The definitions provided here reflect usage within the TValue programs. Interest rate and related terminologies may vary. Our experience with a wide variety of users over the years has led us to this usage. Be alert to possible differences between TValue and your own requirements.
INTEREST COMPUTATION METHODS
Under the actuarial method, unpaid interest becomes part of the principal at the end of each compounding period. With monthly compounding, if the interest for a given month is $100 and only $75 is paid at the end of that month, then the $25 unpaid interest is added to the balance. Interest for the next month will be greater by the amount of interest on the additional $25. This compounding of interest makes sense since the borrower, in effect, receives an additional loan of $25 for a month. The resulting increase in principal due to unpaid interest is referred to as negative amortization.
Some states have laws that forbid charging interest on interest. So, in situations where payments are not large enough to cover current interest, use of the actuarial method may not be legal and it may be necessary to use some other methodologies, such as U.S. Rule.
U.S. Rule (Simple Interest)
Where negative amortization occurs, the U.S. Rule method can be used in place of the actuarial method to avoid charging interest on interest. The U.S. Rule approach, in effect, establishes an account for interest accrued but not paid. No interest is charged on the balance of this account. Payments are applied first to interest. If the payment is greater than the interest due, principal is reduced. If not, the interest account is increased.
Payment schedules on such loans are designed so that the balance in this account is fully paid off over the term of the loan. If you have specified U.S. Rule, TValue will solve for the appropriate payment. In cases where there is negative amortization, this payment will be lower than with the actuarial method. If there is no negative amortization, U.S. Rule and the actuarial method will produce identical results.
The term “simple interest” is often used interchangeably with U.S. Rule. With U.S. Rule, payments are usually applied first to interest, then principal. With TValue, you can apply payments to interest or principal first.
Canadian law and practice regarding interest computations and disclosures differ from that of the United States in at least two respects.
(1) In the United States, the nominal annual rate is pivotal and is used for truth-in-lending disclosures. In Canada, the effective annual rate plays the more important role and is normally a required disclosure.
(2) Canadian mortgages quote interest as if it were compounded semiannually or annually, even though payments may occur more frequently, such as every month. Interest is allocated monthly based on an equivalent monthly rate.
When Canadian is chosen as the Compute method in TValue, the Compound Period field on the main screen will be changed to Computation Interval or Interval. The interval is used to establish the relationship between the nominal annual rate, the effective annual rate, and the periodic rate. The interval usually matches the payment period. For example, a typical Canadian mortgage would show a semiannual Canadian basis with a monthly interval. Interest is compounded at the end of each interval at a rate that is equivalent to the effective annual rate.
When Canadian is the Compute method in TValue, you can select how to treat interest for odd days. The odd days setting in the Compute Setup dialog box controls the calculation of interest for periods less than one compounding period in length. It can be either straight-line or compounded.
For straight-line odd days interest, the interest factor is (n/365) * nom
where n is the number of odd days and nom is the nominal annual rate.
For compounded odd days interest, the interest factor is (l + eff)n/365
where eff is the effective annual rate. This factor is the odd days equivalent to the effective annual rate. Compounded odd days interest, though not always used, is required in Canadian practice.
The term “daily rate” as used in TValue is, under Canadian practice, correct only when odd days compounded interest is used.
Rule of 78
The Rule of 78 is a method of allocating pre-computed interest to time periods. The total interest on a loan is pre-computed. This total interest is considered earned in proportion to the periodic time balances. The Rule of 78 provides a straight-line approximation of compound interest. This approximation is close where the term is short and the interest rates are low but becomes increasingly biased in favor of early recognition of income as the interest rate and/or the term of the loan increases. The largest difference occurs about 1/3 of the way through a loan.
The Rule of 78 was designed solely for loans that are paid off with a series of equal payments equally spaced. For a loan payable in 12 equal installments, interest is allocated as follows: 12/78 of the interest is considered earned in the first month, 11/78 in the second, 10/78 in the third, and so on.
TValue also provides an extension of the Rule of 78 for cases where payments don’t fit the regular pattern just described. In these cases, TValue automatically adjusts the interest allocation in proportion to the balances outstanding and the time that particular balances are outstanding. For example, if a loan is payable in 12 equal monthly installments but payments start two months from the loan date, 24/90 of the interest would be recognized with the first installment, 11/90 in the second, 10/90 in the third, etc. Accountants may recognize this as equivalent to the bonds outstanding method of recognizing interest on serial bonds.
CALCULATING ACCRUED INTEREST
In TValue, amortization schedules are built around event dates (e.g., payment dates). On an event date, an amortization schedule shows the interest that has accrued as of that date. Accrued interest does not display on the schedule if no activity takes place. Since event dates do not always take place regularly or on financial statement dates, interest accruals may not always appear on the dates you need. For those instances, we offer this suggestion:
This section explains how the balance command calculates its results. There are two basic approaches to calculating a balance between payment dates. We will call these the proration approach and the stub period approach. To illustrate the differences between these two approaches, assume the following: Payments on the 15th of each month; Balance after March 15 payment is $10,000; Interest included in April 15 payment is $100; Balance is needed as of March 25 (10 days after the 15th).
The proration approach starts with the balance at the last payment before the payoff date and adds a prorated portion of interest from the next payment. The balance using the proration approach would be calculated as follows: $10,000 + (10/31) x $100 = $10,032.26.
The stub-period approach computes the interest for the stub-period based on the preceding balance, the current interest rate, and the current compounding period. In the above example, the interest rate is 1 percent per month. With monthly compounding, this means an annual rate of 12 percent and a daily rate of .12/365 = 0.03288 percent. The balance using the stub-period approach would be calculated as follows: $10,000 + $10,000 x (10 x 0.0003288) = $10,032.88. The stub-period approach is equivalent to removing all payments after the balance date and solving for a single unknown payment at that date.
The compound period determines how often interest is computed. The compound periods TValue uses in calculations range from Continuous to Annual.
Continuous compounding is a variation of daily compounding. It differs in that instead of compounding each day, it compounds an infinite number of times throughout the day.
A daily compound period calculates interest for each day by multiplying the balance by the daily rate. This interest is then added to the balance and compounded each day. The daily rate is determined by dividing the nominal annual rate by the year length (e.g., 360, 364, or 365). With daily, exact days, and continuous compounding, interest amounts don’t always decrease each month during a standard loan. This is because, for example, there may be two or three more days’ interest in an April 15 monthly payment than in a March 15 monthly payment.
Interest for each period is calculated at the daily rate times the number of days in a period. If payments are monthly, the calculation is based on the number of days in each month. For example, the interest charge for March (31 days), will be greater than the interest charge for April (30 days).
The week-based periods, weekly, biweekly, and 4-week are quite straightforward since all weeks are the same length. They function in much the same way as month-based periods. The periodic rate is determined by dividing the nominal annual rate by the number of periods in a year.
|Compound Period||Period Length||Compounding Occurs|
|Weekly||1/52 of year||52 times/year (every week)|
|Biweekly||1/26 of year||26 times/year (every other week)|
|4-week||1/13 of year||13 times/year (every fourth week)|
The only unusual thing about half-month compounding is, given a starting date, deciding when the other payments fall. Here’s how TValue does it:
|First Payment||Next Payment|
|15th of month||Last day of month|
|Last day of month||15th of month|
|1st through 14th||Add 15 days|
|16th through 30th||Subtract 15 days (not end of month)|
If a date would fall on February 29 in a non-leap year, it gets dropped back to February 28. The description above will not produce February 30.
With monthly compounding, each month is treated as 1/12 of a year. The periodic rate is determined by dividing the nominal annual rate by 12 (the number of periods in a year).
Examples of how months are counted in TValue are described below. Keep in mind that all methods of evenly dividing a year into 12 convenient units have problems.
Month-based periods include 2-month, quarterly, 4-month, semiannual, and annual. These all function in essentially the same way as monthly periods. The periodic rate is determined by dividing the nominal annual rate by the number of periods in a year.
The periodic interest rate is used for each period, although actual period lengths vary slightly. For example, a quarter can range in actual length from 89 to 92 days, but the interest rate for a quarter is not affected by this difference if compounding is quarterly. Each quarter is treated as 1/4 of a year.
|Compound Period||Period Length||Compounding Occurs|
|2-month||1/6 of year||6 times/year|
|Quarterly||1/4 of year||4 times/year|
|4-month||1/3 of year||3 times/year|
|Semiannual||1/2 of year||2 times/year|
|Annual||1 year||1 time/year|
Computing Interest on Odd Days
When irregular payments occur, the days that are not part of a complete compound period must be accounted for. These odd days are collectively referred to as a stub period. TValue recognizes the length of stub periods in addition to the compound periods present in a calculation.
TValue computes interest on odd days similar to how it computes interest using exact days compounding. Interest for each stub period is calculated at the daily rate times the exact number of days in the period.
Interest for a stub period of five days earning 10 percent annual interest on a $20,000 balance using a 365-day year is computed as follows:
$20,000 x .10 / 365 x 5 = $27.40
Daily, exact days, and continuous compounding, do not produce odd days.
Add-on interest is a form of pre-computed interest. The lender starts with a so-called annual rate and divides this by 12 to get a monthly rate. This rate is applied to the original amount advanced for each month in the term of the loan without any reduction for monthly payments. For example, if $1,000 is advanced for 36 months at an add-on rate of 12 percent, the total interest charge is $1,000 x .01 x 36 = $360. This works out to a nominal annual rate under the actuarial method of 21.2 percent. The 21.2 percent rate is the rate that would be disclosed under federal truth-in-lending rules. TValue does not calculate add-on interest.
Amount of a loan, minus points, prepaid interest, and other finance charges entered in the loan detail window.
Annual Percentage Rate (APR)
The annual percentage rate is defined by the Federal Reserve Board in its Regulation Z. It is the interest rate that every lender is required to reveal in its statement to the borrower. Its purpose is to provide comparable interest rate information for all possible consumer financing arrangements. It works much like a computation of the lender’s internal rate of return on a loan except that it reports the nominal annual rate instead of the effective annual rate. If a loan has regular payments with the same compound period, and no points or fees, the APR will usually be the same as the nominal annual rate. The two rates tend to differ from each other when there are points and fees, stub periods, day-based compounding, or U.S. Rule amortization. For more information, refer to Appendix J of Regulation Z. Regulation Z attempts to cut through the terminology and deal with actual cash flows. Appendix J to Regulation Z provides a wide variety of examples of consumer loans. TValue reproduces these official results exactly in almost all of the official examples and within established tolerances for the rest.
Annual Percentage Yield (APY)
The annual percentage yield (APY) measures the total amount of interest paid on an account based on the interest rate and the frequency of compounding. The APY is expressed as an annualized rate based on a 365-day year. Appendix A of Regulation DD states that, in determining the total interest figures in most cases, institutions must assume that all principal and interest remain on deposit for the entire term. Also, that no other transactions occur during the term. For time accounts, institutions must base the number of days on either the actual number of days during the applicable period, or the number of days that would occur for any actual sequence of that many calendar months.
A series of equal payments at equal intervals. The modifiers commonly used with the term “annuity” indicate the value of the annuity at a particular date relative to the cash flows. For example, the “present value of an annuity” refers to the value of an annuity at a date one payment period before the date of the first payment. The “present value of an annuity due” refers to the annuity value at the date of the first payment. TValue handles these relationships automatically based on the dates you enter.
A payment added at the end to pay off a loan or withdraw a remaining balance. If the balloon option is selected from the Rounding dialog box that appears after computing or checking rounding, TValue creates a balloon payment one payment period after the last payment shown in the cash flow data. If no payment period has been established, the balloon payment is placed one compounding period after the last payment. From the Balance dialog box, if the balloon option is selected, TValue creates a final payment on the date shown in the Balance dialog box.
Cash Flows Cash flows can consist of loans, payments, deposits, withdrawals, etc. Cash flows are represented as lines in the cash flow data section of the TValue document.
Cash flows can consist of loans, payments, deposits, withdrawals, etc. Cash flows are represented as lines in the cash flow data section of the TValue document.
A method of computing interest charges in which interest is computed at regular intervals or at payment dates. Interest so computed is added to principal, and payments are deducted in full from the principal amount. When interest is computed and added to principal, unless a simultaneous payment completely covers the interest amount, the principal balance will increase. This is referred to as “negative amortization,” and the lender in such situations is often said to be charging “interest on interest,” which may not be acceptable in some states.
The period (from one day to one year) after which interest is computed and added to principal. This is done for normal (actuarial) and Rule of 78 interest. Unpaid interest under U.S. Rule does not compound but is computed at each payment date and added to a special non-interest-bearing account.
The nominal annual rate divided by the year length in days. This length is 365, 364, or 360, depending on the mode selected using the Compute menu. TValue does not offer a 366-day year.
A loan to be repaid in installments, in which each installment is treated as a separate loan and at its due date is repaid at its face amount plus simple interest on that installment only.
As a simple example, assume that $200 is lent on a direct-reduction loan today, to be repaid in two equal annual installments of $100 principal plus interest at 10 percent. The repayment at the end of the first year will be $110 ($100 principal plus $10 interest), and the repayment at the end of the second year will be $120 ($100 principal plus $20 interest for the two-year period).
In contrast, if a borrower repays $110 at the end of the first year on a $200 U.S. Rule loan (also at 10 percent interest), $121 will be due at the end of the second year. The reason is that under U.S. Rule, payment of all interest to date (in this case, $20) is due when the first payment is made. At the end of the first year only $90 is applied to principal. The extra $1 of interest under U.S. Rule for the second year comes from calculating interest on a principal balance that is $10 larger.
Effective Annual Rate
Calculated as (l + i)m - 1, where i is the periodic rate and m is the number of compounding periods per year. The effective annual rate would show, for example, the cost of borrowing one dollar for one year, where compounding is more frequent than annual. This rate is often referred to as the “yield.”
The date of the last event on a cash flow line. If the number on the cash flow line is 1, the end date is the same as the start date and therefore will not be displayed. If the number of events on the cash flow line is greater than one, the end date indicates when the last event in the series occurs. For example, if 360 mortgage payments are entered on a cash flow line, the end date will display the date of the 360th payment.
The balance on a given date in the future after a number of deposits have been made. It is often the value of the last cash flow in a problem. In TValue, if you are thinking in terms of future value, you will probably be using deposit and withdrawal events, and the future value will be the value of the final withdrawal.
Negative amortization occurs when the interest due at a payment date is more than the amount of the payment. Under normal (actuarial) amortization, the excess interest is added to principal and is subject to interest in succeeding periods. Under U.S. Rule amortization, the excess interest is held in a separate non-interest-bearing account. In either case, the overall balance due increases.
Nominal Annual Rate
The nominal annual rate is the rate typically quoted when talking about interest rates. If compounding is annual, the nominal annual rate is equal to the effective annual rate, which is also equal to the rate per compounding period. If compounding is monthly, the nominal annual rate will be equal to 12 times the periodic rate.
The number of days before the normal starting date for loan payments. For example, monthly payments on a loan taken out on January 10 would normally start on February 10—one payment period after the loan date. If payments are to start on February 15, there are five odd days from January 10 to January 15.
This option in the Rounding dialog box leaves the left over amount as a balance on the amortization schedule. This allows you to run an amortization schedule without forcing a zero balance at the end. This is useful when you are recording payments on an ongoing basis or when you need to show the remaining balance on a note. You cannot do APR calculations if you select this option.
If you re-calculate after selecting open balance, you will need to choose Open Balance again to keep the remaining balance open.
The time period between successive payments specified on a single cash flow line. The payment period is usually the same as the compounding period. The payment period must be equal to or greater than the compound period.
The periodic rate (rate per compounding period) is the rate that is multiplied times the balance at the beginning of a compounding period to find the interest for the period. This rate times the number of compounding periods per year gives the nominal annual rate. For example, if compounding is monthly and the periodic rate is 1 percent, the nominal annual rate is 12 percent.
An up-front finance charge computed as a percentage of the loan amount. A “point” is one percent of the loan amount. For example, if a $100,000 note is negotiated with “two points,” the lender will charge $2,000 as a finance charge to be paid at settlement.
The practice of charging prepaid interest seems to result from a desire to simplify calculations. The odd interest amount is calculated for the period of time between when escrow closes and when the first interest period begins. This odd amount is prepaid, so the first regularly scheduled payment includes interest for only one period and is always charged to the first of the month.
The value of the first cash flow in a problem. In loan or lease situations, the present value would normally be the first loan event. In investment analysis situations, the present value would normally be the first deposit event. We say “normally” because it is quite possible to do loan or lease problems using deposit/withdrawal events and to do investment analysis using loan/ payment events.
An interest computation in which interest is equal to the principal balance times the interest rate per period times the number of periods. Under simple interest, interest is not charged on interest. You can do simple interest calculations with TValue by setting the compute method to U.S. Rule. The program allows you to apply payments to interest first or principal first when using U.S. Rule.
The total of all payments after the inception of the loan. Total payments include finance charges after the start of the loan, but do not include points, prepaid interest, and other finance charges deducted from the loan amount to calculate the Amount Financed.
The exact definition varies with the type of calculation, but it is generally calculated as an annualized rate of return. When using TValue for yield calculations, the yield is generally represented by the nominal annual rate. In some cases, the effective annual rate deals more realistically with interest returned from the investment. It shows a real difference between an investment with monthly compounding and an otherwise identical investment with annual compounding.
Example: Adams compares investments A and B. Each call for an investment of $1,000 today. Investment A returns $10 interest at the end of each month plus $1,000 at the end of the year. Investment B has no monthly returns but returns $1,120 at the end of the year. Both investments have a nominal annual rate of 12 percent, but investors would prefer A to B. Why? Because the returns from A can normally be reinvested at some rate. If we use the nominal annual rate when evaluating investment A, we are assuming a zero reinvestment rate. If Adams can reinvest the monthly returns from investment A under the same (or nearly the same) terms, then the effective annual rate is the more realistic measure.
You can convert from the effective annual rate to the nominal annual rate using the following formula:
NAR=[(1 + EAR)^(1/12) - 1] * 12,
where NAR is the nominal annual rate and EAR is the effective annual rate, based on monthly compounding. For example, if the effective annual rate is 0.12682503 (that's 12.682503 percent), then the nominal annual rate is 0.12 or 12 percent.