III – Interest
The concept of interest should be relatively straightforward: it is simply the cost of borrowing money.
So for example, if you borrow $200,000 from a bank at an annual interest rate of 5% to be repaid at the end of 3 years, you may be inclined to calculate the interest as $30,000, calculated as follows:
$200,000 loan x 5% interest/year x 3 years = $30,000
In this example, you would repay $230,000 at the end of 3 years – the original $200,000 loan plus $30,000 in interest.
Actually, this is incorrect: the actual interest you would have to pay is $31,525! Therefore, you would actually have to pay to the bank $231,525 ($31,525 in interest plus the $200,000 loan).
What accounts for this difference? The first method of calculating interest is called the simple method. Banks and other lending institutions never calculate interest in this manner. Rather, they will calculate interest using the compounding method. Here is the calculation using this method:
Year 1: $200,000 loan x 5% interest = $10,000
Add $10,000 interest to original $200,000 loan = $210,000
Year 2: $210,000 loan balance x 5% interest = $10,500
Add $10,500 interest to beginning $210,000 balance = $220,500
Year 3: $220,500 loan balance x 5% interest = $11,025
Add $11,025 interest to beginning $220,500 balance = $231,525
As you can see, you borrowed $200,000 at 5% per year for 3 years, and you had to repay $231,525. Therefore, your cost of borrowing was $31,525. We can summarize this in the following table:
Beginning | Interest Cost |
Ending |
Balance | (beginning balance x 5%) | Balance |
$ 200,000 | $ 10,000 | $210,000 |
$ 210,000 | $ 10,500 | $220,500 |
$ 220,500 | $ 11,025 | $231,525 |
$ 31,525 |
It is extremely important that you understand the distinction between simple interest and compound interest. With reference to our example: in the former, interest was calculated every year only on the amount borrowed. In the latter, interest was calculated on the amount borrowed and interest calculated in the previous year.
You should now start to see that the cost of borrowing can end up being much higher than you were initially led to believe.
Opening Your Eyes
Now let’s extend our previous of example of calculating compound interest. Again, here are the facts:
Amount borrowed: $200,000 in January 2011
Interest rate: 5% per year
Length borrowed: 3 years, to be repaid end of December 2014
However, instead of calculating interest annually, the bank will calculate interest each month during the 3 year borrowing period. It will do this by multiplying the beginning loan balance by 0.4166% each month (5% divided by 12 months = 0.4166%).
For clarity: the annual interest rate is still 5%, but the interest will be calculated every month instead of every year. Here is the result in the following table:
Beginning | Interest Cost |
Ending | |
Month | Balance | (beginning balance x 0.4166%) |
Balance |
Jan-11 | $ 200,000 | $ 833 | $200,833 |
Feb-11 | $ 200,833 | $ 837 | $201,670 |
Mar-11 | $ 201,670 | $ 840 | $202,510 |
Apr-11 | $ 202,510 | $ 844 | $203,354 |
May-11 | $ 203,354 | $ 847 | $204,202 |
Jun-11 | $ 204,202 | $ 851 | $205,052 |
Jul-11 | $ 205,052 | $ 854 | $205,907 |
Aug-11 | $ 205,907 | $ 858 | $206,765 |
Sep-11 | $ 206,765 | $ 862 | $207,626 |
Oct-11 | $ 207,626 | $ 865 | $208,491 |
Nov-11 | $ 208,491 | $ 869 | $209,360 |
Dec-11 | $ 209,360 | $ 872 | $210,232 |
Jan-12 | $ 210,232 | $ 876 | $211,108 |
Feb-12 | $ 211,108 | $ 880 | $211,988 |
Mar-12 | $ 211,988 | $ 883 | $212,871 |
Apr-12 | $ 212,871 | $ 887 | $213,758 |
May-12 | $ 213,758 | $ 891 | $214,649 |
Jun-12 | $ 214,649 | $ 894 | $215,543 |
Jul-12 | $ 215,543 | $ 898 | $216,441 |
Aug-12 | $ 216,441 | $ 902 | $217,343 |
Sep-12 | $ 217,343 | $ 906 | $218,249 |
Oct-12 | $ 218,249 | $ 909 | $219,158 |
Nov-12 | $ 219,158 | $ 913 | $220,071 |
Dec-12 | $ 220,071 | $ 917 | $220,988 |
Jan-13 | $ 220,988 | $ 921 | $221,909 |
Feb-13 | $ 221,909 | $ 925 | $222,834 |
Mar-13 | $ 222,834 | $ 928 | $223,762 |
Apr-13 | $ 223,762 | $ 932 | $224,694 |
May-13 | $ 224,694 | $ 936 | $225,631 |
Jun-13 | $ 225,631 | $ 940 | $226,571 |
Jul-13 | $ 226,571 | $ 944 | $227,515 |
Aug-13 | $ 227,515 | $ 948 | $228,463 |
Sep-13 | $ 228,463 | $ 952 | $229,415 |
Oct-13 | $ 229,415 | $ 956 | $230,371 |
Nov-13 | $ 230,371 | $ 960 | $231,331 |
Dec-13 | $ 231,331 | $ 964 | $232,294 |
$ 32,294 |
As you may recall, when interest was calculated annually, the interest payable over 3 years was $31,525. But when interest is calculated monthly, the interest payable over 3 years is $32,294.
What accounts for this difference? Answer: the frequency in which interest is calculated during the borrowing period. Given an annual interest rate, the more frequently interest is calculated (or “compounded”) over the borrowing period, the higher the actual interest that has to be paid. Here’s a summary of what we’ve calculated so far:
Amount | “Nominal” | Compounding | Actual | “Real” | |
Borrowed | Rate | Method | Frequency | Interest | Rate |
$200,000 | 5% | Simple | N/A | $30,000 | 5.00% |
$200,000 | 5% | Compounding | Annually | $31,525 | 5.25% |
$200,000 | 5% | Compounding | Monthly | $32,294 | 5.38% |
As you can see, given: (1) a loan amount; and (2) a quoted annual interest rate (also referred to as the “nominal” interest rate), the actual interest you would pay on the loan will be a function of:
- The method of calculating interest (i.e., simple versus compounding); and
- The frequency of compounding during the loan period
So, as you can see from the table above, the “real” annual interest rate is often higher than the “nominal” (i.e., quoted) annual interest rate you see in advertisements for:
- credit cards – interest in compounded daily
- car loans – interest is compounded monthly
- mortgages – interest is compounded monthly
Takeaway
- Never finance anything with a credit card. Annual (i.e., nominal) interest rates for credit cards can be as high as 30% and the interest on outstanding balances is compounded daily, so the real interest rate you’re paying can be astronomical.
- Since interest on mortgages are compounded monthly, pay down your mortgage as quickly as you can. Compounding works both ways – the slower you pay down your mortgage, the more interest you end up paying whereas the more quickly you pay off your mortgage, the less interest you end up paying. Most banks will allow you to prepay 15 to 25 percent of the original mortgage balance once a year during the term of your mortgage.
© Copyright Fong and Partners Inc., 2011.