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Solving Problems with Non-Annual Periods on the HP 12C
Many, perhaps most, time value of money problems in the real world involve other than annual time periods. For example, most consumer loans (e.g., mortgages, car loans, credit cards, etc) require monthly payments. All of the examples in the previous pages have used annual time periods for simplicity. On this page, I’ll show you how easy it is to deal with non-annual problems.
General Considerations
The first thing to understand is that all of the principles that you have learned to apply for annual problems still apply for non-annual problems. In truth, nothing has changed at all. If you try to think in terms of “periods” rather than years, you will be ahead of the game. A period can be any amount of time. Most common would be daily, monthly, quarterly, semiannually, or annually. However, a time period could be any imaginable amount of time (e.g., seven weeks, hourly, three days).
The first, and most important, thing to think about when dealing with non-annual periods is the number of periods in a year. The reason that this is so important is because you must be consistent when entering data into the HP 12C. The numbers entered into the N, i and PMT keys must agree as to the length of the time periods being used. So, if you are working on a problem with monthly compounding, then N should be the total number of months, i should be the monthly interest rate, and PMT should be the monthly annuity payment.
Note
It is not uncommon for the payment frequency to differ from the interest compounding frequency. For example, you may make monthly payments while interest is compounded daily.
Ultimately, the payment frequency governs the length of the time period. So, the interest rate must be adjusted to match the payment frequency. While not difficult, this is beyond the scope of this page. Here we will assume that the payment and compounding frequency always match, which is typical of most textbook problems.
An Example
Very often in a problem, you are given annual numbers but then told that “payments are made on a monthly basis,” or that “interest is compounded daily.” In these cases, you must adjust the numbers given in the problem. Let’s look at an example:
You are considering the purchase of a new home for $250,000. Your banker has informed you that they are willing to offer you a 30-year, fixed rate loan at 7% with monthly payments. If you borrow the entire $250,000, what is the required monthly payment?
Notice that we are told that the loan term is 30 years and the interest rate is 7% per year (that is implied, not explicitly stated). So, you might be forgiven for expecting that a period is one year. However, on further reading you see that the payments must be made every month. Therefore, the length of a period is one month, and you must convert the variables to a monthly basis in order to get the correct answer.
Since there are 12 months in a year, we calculate the total number of periods by multiplying 30 years by 12 months per year. So, N is 360 months, not 30 years. Similarly, the interest rate is found by dividing the 7% annual rate by 12 to get 0.5833% per month. Note that we do not make any adjustments to the PV (\$250,000) because it occurs at a single point in time, not repeatedly. The same logic would apply if there was an FV in this problem. When you solve for the payment, the calculator will automatically give you the monthly (per period to be exact) payment amount.
In this problem, then, we would solve for the payment amount by entering 360 in N, 0.5833 into i, and 250,000 into PV. When you press PMT you will find that the monthly payment is \$1,663.26.
Note
One thing to be careful about is rounding. For example, when calculating the monthly interest rate, you should do the calculation in the calculator and then immediately press the i key, or save it to memory and recall it when needed. Do not do the calculation and then write down the answer for later entry. If you do, you will be truncating the interest rate to the number of decimal places that are shown on the screen, and your answer will suffer from the rounding. The difference may not be more than a few pennies, but every penny matters. Try sending your lender a payment that is consistently three cents less than required and see what happens. It probably won’t be long before you get a nasty letter.
Adjust First, Not After, Solving the Problem
You might be tempted to think that you could treat the problem as an annual one, and then adjust your answer to be monthly. Don’t do that! The math simply doesn’t work that way. To prove it, let’s input annual numbers, and then convert the annual payment to monthly by dividing by 12. Enter 30 into N, 7 into i, and 250,000 into PV. When you press PMT, you will find that the annual payment would be \$20,146.60. However, you have to make monthly payments so if we divide that by 12 we get a monthly payment of \$1,678.88.
Do you see the problem? If you do the problem this way, you get an answer that is \$15.63 too high every month. So, when you make the adjustments matters. Always adjust your variables before solving the problem. The reason for the difference is the compounding of interest. If you have read through my tutorial on the Mathematics of Time Value of Money, then you know that the more frequently interest is compounded, the smaller the payment has to be in order to grow to a particular future value.
I hope that you have found this tutorial to be helpful. If you have any questions or comments, please feel free to contact me.