interest is really multiplying) each month and becomes part of the balance on which future interest is calculated.

Go back to your example Bankrate mortgage of $100,000 at 5% for 30 years. Look closely at the amortization table.

Each month the interest paid is exactly 5%/12 times the previous month's remaining balance. Interest is paid on the principal balance held over the month alone. There is no interest on interest.

Part of the confusion here is because a mortgage loan is a special type of loan which amortizes, which means that, in addition to paying a monthly interest on the balance, you also make a payment that reduces principal. The amortization table creates a path that pays off the entire principal at a fixed date, while also paying simple interest each month on the declining balance.

In your example of a $500 pre-payment you asked where the additional interest savings comes from. It comes from the fact that you have shortened the period of the loan from 360 months to 356 months. A four-month shorter loan period eliminates four months of interest at the average balance value. This produces the extra $900 of interest savings.

Why is the loan duration shortened? It is simply because after your first pre-payment, you are over-paying principal each month for the life of the loan. This is because you haven't re-amortized your loan from $100,000 to $99,500. Instead you are still making monthly payments as if you had a $100,000 loan, which results in an over-payment of principal each month. Each monthly over-payment of principal saves interest over the remaining life of the loan, not just the initial $500.

[mille424]

No you're completely wrong about how mortgages work. Interest is "added" (again "adding" interest is really multiplying) each month and becomes part of the balance on which future interest is calculated.

Go back to your example Bankrate mortgage of $100,000 at 5% for 30 years. Look closely at the amortization table.

Each month the interest paid is exactly 5%/12 times the previous month's remaining balance. Interest is paid on the principal balance held over the month alone. There is no interest on interest.

Different person. No one is debating about the amortization table. I agree the table is a series of monthly simple interest calculations, but my stance is that when you look over the entire loan it behaves more like a compound interest vehicle than a simple interest one.

[JamesSFO]

No you're completely wrong about how mortgages work. Interest is "added" (again "adding" interest is really multipying) each month and becomes part of the balance on which future interest is calculated.

Ok, I'm willing to be "wrong" along with "the Mortgage Professor" (Jack Guttentag, Professor of Finance Emeritus, formerly Jacob Safra Professor of International Banking, at the Wharton School of the University of Pennsylvania. ) who has written books on the topic.

[JamesSFO]

Different person. No one is debating about the amortization table. I agree the table is a series of monthly simple interest calculations, but my stance is that when you look over the entire loan it behaves more like a compound interest vehicle than a simple interest one.

I can actually get behind that: The calculation of a given payment is a simple interest payment, the cumulative lifetime behavior of the loan is comparable to a compound interest vehicle.

[JamesSFO]

Why is the loan duration shortened? It is simply because after your first pre-payment, you are over-paying principal each month for the life of the loan. This is because you haven't re-amortized your loan from $100,000 to $99,500. Instead you are still making monthly payments as if you had a $100,000 loan, which results in an over-payment of principal each month. Each monthly over-payment of principal saves interest over the remaining life of the loan, not just the initial $500.

This is also why the comparison to the savings account does not translate. There is no equivalent of continuing to act as if $100K was in the savings account for the life of the deposit.

[Jack]

The amortization table clearly shows there is never interest on interest. There is only interest on principal. Each month you pay interest on the amount of money you borrowed for that month.

An amortization loan is a simple interest loan in which the amount borrowed declines each month, and therefore the interest paid declines each month. It's simple to calculate the amount of interest due each month. Just multiply the amount borrowed for that month by the interest rate. That's it.

[BrandonBogle]

Quite simply, go to Bankrate.com's auto loan calculator and punch in $100,000 for 30 years at 5% interest. Hit show Amortization Schedule and scroll down to see the cumulative interest over the life of the loan $93,255.78.

Now go to the mortgage calculator and perform the same calculation. You will see that the cumulative interest is again $93,255.78.

Now, unless you want to call an auto loan a compound loan, these loans have the same interest "treatment", as in simple interest. The difference is in how each loan is paid. Auto loans treat payments as if received the day it is processed (exactly one would think). Mortgages treat payments as made on the first day of the month it is due, as long as you aren't delinquent. So making early payments on a mortgage will NOT reduce your principal and thus, earns you nothing. Prepaying an auto loan though will effectively cause you to pay less interest over the life of the loan.

[FNK]

It's a koan. Is there interest on interest if interest is never capitalized?

Why does one care? I guess you want a clear standard for comparing rates. So let's compare rates.

Say, I take a 10% $100K 10 year mortgage and put it into a 10% rate savings account. I make payments from the savings account back into the mortgage. Ignoring taxes and with no rate changes, after 10 years, am I over, under or at zero? What if it's a 10% APY account?

[Jack]

If you wanted to run a mortgage backwards like a savings account, here is what you would do. You start at the bottom of the amortization table and deposit that month's principal payment amount into your savings account. At the end of each month you compute the simple interest on the accumulated principal for that month but you take the interest earned and put it in a separate non-interest bearing account (a mayonnaise jar). You continue to do this for 360 months. At the end you have a savings account with exactly $100,000 in it and a mayonnaise jar that contains the total interest paid over the life of the savings account. This is the exact reverse of the mortgage loan. Note that all of the interest accrued was simple interest. There was no interest paid on the interest in the mayonnaise jar.

[555]

No you're completely wrong about how mortgages work. Interest is "added" (again "adding" interest is really multipying) each month and becomes part of the balance on which future interest is calculated.

Ok, I'm willing to be "wrong" along with "the Mortgage Professor" (Jack Guttentag, Professor of Finance Emeritus, formerly Jacob Safra Professor of International Banking, at the Wharton School of the University of Pennsylvania. ) who has written books on the topic.

It doesn't matter who the guy is. The mathematics speaks for itself, and the mathematics is the mathematics of compound interest.

Amortization calculations are made based on compound interest.

Search the thread for the "^" symbol (exponentiation symbol). That's the post where I explain the amortization calculation, and the exponentiation makes it clear that compound interest is being used.

[555]

If you wanted to run a mortgage backwards like a savings account, here is what you would do. You start at the bottom of the amortization table and deposit that month's principal payment amount into your savings account. At the end of each month you compute the simple interest on the accumulated principal for that month but you take the interest earned and put it in a separate non-interest bearing account (a mayonnaise jar). You continue to do this for 360 months. At the end you have a savings account with exactly $100,000 in it and a mayonnaise jar that contains the total interest paid over the life of the savings account. This is the exact reverse of the mortgage loan. Note that all of the interest accrued was simple interest. There was no interest paid on the interest in the mayonnaise jar.

Irrelevant. They are both calculated using compound interest both going forward in time.

[JamesSFO]

It doesn't matter who the guy is. The mathematics speaks for itself, and the mathematics is the mathematics of compound interest.

Amortization calculations are made based on compound interest.

Search the thread for the "^" symbol (exponentiation symbol). That's the post where I explain the amortization calculation, and the exponentiation makes it clear that compound interest is being used.

I understand exponentiation, FV, PV, PMT, EFFECT, NOMINAL, a variety of other related Excel functions and the underlying concepts. I suspect the mortgage professor does as well.

I really think there is a perspective issue here, when the bank goes to calculate how to allocate a given (fixed) payment between interest and principal, I believe we all agree they use the simple interest formula on the present loan balance.

I suspect we also all agree that when the bank goes to figure out how to set that (fixed) payment they based it on compounding over the period: Payment = Loan Balance [rate (1 + rate)^nper]/[(1 + rate)^nper - 1] or in Excel speak =-PMT(rate,nper,loan balance,0,0).

You would argue therefore the loan is compound, but others (including the mortgage professor who you disdain) would argue that it is simple, because that ALLOCATION formula for payments which is a SIMPLE interest computation.

In contrast in the savings account scenarios, the "amortization" has to be regularly recomputed based on the new balance rather than always on $100K [EDIT to add:] and the formula itself is a compound interest formula based on the average balance during the period. Which is why when you run the savings account vs. the mortgage loan, example the savings account actually ends up drained faster than the mortgage gets paid off:

It's a [l]oan. Is there interest on interest if interest is never capitalized?

Perhaps that is another way of framing the distinction.

Why does one care? I guess you want a clear standard for comparing rates. So let's compare rates.

Say, I take a 10% $100K 10 year mortgage and put it into a 10% rate savings account. I make payments from the savings account back into the mortgage. Ignoring taxes and with no rate changes, after 10 years, am I over, under or at zero? What if it's a 10% APY account?

Ignoring the APY question and ignoring timing issues such as can you get the full month of interest at first, the 10% $100K 10 year mortgage has a $1321.51 payment that never changes. We need to make some assumptions about the bank account, a simple average of the starting balance and the staring balance less the payment is used as the average collected balance for daily compound interest with a 30 day month and a 360 day year. In this example, the bank account ends up drained faster than the loan and there is not enough money in year 10, month 12 to pay the loan off. [aside switching to monthly compounding slightly increased the shortfall] Formula for a month's interest is -FV(10%/360,30,,avg daily balane,0)-avg daily balance.

(copy-paste from Excel below)

Code: Select all

Period	Monthly Payment	Interest	Principal		Loan Balance	 Total Loan Interest 		 Starting Bank Balance 	 Withdrawal 	 Simple Average 	Month Interest	Ending Balance
1	$1,321.51	 $833.33 	 $488.18 		$99,511.82	833.33		 100,000 	 1,322 	 99,339 	$831.17 	 99,509.66 
2	 1,321.51 	829.27	492.24		99019.58	1,662.60		 99,510 	 1,322 	 98,849 	$827.07 	 99,015.22 
3	 1,321.51 	825.16	496.35		98523.23	2,487.76		 99,015 	 1,322 	 98,354 	$822.93 	 98,516.64 
4	 1,321.51 	821.03	500.48		98022.75	3,308.79		 98,517 	 1,322 	 97,856 	$818.76 	 98,013.89 
5	 1,321.51 	816.86	504.65		97518.09	4,125.64		 98,014 	 1,322 	 97,353 	$814.55 	 97,506.93 
6	 1,321.51 	812.65	508.86		97009.24	4,938.30		 97,507 	 1,322 	 96,846 	$810.31 	 96,995.73 
7	 1,321.51 	808.41	513.10		96496.14	5,746.71		 96,996 	 1,322 	 96,335 	$806.03 	 96,480.25 
8	 1,321.51 	804.13	517.38		95978.76	6,550.84		 96,480 	 1,322 	 95,819 	$801.72 	 95,960.46 
9	 1,321.51 	799.82	521.69		95457.07	7,350.66		 95,960 	 1,322 	 95,300 	$797.37 	 95,436.32 
10	 1,321.51 	795.48	526.03		94931.04	8,146.14		 95,436 	 1,322 	 94,776 	$792.99 	 94,907.80 
11	 1,321.51 	791.09	530.42		94400.62	8,937.23		 94,908 	 1,322 	 94,247 	$788.56 	 94,374.85 
12	 1,321.51 	786.67	534.84		93865.78	9,723.90		 94,375 	 1,322 	 93,714 	$784.10 	 93,837.45 
13	 1,321.51 	782.21	539.30		93326.49	10,506.12		 93,837 	 1,322 	 93,177 	$779.61 	 93,295.55 
14	 1,321.51 	777.72	543.79		92782.70	11,283.84		 93,296 	 1,322 	 92,635 	$775.07 	 92,749.11 
15	 1,321.51 	773.19	548.32		92234.38	12,057.03		 92,749 	 1,322 	 92,088 	$770.50 	 92,198.10 
16	 1,321.51 	768.62	552.89		91681.49	12,825.65		 92,198 	 1,322 	 91,537 	$765.89 	 91,642.48 
17	 1,321.51 	764.01	557.50		91123.99	13,589.66		 91,642 	 1,322 	 90,982 	$761.24 	 91,082.22 
18	 1,321.51 	759.37	562.14		90561.85	14,349.03		 91,082 	 1,322 	 90,421 	$756.56 	 90,517.26 
19	 1,321.51 	754.68	566.83		89995.02	15,103.71		 90,517 	 1,322 	 89,857 	$751.83 	 89,947.58 
20	 1,321.51 	749.96	571.55		89423.47	15,853.67		 89,948 	 1,322 	 89,287 	$747.06 	 89,373.13 
21	 1,321.51 	745.20	576.31		88847.15	16,598.86		 89,373 	 1,322 	 88,712 	$742.26 	 88,793.88 
22	 1,321.51 	740.39	581.12		88266.04	17,339.26		 88,794 	 1,322 	 88,133 	$737.41 	 88,209.77 
23	 1,321.51 	735.55	585.96		87680.08	18,074.81		 88,210 	 1,322 	 87,549 	$732.52 	 87,620.79 
24	 1,321.51 	730.67	590.84		87089.23	18,805.47		 87,621 	 1,322 	 86,960 	$727.59 	 87,026.87 
25	 1,321.51 	725.74	595.77		86493.47	19,531.22		 87,027 	 1,322 	 86,366 	$722.62 	 86,427.98 
26	 1,321.51 	720.78	600.73		85892.74	20,252.00		 86,428 	 1,322 	 85,767 	$717.61 	 85,824.09 
27	 1,321.51 	715.77	605.74		85287.00	20,967.77		 85,824 	 1,322 	 85,163 	$712.56 	 85,215.14 
28	 1,321.51 	710.72	610.79		84676.21	21,678.49		 85,215 	 1,322 	 84,554 	$707.47 	 84,601.09 
29	 1,321.51 	705.64	615.87		84060.34	22,384.13		 84,601 	 1,322 	 83,940 	$702.33 	 83,981.91 
30	 1,321.51 	700.50	621.01		83439.33	23,084.63		 83,982 	 1,322 	 83,321 	$697.15 	 83,357.55 
31	 1,321.51 	695.33	626.18		82813.15	23,779.96		 83,358 	 1,322 	 82,697 	$691.92 	 82,727.96 
32	 1,321.51 	690.11	631.40		82181.75	24,470.07		 82,728 	 1,322 	 82,067 	$686.66 	 82,093.10 
33	 1,321.51 	684.85	636.66		81545.09	25,154.92		 82,093 	 1,322 	 81,432 	$681.34 	 81,452.94 
34	 1,321.51 	679.54	641.97		80903.12	25,834.46		 81,453 	 1,322 	 80,792 	$675.99 	 80,807.41 
35	 1,321.51 	674.19	647.32		80255.80	26,508.65		 80,807 	 1,322 	 80,147 	$670.59 	 80,156.49 
36	 1,321.51 	668.80	652.71		79603.09	27,177.45		 80,156 	 1,322 	 79,496 	$665.14 	 79,500.12 
37	 1,321.51 	663.36	658.15		78944.94	27,840.81		 79,500 	 1,322 	 78,839 	$659.65 	 78,838.26 
38	 1,321.51 	657.87	663.64		78281.30	28,498.68		 78,838 	 1,322 	 78,178 	$654.11 	 78,170.86 
39	 1,321.51 	652.34	669.17		77612.14	29,151.03		 78,171 	 1,322 	 77,510 	$648.53 	 77,497.87 
40	 1,321.51 	646.77	674.74		76937.40	29,797.80		 77,498 	 1,322 	 76,837 	$642.90 	 76,819.26 
41	 1,321.51 	641.14	680.37		76257.03	30,438.94		 76,819 	 1,322 	 76,159 	$637.22 	 76,134.97 
42	 1,321.51 	635.48	686.03		75571.00	31,074.42		 76,135 	 1,322 	 75,474 	$631.49 	 75,444.95 
43	 1,321.51 	629.76	691.75		74879.24	31,704.17		 75,445 	 1,322 	 74,784 	$625.72 	 74,749.16 
44	 1,321.51 	623.99	697.52		74181.73	32,328.17		 74,749 	 1,322 	 74,088 	$619.90 	 74,047.54 
45	 1,321.51 	618.18	703.33		73478.40	32,946.35		 74,048 	 1,322 	 73,387 	$614.03 	 73,340.06 
46	 1,321.51 	612.32	709.19		72769.21	33,558.67		 73,340 	 1,322 	 72,679 	$608.11 	 72,626.66 
47	 1,321.51 	606.41	715.10		72054.11	34,165.08		 72,627 	 1,322 	 71,966 	$602.14 	 71,907.28 
48	 1,321.51 	600.45	721.06		71333.05	34,765.53		 71,907 	 1,322 	 71,247 	$596.12 	 71,181.89 
49	 1,321.51 	594.44	727.07		70605.98	35,359.97		 71,182 	 1,322 	 70,521 	$590.05 	 70,450.43 
50	 1,321.51 	588.38	733.13		69872.85	35,948.35		 70,450 	 1,322 	 69,790 	$583.93 	 69,712.85 
51	 1,321.51 	582.27	739.24		69133.62	36,530.63		 69,713 	 1,322 	 69,052 	$577.76 	 68,969.10 
52	 1,321.51 	576.11	745.40		68388.22	37,106.74		 68,969 	 1,322 	 68,308 	$571.53 	 68,219.12 
53	 1,321.51 	569.90	751.61		67636.61	37,676.64		 68,219 	 1,322 	 67,558 	$565.26 	 67,462.87 
54	 1,321.51 	563.64	757.87		66878.74	38,240.28		 67,463 	 1,322 	 66,802 	$558.93 	 66,700.30 
55	 1,321.51 	557.32	764.19		66114.56	38,797.61		 66,700 	 1,322 	 66,040 	$552.55 	 65,931.34 
56	 1,321.51 	550.95	770.56		65344.00	39,348.56		 65,931 	 1,322 	 65,271 	$546.12 	 65,155.95 
57	 1,321.51 	544.53	776.98		64567.02	39,893.09		 65,156 	 1,322 	 64,495 	$539.63 	 64,374.07 
58	 1,321.51 	538.06	783.45		63783.57	40,431.15		 64,374 	 1,322 	 63,713 	$533.09 	 63,585.64 
59	 1,321.51 	531.53	789.98		62993.59	40,962.68		 63,586 	 1,322 	 62,925 	$526.49 	 62,790.63 
60	 1,321.51 	524.95	796.56		62197.03	41,487.63		 62,791 	 1,322 	 62,130 	$519.84 	 61,988.96 
61	 1,321.51 	518.31	803.20		61393.83	42,005.94		 61,989 	 1,322 	 61,328 	$513.13 	 61,180.58 
62	 1,321.51 	511.62	809.89		60583.93	42,517.55		 61,181 	 1,322 	 60,520 	$506.37 	 60,365.44 
63	 1,321.51 	504.87	816.64		59767.29	43,022.42		 60,365 	 1,322 	 59,705 	$499.55 	 59,543.47 
64	 1,321.51 	498.06	823.45		58943.84	43,520.48		 59,543 	 1,322 	 58,883 	$492.67 	 58,714.63 
65	 1,321.51 	491.20	830.31		58113.53	44,011.68		 58,715 	 1,322 	 58,054 	$485.74 	 57,878.86 
66	 1,321.51 	484.28	837.23		57276.30	44,495.96		 57,879 	 1,322 	 57,218 	$478.74 	 57,036.09 
67	 1,321.51 	477.30	844.21		56432.09	44,973.26		 57,036 	 1,322 	 56,375 	$471.69 	 56,186.28 
68	 1,321.51 	470.27	851.24		55580.85	45,443.53		 56,186 	 1,322 	 55,526 	$464.58 	 55,329.35 
69	 1,321.51 	463.17	858.34		54722.51	45,906.70		 55,329 	 1,322 	 54,669 	$457.41 	 54,465.25 
70	 1,321.51 	456.02	865.49		53857.02	46,362.72		 54,465 	 1,322 	 53,804 	$450.18 	 53,593.92 
71	 1,321.51 	448.81	872.70		52984.32	46,811.53		 53,594 	 1,322 	 52,933 	$442.89 	 52,715.30 
72	 1,321.51 	441.54	879.97		52104.35	47,253.07		 52,715 	 1,322 	 52,055 	$435.54 	 51,829.33 
73	 1,321.51 	434.20	887.31		51217.04	47,687.27		 51,829 	 1,322 	 51,169 	$428.13 	 50,935.95 
74	 1,321.51 	426.81	894.70		50322.34	48,114.08		 50,936 	 1,322 	 50,275 	$420.65 	 50,035.09 
75	 1,321.51 	419.35	902.16		49420.18	48,533.43		 50,035 	 1,322 	 49,374 	$413.11 	 49,126.69 
76	 1,321.51 	411.83	909.68		48510.51	48,945.27		 49,127 	 1,322 	 48,466 	$405.51 	 48,210.70 
77	 1,321.51 	404.25	917.26		47593.25	49,349.52		 48,211 	 1,322 	 47,550 	$397.85 	 47,287.04 
78	 1,321.51 	396.61	924.90		46668.35	49,746.13		 47,287 	 1,322 	 46,626 	$390.12 	 46,355.65 
79	 1,321.51 	388.90	932.61		45735.74	50,135.03		 46,356 	 1,322 	 45,695 	$382.33 	 45,416.47 
80	 1,321.51 	381.13	940.38		44795.36	50,516.16		 45,416 	 1,322 	 44,756 	$374.47 	 44,469.43 
81	 1,321.51 	373.29	948.22		43847.15	50,889.46		 44,469 	 1,322 	 43,809 	$366.55 	 43,514.46 
82	 1,321.51 	365.39	956.12		42891.03	51,254.85		 43,514 	 1,322 	 42,854 	$358.56 	 42,551.51 
83	 1,321.51 	357.43	964.08		41926.95	51,612.28		 42,552 	 1,322 	 41,891 	$350.50 	 41,580.50 
84	 1,321.51 	349.39	972.12		40954.83	51,961.67		 41,580 	 1,322 	 40,920 	$342.37 	 40,601.36 
85	 1,321.51 	341.29	980.22		39974.61	52,302.96		 40,601 	 1,322 	 39,941 	$334.18 	 39,614.04 
86	 1,321.51 	333.12	988.39		38986.22	52,636.08		 39,614 	 1,322 	 38,953 	$325.92 	 38,618.45 
87	 1,321.51 	324.89	996.62		37989.60	52,960.97		 38,618 	 1,322 	 37,958 	$317.59 	 37,614.53 
88	 1,321.51 	316.58	1004.93		36984.67	53,277.55		 37,615 	 1,322 	 36,954 	$309.19 	 36,602.21 
89	 1,321.51 	308.21	1013.30		35971.36	53,585.75		 36,602 	 1,322 	 35,941 	$300.72 	 35,581.42 
90	 1,321.51 	299.76	1021.75		34949.61	53,885.51		 35,581 	 1,322 	 34,921 	$292.18 	 34,552.09 
91	 1,321.51 	291.25	1030.26		33919.35	54,176.76		 34,552 	 1,322 	 33,891 	$283.57 	 33,514.15 
92	 1,321.51 	282.66	1038.85		32880.50	54,459.42		 33,514 	 1,322 	 32,853 	$274.88 	 32,467.53 
93	 1,321.51 	274.00	1047.51		31832.99	54,733.42		 32,468 	 1,322 	 31,807 	$266.13 	 31,412.14 
94	 1,321.51 	265.27	1056.24		30776.76	54,998.70		 31,412 	 1,322 	 30,751 	$257.30 	 30,347.93 
95	 1,321.51 	256.47	1065.04		29711.72	55,255.17		 30,348 	 1,322 	 29,687 	$248.39 	 29,274.81 
96	 1,321.51 	247.60	1073.91		28637.81	55,502.77		 29,275 	 1,322 	 28,614 	$239.41 	 28,192.71 
97	 1,321.51 	238.65	1082.86		27554.95	55,741.42		 28,193 	 1,322 	 27,532 	$230.36 	 27,101.56 
98	 1,321.51 	229.62	1091.89		26463.06	55,971.04		 27,102 	 1,322 	 26,441 	$221.23 	 26,001.28 
99	 1,321.51 	220.53	1100.98		25362.08	56,191.57		 26,001 	 1,322 	 25,341 	$212.02 	 24,891.80 
100	 1,321.51 	211.35	1110.16		24251.92	56,402.92		 24,892 	 1,322 	 24,231 	$202.74 	 23,773.03 
101	 1,321.51 	202.10	1119.41		23132.51	56,605.02		 23,773 	 1,322 	 23,112 	$193.38 	 22,644.90 
102	 1,321.51 	192.77	1128.74		22003.77	56,797.79		 22,645 	 1,322 	 21,984 	$183.94 	 21,507.33 
103	 1,321.51 	183.36	1138.15		20865.62	56,981.15		 21,507 	 1,322 	 20,847 	$174.42 	 20,360.24 
104	 1,321.51 	173.88	1147.63		19717.99	57,155.03		 20,360 	 1,322 	 19,699 	$164.83 	 19,203.56 
105	 1,321.51 	164.32	1157.19		18560.80	57,319.35		 19,204 	 1,322 	 18,543 	$155.15 	 18,037.19 
106	 1,321.51 	154.67	1166.84		17393.96	57,474.02		 18,037 	 1,322 	 17,376 	$145.39 	 16,861.07 
107	 1,321.51 	144.95	1176.56		16217.40	57,618.97		 16,861 	 1,322 	 16,200 	$135.55 	 15,675.11 
108	 1,321.51 	135.15	1186.36		15031.04	57,754.12		 15,675 	 1,322 	 15,014 	$125.62 	 14,479.23 
109	 1,321.51 	125.26	1196.25		13834.79	57,879.38		 14,479 	 1,322 	 13,818 	$115.62 	 13,273.33 
110	 1,321.51 	115.29	1206.22		12628.57	57,994.67		 13,273 	 1,322 	 12,613 	$105.53 	 12,057.35 
111	 1,321.51 	105.24	1216.27		11412.30	58,099.91		 12,057 	 1,322 	 11,397 	$95.36 	 10,831.20 
112	 1,321.51 	95.10	1226.41		10185.89	58,195.01		 10,831 	 1,322 	 10,170 	$85.10 	 9,594.79 
113	 1,321.51 	84.88	1236.63		8949.26	58,279.89		 9,595 	 1,322 	 8,934 	$74.75 	 8,348.03 
114	 1,321.51 	74.58	1246.93		7702.33	58,354.47		 8,348 	 1,322 	 7,687 	$64.32 	 7,090.84 
115	 1,321.51 	64.19	1257.32		6445.00	58,418.65		 7,091 	 1,322 	 6,430 	$53.80 	 5,823.13 
116	 1,321.51 	53.71	1267.80		5177.20	58,472.36		 5,823 	 1,322 	 5,162 	$43.19 	 4,544.81 
117	 1,321.51 	43.14	1278.37		3898.84	58,515.51		 4,545 	 1,322 	 3,884 	$32.50 	 3,255.80 
118	 1,321.51 	32.49	1289.02		2609.82	58,548.00		 3,256 	 1,322 	 2,595 	$21.71 	 1,956.00 
119	 1,321.51 	21.75	1299.76		1310.05	58,569.74		 1,956 	 1,322 	 1,295 	$10.84 	 645.33 
120	 1,321.51 	10.92	1310.05		0.00	58,580.66		 645 	 1,322 	 (15)	($0.13)	 (676.31)

[Jack]

It doesn't matter who the guy is. The mathematics speaks for itself, and the mathematics is the mathematics of compound interest.

Amortization calculations are made based on compound interest.

Search the thread for the "^" symbol (exponentiation symbol). That's the post where I explain the amortization calculation, and the exponentiation makes it clear that compound interest is being used.

There is no compound interest. You cannot show any point at which interest is charged on interest. The exponentiation comes from the fact that it is an amortizing loan, not because there is compound interest. Just because there is mathematically exponentiation of interest does not mean that interest is paid on interest. That is just an artifact of amortization.

http://en.wikipedia.org/wiki/Amortizati ... he_formula

Look here at the derivation for the amortization formula. Here are the first three terms in the derivation.

p(0) = P
p(1) = p(0) r - A = P r - A
p(2) = p(1) r - A = P r^2 - A r - A
p(3) = p(2) r - A = P r^3 - A r^2 - A r - A

It is quite clear that each payment is a simple interest calculation. When you sum up the simple interest payments, you can make a mathematical substitution that includes a geometric progression. That mathematical substitution does not convert a simple interest loan into a compound interest loan. You can say that an amortization loan can be mathematically calculated using a compound interest rate formula, but that is just a mathematical simplification. At no point is interest paid on interest. A mathematical simplification does not convert simple interest into compound interest. It is just a mathematical transformation for deriving a simple interest summation.

It is Is it not true that you can replace a mortgage loan with 360 individual monthly loans which pay simple interest? At no point have you paid interest on interest.

Corrected typo.

[madsinger]

Against my better judgment, I'm going to offer some thoughts on this thread.

Think of a bank account that pays 1% interest on your balance every month (hey....that happened in the 80's...), and we put $100,000 at the start. After one month, we get $1000 (1% of 100,000) and we have a balance of $101,000. After two months, we get $1010 in interest and have a balance of $102,010. I think we all agree this is "classic compound interest". (if we don't agree, don't bother reading further!)

Now, same situation, but at the end of each month, right after interest is credited, we withdraw $1500.

Start with $100,000. Get $1000 in interest after first month. Withdraw $1500. After the withdrawal, we have $99,500:

$100,000 + $1,000 - $1,500 = $99,500

Go on to the next month. Get $995 in interest, withdraw $1500, result after two months is $98,995.

$99,500 + $995 - $1500 = $98,995

Every month, we get less interest, because we keep withdrawing more ($1500) than we ever get in interest. By month 110:

$2,089.38 + $20.89 - $1500 = $610.27

This (of course) is exactly a mortgage. I "lent" the bank $100,000. Every month, they paid me 1% on my outstanding balance, and I withdrew $1500. After 110 months, I have only $610.27 left in the bank. They'll pay me $6.10 in interest for that last month, and I'll take out what is left. Mortgage paid off.

So...is my bank account (mortgage) an account that pays compound interest?

Yes, the account pays interest on the balance every month, and IF I did not withdraw $1500 every month, it would "compound".

Or,

No, my account never paid "interest on interest".

Don't let semantics get in the way of math. As 555 says, the math is absolutely correct. As others said, there's never "interest on interest".

I don't care if it's "compound" or "simple", those words do not capture the essence of the math behind a mortgage...which, frankly, is pretty "simple" math

-Brad.

[rkhusky]

So...is my bank account (mortgage) an account that pays compound interest?

Yes, the account pays interest on the balance every month, and IF I did not withdraw $1500 every month, it would "compound".

Or,

No, my account never paid "interest on interest".

Don't let semantics get in the way of math. As 555 says, the math is absolutely correct. As others said, there's never "interest on interest".

I don't care if it's "compound" or "simple", those words do not capture the essence of the math behind a mortgage...which, frankly, is pretty "simple" math

-Brad.

So, the bank pays "compound interest", but you don't receive "compound interest".

[rkhusky]

I don't know what you call it, but I know it isn't simple interest. I work in a bank, and one of my jobs is to make sure the new home mortgage loans are NOT set to simple interest. What this means is that I make sure the little box in front of the simple interest box is not checked in the loan's profile in our data system. I don't remember if "compound interest" is one of the other choices - I am on vacation this week and everyone will have to wait until next week when I can look. I have found a few home mortgage loans that were set to simple interest, and the interest calculations were not what they should be (but the difference was usually only a few dollars a month). The formula used is: principal balance times (interest rate / 12).

Home mortgage loans are not considered late unless your payment arrives AFTER the grace period date. Beware about paying at a bank branch - sometimes those don't count as accepted until the check actually arrives at the loan service department.

"Simple interest" can be computed on a monthly or daily basis. It seems like mortgages are computed and paid monthly, but home equity loans are computed on a daily basis (but not compounded) and paid monthly. For mortgages, the amount paid on principal increases in a regular fashion irrespective of the number of days in the month, whereas the HEL's jumps around a bit, depending on the number of days. Like you say, the difference is generally just a few dollars.

[Jack]

I don't care if it's "compound" or "simple", those words do not capture the essence of the math behind a mortgage...which, frankly, is pretty "simple" math

You could come up with an even simpler case. Assume you have a $100,000 loan at 10% interest and you make a $10,000 payment each year.

Is that a compound or simple interest loan? It could be either -- a simple interest or compound interest loan. The results are identical. The best you could say is that it might be a compound interest loan that never compounds because interest is paid in full each year.

So the question boils down to whether a compound interest loan that never compounds is really compound. Or is a compound interest loan that never compounds the equivalent of a simple interest loan? Is it the actual compounding of interest that makes a loan compound or is it merely the potential of compounding that makes a loan compound?

I think the best you can say is that an amortization loan is a compound interest loan that never compounds interest. Or one could say that it is a compound interest loan in theory but a simple interest loan in practice. If you pay interest in full on a compound interest loan, it reduces to a simple interest loan.

The simple interest loan is a sub-set of the the more general class of compound interest loans in which full interest is paid each period. If interest is not accrued, there is no compounding. If you believe that, then both claims would be true. The amortization loan is a compound interest loan in general which reduces to the more specific case of a simple interest loan.

[Abe]

I'll take a stab at it. I am using 2 examples. I would consider loan #1 to be simple interest and I would consider loan #2 to be compound interest.

Loan #1
10 year, $10,000.00 mortgage at 10%, monthly payment $132.15 for 120 months. After I make the first payment of $132.15, $83.33 would be applied to interest and $48.82 would be applied to principal. The following month, I would pay $132.15 of which $82.93 would apply to interest and $49.22 would apply to principal and so on. I would say this is simple interest because the interest is not added back to the principal balance. In other words, I am only paying interest on the unpaid principal balance. The total interest I would pay on this loan over the 10 year period would be $5,858.00.

Loan #2
10 year, $10,000.00 mortgage at 10% compounded monthly with interest and principal due in full at the end of 10 years. At the end of 10 years, I would owe $27,070.41 of which $10,000.00 would be principal and $17,070.41 would be interest.

Both loans are for the same dollar amount, the same interest rate and the same term, but I would be paying almost 3 times more interest on loan #2 (compound interest) as I would be paying on loan #1 (simple interest). I think.

[555]

Compound interest is interest that is computed by multiplying (the balance) by an appropriate factor. If you multiply by r after 1 time period, then you multiply by r^n after n time periods. This exponential growth (assuming r>1) is a hallmark of compound interest. It is the effect of iterating the interest calculation that is the key feature. It is compound interest regardless of if/when amounts are added or subtracted to the balance of a loan or deposit. The effect of any payment/withdrawal/deposit grows exponentially with time (by a factor of r^n after n time periods). Classic compound interest. It doesn't matter if the payment/withdrawal/deposit is positive or negative (in an idealized case without penalties or payment dependent interest). As far as the mathematics is concerned, it certainly doesn't matter whether a loan payment "covers" the "accrued interest" (and it doesn't matter how the payment is "applied" to "principal and interest") because all that's just part of an accounting shell game that is irrelevant to the underlying mathematics. Apparently this accounting shell game has been contrived to make it look like interest is not being charged on interest, and they may be able to hoodwink the politicians who make laws against "interest on interest", but all of us here should know better. Interest is charged on the whole balance which includes all previous interest, and all that previous interest is certainly there in the balance because the balance is higher than it would have been if the interest were zero.

[Kosmo]

This topic has gone well beyond what I anticipated. I think maybe the difficult to grasp issue is the frequency of interest calculation (which could be considered compounding?). Simple interest: 1 interest calculation. Compound interest: interest calculated at some compounding frequency.

So in hopefully clearer terms, a mortgage is not simple interest (in the strictest sense) since the total interest due is not the interest rate times the mortgage amount. But the monthly interest is calculated using a simple interest calculation.

Quick comparison:

Mortgage case:
Interest is calculated each month based on the rate and outstanding principal. Therefore it is a simple interest calculation. The amortization table shows the interest /principal breakdown for every payment (assuming no pre-payments). But you also pay down the principal every month, so the subsequent month's interest is less. The interest does not compound because the value on which the interest is calculated is decreasing.

Savings account case:
Interest is calculated each month based on the rate and account balance. Therefore it is a simple interest calculation. The difference here is that you leave the interest in the bank account and the value on which the interest is calculated is increasing. The bank doesn't pay compound interest, you allow the interest to compound by leaving it in the bank. As someone described it: Would you say the bank switched from paying compound interest to simple interest if the interest was placed in a separate account? No, because the bank never paid compound interest to start with.

[rkhusky]

Compound interest is interest that is computed by multiplying (the balance) by an appropriate factor. If you multiply by r after 1 time period, then you multiply by r^n after n time periods. This exponential growth (assuming r>1) is a hallmark of compound interest. It is the effect of iterating the interest calculation that is the key feature. It is compound interest regardless of if/when amounts are added or subtracted to the balance of a loan or deposit. The effect of any payment/withdrawal/deposit grows exponentially with time (by a factor of r^n after n time periods). Classic compound interest. It doesn't matter if the payment/withdrawal/deposit is positive or negative (in an idealized case without penalties or payment dependent interest). As far as the mathematics is concerned, it certainly doesn't matter whether a loan payment "covers" the "accrued interest" (and it doesn't matter how the payment is "applied" to "principal and interest") because all that's just part of an accounting shell game that is irrelevant to the underlying mathematics. Apparently this accounting shell game has been contrived to make it look like interest is not being charged on interest, and they may be able to hoodwink the politicians who make laws against "interest on interest", but all of us here should know better. Interest is charged on the whole balance which includes all previous interest, and all that previous interest is certainly there in the balance because the balance is higher than it would have been if the interest were zero.

Simple interest is just compound interest with no compounding. Linear growth is just exponential growth with rate=epsilon.

[madsinger]

Compound interest is interest that is computed by multiplying (the balance) by an appropriate factor. If you multiply by r after 1 time period, then you multiply by r^n after n time periods. This exponential growth (assuming r>1) is a hallmark of compound interest. It is the effect of iterating the interest calculation that is the key feature. It is compound interest regardless of if/when amounts are added or subtracted to the balance of a loan or deposit. The effect of any payment/withdrawal/deposit grows exponentially with time (by a factor of r^n after n time periods). Classic compound interest. It doesn't matter if the payment/withdrawal/deposit is positive or negative (in an idealized case without penalties or payment dependent interest). As far as the mathematics is concerned, it certainly doesn't matter whether a loan payment "covers" the "accrued interest" (and it doesn't matter how the payment is "applied" to "principal and interest") because all that's just part of an accounting shell game that is irrelevant to the underlying mathematics. Apparently this accounting shell game has been contrived to make it look like interest is not being charged on interest, and they may be able to hoodwink the politicians who make laws against "interest on interest", but all of us here should know better. Interest is charged on the whole balance which includes all previous interest, and all that previous interest is certainly there in the balance because the balance is higher than it would have been if the interest were zero.

This is absolutely correct.

-Brad.

[billyt]

There is absolutely no question at all that 555 is correct. It's simple math. How did this thread go on for so long?

[Phineas J. Whoopee]

I think I got it!

Nevertheless, let nobody misconstrue this post as me putting words into 555's mouth (or keyboard, as the case may be). I'm checking to see if I understand the position.

$100,000 loan. 5% annualized interest charged at the end of the month. $1,000 monthly payment.

First month: $100,000 loan, $1,000 payment, principal only reduced by $583.33, leaving $99,416.67. But if all of the first month's $1,000 payment had gone to principal, and none to interest, then for the second month the remaining balance would be $99,000.

Second month: $99,416.67 loan, $1,000 payment, principal only reduced by $585.76, leaving $98,830.91. But if all of the second month's $1,000 had gone to principal, and none to interest, then for the third month the remaining loan would be $98,000.

And so on through however many months: only 100, if there never was any interest to pay; many more if we imagine all the accumulated interest still left at the end of the hundred $1,000 principal payments and feeding on itself.

Can that be the compound interest argument?

PJW

[LadyGeek]

For the final touch, see this tutorial: Loan Amortization with Microsoft Excel

To the OP - Did you get your question answered?

[CantPassAgain]

There is absolutely no question at all that 555 is correct. It's simple math. How did this thread go on for so long?

Yeah. Wouldn't a simple interest mortgage have the same split between principal and interest for every payment (so principal is reduced in a straight-line fashion)? That is certainly not how any mortgage I have ever had works. I always considered mortgage amortization schedules to be based on compound interest math.

[Jack]

As far as the mathematics is concerned, it certainly doesn't matter whether a loan payment "covers" the "accrued interest" (and it doesn't matter how the payment is "applied" to "principal and interest") because all that's just part of an accounting shell game that is irrelevant to the underlying mathematics. Apparently this accounting shell game has been contrived to make it look like interest is not being charged on interest.

Yet a mortgage is simply 360 monthly loans in which simple interest is charged on the principal borrowed each month. Where is the interest on interest? It that an illusion?

Or the simpler case of a $100,000 loan at 10% interest and there is a payment of $10,000 each year, where is the interest on interest? Is that an illusion too?

[Phineas J. Whoopee]

Thinking further: I'm convinced.

555 was right and I was wrong. It's a matter of conflating one period with the entire term.

Boy, do I feel foolish.

Thanks 555!

PJW

[dm200]

My goodness - I believe this thread demonstrates that "Bogleheads" "think" differently than most "normal" folks!

I wonder what the thread would be if someone asked "What time is it?"

[Phineas J. Whoopee]

My goodness - I believe this thread demonstrates that "Bogleheads" "think" differently than most "normal" folks!

I wonder what the thread would be if someone asked "What time is it?"

Sideraeal, solar, Earth-bound, UDC vs. GMT (wanna buy a bridge?), Stardate? Whatever my boss's watch happens to say which defines time for all of space and, well, time?

18:09 is my answer, but only believe me if you're a relative.

PJW

[555]

Or the simpler case of a $100,000 loan at 10% interest and there is a payment of $10,000 each year, where is the interest on interest? Is that an illusion too?

Use the fact that the limit as r approaches 1 of [(r^n-1)/(r-1)] equals n.

[rkhusky]

If I make a 10%, 10 year, $100,000 mortgage-type loan to someone, my investment grows like I = -100,000 + 1321.51m, m=1:120. This is a linear function of time, implying simple interest. Compound interest produces exponential growth.

[pop77]

You're dead wrong on this. Paying during the grace period constitutes timely payment in the eyes of the bank and according to the terms of the note. No red flags are set off at the bank by using the grace period.
Bruce

+1 -Paying within grace has not affected my credit(based on credit report). Not aware what red flag means and where it is recorded. However delaying the payment just for heck of though you have cash on bank does not give you any economic benefit except the satisfaction that you have squeezed the max time from bank. There is some psychological pleasure in that.

[Phineas J. Whoopee]

The amount of the debtor's payment that was interest, remains principal for the next month.

The interest for the next month is calculated on the remaining principal, which includes the interest paid in the previous month which remains principal as stated in the prior sentence.

For me it was a matter of shifting perspective; like imagining the Sun, not the Earth, as the center of the Solar System.

555 is right. OK, not Galileo right. Not even Leonardo right. Maybe not even Leibniz right. Perhaps a little below Schwed level. Certainly higher than Mr. Lizard the Wizard.

At any level, right is right.

PJW

[Jack]

Or the simpler case of a $100,000 loan at 10% interest and there is a payment of $10,000 each year, where is the interest on interest? Is that an illusion too?

Use the fact that the limit as r approaches 1 of [(r^n-1)/(r-1)] equals n.

So how much interest has been paid on interest? The answer is zero.

[Kosmo]

The amount of the debtor's payment that was interest, remains principal for the next month.

The interest for the next month is calculated on the remaining principal, which includes the interest paid in the previous month which remains principal as stated in the prior sentence.

Not correct. The amount of the debtor's payment that was interest covers the interest due for that month. The remaining portion of the payment is used to reduce principal.

The interest for the next month is calculated on the remaining principal (you had that correct), which does not include any previous interest paid because it has already been paid. That's how amortized loans work.

Everyone seems to understand compound interest when it is positive. Any interest paid to the account will compound at a rate of r^n, provided that it remains in the account. But in the case of a mortgage, the principal value is decreasing with every payment, so interest accrued doesn't compound. For lack of a better word, it "de-compounds", if it does anything. The interest owed gets less and less as time goes on. (Again, it's similar to if you withdrew the interest +$1 from your savings account every month...it would slowly decline to nothing.) Does this periodic interest calculation make it "compound interest"? I think at this point you're arguing semantics. Regardless of that, for each individual month, the interest calculation is a simple interest calculation based solely on the rate and principal. It does not account for past or future payments or paid or unpaid interest.

[rkhusky]

If I make a 10%, 10 year, $100,000 mortgage-type loan to someone, my investment grows like I = -100,000 + 1321.51m, m=1:120. This is a linear function of time, implying simple interest. Compound interest produces exponential growth.

If the person to whom I made the loan puts the money into a savings account for the same term and rate and reinvests the interest, their investment grows like I = 100,000 (1.0083333)^m, m=1:120. This is exponential growth since log(I)=11.513 + 1.0083333m, i.e. log of investment is a linear function of time. After 10 years, I would receive a profit of $58,581 for making a mortgage-like loan, whereas the person that invested in the savings account would receive a profit of $170,703.

If instead, the person to whom I made the loan found a savings account paying 4.62% interest, they would still experience exponential growth, with a profit of $58,581, equal to the 10% mortgage.

For an even simpler case, consider an interest-only mortgage-like loan, 10% for 10 years with a final balloon payment of the remaining amount. I = -100,000 + 833.33m, m=1:119; I = 200,000, m=120. Again, this is a linear function of time. Clearly a simple interest calculation.The return on investment is the interest of $833.33 per month, every month. There is no compounding, no exponential growth. The total profit is $100,000.

[555]

Or the simpler case of a $100,000 loan at 10% interest and there is a payment of $10,000 each year, where is the interest on interest? Is that an illusion too?

Use the fact that the limit as r approaches 1 of [(r^n-1)/(r-1)] equals n.

So how much interest has been paid on interest? The answer is zero.

After n years the "interest paid on interest" is not zero, instead it is

{n choose 2}*$1000

[Jack]

Or the simpler case of a $100,000 loan at 10% interest and there is a payment of $10,000 each year, where is the interest on interest? Is that an illusion too?

Use the fact that the limit as r approaches 1 of [(r^n-1)/(r-1)] equals n.

So how much interest has been paid on interest? The answer is zero.

After n years the "interest paid on interest" is not zero, instead it is

{n choose 2}*$1000

So you would claim that if I paid full interest every year for 10 years, that I would have paid $45,000 in interest on interest. This is clearly gibberish nonsense.

By your same logic, you would claim that if I put $100,000 into a savings account that compounds annually at 10% and withdrew $10,000 each year, that I would have received $45,000 of interest on interest in 10 years. How can I receive interest on interest if the interest is withdrawn each year?

Complete nonsense.

[Abe]

I'm not trying to be argumentative. If someone can show me where I'm wrong, I'll be happy to admit I was wrong.
I posted one time previously on this thread using this example: $10,000.00 loan, 10% interest, 120 monthly payments, payment amount $132.15.
After the first payment of $132.15, $83.33 (the interest on $10,000 for one month) is deducted from the $132.15 leaving $48.82 which is deducted from the $10,000. principal balance leaving a principal balance of $9,951.18. The interest part of the payment of $83.33 is one months interest on $10,000.00 and it is paid at the end of the first month of the loan. At the beginning of the second month the principal balance is $9,951.18. The interest has already been paid for the previous month, so there is no interest included in the principal balance. The second month everything is the same except the interest part of the payment would be a little less and the principal part would be a little more and this would continue to be the same until the loan is paid off. So, in effect, this could be considered 120 separate simple interest loans.

The definition of compound interest is: interest calculated on both the principal and the accrued interest. Since the interest on the above loan is paid in full each month, there is no interest carried over to the principal balance. The interest is paid each month and any money remaining is principal only (no interest included).

As I said, I'll be happy to admit I'm wrong, if anyone can show me this is incorrect.
Best wishes and have a good day.

[madsinger]

So, in effect, this could be considered 120 separate simple interest loans.

The definition of compound interest is: interest calculated on both the principal and the accrued interest. Since the interest on the above loan is paid in full each month, there is no interest carried over to the principal balance. The interest is paid each month and any money remaining is principal only (no interest included).

As I said, I'll be happy to admit I'm wrong, if anyone can show me this is incorrect.
Best wishes and have a good day.

Hi Abe,

I think this whole discussion comes down to semantics, not math. We all seem to agree on the math of mortgages. We just don't know what to call it. I looked up one definition of mortgage, and it said it is neither simple interest or compound interest, but an amortization calculation.

I think of all compound interest as a series of simple interest calculations computed with a new principal for each period. If it's a bank account at 1% per month, then it's a P=$1000, r = 1%, i = $10. Next month it's P = $1010, r = 1%, i = $10.10, etc. It's "compound interest", but it's just a bunch of simple interest calculations starting with a different principal each month. If we add a "withdrawal" every month to our equation, and that withdrawal exceeds the interest we earn, then it's a mortgage. For me, this is "compound interest" --- ...a bunch of simple interest calculations starting with a different principal each month.

There is nothing "incorrect" in the math you presented, it's just a matter of defining what is meant by the terms "simple" and "compound" interest.

-Brad.

[Abe]

So, in effect, this could be considered 120 separate simple interest loans.

The definition of compound interest is: interest calculated on both the principal and the accrued interest. Since the interest on the above loan is paid in full each month, there is no interest carried over to the principal balance. The interest is paid each month and any money remaining is principal only (no interest included).

As I said, I'll be happy to admit I'm wrong, if anyone can show me this is incorrect.
Best wishes and have a good day.

Hi Abe,

I think this whole discussion comes down to semantics, not math. We all seem to agree on the math of mortgages. We just don't know what to call it. I looked up one definition of mortgage, and it said it is neither simple interest or compound interest, but an amortization calculation.

I think of all compound interest as a series of simple interest calculations computed with a new principal for each period. If it's a bank account at 1% per month, then it's a P=$1000, r = 1%, i = $10. Next month it's P = $1010, r = 1%, i = $10.10, etc. It's "compound interest", but it's just a bunch of simple interest calculations starting with a different principal each month. If we add a "withdrawal" every month to our equation, and that withdrawal exceeds the interest we earn, then it's a mortgage. For me, this is "compound interest" --- ...a bunch of simple interest calculations starting with a different principal each month.

There is nothing "incorrect" in the math you presented, it's just a matter of defining what is meant by the terms "simple" and "compound" interest.

-Brad.

Thanks for responding Brad. I agree; I think it comes down to semantics also. Some say it's compound and some say it's simple, and I don't think that's going to change. I don't think it's important what you call it. What is important is understanding how it works. This was the OP's original post:
"Can someone explain how a home mortgage loan, fixed rate, works please? Is it simple interest or compund and if its compund would that be daily compound or something else? Thank you"
So, to answer the OP's question as to how a home mortgage works I think the best way to understand how it works it to do one by hand as opposed to doing it on a computer. That way you know exactly what happening. Most home mortgages use monthly interest. So, using the $10,000. loan, 10% interest, 120 monthly payments, $132.15 payment amount . To calculate the interest for the first month, multiply $10,000 by 10% which is the interest for 1 year or $1,000. Then divide $1,000. by 12 to get the interest for 1 month which is $83.33. Then subtract the interest $83.33 from $132.15 to get the amount of principal payment which is $48.82. When you subtract $48.82 from $10,000.00, this will give you the principal balance for the next month which is $9,951.18. Then you do the same calculations for each month until the loan is paid off. There are other ways to do the calculations but this way is as good as any. I won't get in to whether the interest is simple or compound. As I said, I think the important thing is to understand how it work. Also, technically when people say they are paying on their mortgage, they are not actually paying on the mortgage; they are paying on the note. The mortgage is just the instrument that creates a lien on the property. I hope this is my last post on this thread.

[555]

Or the simpler case of a $100,000 loan at 10% interest and there is a payment of $10,000 each year, where is the interest on interest? Is that an illusion too?

Use the fact that the limit as r approaches 1 of [(r^n-1)/(r-1)] equals n.

So how much interest has been paid on interest? The answer is zero.

After n years the "interest paid on interest" is not zero, instead it is

{n choose 2}*$1000

So you would claim that if I paid full interest every year for 10 years, that I would have paid $45,000 in interest on interest. This is clearly gibberish nonsense.

By your same logic, you would claim that if I put $100,000 into a savings account that compounds annually at 10% and withdrew $10,000 each year, that I would have received $45,000 of interest on interest in 10 years. How can I receive interest on interest if the interest is withdrawn each year?

Complete nonsense.

After n-1 steps (n-1)*$10000 interest has been charged, so at the n'th step the interest on that interest is (n-1)*$1000. Summing that from 1 to n gives the answer I wrote.

[555]

@Abe and others.
Interest is calculated by multiplication, not addition.

[Jack]

Or the simpler case of a $100,000 loan at 10% interest and there is a payment of $10,000 each year, where is the interest on interest? Is that an illusion too?

Use the fact that the limit as r approaches 1 of [(r^n-1)/(r-1)] equals n.

So how much interest has been paid on interest? The answer is zero.

After n years the "interest paid on interest" is not zero, instead it is

{n choose 2}*$1000

After n-1 steps (n-1)*$10000 interest has been charged, so at the n'th step the interest on that interest is (n-1)*$1000. Summing that from 1 to n gives the answer I wrote.

Please tell us how many dollars of interest on interest one pays in the case above over 10 years in which not one dollar of interest is deposited in the account. Please tell us how many dollars of interest on interest are collected by the savings account in the case above over 10 years in which not one dollar of interest is ever deposited in the account. Dollars, please -- you know, a decimal number, not a formula.

[grunj]

In my opinion a mortgage is CLOSER to compound interest than simple interest when
viewed as a whole.

(I have lurked on this awesome website for 2 years and this is my first post

Firstly, the classical compound interest formula does not account for
payments or withdrawals (a mortgage obviously has payments). So, the best one can say
about a mortgage is that it is more like compound interest or more like
simple interest.

For me there are two things that convinced me that it is more compound-like
than simple-like.

1) Imagine an investor who has a choice of three investments

a) A simple interest investment ($400,000, 6% annual, 30 years)
b) A compound interest investment ($400,000, 6% annual, 30 years, compounded monthly)
c) A hybrid investment
Lend someone $400,000 via a 30 year, 6% annual, compounded monthly mortgage and
then (this is key) reinvest the monthly fixed payments you receive into
a compound interest interest investment ($400,000, 6%, compounded monthly).
The reinvestment is key, because you are not going to just leave your monthly
payments you receive just 'hanging about'.

Interestingly 'b' and 'c' both end at the EXACT same point, viz. approx $2,395,000! Additionally,
at every point the 'b' investment is higher than the 'c' investment
by EXACTLY the principal owed. In other words, in 'c' if the mortgage is payed off early,
the investment converts exactly to 'b'.

The graph below shows the picture.



2) Now what about the argument that you do not pay interest-on-interest in a mortgage?
Strictly speaking this is true. And by THAT definition one could argue that a mortgage
is not compound interest. However, consider the following graph




In this graph I have compared the outstanding balance on a 6% mortgage vs. a 0%
mortgage. At any given point in time the outstanding balance on the 6% line is
above that of the 0% line. Why is this? Why does the presence of interest cause
the principal in any given month to be above that in the absence of interest?

The reason is that the fixed monthly payment is "allocated" more to interest than
principal initially. So, while you are not strictly paying interest-on-interest, your principal
for your series of 360 (30 years) "simple interest loans" is higher in the presence
of interest. it turns out that the way this allocation is done causes the overall loan
to behave exactly like a "compound interest loan" in aggregate.

So, what would a "simple interest mortgage" look like? Obviously this is somewhat
speculative. But, in my opinion, for the loan
to behave in aggregate like a simple interest loan, the principal should be "flattened"
(made linear) to draw down like in the zero interest case. If the principal draws
in a linear manner AND interest is paid off every month, then it turns out that
there is no way to do this with a fixed monthly payment.

The graph below shows the monthly payments for such a loan.



Note that even though this "simple interest mortgage" starts at a higher
monthly payment, the total payment in aggregate is less ($761,000 vs.
$863,352).

So, in my opinion you are better of thinking of a mortgage as compound
interest than simple interest because you are effectively paying (in aggregate) like
in a compound interest situation which at the end of the day is
what matters. While the interest-on-interest argument is strictly speaking
accurate, I think it is misleading relative to the big picture.

[555]

Or the simpler case of a $100,000 loan at 10% interest and there is a payment of $10,000 each year, where is the interest on interest? Is that an illusion too?

Use the fact that the limit as r approaches 1 of [(r^n-1)/(r-1)] equals n.

So how much interest has been paid on interest? The answer is zero.

After n years the "interest paid on interest" is not zero, instead it is
{n choose 2}*$1000

After n-1 steps (n-1)*$10000 interest has been charged, so at the n'th step the interest on that interest is (n-1)*$1000. Summing that from 1 to n gives the answer I wrote.

Please tell us how many dollars of interest on interest one pays in the case above over 10 years in which not one dollar of interest is deposited in the account. Please tell us how many dollars of interest on interest are collected by the savings account in the case above over 10 years in which not one dollar of interest is ever deposited in the account. Dollars, please -- you know, a decimal number, not a formula.

After 10 time step (years in this case), the amount of interest is $100,000. (This includes interest on interest (on interest etc.).)
The amount of interest on interest is $45,000. (This includes interest on interest on interest (on interest etc.).)

Think of it this way.
Start with P_0=100000, then for each n=0,1,2,3,... let
P_{n+1}=P_n(1+0.1x)-10000.
For each k=0,1,2,3,..., call the coefficient of x^k "the balance of account k at time step n".
At each time step 10% of the balance of account k is added to account k+1, and then 10000 is subtracted from account 0.
At each time step the "total balance" is the total of all accounts 0 and higher,
the "interest" is the total of all accounts 1 and higher,
and the "interest on interest" is the total of all accounts 2 and higher.

So
P_0=100000 (initial balance)
P_1=90000+10000x (total balance is 100000, interest is 10000)
P_2=80000+19000x+1000x^2 (total balance is 100000, interest is 20000, interest on interest is 1000)
P_3=70000+27000x+2900x^2+100x^3 (total balance is 100000, interest is 30000, interest on interest is 3000, interest on interest on interest is 100)
P_4=60000+34000x+5600x^2+390x^3+10x^4 (total balance is 100000, interest is 40000, interest on interest is 6000, interest on interest on interest is 400, interest on interest on interest on interest is 10)
etc.