Chapter 2: Rate of Interest
Interest is payment for the use of borrowed money. The actual amount of money that is borrowed is called the principal amount. The percent of interest charged for the use of principal for one year is called the rate (or rate of interest). The period of time for which interest is paid is called the term of a loan. Interest is expressed as a percentage, the principal is considered the base amount of the loan and the rate percent = the rate of interest * the term of the loan.
Many different methods of computing interest have been developed. Simple interest, compound interest, and compound discount form the basis for most business and finance applications that involve the evaluation or computation of obligations. In calculating interest, the particular method of computation and the variables used in the computation must be understood. The interpretation of the specific terms, method of repayment, and rate will impact the loan.
Simple Interest
Simple interest is interest paid on the principal amount only. Simple interest is usually charged on loans that are extended for only a short period of time or on the balance of accounts that are expected to be paid within a short period of time. The term of a simple interest loan is usually a fractional part of the year. The calculation of simple interest involves some commonly used symbols as follows:
P = principal, expressed in dollars
r = annual rate, expressed as a percent
t = term, expressed in years or a fractional part of a year
I = total interest, expressed in dollars
S = amount or sum of the principal and interest, expressed in dollars
The computation for simple interest is, I = Prt. Some of the most common problems involving simple interest include finding the percentage, the principal, the rate, and the rate of interest. Formulas for solving such problems are as follows:
S = P + I
Since I = Prt, then
S = P + Prt = P (1 + rt)
and
P = S ÷ (1 + rt)
With these basic formulas, any type of simple interest problem can be solved.
Example:
The interest on $1,750 at 4(½)% for 132 days is calculated as follows:
P = $1,750
r = 4 (½)% = 4.5%
t = 132 days = 132 ÷ 365 = 0.36 yrs
I = Prt
I = $1,750 * 4.5% * 0.36 yrs
I = $28.35
The term of a loan is expressed in years or a fraction of a year, but not everyone interprets a year in the same manner. Some computations include a 360-day year and other computations include a 365-day year (or 366 day leap year). For most computations of interest for fractional parts of a year, the amount calculated will not vary significantly whether 12 months, 365 days, or 360 days are used as the basis for computation.
If a year is considered to have 365 days, the interest per day is less than the interest per day if a year is considered to have 360 days since 1/365 is less than 1/360. When interest is calculated based on a 360-day year, each month is considered to be 30 days or 1/12 of a year. The calculated interest is considered ordinary interest. When interest is calculated based on a 365-day (or 366-day leap year), the interest is considered exact interest or accurate interest since its calculation is based on the exact number of days in a year. Interest per day on an exact basis is always less than interest, applied at the same rate, on an ordinary basis. Another form of simple interest that is based on a 360-day year is called banker’s interest. In calculating simple interest, there must be some method of determining whether ordinary, exact, or banker’s methods are used. The specified rate should also be expressed as some multiple of 100.
Exact Interest and Exact Number of Days
It is customary to count the time between dates using the first or last day of the month or year, but not both. When days in a year are numbered from 1 to 365, the number of days between any given period of time is calculated as the difference between the number of the first date and the number of the last date. This difference is called the exact number of days. The numbered days in a 365-day calendar year can be found in a chart of numbered days, such as the one shown at Appendix 1. The time frame between January 1 and January 31 is actually 30 days, not 31 days, since 31 – 1 = 30. Likewise, the exact number of days between May 15 and November 4 is calculated as follows:
From the chart of numbered days, it is determined that
May 15 is the 135th day in a 365-day year and
November 4 is the 308th day in a 365-day year.
The difference:
308 days – 135 days
= 173 days, which is the same as above
The table showing numbered days in a year may be used to find the numbered day that corresponds to any particular date. However, if such a chart is not available, the exact time between dates can be calculated as follows:
|
Term: May 15 through November 4 |
|
|
Days remaining in May ( 31 – 15 = 16 ) |
16 |
|
Days in June |
30 |
|
Days in July |
31 |
|
Days in August |
31 |
|
Days in September |
30 |
|
Days in October |
31 |
|
Days in November |
4 |
|
Total days: |
173 |
The time interval between dates may span two different calendar years, even though the time frame is less than one year. To calculate the time between dates in different calendar years, find the days remaining in the first year and add it to the days in the second year. If a time frame is specified as November 4 of one year to May 15 of the next year, the number of days is calculated as follows:
November 4 is the 308th day of the year
Days remaining in the first year = 365 – 308 = 57
May 15 is the 135th day of the year
Days in the second year = 135
So, the total number of days in the period = days remaining in the first year + days in the second year
= 57 + 135
= 192 days
Ordinary Interest and the Approximate Number of Days
When each month is considered to have 30 days, the time between dates is calculated as an approximate number of days. This type of count is customarily used in the bond market in determining how long a bond is held and how much interest has accrued on such bonds. This type of ordinary interest may also be referred to as bond interest. Since, in actuality, only one month of the year has less than 30 days and seven months have 31 days, the exact number of days between any two dates is usually greater than the approximate number of days for that same time frame. As such, the calculation of exact number of days favors the party receiving interest payments since interest will be higher.
Banker’s Interest
Banker’s interest is a form of simple interest where the year is considered to have 360 days, but the actual number of days in a month is counted or exact. Banker’s interest does not differ from exact or ordinary interest when a full year is used to compute interest. Differences arise when fractional parts of a year are used.
Banker’s interest for one day is equal to 1/360 of a year’s interest whereas exact interest is equal to 1/365 of a year’s interest (1/366 for leap years). The relationship between banker’s interest and exact interest equates to the proportion:
(1 ÷ 365) ÷ (1 ÷ 360) = 360 ÷ 365 = 72/73
Therefore, exact interest is 72/73 of banker’s interest. Exact interest is equal to banker’s interest decreased by 1/73 of itself. Conversely, banker’s interest is 73/72 of exact interest. Banker’s interest is equal to exact interest increased by 1/72 of itself.
The calculation of banker’s interest equates to more than exact interest. When a lender draws up a contract, banker’s interest is used since it results in greater returns for the lender. The return may only amount to a few cents for an individual borrower, but lenders with thousands of accounts, worth millions of dollars, realize sizable advantages by using the banker’s interest method. Governments, which pay interest on billions of dollars for short periods of time, favor the exact method since it results in less interest payments. When a borrower is the party to draw up a contract, the borrower is most likely to use the exact interest method. As such, the exact interest method is customary for municipal securities, U.S. securities, and loans by the Federal Reserve Bank.
The 6% Method
The fact that 6 is a multiple of 360 and 360 is used in calculating ordinary and banker’s interests has led to the development of shortcut methods in calculating those interests. The ordinary and banker’s interest on $1,000 at 6% for 60 days is calculated as follows:
I = Prt
|
I |
= |
1000 |
* |
6 |
* |
60 |
|
100 |
360 |
|||||
|
= |
1000 |
|||||
|
100 |
||||||
|
= |
10 |
|
After the possible cancellations are performed, I is equal to 1/100 of the principal amount or $1,000 ÷ 100. As such, the 6% rule specifies that the interest on any amount at 6% for 60 days can be determined by moving the decimal point two places to the left of the principal.
Since 6 days is 1/10 of 60 days, it also follows that the interest for 6 days at 6% can be determined by moving the decimal point three places to the left of the principal. Further, if 6 is then divided into that amount, the resulting quotient equates to the interest at 6% for 1 day.
This method may be applied for other rates, such as 3% and 9%, to reduce the complexity in solving problems. These shortcut methods are intended to save time using the principles of fractions and cancellation. There are some circumstances, however, where these short cuts will not save much time. With the advent of calculators and computers, any method of calculating interest can be simplified by automation.
Simple Discount
In simple interest, the amount paid in the future is greater than the amount borrowed or invested. Discount is defined as the difference between the present value of a debt and the value of the debt at its maturity. As such, the future amount (sum S) is greater then the present value (principle P). Debts and their obligation are often expressed in terms of their future value.
Example:
The present value (PV) of what amount, invested at 5%, results in a future value (FV) of $5,000 in one year.
r = 5%
t = 1 year
S = $5,000
We seek to find the value of P.
|
P |
= |
S |
|
1 + rt |
||
|
= |
5000 |
|
|
1 + (5% * 1) |
||
|
= |
5000 |
|
|
1.05 |
= $4,761.90
A 5% return on $4,761.90 will result in $5,000 in one year. Likewise, the discount on $5,000 for one year is $5,000 – $4,761.90 = $238.10. A simple discount is the difference between the present value of a debt and the value of the debt at maturity. The simple interest on P is equal to the simple discount on S.
The simple interest on $4,761.90 at 5% is $238.10.
The simple discount on $5,000 at 5% is $238.10.
Interest Bearing and Non-Interest Bearing Debts
The present value (or principal) in simple interest assumes that debt does not bear interest. However, it is often necessary to determine the present value of an interest bearing debt. When the discount rate of a note equals the interest rate of the note, the present value is equal to the face amount of the note. However, the two rates are not usually the same. The present value of a debt usually involves one rate, and the debt bears interest at another rate. Thus, two mathematical problems are involved in calculating the present value of an interest bearing debt. The first mathematical problem involves computing the maturity value of the debt using the rate of interest on the note. The second mathematical problem involves computing the present value of the maturity value using a rate of discount. A rate of discount is expressed as either the discount rate or it is designated by the expression “money is worth.” The two problems may differ not only in the interest rates applied, but also in the number of periods of time. The steps to solving interest-bearing debt problems are as follows:
Step 1: Determine the date of maturity for the note.
Step 2: Compute the value of the debt at maturity.
Step 3: Determine the discount period.
Step 4: Compute the proceeds (or sum received).
|
P |
|
S |
|
Original value of debt (principal) |
Rate of note |
Maturity value of debt (sum) |
|
S |
|
P |
|
Maturity value of debt at the rate of note |
Rate money is worth |
Present value of debt at the rate money is worth |
Example:
A 90-day note is dated May 18 for $800 at 7%. Money is worth 7(½)%. The note is discounted on July 6. The proceeds are computed as follows:
Step 1:
The date of maturity is 90 days from May 18, which is day 138 of the calendar year. The calendar day corresponding to day 138 + 90 = day 228, which is August 16.
Step 2:
Calculate S = P (1 + rt) where:
P = $800
r = 7%
t = 90/365 = 0.25 days
S = P (1 + rt)
= $800 * [ 1 + (7% * 0.25)]
= $800 * [1 + (.07 *0.25)]
= $800 * (1 + 0.0175)
= $800 * 1.0175
= $814 (the debt at maturity)
Step 3:
The discount period is from July 6 to August 16. July 6 is day 187 of the calendar year and August 16 is day 228. So, day 228 – day 187 = 41 days.
Step 4:
Calculate P = S + (1 + rt), where:
S = $814
r = 7.5%
t = 41/365 = 0.1123287
|
P |
= |
S |
|
1 + rt |
||
|
= |
814 |
|
|
1 + (.075 * .1123287) |
= $814 ÷ (1 + .0084246)
= $814 ÷ (1.0084246)
= $807.20
Bank Discounts
Banks do not use true discount. Instead, they figure interest on the maturity value of a debt and deduct the interest from the maturity value of the debt. The amount that is deducted is called a discount. As an example, a bank discounts a $200, 90 day note at 6%. Simple interest is calculated as I = Prt = $200 * .06 * 90/365 = $3.00.
The amount received by the borrower is $200 – $3 = $197, which is the proceeds or bank proceeds. The $3 deduction from the value of the note is the discount or bank discount. It may be said that the bank discounted the borrower’s note or the borrower discounted a note at the bank.
The bank discount calculation is similar to the calculation for simple interest, with the principle replaced by the maturity value of the note and the other variables defined as follows:
S = maturity value of the note
t = number of years, or fractional part of a year between the date of discount and the maturity date of the note
d = the annual bank discount
D = the bank discount
Pb = bank proceeds
D = Sdt
The bank proceeds to the borrower are calculated as:
Pb = S – D
= S – Sdt
= S (1 – dt)
and
S = Pb ÷ (1 – dt )
Notes discounted by a bank may be either interest bearing or non-interest bearing. In the last example above, a non-interest bearing note was discounted. As such, the maturity value of that note was equivalent to the face value of the note. The period of discount was equal to the time of the note. The proceeds were equal to the difference between the maturity value of the note and the discount.
Example:
The bank discounts a customer’s note for 90 days at 6%. The bank proceeds to the customer are $2,800. The face value of the note is:
S = Pb ÷ (1 – dt)
= $2,800 ÷ (1 – .06 * .25)
= $2,842.64
Bank Discounts on Interest Bearing Notes
With deferred payments, a debtor may agree to pay interest on a loan amount with the terms of the loan specified in a promissory note. The promissory note will specify, among other things, the interest rate to be applied. The face value of the note is equal to the debt, but the value of the debt at maturity is greater than the debt. Rather than wait for the note to mature, a creditor may discount the note at a bank at a specified rate of interest. By acquiring a bank discount, the creditor receives proceeds immediately.
The bank discount of an interest-bearing note requires two computations: a simple interest computation and a bank discount computation. The simple interest computation is necessary to compute the maturity value of a note. This computation is not necessary for non-interest bearing notes. Bank discount computations use the same symbols that are used in simple interest computations, but a bank discount usually involves different values for time and rate than are specified in the original note.
Example:
The banks accepts a 120-day, 6%, $1,000 note from a customer. The customer discounts the note 90 days before the maturity date, and the bank discounts the note at 5%. The bank proceeds to the customer are:
The maturity value of the note is the face value plus simple interest.
S = $1,000 + $1,000 * .06 * 120/360
= $1,020
The bank discount on the maturity value is:
D = $1,020 * .05 * 90/360
= $12.75
The bank proceeds to the customer are:
Pb = $1,020 – $12.75
= $1,007.25
Bank discounts are usually computed using the exact number of days. In some instances, the first day of the discount is not included in the time of discount, but the last day of maturity is included. In other instances, both days are included.
Example:
The bank proceeds from the following notes are computed using the steps outlined in the preceding section, Interest Bearing and Non-Interest Bearing Debts.
|
Face Value of Note |
Date of Note |
Interest Rate of Note |
Term |
Date of Bank Discount |
Bank Discount Rate |
Bank Proceeds |
|
$2,500 |
4/16 |
4% |
60 days |
5/14 |
5% |
$2,505.48 |
|
Step 1: 106 + 60 = 166 |
||||||
|
Face Value of Note |
Date of Note |
Interest Rate of Note |
Term |
Date of Bank Discount |
Bank Discount Rate |
Bank Proceeds |
|
$1,800 |
5/14 |
5% |
90 days |
5/24 |
7% |
$1,794.15 |
|
Step 1: 134 + 90 = 224 |
||||||
Due Date
The average due date, called equated date, is the date on which a series of debts may be equitably discharged by a single payment. This payment must be equal to the face value of the sum of debts. Any past due debts may cause the average due date to be a date that has already passed. However, once the date is known, adjustments may be made from that passed date by adding interest to the amount due. If the payment is to be made earlier than the due date, adjustments are made to discount the amount.
In calculating the average due date, any convenient date may be used as the focal date. However, the earliest due date of all debts is often selected. Since due dates are under consideration, the date of purchase or the date on which an obligation incurred are not taken into consideration. A weighted dollar, known as a dollar day, is used for the calculation. Dollar days are based on the premise that, for any given interest rate, one dollar for one day is just as valuable as another dollar for one day. Debts are converted into dollar days. Payments are also converted into dollar days by multiplying the amount of each debt by the number of days from the due date or focal date. To find the number of days from the focal date, the difference between the dollar days of the debt and the dollar days of payments is divided by the net balance of the debt. The average due date is then calculated by adding or subtracting this calculated number of days from the focal date.
Example:
The following purchases were made on the specified date:
|
July 30 |
$750 |
|
August 15 |
$400 |
|
August 25 |
$300 |
A focal date of July 30 is chosen. The due date is established as July 30, the earliest of all of the purchase dates, and the payments are converted to dollar days.
|
Due Date |
Amount |
Days From Focal Date |
Dollar Days |
|
July 30 |
$750 |
0 |
0 |
|
August 15 |
$400 |
16 |
6,400 |
|
August 25 |
$300 |
26 |
7,800 |
|
Totals: |
$1,450 |
14,200 |
Due Date
= $14,200 ÷ $1,450
= 10
The average due date is 10 days beyond the focal date of July 30. Since July has 31 days, there is one day left in the month of July. So the due date is August 9. If payment is made before August 9, the amount may be discounted, and if it is made after August 9, interest may be applied to the amount.
Compound Interest
Compound interest is an important part of investment and business decisions. When money is borrowed for a short period of time, the lender anticipates receiving the amount of the loan plus interest at the maturity of the loan. At the time of payment, the lender has the option of re-lending, investing, or spending the principal amount, as well as any income received as interest on the principal. If the lender chooses to re-lend the sum of principal and interest for another period, the lender, in effect, receives interest on the original principle and its interest.
Example 1:
A $1,000 loan is made at 6% interest for 6 months, and the lender receives $1,030 when the loan is paid.
I = Prt, where P = $1,000, r = 6% = .06%, and t = 6 months = 6 ÷ 12 yrs = 0.5 yrs
I = $1,000 * 0.6 * 0.5 = $30.00
The sum: S = IP = $1,000 + $30
= $1,030
If the lender decides to re-lend the sum to another borrower at 6 % for 6 months, the lender will receive $1,060.90 when this new loan is repaid.
I = Prt, where P = $1,030, r = 6% = .06%, and t = 6 months = 6 ÷ 12 yrs = 0.5 yrs
I = $1,030 * 0.6 * 0.5 = $30.90
The sum: S = IP = $1,030 + $30.90
= $1,060.90
In the two 6 month periods, the lender receives $30 + $30.90 = $60.90 in interest. If, instead, the lender had loaned $1,000 at 6% for 1 year, he would receive only $60 in interest at maturity. The lender receives less return for the longer-term loan than the two short-term loans. To make the two loan scenarios equitable, the lender would need to make the long-term loan at a higher interest rate or require that interest be paid periodically. In our example, the lender would need to raise the interest rate on the longer term loan to 6.09% to receive the same $1,060.90 return that was received from the two short term loans or require payment at the end of the first 6-month period and then re-loan or invest the return at the same rate.
Compound interest is interest received on a principal amount, which is increased periodically by interest incurred during the period. Compound interest differs from simple interest since the principal does not remain constant, but varies during the term of a loan. Compound interest essentially allows interest to be earned on interest. Symbols used in calculating compound interest are as follows:
P = principal or present value. The initial amount lent, borrowed, or invested, expressed in dollars.
i = interest rate per period, expressed as a percent. This rate may be stated as an annual rate or the rate for any given period.
n = number of interest periods, expressed as an integer.
I = total interest, expressed in dollars.
S = compound amount or sum of the compounded principal and interest, expressed in dollars.
m = frequency of conversion, expressed as an integer.
j = nominal (annual) interest rate, expressed as a percent.
Compound vs. Simple Interest
When interest is paid or added to the principal once per year, the interest is considered compounded or converted annually. However, interest may also be converted at other regular periods, such as monthly, quarterly, or semiannually. The number of times for which interest is converted per year is called the frequency of conversion. The frequency of conversion is represented by the symbol m. If interest is converted quarterly, m = 4 and the conversion period is 4 months.
In simple interest calculations, the interest rate is represented by r and it always signifies an annual rate. When interest rates are calculated annually, the rate is referred to as a nominal interest rate (or nominal rate). In calculating compound interest, a rate per conversion period, also called a periodic rate, is used. The symbol used for periodic rate is i. Periodic rate is calculated by dividing the nominal rate by the frequency of conversion. Only when interest is compounded annually does the periodic rate equal the nominal rate. The nominal interest rate is often represented by the symbol j. Hence i = j ÷ m.
In compound interest, the actual rate of increase during the year is called the effective rate. If the frequency of conversion is greater than one, the effective rate of interest will always exceed the nominal rate of interest. If interest is converted annually, the effective rate and nominal rate are equal. When a frequency of conversion is not stated, it is assumed to be annual.
In simple interest calculations, time is represented by t and it signifies years or the fractional part of a year. Since interest is not always converted annually in compound interest, time is measured by the number of conversion periods in a year rather than years or fractional parts of years. In the computation of compound interest, the rate per period and number of periods must be known. The number of periods is found by multiplying the time in years by the frequency of conversion. Hence, n = t * m. If compound interest is used to compute the interest on $5,000 for 10 years at 6% interest converted quarterly, the value of n is 10 * 4 = 40, not 4. The frequency of conversion over the 10-year period is 4. The conversion period is 3 months, and the interest per period is 6% ÷ 4 = 1.68% (effective).
Determining the Compound Sum
In compound interest calculations, the sum of principal and interest, S =P(1 + i). As such, the sum at the end of the first period is defined as:
S = P(1 + i)
At the end of the second period, the amount is calculated as the sum at the end of the first period times (1+ i).
S2 = S + (1 + i)
S2 = P (1 + i) + (1 + i)
S2 = P (1 + i)2
At the end of the third quarter, the sum is calculated as:
S3 = S2 + (1 + i)
S3 = P (1 + i)2 + (1 + i)
S3 = P (1 + i)3
Since n represents the number of periods, the sum at the end of any period could be expressed as:
S = P (1 + i)n
Example 1:
If the principal P = $2,000, n = 5 periods, and i = 2(½)% = 2.5%, the accumulated sum is calculated as follows:
S = P (1 + i)n
S = $2,000 (1 + 2.5)5
S = $2,000 (1.025)5
=$2,262.82
Example 2:
The interest on $326.40 compounded semiannually at 6% for 10 years is calculated as follows:
The principal: P = $326.40
The number of periods: n = 2 * 10 = 20
The frequency of conversion: m = 2
The nominal rate of interest: j = 6 %
The periodic rate of interest: i = j ÷ m = 6% ÷ 2 = 3%
S = P (1 + i)n
= $326.40 * (1 + 3%)20
= $326.40 * (1 + .03)20
= $326.40 * (1.03)20
= $326.40 * 1.806111235
= $589.51
This amount includes both the original principal and the compounded interest over the 10-year time frame. In order to determine only the compound interest amount, the original principal amount must be subtracted from this sum. So, the compound interest amount is:
$589.51 – $326.40 = $263.11
Compound Interest Tables
Computations of compound interest may be simplified with the use of a compound interest table, such as that included in the table at Appendix 4, or some method of digital computation. Compound interest tables are constructed to provide a compound amount, given the rate per period and nominal interest rate. The table is constructed such that compound interest amounts fall under the column headings (1+i)n. The table is constructed as follows:
• Each section of the table provides data for various rates of interest, for example i = ¼%.
• The column labeled n lists the number of periods.
• The column labeled (1 + i)n, which is the formula for a compound sum with principal, P = zero, contains tabular data for the amount of one. The amount of one is defined as how $1 dollar will grow at compound interest.
The values in the table are shown to eight decimal places. If these values are rounded to include the same number of decimal places as there are significant places (dollars and cents) in the multiplier, answers correct to the nearest cent can be obtained for amounts up to $100,000.00. However, in all likelihood, a calculator will be used for these types of computations, and the results provided by such electronic devices are often sufficient for most applications, even though the result may or may not offer the same accuracy.
To find a compound amount, the section corresponding to the particular rate per period is located. The tabular value, which corresponds to the number of periods n, is then located. The tabular value is then multiplied by the principal amount P.
Example 1:
We compute the compound sum for the previous example using the compound interest table for i = 3%. We find the tabular value that corresponds to n = 20 is 1.8061112347. If we multiply this value by the principal P = $3,276.40, we have the sum of $589.51, as before.
Example 2:
An investor borrows $10,000 at 9% with interest to be compounded and paid monthly. The investor fails to make the monthly payments on interest. Instead, the principal and interest continue to be compounded monthly and must be paid at the end of the year. The amount due at the end of the year is calculated as follows:
P = $10,000
i = 9% ÷ 12 = 0.75%
n = 12
S = P (1 + i)n
= $10,000 * (1 + 0.75%)12
= $10,000 * (1 + .0075)12
= $10,000 * (1.0075)12
= $10,000 * 1.093806898
= $10, 938.07
Example 3:
We compute the compound sum for the previous example using the compound interest table for i = 3/4%. We find the tabular value that corresponds to n = 12 is 1.0938068977. If we multiply this value by the principal P = $10,000, we have the sum of $10,938.07, as before.
Determining the Number of Periods
In problems involving compound interest, it may be necessary to determine the periodic rate or number of periods rather than the amount of accumulated interest. Since S = P (1 + i)n, the value of n may be found using logarithms. Given that log (a * b) = log a + log b and the log ab = b log a, the number of periods is calculated as follows:
log S = log [ P (1 + i)n]
log S = log P + log (1 + i)n
log S = log P + [n * log (1 + i)]
and
|
n = |
log S – log P |
|
log (1 + i) |
A table of logarithms could be used to solve this equation, but that would require a very laborious process, which will not be discussed here. A simpler method of finding the solution is to make use of a scientific calculator.
Example:
The time required for any amount of money to double itself at 5% interest compounded annually is calculated as follows:
S = 2 * P
i = 5% per period
n =1 year
|
n = |
log S – log P |
|
log (1 + i) |
|
|
= |
log (2P) - log P |
|
log (1 + .05) |
|
|
= |
log 2 + log P – log P |
|
log (1.05) |
|
|
= |
log 2 |
|
log (1.05) |
= 0.301029995 ÷ 0.021189299
= 14.21026
= 14.21 years
which equates to 14 years and approximately 2(1/2) months
since 12 * .21 = 2.52
Determining the Periodic Rate
When making decisions regarding investment opportunities, most investors will consider the rate of interest to be paid or received. A decision may be made based on the compound amount as computed at the end of a period. This decision, however, is complicated when the sums, number of periods, and frequency of conversion differ among the investment opportunities. To simplify such comparisons, investors may change all rates under consideration to a comparable basis. This is achieved by computing an effective interest rate, which is a measure of the actual rate of increase during a one-year period.
If interest is converted more than once per year, an effective rate should be determined. The effective rate is determined by subtracting 1 from the sum (1 + i) for the number of periods specified in the frequency of conversion.
Example:
The effective rate that is equivalent to 4% converted quarterly is computed as follows:
The frequency of conversion: m = 4
The periodic interest: i = 4% ÷ 4 =1%
So,
(1 + i)n – 1
= (1 + 1%)4 – 1
= (1.01)4 – 1
= 1.040660401 – 1
= 0.0406
= 4.1%
Non-Integral Conversion Periods
Conversion periods in compound interest do not necessarily have to be integral values. Often times when interest is compounded, the conversion period may be specified as some number of years and/or months.
Example:
The amount on $3,265 compounded semiannually for 20 years and 4 months at 8% is computed as follows:
P = $3,265
r = 8% and i = 4% per period
n = 40(4/6) = 40(2/3)
S = P (1 + i)n
= $3,265 * (1 + 4%)122/3
= $3,265 * (1.04)40.67
= $3,265 * 4.928853494
= $16,092.70666
= $16,092.71
If the table of compound interest is used, the solution involves determining the compound interest for the integral number of periods and then calculating and adding a simple interest computation on this amount for the fractional part of the period.
Using the same example, the tabular value of compound interest for 40 periods is 4.80102063 and the sum = 4.80102063 * $3,265 = $15,675.33.
The simple interest on this amount for 4/12 years is:
I = Prt
= $15,675.33 * .08 * 1/3
= $418.0088
=$418.01
$15,675.33 + $418.01 = $16,093.34
This is the same as above with some error due to rounding.
Effect of Frequency Conversion on Effective Rates
In compound interest, a periodic rate is used as the basis for computations. The more frequent the number of conversions, the sooner interest is paid on accrued interest. As such, the effective rate of interest is increased as the frequency of conversion is increased. However, as the number of conversions is increased, only a progressively smaller increase in the effective rate is achieved.
Example:
A nominal rate of 6% compounded at the following frequencies produces the following effective rates:
|
6% Compounded |
Number of Conversions Per Year |
Effective Rate of Interest |
|
Annually |
1 |
6% |
|
Semiannually |
2 |
6.09% |
|
Quarterly |
4 |
6.13636% |
|
Monthly |
12 |
6.16778% |
|
Weekly |
52 |
6.17998% |
|
Daily |
365 |
6.18313% |
|
Continuously |
Infinity |
6.18365% |
These values indicate that little gain is achieved by increasing the frequency of conversion beyond one month. Though the concept of continuous compounding has no real mathematical application in finance, real estate, or investments, it is presented here for clarity to indicate the limit of annual growth. It should be noted that continuous compounding does have application in nature as a rate of growth.
Discounts in Compound Interest
Compound interest is used to determine the future value of a present sum. It also may be used to determine the present value of a future sum. Even though present value is thought to indicate the value now and future value is thought to indicate a value sometime in the future, mathematically, it does not matter whether present value is used to indicate now, some time in the past, or some time in the future. Likewise, it does not matter whether future value is used to indicate now, some time in the past, or some time in the future. It is relationship between any two chosen dates that is mathematically significant.
The difference between an amount S and principal P compounded at 9% for 5 years is the same regardless of whether either date is used to represent now, the past, or the future.
If S = P (1 + i)n
then
P = S ÷ (1 + i)n
and
P = S * (1 + i)-n
To find the present value of a future sum, the future value must be discounted. The difference between the future value and the present value is called the compound discount. The term (1 + i)n is called the discount factor. The formula for present value is derived from the formula for compound amount, with all symbols having the same meaning.
P = principle, which represents the value of an obligation at some date
S = sum, which represents the amount of that obligation, n periods in the future
|
P |
|
S |
|
(value now) |
n periods |
(value in n periods) |
The difference between P and S can be thought of as either
• The compound interest on P in n periods or
• The discount, at the compound interest rate, on S
The present value of a non-interest bearing note is found by computing the compound amount until maturity. However, the present value of an interest-bearing note requires a calculation of the compound amount and a calculation of the discount on the maturity value of the note.
Example 1:
A note for $3,000 is due in 4 years with 5% interest. The current interest rate charged on similar types of loans is 4%. The present value of the note is calculated as follows:
The value of the note at maturity is calculated as S = P (1 + i)n where:
P = $3,000 i = .05 n = 4
S = $3,000 (1 + .05)4
= 3000 * 1.21550625
= $3,646.52
The present value of $3,646.52 due in 4 years at 4% is calculated as P = S ÷ (1 + i)n where:
S = 3,646.52
i = .04
n = 4
P = S ÷ (1 + i)n
= 3,646.52 ÷ (1 + .04)4
= 3,646.52 ÷ 1.16985856
= $3,117.06
The present value of the note is $3,117.06.
The discount applied to the note is the difference between the value of the note at maturity and the present value of the note. The discount is $3,646.52 – $3,117.06 = $529.46.
Example 2:
The 2-year discount on $5,000 at 4% is determined for each of the following types of discount:
|
Simple Discount |
P = S ÷ (1 + rt) |
|
Compound Discount |
P = S ÷ (1 + i)n |
|
Bank Discount |
Pb = S * (1 – dt) |
|
Compound Discount Converted Semiannually |
P = S ÷ (1 + i)n |
Value of an Obligation
Many times it will be necessary to determine the value of an obligation either before or after a specified payment date. The value of a non-interest bearing amount of $1,000 compounded at 6% annually and due in the year 2010 may be represented as follows:
|
YEAR |
AMOUNT |
|
|
2000 |
$1,000 * (1 + 6%)-10 |
$558.39 |
|
2005 |
$1,000 * (1 + 6)-3 |
$-747.26 |
|
2009 |
$1,000 * (1 + 6%)-1 |
$943.40 |
|
2010 |
$1,000 |
$1,000 |
|
2011 |
$1,000 * (1 + 6%) |
$-1,060.00 |
|
2015 |
$1,000 * 1 + 6%)+5 |
$-1,338.23 |
|
2020 |
$1,000 * 1 + 6%)+10 |
$1,790.85 |
The table shows that $1,000 due in 2010 is worth only $558.39 ten years earlier and $1,790.85 ten years in the future.
To find the value of an amount before the due date, the discount factor, (1 + i)n, is used to discount the amount for the particular amount of time. To find the value of an amount after the due date, the compound interest must be determined.
Example:
An investor owes a debt of $5,000, which is due in 4 years. The investor expects to sell two homes, the first of which is expected to sell in 2 years with a profit of $2,500. This amount will be used to reduce the $5,000 debt. Money is worth 5%. The balance of the debt at the end of the fourth year is calculated as follows:
The value of $2,500 compounded at 5% for 2 years is:
S = $2,500 (1 + .05)2
= $2,500 * 1.1025
= $2,756.25
The original $5,000 debt will be reduced by $2,756.25, leaving a balance of $2,243.75 at the end of the fourth year.
Equation of Payment
Many times it is necessary to find one amount that is equivalent to two or more obligations. For example, two separate notes are owned. One note is due in two years and the other note is due in five years. Equation of payment is used to determine a sum of money that is equivalent to the two separate obligations that are due at different times. In compound interest, it may be necessary to commute one set of obligations into another set. In other words, it may be necessary to substitute one set of obligations to be paid in one manner for another set of obligations to be paid in a different manner. The old set of obligations is determined and a new set of obligations is established as equivalent to the old set of obligations on a given date. This date is the focal date in compound interest. If one set of obligations is equivalent to another set of obligations on a given date, the obligations remain equivalent on any other date. The focal date is usually chosen as the date on which payment is made, but may be any chosen date. The steps to solving equation of payment problems at compound interest are as follows:
Step 1: Select the date of the first unknown payment as the focal date.
Step 2: Compute the equivalent value of each of the original sets of obligations on the focal date.
Step 3: Compute the equivalent value of each new obligation on the focal date.
Step 4: Equate the sum of the two sets of equivalent values.
Step 5: Solve the algebraic equation for the value of the unknown payment or for each of the unknown payments.
Example 1:
We want to commute the debts of $500 due in 2 years and $1,000 due in 3 years to two equal payments due in 2 and 3 years, respectively. Money is worth 5%, annually.
Step 1: The focal point is chosen as 2 years.
Step 2: The original $500 debt is already established at the focal date.
The original $1,000 debt must be commuted to the focal date. The 3-year debt is discounted for one year:
S = $1,000 * (1 + .05)-1
= $1,000 * .952380952
= $952.3809524
Step 3: The new obligations require that x be paid in 2 years and x be paid in 3 years. The second payment of x must be commuted to the focal date. As such, the second payment of x must be discounted by 1 year.
Step 4: The sum of the new obligations on the focal date must be equal to the sum of the old obligations on the focal date. As such:
500 + $1,000 * (1+ .05)-1 = x + x (1+ .05)-1
500 + 952.3809524 = x + (x * 0.952380952)
1.952380952 * x = 1,452.380952
x = $743.90
Two payments of $743.90 will be made in years 2 and 3, respectively.
Example 2:
An investor has debt totaling $28,000 that is distributed as follows:
A $4,000 note at 5% that is 1 year past due.
A 3-year, $10,000 note at 5% that is due in 1 year.
A $14,000 non-interest bearing note that is due in 5 years.
The investor would like to commute these debts such that the $10,000 note is due in 2 years and the two other non-interest bearing notes for the same amount are due in 4 and 6 years. Money is worth 4(½)% per year. The face value of the two notes is calculated as follows:
Step 1: The focal point is chosen as 4 years.
Step 2: (a) The $4,000 note is past due by 1 year and the present value is:
S = P (1 + i)n
= $4,000 (1 + 5%)1
= $4,000 * 1.05
= $4,200
$4,200 commuted to the focal date of 4 years:
S = P (1 + i)n
= $4,200 (1 + 4.5%)4
= $4,200 * 1.192518601
= $5,008.578123
= $5,008.58
(b) The $10,000 note has been compounded for 3 years at 5(1/2)% and the present value is:
S = P (1 + i)n
= $10,000 (1 + 5.5%)3
= $10,000 * 1.17241375
= $11,742.41375
= $11,742.41
$11,742.41, commuted to the focal date, which is in 3 years:
S = P (1 + i)n
= $11,742.41 (1 + 4.5%)3
= $11,742.41 * 1.141166125
= $13,400.04052
= $13,400.04
(c) The $14,000 note is due in 5 years and is discounted for 1 year to commute it to the focal date:
S = P (1 + i)n
= $14,000 (1 + 4.5%)-1
= $14,000 * 0.956937799
= $13,397.12919
= $13,397.13
Step 3: The new obligations on the focal date are x to be paid in 4 years and x to be paid in 6 years.
(a) The new obligations require that the $1,000 note be due in 2 years:
S = P (1 + i)n
= $10,000 (1 + 4.5%)2
= $10,000 * 1.092025
= $10,920.25
(b) x is to be paid in 4 years
(c) x is to be paid in 6 years. As such, the second payment of x must be discounted by 2 years:
x (1 + 4.5%)-2
= x * 0.915729951
Step 4: The sum of the new obligations on the focal date must be equal to the sum of the old obligations on the focal date. As such:
$4,200 (1 + 4.5%)4 + $11,742.41 (1 + 4.5%)3 + $14,000 (1 + 4.5%)-1 = $10,000 (1 + 4.5%)2+ x + x (1 + 4.5%)-2
$5,008.58 + $13,400.04 + $13,397.13 = $10,920.25 + x + 0.915729951x
$20,885.5 = 1.915729951x
x = 10,902.11
Making a payment of $10,000 in 2 years, a payment of $10,902.11 in 4 years, and a final payment of $10,902.11 in 6 years will pay the three original obligations.