principal = 100000 # mortgage loan
interest_rate = 0.025 # for the loan
m_payment = 5000 # yearly amount to pay for the mortgage (interest and principal)
reference_rate = 0.028 # interest rate for the non-risky investment
lump_sum = 15000 # money that can be invested or used to pay off the loan
ls_year = 2 # year when the lump sum is invested or used as an extra payment for the loan
appreciation = 0.02 # value gain of the house per year
Warning
Disclaimer: This blog post is not a financial advice! This is a toy example. This blog post is full of unrealistic assumption. All numbers are made up. Reach out to a professional financial advisor you trust, if you need financial advise. And most important, take a spreadsheet and do the math yourself with your numbers and your assumptions.
A tricky question
Last weekend I had a discussion about repaying a mortgage early or instead investing the money in a non-risky asset.
👨: I just did an extra payment to reduce my mortgage.
👤: How much is your interest rate?
👨: 2.5% - again, numbers are made up
👤: But didn’t you consider to invest the money instead. Even if you are risk averse, you would get 2.8% in a fixed deposit?
👨: I thought about it. But it didn’t make sense.
👤: But why? You would have gained 0.3% per year.
Who is right in this scenario? The answer is ambiguous.
Making up some numbers
Before we take it apart, let’s make up some numbers:
Besides, we make some assumption for the ease of interpretation.
- No fees, no taxes.
- The yearly payment for the mortgage stayes the some.
- As soon as the mortgage is payed off, the yearly payment stays the same and will be invested.
- Interest rates for loan and investment are not changing during the entire time.
- The lump sum can either be used to invest or to make an extra payment to reduce the loan.
- The non-risky investment can’t be withdrawn before the house is payed of.
- Interest income is being reinvested (accumulating).
Important
These assumptions are unrealistic and incomplete for a real live scenario. Also, changing the numbers can lead to other conclusions than those described in this blog post.
The simple case
First, let’s have a look at the thoughts of 👤:
An extra payment towards the mortgage will reduces the amount I have to pay on interest every year by 2.50%. That’s a value gain of 2.50% each year.
If instead the money is invested in the fixed deposit with an interest rate of 2.80%, I had an value gain of 2.80% each year.
loan interest rate - reference rate = 2.50% - 2.80% = -0.30%
The investment has a yearly gain of 0.30% compared to the extra payment.
Both payments are compounding, as can be seen in the example:
| Payment | Interest savings through loan reduction | Gains from investment | Difference | |
|---|---|---|---|---|
| 0 | 0 | 0.0 | 0.0 | 0.0 |
| 1 | 0 | 0.0 | 0.0 | 0.0 |
| 2 | 15000 | 0.0 | 0.0 | 0.0 |
| 3 | 0 | 375.0 | 420.0 | 45.0 |
| 4 | 0 | 759.0 | 805.0 | 46.0 |
| 5 | 0 | 1153.0 | 1201.0 | 48.0 |
| 6 | 0 | 1557.0 | 1606.0 | 49.0 |
| 7 | 0 | 1971.0 | 2021.0 | 50.0 |
In this simple example we have a value gain of 50.0 € after the fifth year of investing the extra money compared to an extra payment for the loan.
Amortization plan
Now let’s get familiar with the numbers and draw the entire amortization plan without extra payment or investments.
| Interest | Principal | Loan Balance | |
|---|---|---|---|
| 0 | -100000 | ||
| 1 | -2500 | 2500 | -97500 |
| 2 | -2438 | 2562 | -94938 |
| 3 | -2373 | 2627 | -92311 |
| 4 | -2308 | 2692 | -89619 |
| 5 | -2240 | 2760 | -86859 |
| 6 | -2171 | 2829 | -84031 |
| 7 | -2101 | 2899 | -81131 |
| 8 | -2028 | 2972 | -78160 |
| 9 | -1954 | 3046 | -75114 |
| 10 | -1878 | 3122 | -71992 |
| 11 | -1800 | 3200 | -68791 |
| 12 | -1720 | 3280 | -65511 |
| 13 | -1638 | 3362 | -62149 |
| 14 | -1554 | 3446 | -58703 |
| 15 | -1468 | 3532 | -55170 |
| 16 | -1379 | 3621 | -51549 |
| 17 | -1289 | 3711 | -47838 |
| 18 | -1196 | 3804 | -44034 |
| 19 | -1101 | 3899 | -40135 |
| 20 | -1003 | 3997 | -36138 |
| 21 | -903 | 4097 | -32042 |
| 22 | -801 | 4199 | -27843 |
| 23 | -696 | 4304 | -23539 |
| 24 | -588 | 4412 | -19127 |
| 25 | -478 | 4522 | -14606 |
| 26 | -365 | 4635 | -9971 |
| 27 | -249 | 4751 | -5220 |
| 28 | -130 | 4870 | -350 |
| 29 | -9 | 350 | 0 |
| Total: | -40359 | 100000 | 0 |
With a yearly fixed payment of 5,000 € the mortgage of 100,000 € will be payed of after 29 years. The amount of interest paid for the loan is 40,359 €
The amortization schedule nicely shows how the interest decreases and principal increases over time.
Investment payment
Next, we extend the amortization plan with columns for
- Extra payment for the loan
- Payment on investment
- Compounding balance for investment
- Appreciation of the principal payed sofar
- The Net Worth (asset values - liabilities)
- The Net Worth including appreciation
In the first scenario we invest the lump sum.
| Mortgage | Investment | Appreciation | Net Worth | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Year | Interest | Principal | Extra Payment | Loan Balance | Payment | Balance | Balance | w/o appr. | with appr. |
| 0 | -100000 | 0 | 0 | ||||||
| 1 | -2500 | 2500 | 0 | -97500 | 0 | 0 | 50 | 2500 | 2550 |
| 2 | -2438 | 2562 | 0 | -94938 | 15000 | 15000 | 205 | 20062 | 20267 |
| 3 | -2373 | 2627 | 0 | -92311 | 0 | 15420 | 471 | 23109 | 23580 |
| 4 | -2308 | 2692 | 0 | -89619 | 0 | 15852 | 856 | 26233 | 27089 |
| 5 | -2240 | 2760 | 0 | -86859 | 0 | 16296 | 1368 | 29436 | 30804 |
| 6 | -2171 | 2829 | 0 | -84031 | 0 | 16752 | 2015 | 32721 | 34736 |
| 7 | -2101 | 2899 | 0 | -81131 | 0 | 17221 | 2805 | 36090 | 38895 |
| 8 | -2028 | 2972 | 0 | -78160 | 0 | 17703 | 3749 | 39543 | 43293 |
| 9 | -1954 | 3046 | 0 | -75114 | 0 | 18199 | 4855 | 43085 | 47940 |
| 10 | -1878 | 3122 | 0 | -71992 | 0 | 18708 | 6134 | 46717 | 52851 |
| 11 | -1800 | 3200 | 0 | -68791 | 0 | 19232 | 7595 | 50441 | 58036 |
| 12 | -1720 | 3280 | 0 | -65511 | 0 | 19771 | 9251 | 54260 | 63511 |
| 13 | -1638 | 3362 | 0 | -62149 | 0 | 20324 | 11113 | 58175 | 69289 |
| 14 | -1554 | 3446 | 0 | -58703 | 0 | 20893 | 13194 | 62191 | 75384 |
| 15 | -1468 | 3532 | 0 | -55170 | 0 | 21478 | 15505 | 66308 | 81813 |
| 16 | -1379 | 3621 | 0 | -51549 | 0 | 22080 | 18062 | 70530 | 88592 |
| 17 | -1289 | 3711 | 0 | -47838 | 0 | 22698 | 20877 | 74860 | 95737 |
| 18 | -1196 | 3804 | 0 | -44034 | 0 | 23334 | 23967 | 79299 | 103267 |
| 19 | -1101 | 3899 | 0 | -40135 | 0 | 23987 | 27347 | 83852 | 111199 |
| 20 | -1003 | 3997 | 0 | -36138 | 0 | 24659 | 31033 | 88520 | 119554 |
| 21 | -903 | 4097 | 0 | -32042 | 0 | 25349 | 35044 | 93307 | 128351 |
| 22 | -801 | 4199 | 0 | -27843 | 0 | 26059 | 39396 | 98216 | 137612 |
| 23 | -696 | 4304 | 0 | -23539 | 0 | 26788 | 44110 | 103249 | 147360 |
| 24 | -588 | 4412 | 0 | -19127 | 0 | 27538 | 49206 | 108411 | 157617 |
| 25 | -478 | 4522 | 0 | -14606 | 0 | 28310 | 54704 | 113704 | 168408 |
| 26 | -365 | 4635 | 0 | -9971 | 0 | 29102 | 60627 | 119131 | 179759 |
| 27 | -249 | 4751 | 0 | -5220 | 0 | 29917 | 66999 | 124697 | 191696 |
| 28 | -130 | 4870 | 0 | -350 | 0 | 30755 | 73843 | 130404 | 204247 |
| 29 | -9 | 350 | 0 | 0 | 4641 | 36257 | 77584 | 136257 | 213841 |
| Total: | -40359 | 100000 | 0 | 0 | 19641 | 36257 | 77584 | 136257 | 213841 |
The amortization duration is 29 years.
During the time when the loan is paid off, the investments compounds. In the same time the value of the house raises as shown in the appreciation column.
Extra loan payment
In the second scenario the lump sum is not invested but instead used as an extra payment to lower the loan.
| Mortgage | Investment | Appreciation | Net Worth | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Year | Interest | Principal | Extra Payment | Loan Balance | Payment | Balance | Balance | w/o appr. | with appr. |
| 0 | -100000 | 0 | 0 | ||||||
| 1 | -2500 | 2500 | 0 | -97500 | 0 | 0 | 50 | 2500 | 2550 |
| 2 | -2438 | 2562 | 15000 | -79938 | 0 | 0 | 811 | 20062 | 20873 |
| 3 | -1998 | 3002 | 0 | -76936 | 0 | 0 | 1412 | 23064 | 24476 |
| 4 | -1923 | 3077 | 0 | -73859 | 0 | 0 | 2155 | 26141 | 28295 |
| 5 | -1846 | 3154 | 0 | -70706 | 0 | 0 | 3049 | 29294 | 32343 |
| 6 | -1768 | 3232 | 0 | -67473 | 0 | 0 | 4104 | 32527 | 36630 |
| 7 | -1687 | 3313 | 0 | -64160 | 0 | 0 | 5329 | 35840 | 41169 |
| 8 | -1604 | 3396 | 0 | -60764 | 0 | 0 | 6735 | 39236 | 45971 |
| 9 | -1519 | 3481 | 0 | -57283 | 0 | 0 | 8334 | 42717 | 51050 |
| 10 | -1432 | 3568 | 0 | -53716 | 0 | 0 | 10136 | 46284 | 56421 |
| 11 | -1343 | 3657 | 0 | -50058 | 0 | 0 | 12155 | 49942 | 62096 |
| 12 | -1251 | 3749 | 0 | -46310 | 0 | 0 | 14402 | 53690 | 68092 |
| 13 | -1158 | 3842 | 0 | -42468 | 0 | 0 | 16892 | 57532 | 74424 |
| 14 | -1062 | 3938 | 0 | -38529 | 0 | 0 | 19639 | 61471 | 81109 |
| 15 | -963 | 4037 | 0 | -34493 | 0 | 0 | 22657 | 65507 | 88164 |
| 16 | -862 | 4138 | 0 | -30355 | 0 | 0 | 25963 | 69645 | 95608 |
| 17 | -759 | 4241 | 0 | -26114 | 0 | 0 | 29572 | 73886 | 103459 |
| 18 | -653 | 4347 | 0 | -21767 | 0 | 0 | 33503 | 78233 | 111737 |
| 19 | -544 | 4456 | 0 | -17311 | 0 | 0 | 37773 | 82689 | 120463 |
| 20 | -433 | 4567 | 0 | -12743 | 0 | 0 | 42402 | 87257 | 129659 |
| 21 | -319 | 4681 | 0 | -8062 | 0 | 0 | 47409 | 91938 | 139347 |
| 22 | -202 | 4798 | 0 | -3264 | 0 | 0 | 52816 | 96736 | 149552 |
| 23 | -82 | 3264 | 0 | 0 | 1655 | 1655 | 57690 | 101655 | 159345 |
| 24 | 0 | 0 | 0 | 0 | 5000 | 6701 | 60844 | 106701 | 167545 |
| 25 | 0 | 0 | 0 | 0 | 5000 | 11889 | 64061 | 111889 | 175949 |
| 26 | 0 | 0 | 0 | 0 | 5000 | 17222 | 67342 | 117222 | 184563 |
| 27 | 0 | 0 | 0 | 0 | 5000 | 22704 | 70689 | 122704 | 193393 |
| 28 | 0 | 0 | 0 | 0 | 5000 | 28340 | 74102 | 128340 | 202442 |
| 29 | 0 | 0 | 0 | 0 | 5000 | 34133 | 77584 | 134133 | 211718 |
| Total: | -28345 | 85000 | 15000 | 0 | 31655 | 34133 | 77584 | 134133 | 211718 |
The amortization duration is 23 years.
The amortization plan shows that the loan is payed back earlier. So that the entire rate could afterwards be invested.
If we compare the Total row values of both scenarios we see that the interest rates paid to the bank is less, when the extra payment is done. Nonetheless both Net Worth values are bigger in the first scenario, when the money is invested.
So, 👤 was right! But …
Comparing the strategies
Finally let’s compare the Net Worh values of both scenarios by calculating their difference (<>).
| Net Worth (w/o appreciation) | Net Worth (with appreciation) | |||||
|---|---|---|---|---|---|---|
| Year | Investment | <> | Extra Payment | Investment | <> | Extra Payment |
| 0 | 0 | 0 | 0 | |||
| 1 | 2500 | 0 | 2500 | 2550 | 0 | 2550 |
| 2 | 20062 | 0 | 20062 | 20267 | 606 | 20873 |
| 3 | 23109 | -45 | 23064 | 23580 | 896 | 24476 |
| 4 | 26233 | -92 | 26141 | 27089 | 1207 | 28295 |
| 5 | 29436 | -142 | 29294 | 30804 | 1539 | 32343 |
| 6 | 32721 | -195 | 32527 | 34736 | 1894 | 36630 |
| 7 | 36090 | -250 | 35840 | 38895 | 2274 | 41169 |
| 8 | 39543 | -308 | 39236 | 43293 | 2678 | 45971 |
| 9 | 43085 | -369 | 42717 | 47940 | 3110 | 51050 |
| 10 | 46717 | -432 | 46284 | 52851 | 3570 | 56421 |
| 11 | 50441 | -499 | 49942 | 58036 | 4060 | 62096 |
| 12 | 54260 | -569 | 53690 | 63511 | 4581 | 68092 |
| 13 | 58175 | -643 | 57532 | 69289 | 5136 | 74424 |
| 14 | 62191 | -720 | 61471 | 75384 | 5725 | 81109 |
| 15 | 66308 | -801 | 65507 | 81813 | 6351 | 88164 |
| 16 | 70530 | -885 | 69645 | 88592 | 7016 | 95608 |
| 17 | 74860 | -974 | 73886 | 95737 | 7721 | 103459 |
| 18 | 79299 | -1066 | 78233 | 103267 | 8470 | 111737 |
| 19 | 83852 | -1163 | 82689 | 111199 | 9264 | 120463 |
| 20 | 88520 | -1264 | 87257 | 119554 | 10105 | 129659 |
| 21 | 93307 | -1369 | 91938 | 128351 | 10996 | 139347 |
| 22 | 98216 | -1480 | 96736 | 137612 | 11940 | 149552 |
| 23 | 103249 | -1595 | 101655 | 147360 | 11985 | 159345 |
| 24 | 108411 | -1710 | 106701 | 157617 | 9928 | 167545 |
| 25 | 113704 | -1815 | 111889 | 168408 | 7541 | 175949 |
| 26 | 119131 | -1910 | 117222 | 179759 | 4805 | 184563 |
| 27 | 124697 | -1993 | 122704 | 191696 | 1697 | 193393 |
| 28 | 130404 | -2065 | 128340 | 204247 | -1805 | 202442 |
| 29 | 136257 | -2124 | 134133 | 213841 | -2124 | 211718 |
| Total: | 136257 | -2124 | 134133 | 213841 | -2124 | 211718 |
As seen before, the Net Worth of the investment scenario is higher at the very end. But what if we take appreciation into account and sold the house right after it’s been payed of or even before. In this case the scenario with the extra payment can be superior (depending of the value of the reference rate).
By the way, in case the house looses value (negative appreciation) the extra payment and selling the house early can have a very negative effect.
Conclusion
Even when the math seems easy it’s always a good idea to open a spreadsheet and plot some charts to see what’s going on and how different scenarios play out.
Ressources
Interactive notebook
- Play with the numbers:
- Downloading the notebook:
