SIMPLE INTEREST Some of the oldest documents in existence – clay tablets dating back almost 5000 years - show the calculation of interest charges. Interest, a fee for borrowing money, is about as old as civilization itself. The largest and financially most secure companies, such as IBM and AT & T, borrow at or near the most favourable interest rate for short-term loans, known as the prime rate. The prime rate is an important factor in determining the rates of interest paid to depositors on savings and the rates fluctuates widely, although it typically remains between 6% and 15%. Most short-term interest rates move up and down with the prime rate. For example, when the prime rate moves up, it will likely cost you more to finance a car. Long-term rates such as on home loans also move up and down but these rates differ from short term rates. A good understanding of interest is important since interest charges can represent a significant cost for both individuals and firms. Two basic types of interest are in common use today: simple interest and compound interest. Simple interest is interest paid on only the principal. Compound interest is interest paid on both principal and past interest.
Basics of Simple Interest 1. Find simple interest 2. Find interest for less than a year 3. Find principal if given rate and time 4. Find rate if given principal and time 5. Find time if given principal and rate. Interest is the price paid for borrowing money. Interest rates are usually expressed as a percent of the amount borrowed. For example, in some states, retail companies such as Sears, The Home Depot and J.C. Penney charge up to 21% interest per year on money owed to them. The amount borrowed is called the principal. The percent of interest charged is called the rate or rate of interest. The nu8mber of years or the fraction of a year for which the loan is made is called the time. Objective Find Simple Interest simple interest is interest charged on the entire principal for the entire length of the loan and is usually used for short-term loans that last less than a year. It is found
using a modification of the basic percent formula. The simple interest I, on a principal of P dollars, at a rate of interest R percent per year, for T years is given as follows: I = P . R . T = PRT The rate R, is expressed as a decimal or fraction and time T is expressed as the number of years, or the fraction of a year. Note: It is important to remember that time is in years. This means that a time period given in months or days must be converted to a fraction of a year before being substituted into the formula for T. Example finding simple interest In addition to using some of their own money, Gilbert Construction Company must borrow $60,000 to build an 1800-square-foot home. The owner, Susan Gilbert, is considering whether she should borrow the funds at (a) 8% per year for 1 year or (b) 8½% per year for 1½ years. Find the simple interest on both loans. Solution (a) Use the formula I = PRT. Substitute $60,000 for P, 0.08 (the decimal form of 8%) for R, and 1 for T. I = PRT = $60,000 X 0.08 X 1 = $4800
Simple Interest
Check the answer by dividing $4800 by $60,000 to find .08 or 8% interest. (b) Again use the simple interest formula. However, now use R = 0.085 (8½%) and T = 1.5 (1½ years). I = PRT =$60,000 x 0.085 x 1.5 = $7650 Simple Interest Gilbert chooses the 1-year loan since she believes that she can build and sell the home within 1 year and she wishes to keep interest costs low. Find interest for less than a year Notice in part (a) of the next example how 9 months is written as 9/12 of a year and in part (b) that 13 months (obviously more than 1 year) is written as 13/12 of a year. Also notice that
sometimes it is necessary to round interest to the nearest cent, as shown next in part (b)of example 2.
Finding Simple Interest Using Months Jodie Luk needs to borrow $2800 and her uncle offered her 8% simple interest for a period of (a) 9 months or for (b)13 months. Find the interest for both. Solution (a) Since there are 12 months in a year, 9 months is 9/12 or 0.75 of a year. Find the interest as follows: I = PRT = $2800 X 0.08 X 9/12 = $168 (b) Use 13/12 to represent 13 months in terms of number of years. I = PRT = $2800 X 0.08 X 13/12 = $242.67
rounded
Scientific calculator approach The calculator solution to part (a) of example 2 follows. 2800 x .08 x 9 12 = 168
Find Principal if given Rate and Time Sometimes the amount of interest is known, but the principal, rate, or time must be found. Do this with the following modifications of the formula for simple interest: I = PRT =
Divide both sides by RT. = P or P =
Similarly, divid8000ing both sides of I = PRT by PT gives the formula for R.
R= And dividing both sides of I = PRT by PR gives the formula for T. T= Note: You do not have to remember all of the formulas above. Just remember I = PRT and use algebra to solve for the unknown (Section 2.3). Do not forget that the rate must be a decimal or fraction and time must be in years. Finding the Principal Gilbert Construction Company borrows funds at 10.5% for 10 months to build a home. Find the principal that results in interest of $8000. Solution Find the principal by dividing both sides of I = PRT by RT. P= =
Substitute values for variables.
= $91,428.57 rounded The principal or loan amount is $91,428.57. Scientific calculator approach For the calculator solution to example 3, divide using the chain calculation. 8000 + (.105 x 10 ÷ 12 ) = 91428.57
Find Rate if given Principal and Time The next example shows how to find the interest rate if the principal and time are given. Finding the Rate After a large down payment, Jessica Warren borrowed $9000 from her credit union to purchase a previously owned Toyota Camry. Find the interest rate if the loan was for 9 months and the interest was $540. Solution
Divide both sides of I = PRT by PT to get the following: R= =
Substitute values for variables
= 0.08 The rate was 8%. Note: In order to avoid rounding errors, it is important not to round off any calculations until the very end when solving these types of problems. Find Time if given Principal and Rate The simple interest formula can also be used to find the time of a loan. Finding the Time Eric Thomas, a loan officer at Midwest Bank made a loan of $4800 at 10%. How many months will it take to produce $280 in interest? Solution Use I = PRT and divide both sides by PR to get T (in years) = We included “(in years)” to emphasize this important point – time of the loan is measured in years. Now substitute $280 for I, $4800v for P and 0.10 for R. T (in years) =
substitute values for variables.
T (in years) = Reduced to lowest terms T (in years) =
=
year = 7 months
Note: In all the examples of this section, time has been expressed in years or months. The next section will deal with interest problems in which the time is expressed in days. Simple Interest for a given Number of Days
Objectives 1. Find the number of days from one date to another using a table. 2. Find the number of days from one date to another using the actual number of days 3. Find exact and ordinary interest. The previous section showed how to find simple interest for loans of a given number of months or years. In this section, loans for a given number of days are discussed. In business, it is common for loans to be for a given number of days, such as “due in 90 days,” or else to be due at some fixed date in the future, such as “due on April 17”. You will find this topic useful if you own your own business or if you are involved in the finances of a business. 1.
Find the number of Days from One Date to Another Using a Table. There are two ways to find the number of days from one date to another. One way is by the use of Table 11.1 on page 411. This table assigns a different number to each day of the year. For example, the number for June 11 is found by locating 11 at the left, and June across the top. You should find that June 11 is day 162. Also, December 29 is day 363. The number of days from June 11 to December 29 is found by subtracting. December 29 is day 363 June 11 is day - 162 subtract 201
There are 201 days from June 11 to December 29. (Throughout this book, ignore leap years unless otherwise stated). Note When counting the number of days of a loan, do not count the day the loan is made, but do count its due date. The number of each of the days of the year (add 1 to each date after February 29 for a leap year) Day of month 1 2 3 4 5 6 7 8 9
Jan
Feb
March April May June July Aug Sept Oct
Nov Dec
1 2 3 4 5 6 7 8 9
32 33 34 35 36 37 38 39 40
60 61 62 63 64 65 66 67 68
305 306 307 308 309 310 311 312 313
91 92 93 94 95 96 97 98 99
121 122 123 124 125 126 127 128 129
152 153 154 155 156 157 158 159 160
182 183 184 185 186 187 188 189 190
213 214 215 216 217 218 219 220 221
244 245 246 247 248 249 250 251 252
274 275 276 277 278 279 280 281 282
335 336 337 338 339 340 341 342 343
Day of month 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59
69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90
100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120
130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151
161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181
191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212
222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243
253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273
283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304
314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334
344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
Example : Finding the Number of Days Find the number of days from (a) March 24 to July 22 and (b) November 8 to February 17 of the next year. Solution (a) The number of days can be estimated. There are about 4 months from March 24 to July 22. Assume 30 days per month and multiply to find about 120 days. Now to find the exact number of days, note that March 24 is day 83 and July 22 is day 203. July 22 is day March 24 is day
203 - 83 120
There are 120 days from March 24 to July 22. It turns out that our estimate was the exact number of days. Estimates rarely provide the exact answer, but they can help minimize errors by providing approximate values. (b) Since November 8 is in one year and February 17 is in the next year, first find the number of days from November 8 to the end of the year. Last day of the year is number 365
November 8 is day
- 312 53
There are 53 days from November 8 to the end of the year. Then find the number of days from the beginning of the next year to February 17. According to the chart, February 17 is the 48th day of the year. The total number of days is found as follows: November 8 to end of year January 1 to February 17
53 + 48 101 There are 101 days from November 8 to February 17 of the next year. 2. Find the number of Days from One Date to Another Using the Actual Number of Days. The number of days from one date to another can also be found using the number of days in each month of the year as shown in the Table below:
Table: THE NUMBER OF DAYS IN EACH MONTH 31 days 30 Days 28 Days January August April February March October June (29 days in leap year) May December September July November Finding the number of Days from One Date to Another Using Actual Days Find the number of days from (a) March 12 to June 7 and (b) November 4 to February 21. Solution (a) Since March has 31 days, there are 31 – 12 = 19 days left in March, then 30 days in April 31 in May and an additional 7 days in June for a total as follows: 19 Remaining in March 30 April 31 May + 7 June 87 Days from March 12 to June 7 (b) Add the days in each of the months. 26 31
Remaining in November December
31 January + 21 February 109 There are 109 days from November 4 to February 21. Two other ways of remembering the number of days in each month are the rhyme method and the knuckle method, as seen below: Rhyme Method: 30 days hath September Dec April, June and November. All the rest have 31, except February, which has 28 and in a leap year 29.
Jan Mar May July 31 days Aug Oct Feb April June 30 days Sept (28 in Feb)
Nov
Finding Specific Dates Find the date that is 90 days from (a) March 25 and (b) November 7.
Solution (a) From table 11.1, March 25 is day 84. Add 90. March 25 is day 84 + 90 days 174 As shown in table 11.1 day 174 is June 23, so 90 days from March 25 is June 23. Alternatively, work as follows: March 25 to end of month 6 April 30 May + 31 67 Since 90 – 67 = 23, an additional 23 days in June are needed, giving June 23. (b) November 7 is day 311. Add 90 days to get the following: 311 + 90
401 Since there are only 365 days in a year, subtract 365. 401 - 365 36 Day of the following year. Day 36 of the following year is February 5, so that 90 days from November 7 is February 5 of the following year. Find Exact and Ordinary Interest In the formula for simple interest, time is always measured in years or parts of years. In the examples of the previous section, time was in months, with T in the formula I = PRT written as T= Things are not so simple when the loan is given in days. There are two common methods for calculating simple interest for a given number of days: exact interest and ordinary (or banker’s interest). In the formula I = PRT, the fraction for time is found as follows: Exact interest
T=
Ordinary or banker’s interest
T=
Government agencies and the Federal Reserve Bank use exact interest as do many credit unions and banks. However, there are still many banks that use ordinary interest for loans. Exact interest is used for savings accounts. Ordinary interest may have been used originally because it was easier to calculate than exact interest. With the modern use of calculators and computers, however, ordinary interest is probably used today out of tradition and because it produces a greater dollars amount of interest than does exact interest, as shown in the next example. Finding Exact and Ordinary Interest Mohawk Radio Shop borrowed $17,650 on May 12. The loan, at an interest rate of 12%, is due on August 27. Find the interest on the loan using (a) exact interest and (b) ordinary interest. Solution Using the table or calculating the number of days in each month, there are 107 days from May 12 to August 27. (a) The exact interest is found from I = PRT with P = $17,650, R = 0.12 and T = (Remember to use 365 as the denominator with exact interest.)
.
I = PRT = $17,650 X 0.12 X = $620.89 rounded (b) Find ordinary interest with the same formula and values, except that T =
.
I = PRT I = $17,650 x 0.12 x I = $629.52 rounded In this example, the ordinary interest is $629.52 - $620.89 = $8.63 more than the exact interest. Note: If P and R are the same, more interest is generated using ordinary interest than using exact interest. The formula from Section 11.1 and 11.2 are repeated here for your convenience. Interest
I = PRT
Principal
P=
Rate Time T (in years) Time T (in months)
T is in years.
R = T= T=
X 12 T is a fraction of a year.
Time T (in days)
T=
X 360 Use 360 days for banker’s (ordinary) interest.
Time T (in days)
T=
X 365 Use 365 days for exact interest.
Note: Throughout the balance of this book, assume ordinary (or banker’s) interest unless stated otherwise. Maturity Value Objectives 1. Find maturity value. 2. Find principal if given maturity value, time and rate. 3. Find rate if given principal, maturity value and time. 4. Find time if given maturity value, principal and rate. Suppose you borrow $8000 at 11% interest for 9 months. The interest you owe on the loan is calculated as follows:
I = PRT = $8000 X 0.11 X = $660 Objective Find Maturity Value The total amount that must be repaid in 9 months is the sum of the principal and the interest. Principal + Interest = $8000 + $660 = $8660 This amount, $8660, is called the maturity value or future value of the loan. The date the loan is paid off is the maturity date. The principal or present value is the amount received by the borrower today. The formula for maturity value is as follows. A loan with a principal of P dollars and interest of I dollars has a maturity value M given as follows: M=P+I Note: The maturity value of a loan always exceeds the principal (the original loan amount) since maturity value is principal plus interest. Example : Finding Interest and Maturity Value Jim Wilcox would like to remodel his small bookstore so that he can serve customers coffee and allow them to sit and browse. To remodel the store, he borrows $7200 for 21 months at 9.25% interest. Find the interest due on the loan and the maturity value. Solution Interest due is found using I = PRT, where T is in years (21 months = years). I = PRT = $7200 x 0.0925 x
= $1165.50
Find the maturity value using M = P + I, where P = $7200 and I = $1165.50. M=P+I = $7200 + $1165.50 = $8365.50 The formula for maturity value, M = P + I, can be written in a different way if I is replaced with PRT (Since I = PRT). M=P+I = P + PRT Substitute PRT for I. = P(1 + RT) Use the distributive property. Therefore, the maturity value M, of a principal of P dollars, at a rate of interest R, for T years can be written in either of the following forms. M = P + I or M = P(1 + RT)
Note Do not round off values too soon when doing interest rate problems. Round after finding the final value. Example Finding Maturity Value Use the formula M = P(1 + RT) to find the maturity value for a loan of $6000 for 120 days at 9% interest. Solution Substitute $6000 for P, 9% or 0.09 for R and The maturity value is as follows: M = P(1 + RT) = $6000 X [ (
for T (since 120 days is
of a year).
)]
Note Parentheses were placed around 0.09 X as a first step.
to emphasize that these numbers are multiplied
After multiplying 0.09 and , add 1. M = $6000 X (1 + 0.03) = $6000 X 1.03 = $6180 The interest can be found by subtracting the principal from the maturity value. I = $6180 - $6000 = $180 Objective Find Principal if given Maturity Value, Time and Rate. Sometimes the maturity value is given and either the principal, rate or time must be found. For example, given the maturity value, rate and time find principal as follows: M = P(1 +RT)
(
)
(
)
= =
( ( ( (
) ) ) )
Divide both sides by (1 + RT). Divide out common factors.
= P or P = Principal is also called the present value of the loan.
Example Finding Principal given Time, Rate and Maturity Value Find the principal that would produce a maturity value of $15,300 in 4 months at 6% interest. Solution Use the formula above and substitute $15,300 for M, 0.06 for R, and
for T.
P = =
(
)
As shown by the parentheses, first multiply 0.06 and
, and then add 1.
P= = = $15,000 The principal is $15,000 and the interest is $15,300 - $15,000 = $300. Objective Find Rate if given Principal, Maturity Value and Time. If principal, maturity value and time are given, rate can be found as follows:
M P 1 RT M P PRT Use the distributive property. M P PRT Subtract P from both sides. M P PRT Divide both sides by PT. PT PT M P M P R or R= PT PT Finding Rate given Principal, Maturity Value and Time Lin Pao invests a principal of $7200 and receives a maturity value of $7540 in 200 days. Find the interest rate. Solution Use the formula above and substitute values.
R
M P PT
$7540 $7200 200 $7200 x 360
=
= 0.085
or
8.5%
Notice that maturity value minus principal (M - P) is interest I. Therefore, M P I PT PT and Example 4 can be solved using this formula. R
Interest = $7540 - $7200 = $340 R
I PT
$340 200 $7200 x 360 = 0.085 or
8.5%
Objective 4 Find Time if given Maturity Value, Principal and Rate. Given maturity value, principal and rate, time can be found as follows:
M P 1 RT Use the distributive property. M P PRT Subtract P from both sides. M P PRT M P PRT Divide both sides by PR. PR PR M P M P T (in years) or T= PR PR This gives a value for time T in years. To convert to days multiply by 360. For example, ½ of a year is ½ x 360 = 180 days.
M P x360 PR Since I=M – P, this is the same as I T (in days) = x360 PR
T (in days) =
Example : Finding the Time in Days City Lights Inc. borrowed $18,250 at 10 % interest for the construction of new signs and agreed to repay $19,687.19. Find the time in days and round to the nearest day if necessary. Solution Use the formula above and substitute values. T
M P x360 PR
$19, 687.19 $18, 250 x360 $18, 250 x0.10125
1437.19 x360 1847.81 280 days Rounded
The formulas from Section 11.3 are repeated here for your convenience. Interest
I PRT M P(1 RT )
Maturity Value
Principal
P
I M P M RT RT 1 RT
Rate
R
I M P PT PT
Time (in years)
T
I M P PR PR
Time (in days)
T
I M P x360 x360 PR PR
COMPOUND INTEREST Simple Interest is paid on the Principal – not on any past interest. However, bank deposits a nd many other investments commonly earn compound interest. Compound interest is calculated on any interest previously credited (paid) to the account in addition to the original principal. This chapter is about compound interest, which can have a significant effect when applied over long periods of time. For example, assume $1 was invested in an account paying 3% compounded annually in 1492, the year Christopher Columbus arrived in the Americas. Figure 13.1 shows that the $1 investment would have grown to over $4,000,000 by 2007. If your parents invested $1 for you when you were born, it would grow to $79.06 by your 75th birthday assuming 6% per year compounded annually. Or, it would grow to $1271.90 by your 75th birthday assuming a growth rate of 10% per year compounded annually. How can compound interest help you meet your personal financial goals? DIAGRAM
Note: Every wealthy individual has used the concept of compounding. OBJECTIVES 1. 2. 3. 4. 5.
Find compound interest and compound amount; Determine the number of periods and rate per period; Find values in the interest table. Use the formula for compound interest to find the compound amount; Find the effective rate of interest.
1. Find compound interest and compound amount Compound interest includes interest on principal and also interest on interest already paid. Assume that $1000 is deposited into a mutual fund paying 8%per year. Use the formula for simple interest at the end of each year for 3 years to find the compound amount in 3 years. Interest at end of year 1 = $1000 X 0.08 X 1 = $80 Now add the interest to the principal to get the new principal at the beginning of year 2. Principal at end of year 1 = $1000 + $80 = $1080
This principal plus interest will earn interest for year 2. Do these same calculations for years two and three. Interest at end of year 2 = $1080 X 0.08 X 1 = $86.40 Principal at end of year 2 = $1080 + $86.40 = $1166.40 Interest at end of year 3 = $1166.40 X 0.08 X 1 = $93.31 Principal at end of year 3 = $1166.40 + $93.31 = $1259.71 The final amount of $1259.71 is the compound amount or future amount. The interest earned during the three years is found as follows: Interest = Compound amount – Original principal Interest = $1259.71 - $1000 = $259.71 Simple interest on $1000 for 3 years at 8% would be: $1000 X 0.08 X 3 = $240. So, compound interest resulted in an extra $19.71 ($259.71 - $240) interest in 3 years. Figure 13.2 shows the advantage of compound interest over simple interest for a $1000 investment as the number of years increase. Note The compound amount is also referred to as maturity value or as future value. The future amount using compound interest can be found by multiplying the principal times the quantity (1 + Rate) once for each period the investment is to be compounded. Compound Interest versus simple Interest
Diagram Rate is the annual interest rate. Therefore, the future amount of $1000 invested at 8% compounded annually for three years is shown as follows: Future value
$1000 x(1 0.08) x(1 0.08) x(1 0.08)
$1000x1.08x1.08x1.08 $1259.71 Note that 1.08 x 1.08 x 1.08 can be written as (1.08)3 or 1.083, using an exponent for the number of times 1.08 is multiplied times itself. Thus, the future amount of $1000 invested at 8% compounded annually for three years can also be shown as follows:
Future value = $1000 x 1.083 = $1259.71 (rounded) The number 3 is the exponent and is the number of years this investment is compounded. The future value of an investment of $1000 Invested at 10%$2000 at 8.5% for several different annual periods is shown.
Number of years 5 10 30
Future value at 8.5% $2000 x 1.0855 = $3007.31 $2000 x 1.08510 = $4521.97 $2000 x 1.08530 = $23,116.50
Example: Finding Compound Interest In 4 years, Tony and Lynn Jameson will need $8000 for a down payment on a house. They have $5500 that they invest in an account earning 5% per year compounded annually. Find (a) the future amount in 4 years, (b) the amount of compound interest earned, (c) the excess of compound interest over simple interest for the four years and (d) the additional amount needed to achieve their goal. Solution (a) Future amount = $5500 X 1.054 = $6685.28 Rounded (b) Compound interest = Future amount – Original investment = $6685.28 - $5500 = $1185.28 (c) Simple Interest = $5500 X 0.05 X 4 = $1100 Compound interest – Simple interest = $1185.28 - $1100 = $85.28 (d) Amount still needed = $8000 - $6685.28 = $1314.72
2. Determine the Number of Periods and Rate per Period Interest is often credited to an account more than once a year when calculating compound interest. The interest rate used to find the amount of interest credited at the end of each compounding period is the nominal or stated annual rate divided by the number of compounding periods in one year.
Compounding
Interest Credited at the
Number of Times interest is credited
Number of Times Interest would
Rate per Compounding Period If R is
Period Annual Semiannual Quarterly Monthly
End of Each
per year
Year 6 months Quarter Month
1 2 4 12
be Credited over 5 years 5X1=5 5 X 2 = 10 5 X 4 = 20 5 X 12 = 60
Rate per Year R
Example: Finding Number of Periods and Rate per Period (a) A bank pays interest of 8%, compounded semiannually. This means that semiannually, or twice a year, interest of 8% ÷ 2 = 4% is added to all money that has been on deposit for 6 months or more. (b) An interest rate of 12% per year, compounded quarterly, means that every 3 months (quarterly), interest of 12% ÷ 4 = 3% is added to all money that has been on deposit for at least a quarter. Note: In Example 2(b) the period of compounding is semiannual (every 6 months), while it is quarterly (every 3 months) in Example 2(b). The formula for compound interest is often written using algebraic notation. If P dollars are deposited at a rate of interest I per period for n periods, then the compound amount and interest earned are found as follows: Compound amount = M = P(1 + i)n Interest earned = I = M – P Note: it is important to keep in mind that I is the interest rate per compounding period and not per year and n is the total number of compounding periods not the number of years. Example: Finding Compound Interest A Canadian invests $2500 in an account paying 6% compounded semiannually for 5 years. (a) Estimate the future value using simple interest. Then find (b) the compound amount, (c) compound interest and (d) the amount by which simple interest calculations underestimate the compound interest that is earned. Solution (a) Simple interest calculations can be used to estimate the compound amount but they will always underestimate the actual amount. Use 6% ÷ 2 = 3% per period and 5 years x 2 = 10 periods for the calculations.
Simple interest = PRT = %2500 X 0.03 X 10 = $750 Compound of future amount = P + I = $2500 + $750 = $3250 (b) Compound amount = P(1 + i)n = $2500 X (1 + 0.03)10 = $2500 X 1.0310 = $3359.79 Rounded (c) Compound interest = M – P = $3359.79 - $2500 = $859.79 (d) Underestimating when simple interest is used = Compound interest – simple interest = $859.79 - $750 = $109.79 Objective 3: Find Values in the Interest Table The value of (1 + i)n can be found by direct calculation using scientific calculators or from tables. One such table is given in the compound interest column (column A) of the interest table in Appendix D. The interest rate per compounding period is on the upper left-hand corner of each page. Example 4 Using the Interest Table Find the following values in the compound interest table in Appendix D. (a) (1 + 5%)12 or (1 + 0.05)12 = (1.05)12 ……………………………… Find the 5% page of the interest table. Look in column A, for compound interest and find 12 (or 12 periods) at the left side. You should find 1.79585633. (b) (1 + 8%)27 = (1.08)27 = 7.98806147 ………………………………. Find the 8% page. Then look in column A and find 27 at the left. Objective 4: Use the formula for Compound Interest to Find the Compound Amount. The evaluation of (1 + i)n using tables or calculators can now be used to find the compound amount and interest. Example 5: Finding Compound Interest John Smith inherits $15,000, which he deposits in a retirement account that pays interest compound semiannually. How much will he have after 25 years if the funds grow (a) at 6%, (b)at 8% and (c) at 10%? Round to the nearest cent.
Solution In 25 years, there are 2 X 25 = 50 semiannual periods. The semiannual interest rates are (a) + 3%, (b) = 4%, and (c) = 5%. Using factors from the table or using the formula n M = P(1 + i) , (a) $15,000(1.03)50 = $15,000 X 4.38390602 = $65,758.59 (b) $15,000(1.04)50 = $15,000 X 7.10668335 = $106.600.25 (c) $15,000(1.05)50 = $15,000 X 11.46739979 = $172,011.00 The $15,000 that John Smith inherits is the present value he has today. The future value is the amount he will have in 25 years. Note Simple interest rate calculations are usually indicated by phrases such as simple interest, simple interest note, or discount rate. Compound interest rate calculations are usually indicated by phrases such as compounded annually, 6% per quarter or compounded daily. The more often interest is compounded, the more interest is earned. Use a financial calculator or a compound interest table more complete than the one in this text and use the compound interest formula to get the results shown in Table 13.1 (Leap years were ignored in finding daily interest.) Table 13.1 INTEREST ON $1000 AT 6% PER YEAR FOR 10 YEARS Frequency of Compounding Not at all (Simple Interest) Annually Semiannually Quarterly Monthly Daily Hourly Every Minute
Interest per Compounding Period 6% 6%/2 6%/4 6%/12 6%/365 6%/8760 6%/525,600
Number of Compounding Periods 10 20 40 120 3650 87,600 5,256,000
Interest $600.00 $1790.85 $1806.11 $1814.02 $1819.40 $1822.03 $1822.12 $1822.12
Table 13.1 shows that compounding produces significantly more interest than does simple interest. However, increasing the frequency of compounding makes smaller and smaller differences in the amount earned with an increasing number of compounding periods. Example 6: Finding Compound Amount Ben Fitzgerald is comparing two different investment options. Find the compound amount on a deposit of $8000 for 3 years at (a) 6% compounded quarterly and (b)6% compounded monthly.
Solution (a) Number of compounding periods = 3 years X 4 quarters per year = 12 and interest of 6% ÷ 4 = 1.5% is credited at the end of each quarter. $8000(1.015)12 = $8000 X 1.19561817 = $9564.95 Rounded (b) Number of compounding periods = 3 years X 12 months/year = 36 and interest of 6% ÷ 12 = 0.5% is credited at the end of each quarter. $8000(1.005)36 = $8000 X 1.19668052 = $9573.44 Rounded Another use of compound interest calculations is to find the effect of inflation on real estate values. Example 7: Finding the Effect of Inflation on Real Estate Values Bill and Joy Lopez purchase a home for $167,200. If the house increases in value by 3% per year, find its value at the end of 4 years. Solution The value of the house compounds at 3% for 4 years. Use the table to find 1.12550881. You can also use your calculator to work this problem. Future value of home = $167,200 X 1.12550881 = $188,185 Rounded The home is expected to increase in value by $188,185 - $167,200 = $20,985 during the 4 years. Objective 5: Find the Effective Rate of Interest. If interest is compounded more often than annually then the actual rate of interest is greater than the nominal or stated rate of interest. For example, depositing $1000 for 1 year at 12% compounded quarterly produces a compound amount as follows: M = $1000(1.12550881) = $1125.51 Rounded The interest earned on this deposit is $125.51, which is 12.551% of the original deposit of $1000. This amount of interest is the same as if the $1000 were invested at simple interest of 12.551% for one year. Although the stated rate of interest was 12%, the actual increase in the investment was 12.551%, the effective rate of interest. It is important to know the effective rate of interest so that you can compare one investment or loan to another.
Finding the Effective Rate of Interest Step 1 Find the entry in column A of the interest table that corresponds to the proper rate per period and the proper number of periods. Step 2 Subtract 1 from the number Step 3 Round to the nearest hundredth of a percent. Example 8: Finding the Effective Rate of Interest James Suhr is comparing two loans. Loan A has a nominal rate of 10% compounded quarterly and Loan B has a nominal rate of 9% compounded monthly. Find the effective rate of interest for each loan. Solution Loan A: 10% ÷ 4 = 2.5% per quarter for 4 quarters. Look in column A of the interest table for 2½% and 4 periods to find the effective interest rate as follows: 1.10381289 - 1.00000000 0.09380690 or 9.38% Note In step 2, be sure to subtract 1 from the number in the table to find the interest.
