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Interest and Loans

Nominal interest rate is the simplest type of interest rate as it is the monetary price that borrowers pay to lenders to use their money (for a single period). There is no defined period of time. So for example. If you agree to a loan where you borrow $100 USD and after 3 years you have to pay $110 and the debt is cleared, then the nominal interest rate of the loan was 10%. However, the interest rate charged per month can be referred to as the nominal interest rate (for that period), which is compounded over the life of the loan.

Annual percentage rate (APR) is the annual rate charged on a loan and expressed as a percentage that represents the yearly cost of funds over the term of a loan. This includes all applicable fees, but does not factor in compound interest. If you had a simple 1 year loan of $100 with no interest but a fee of $10, then the APR of that loan is 10%. However, that same loan of $100 with a monthly interest rate of 0.833333333% (which is 10% / 12 months). That loan would cost you $10.47, assuming you made no payments until the very end of the year when you paid the loan in full. Both loans have a 10% APR. If you think I made a mistake by simply dividing the APR by 12 to calculate the monthly interest rate then like me, you had thought that APR was what is actually EAR. This will become clearer as you read on.

Effective Interest Rate (EAR) takes compound interest into account after several nominal interest rates might have been applied. This is similar to the effective annual rate (EAR) except that this does not have a defined time period, whereas the EAR is for a year.

Savings accounts are advertised with interest rates in EAR whereas loans are advertised in APR, that way the numbers "look better" to the consumer but work slightly better in the banks favour. If you were to take a loan from a bank at 3% APR, and then put it into a savings account of 3% EAR, you would be losing money due to compound interest. Personally, I think both should be advertised in EAR.

Annual Percentage Yield (APY) is APR but takes compound interest into account. This is the figure you actually want to know but is rarely advertised. If you were to lend $100 to somebody who never made any payments, with an interest rate of 10% being applied per month (so 12 periods in a year), then the APY would be 213.843% but the advertised APR would only be 120%. They now owe you $313.84 which is the $213.84 in profit placed on top of the original loan of $100.

Effective Annual Rate (EAR) is the same thing as Annual Percentage Yield, but "on the other side of the table". If you are investing in loans, then you receive an annual percentage yield. If you are a borrower making payments, then you are paying the effective annual rate. Your EAR is someone else's APY.

If you want to convert an APR to an EAR/APY or vice-versa then use this site.

Calculations

In the sections below, we will show you how to use LibreCalc to calculate various financial problems. Most, if not all, of the equations will work the same in Excel.

Calculating Monthly Payment

Calculating a static value you would need to pay each month to clear a debt over a fixed time period, where compound interest applies could be difficult. Luckily for us, there is an in-built function (PMT) in LibreCalc that will take care of this for us. In this example, we will use a mortgage debt of £100,000 over 10 years with an APR of 3%. For this we would use the equation:

=PMT((0.03/12), 20*12, 100000 )

The first parameter in the PMT function is the rate, the amount of interest that is applied per period. For our example, this is the APR divided by 12 because the rate is applied monthly and there are 12 months in a year (don't forget APR doesn't take compound interest into account which simplifies things). The second parameter is the number of periods or payments that will need to be made. Since we are paying monthly over 20 years there are 20 x 12 payments to be made. Finally, the third value is the initial debt that was taken on at the start, which is the 100k we borrowed for our mortgage.

Alternatively, this mortgage calculator is pretty effective and can be applied to any kind of debt, not just mortgages.

Calculating AER/EAR From APR

You can convert from APR to AER with:

=POWER((1+($APR/$NUM_PERIODS)),$NUM_PERIODS)-1

If interest is applied monthly, $NUM_PERIODS needs to be 12. If daily, then 365.

This may be useful if you want to create a loan with an advertised APR and want to figure out what percentage profit you are going to make (assuming no payments are made).

Calculating Effective Interest Rate

If you have a savings account with an advertised EAR and want to figure out how much interest gets applied monthly/daily, then the equation you want is:

=POWER((1+($EAR)), (1/$NUM_PERIODS))-1

In this case, if your EAR was 3% then $EAR needs to be 0.03. If your savings account has the interest calculated monthly then $NUM_PERIODS neds a value of 12 for 12 months in a year. If interest is calculated daily, then $NUM_PERIODS needs to be 365 for 365 days in a year.