I S imple nterest Chapter 6 McGraw-Hill Ryerson©

Learning Objectives Present Calculate
After completing this chapter, you will be able to: Calculate LO-1 … interest, maturity value, future value, and present value in a simple interest environment LO-2 … the equivalent value on any date of a single payment or a stream of payments, and Present … details of the amount and timing of payments in a time diagram

Basic Concepts Parties LO-1 Example: Loan Lender Borrower
Lends the Principal Borrower OWES (Debt) to Lender

Basic Concepts Simple Interest … Rate of Interest: Example: Loan
Lender Borrower Earns (Income) from Borrower i.e. Interest on the Principal Borrower pays Interest to Lender Simple Interest … Calculated on an ANNUAL or per annum (pa) basis Rate of Interest:

Invest $1000 at 10% simple interest for one year. Interest earned is?
Examples 1. Invest $1000 at 10% simple interest for one year Interest earned is? = $1000 * 10% = Principal X Interest Rate $100 2. Invest $1000 at 10% simple interest for six months Interest earned is? $1000 * /2 = Principal X Interest Rate = 10% $50

Invest $1000 at 10% simple interest for one month. Interest earned is?
Examples 3. Invest $1000 at 10% simple interest for three months Interest earned is? /4 = $1000 X 10% $25 Principal X Interest Rate = 4. Invest $1000 at 10% simple interest for one month Interest earned is? /12 = Principal X Interest Rate = $1000 X 10% $8.33

Up to this point we have taken months to represent 1/12th of a year, i
Up to this point we have taken months to represent 1/12th of a year, i.e. each month is treated as having the same number of days! Question Would it not be more accurate to calculate the interest due or payable based on the actual number of days in each month?

Yes, it would! In fact, interest continues to accumulate as each day passes! Example Invest $1000 at 10% simple interest for 30 days! Interest earned is? Answer $1000 *10% * 30 $8.22 = Principal x Interest Rate = 365 Year = 365 days or 366 in a Leap Year

Time period …expressed as a fraction or a multiple of a year
“What is the formula that can be used to calculate SI?” Four Elements are involved … Interest Principal Interest Rate Time Time period …expressed as a fraction or a multiple of a year Amount (paid or received) Principal Amount (loan or investment) Annual Rate of SI I P r t I = Prt Formula

Calculate the Interest earned on $5000 invested at 4%
Formula I = Prt Example Calculate the Interest earned on $5000 invested at 4% for 7 months. I = P * r * t $5000 * .04 * 7 /12 I = $116.67

“I may need to invest or need a loan for a number of days rather than a complete month.”
Question “How do I calculate the time between the starting date and the ending date?”

1. Answer 2. 3. Add up the days according to the
There are three methods that can be used to calculate the number of days: 1. Add up the days according to the days covered by the specific months. Include either the FIRST date or the LAST date when doing this calculation! Answer 2. Use a Number of Days TABLE Use the TI BAII Plus calculator! 3.

Calculating the Number of Days
What is the interest earned on $5000 invested from Oct. 11th to Dec. 29th at 4.5%? Days Oct Method 1. 11th to end of month = 20 Nov Total month = 30 Dec From 1st to 29th of month = 29 Includes the last date 79

Also... The “Hand Calculator!” Jun July Sept Apr Aug May Nov Feb Oct
Mar Oct Feb Dec Jan Knuckles are ALL 31 days Spaces are ALL 30 days except for February which can be either or 29 days in a Leap year! Also...

plus one more in each leap year!
Olde Rhyme… Thirty days has September, April, June and November. All the rest have 31, but February with just 28 days clear plus one more in each leap year!

363 284 = 79 Days Table Dec 29 Oct 11 Method 2. T 6.2 Jan Feb Mar Apr
Day of Month T ABLE 6.2 The Serial Number of Each Day of the Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1 32 60 91 121 152 182 213 244 274 305 335 2 33 61 92 122 153 183 214 245 275 306 336 3 34 62 93 123 154 184 215 246 276 307 337 4 35 63 94 124 155 185 216 247 277 308 338 5 36 64 95 125 156 186 217 248 278 309 339 6 37 65 96 126 157 187 218 249 279 310 340 7 38 66 97 127 158 188 219 250 280 311 341 8 39 67 98 128 159 189 220 251 281 312 342 9 40 68 99 129 160 190 221 252 282 313 343 10 41 69 100 130 161 191 222 253 283 314 344 11 42 70 101 131 162 192 223 254 284 315 345 12 43 71 102 132 163 193 224 255 285 316 346 13 44 72 103 133 164 194 225 256 286 317 347 14 45 73 104 134 165 195 226 257 287 318 348 15 46 74 105 135 166 196 227 258 288 319 349 16 47 75 106 136 167 197 228 259 289 320 350 17 48 76 107 137 168 198 229 260 290 321 351 18 49 77 108 138 169 199 230 261 291 322 352 19 50 78 109 139 170 200 231 262 292 323 353 20 51 79 110 140 171 201 232 263 293 324 354 21 52 80 111 141 172 202 233 264 294 325 355 22 53 81 112 142 173 203 234 265 295 326 356 23 82 113 143 174 204 235 266 296 327 357 24 55 83 114 144 175 205 236 267 297 328 358 25 56 84 115 145 176 206 237 268 298 329 359 26 57 85 116 146 177 207 238 269 299 330 360 27 58 86 117 147 178 208 239 270 300 331 361 28 59 87 118 148 179 209 240 271 301 332 362 29 88 119 149 180 210 241 272 302 333 363 30 89 120 150 181 211 242 273 303 334 364 31 90 151 212 243 304 365 363 Dec 29 284 Oct 11 Method 2. = Days Table 54

Solve Using… Method 3. DBD = 79 Example 1. Texas Instruments BAII PLUS
Start 2nd DBD = Example 1. Calculate… the interest earned on $5000 invested from Oct. 11th to Dec. 29th, 02 at 4.5%. Date Enter Enter CPT Days Between Dates Solve

Calculate the Interest earned on $5000 invested at 4.5%
Formula I = Prt Example 1 Calculate the Interest earned on $5000 invested at 4.5% for ? . 79 Days I = $5000*.045 * 79/365 I = $48.70

What is the interest earned on $5000 invested from Nov 30th, 02 to Jan 6th, 03 at 4.5%? Days Nov Method 1. 30th to end of month = Dec Total month = 31 Jan From 1st to 6th of month = 6 37 Includes the last date

365 334 6 = 31 Days Plus Table = 37 Days Dec 31 Nov 30 Jan 6 Method 2.
Day of Month T ABLE 6.2 The Serial Number of Each Day of the Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1 32 60 91 121 152 182 213 244 274 305 335 2 33 61 92 122 153 183 214 245 275 306 336 3 34 62 93 123 154 184 215 246 276 307 337 4 35 63 94 124 155 185 216 247 277 308 338 5 36 64 95 125 156 186 217 248 278 309 339 6 37 65 96 126 157 187 218 249 279 310 340 7 38 66 97 127 158 188 219 250 280 311 341 8 39 67 98 128 159 189 220 251 281 312 342 9 40 68 99 129 160 190 221 252 282 313 343 10 41 69 100 130 161 191 222 253 283 314 344 11 42 70 101 131 162 192 223 254 284 315 345 12 43 71 102 132 163 193 224 255 285 316 346 13 44 72 103 133 164 194 225 256 286 317 347 14 45 73 104 134 165 195 226 257 287 318 348 15 46 74 105 135 166 196 227 258 288 319 349 16 47 75 106 136 167 197 228 259 289 320 350 17 48 76 107 137 168 198 229 260 290 321 351 18 49 77 108 138 169 199 230 261 291 322 352 19 50 78 109 139 170 200 231 262 292 323 353 20 51 79 110 140 171 201 232 263 293 324 354 21 52 80 111 141 172 202 233 264 294 325 355 22 53 81 112 142 173 203 234 265 295 326 356 23 82 113 143 174 204 235 266 296 327 357 24 55 83 114 144 175 205 236 267 297 328 358 25 56 84 115 145 176 206 237 268 298 329 359 26 57 85 116 146 177 207 238 269 299 330 360 27 58 86 117 147 178 208 239 270 300 331 361 28 59 87 118 148 179 209 240 271 301 332 362 29 88 119 149 180 210 241 272 302 333 363 30 89 120 150 181 211 242 273 303 334 364 31 90 151 212 243 304 365 54 365 Dec 31 334 Nov 30 Method 2. = Days Plus Table 6 Jan 6 = Days

Solve Using… Method 3. DBD = 37 Example 1. Texas Instruments BAII PLUS
Start 2nd DBD = Example 1. Calculate… the interest earned on $5000 invested from Nov. 30th,, 02 to Jan 6th. 03 at 4.5%. Date Enter Enter CPT Days Between Dates Solve

Formula I = Prt Example 2 Calculate the Interest earned on $5000 invested at 4.5% for ? 37 Days I = $5000*.045 * 37/365 I = $22.81

What is the interest earned on $5000 invested from Oct 11th , 02 to Mar 11th, 03 at 4.0%? Days Oct 11th to end of month = Nov & Dec 30 31 Total = Method 1. Jan & Feb 28 31 Total = Mar 11 To 11th of month = 151 Includes the last date

Table Dec 31 365 = 81 Days Oct 11 284 70 = 70 Days =151 Days Method 2.
Day of Month T ABLE 6.2 The Serial Number of Each Day of the Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1 32 60 91 121 152 182 213 244 274 305 335 2 33 61 92 122 153 183 214 245 275 306 336 3 34 62 93 123 154 184 215 246 276 307 337 4 35 63 94 124 155 185 216 247 277 308 338 5 36 64 95 125 156 186 217 248 278 309 339 6 37 65 96 126 157 187 218 249 279 310 340 7 38 66 97 127 158 188 219 250 280 311 341 8 39 67 98 128 159 189 220 251 281 312 342 9 40 68 99 129 160 190 221 252 282 313 343 10 41 69 100 130 161 191 222 253 283 314 344 11 42 70 101 131 162 192 223 254 284 315 345 12 43 71 102 132 163 193 224 255 285 316 346 13 44 72 103 133 164 194 225 256 286 317 347 14 45 73 104 134 165 195 226 257 287 318 348 15 46 74 105 135 166 196 227 258 288 319 349 16 47 75 106 136 167 197 228 259 289 320 350 17 48 76 107 137 168 198 229 260 290 321 351 18 49 77 108 138 169 199 230 261 291 322 352 19 50 78 109 139 170 200 231 262 292 323 353 20 51 79 110 140 171 201 232 263 293 324 354 21 52 80 111 141 172 202 233 264 294 325 355 22 53 81 112 142 173 203 234 265 295 326 356 23 82 113 143 174 204 235 266 296 327 357 24 55 83 114 144 175 205 236 267 297 328 358 25 56 84 115 145 176 206 237 268 298 329 359 26 57 85 116 146 177 207 238 269 299 330 360 27 58 86 117 147 178 208 239 270 300 331 361 28 59 87 118 148 179 209 240 271 301 332 362 29 88 119 149 180 210 241 272 302 333 363 30 89 120 150 181 211 242 273 303 334 364 31 90 151 212 243 304 365 54 Dec 31 365 Oct 11 = 81 Days 284 Method 2. Table Mar 11 70 = 70 Days =151 Days

Solve Using… Method 3. DBD = 151 Example 1. Texas Instruments
BAII PLUS i Method 3. Start 2nd DBD = Example 1. Calculate… the interest earned on $5000 invested from Oct. 11th,, 02 to Mar 11th. 03 at 4.0%. Date Enter Enter CPT Days Between Dates Solve

Formula I = Prt Example 2 Calculate the Interest earned on $5000 invested at 4.0% for ? 151 Days I = $5000* .04 * 151/365 I = $82.74

I = Prt Principal Rate Time Formula
We can ‘reorganize’ the formula to also get each of the following separately: Principal Rate Time Shortcut Tool!

I I = Prt P*r*t Prt I Prt Formula Using this tool! The Triangle
…another useful non-calculator! Formula I = Prt To help remember this… we can place the formula into a triangle as follows… Where variables are BESIDE EACH OTHER this means to MULTIPLY! P*r*t I Prt Where a variable is ABOVE ANOTHER this means to DIVIDE! I Prt Using this tool!

/rt /Pt /Pr I Prt Using this tool! then I then I then I The Triangle
…another useful non-calculator! /rt If you want to find P then I Using this tool! I Prt /Pt If you want to find r then I /Pr If you want to find t then I

…Indicates the steps needed to implement the solution
Using a Time Line What is a Time Line? A Time Line is, as the name suggests, a line that shows the various points of time along which a loan or investment travels to maturity. It is used for diagramming problems involving multiple payments or investments. …Helps organize data Two Benefits …Indicates the steps needed to implement the solution Example

Using a Time Line Example Table 151 days
Calculate the interest earned at 4% on $5000 invested from Oct. 11th to March 11th Example Draw a line for the entire period Step 1 Start Finish Oct 11 Dec 31 March 11 Enter the key dates on the line Step 2 Look up Table $5000 284 365 70 Enter the Investment Step 3 81 days 70 days 151 days Enter the number of days between each date and add Step 4 The calculation can now be restated as follows…

Calculate the interest earned at 4% on $5000 invested for 151 days.
Using a Time Line Calculate the interest earned at 4% on $5000 invested for 151 days. Example I = Prt Using this tool! I Prt I = 5000 * .04 * 151/365 = $82.74

I Prt You can now use this tool! In the next few examples you have to…
(a) First identify which variable you are being asked to solve for, and (b) Reorganize the formula in order to meet the requirement in (a).

P= I P = P = Principal rt P = $9,038.10 Example $195 .0525 *(150/365)
Calculating the Principal Formula I rt P= $195 interest is earned on a 150 day GIC at 5.25%. Example Find the initial investment . P = $195 .0525 *(150/365) P = $195 P = $9,038.10 Calculator

P = $195 .0525 *(150/365) .0525 * 150 / 365 = / 195 = 1/x $

I Pt r = r = r = r = Example /($5000*180/365) /2465.75
Calculating the Rate Formula r = I Pt What Rate of Interest is needed to earn $200 on a $5000 investment invested for 180 days? Example r = /($5000*180/365) $200 r = $200 / r = or 8.11% Calculator

/($5000*180/365) r = $200 5000 * 180 / 365 = / 200 = 1/x 8.11%

t I P = r Example P + I = Sum I Calculating the Time Formula
What is the length of Time required for $2000 to grow to $2100 when invested at 5.6%? Example Find the amount of Interest Step 1 P + I = Sum I P I = $100 $2100 - $2000 More

t I P t = t = *365 days t = t = = r Example $100 / ($2000*.056) $100/
Calculating the Time Formula t I P = r What is the length of Time required for $2000 to grow to $2100 when invested at 5.6%? Example t = $100 / ($2000*.056) Step 2 Calculate t = $100/ 112 *365 days t = Years t = 326 days Calculator

/ ($2000*.056) $100 t = 100 / 325.89 2000 / 0.056 = Years * 365 = 326 days

Future Value

P + I P + Prt P(1+rt) I = Prt Sum = Future Value Sum = Sum =
Up to this point we have used two formulae: & Sum = P + I Formula I = Prt We can combine them as follows: P + I Sum = Now, we substitute for I Step 1 P + Sum = Prt with Step 2 Collecting like terms Sum = P(1+rt) Formula Future Value

P(1+rt) S = S = $17000(1.0267123) S = $17454.11 Future Value Sum =
I place $17000 in a 150 day term deposit on Jan. 6 paying 6.5%. S = $17000 (150) 365 S = $17000( ) How much will the bank pay me on the maturity date? S = $ Calculator

* * S = / = + 1 = = Future Value Next $17000 1+ .065(150) 365 .065 150
* 150 / 365 = $ + 1 = * 17000 = Date of Maturity Next

I place $17000 in a 150 day term deposit on Jan. 6 paying 6.5% pa.
Future Value Determine the Maturity Date I place $17000 in a 150 day term deposit on Jan. 6 paying 6.5% pa. = 6 days + 150 = 156 days Day of Month T ABLE 6.2 The Serial Number of Each Day of the Year Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 1 32 60 91 121 152 182 213 244 274 305 335 2 33 61 92 122 153 183 214 245 275 306 336 3 34 62 93 123 154 184 215 246 276 307 337 4 35 63 94 124 155 185 216 247 277 308 338 5 36 64 95 125 156 186 217 248 278 309 339 6 37 65 96 126 157 187 218 249 279 310 340 7 38 66 97 127 158 188 219 250 280 311 341 Look up Table Find 156 The Term Deposit will mature on June 5th.

Future Value Problem What Total amount will I have on December 29th?
During the year, I invested the following funds at a constant 4% p.a. (The second and third amounts were added to the Feb 14th amount.) Problem Feb 14th $5,000 Mar 17th $3,000 July 1st $2,000 5,000 8,000 10,000 What Total amount will I have on December 29th?

Future Value Solve Problem
Three Steps are required to solve this problem: Step 1 Draw a time line, including the dates and dollar amounts. Problem Solve Step 2 Determine the time between each of the dates Step 3 Calculate the interest amounts, and add together.

Future Value Table Look up Step 1 Draw a time line $5000
45 76 182 363 Feb 14 March 17 July 1 Dec 29 $5000 363 – 45 = 318 Days Step 2 Determine the Time $3000 363 – 76 = 287 Days $2000 363 – 182 = 181 Days

Calculate the Interest amounts, and
Future Value Formula I = Prt Step 3 Calculate the Interest amounts, and add together. Interest I1 = 5000 *.04 *318/365 $174.25 94.36 I2 = 3000 *.04 *287/365 39.67 I3 = 2000 *.04 *182/365 $308.28 $ $308.28 S = $ Total Amount

Future Value Question Answer
“During the year I made an investment that had changes in the rate of interest. Question How do I determine the total interest earned during the period under review?” Every time the interest rate changes,you must stop and make a calculation up to that point. Answer

The changes in interest rates to July 4th are as follows:
Future Value I invest $1,000 on Feb. 14th at 6%. The changes in interest rates to July 4th are as follows: Investment Date rate $1,000 Feb 14th 6% April 20th 6.8% May 18th 7.1% Problem How much Interest did I earn up to July 4th ? Solve

… by finding the number of days between each rate change!
Future Value Solve … by finding the number of days between each rate change! Investment Date rate $1,000 Feb 14th 6% April 20th 6.8% May 18th 7.1% July 4th Look up for Days Table

Future Value Table = 45 = 110 = 65 Days = 138 = 28 Days = 185
Table reading Number of Days Feb 14th April 20th May 18th July 4th = 45 = 110 = 65 Days Look up for Days Table = 138 = 28 Days = 185 = 47 Days Interest Earned

I = Prt $25.04 Future Value Total Interest I1 = 1000 * .06 * 65/365
Interest Earned Formula I = Prt Total Interest Feb 14th April 20th I1 = 1000 * .06 * 65/365 = $10.68 65 Days May 18th I2 = 1000 * .068 * 28/365 = 28 Days July 4th I3 = 1000 * .071 * 47/365 = 47 Days $25.04

Identifing Data

“How much must I invest… …this suggests finding the Principal
Future Value Identifing Data Example How much must I invest in order for it to grow to $5000 within 6 months @ 4.4% simple interest? What are we being asked to provide? “How much must I invest… to grow…” …this suggests finding the Principal Need

I = Prt & Sum = P(1+rt) Future Value Identifing Data Have
How much must I invest in order for it to grow to $5000 within 6 4.4% simple interest? What data do we need? We need the appropriate data to be able to use the appropriate formula… Formulae I = Prt & Sum = P(1+rt) Have

I = Prt & Sum = P(1+rt) Future Value Identifing Data Sum = $5000 To Do
How much must I invest in order for it to grow to $5000 within 6 4.4% simple interest? Formulae I = Prt & Sum = P(1+rt) What data do we have? r = 4.4% t = 6 months = .5 Sum = $5000 To Do

P(1+rt) P = Sum/(1+rt) I = Prt & Sum = P(1+rt) Future Value
Identifing Data How much must I invest in order for it to grow to $5000 within 6 4.4% simple interest? Formulae I = Prt & Sum = P(1+rt) Using Sum = P(1+rt) Solve for P As we know the Sum, the formula now becomes… P = Sum/(1+rt) Solve

P = Sum/(1+rt) P =5000/ P = $4892.37 Future Value Identifing Data
How much must I invest in order for it to grow to $5000 within 6 4.4% simple interest? Formula P = Sum/(1+rt) Solve for P P =5000/ (.5) P = $

Simple Interest 6 LO-2 Equivalent Value

Equivalent Value Table Sum = P(1+rt) Calculate Sum = = $2022.68
What is the equivalent value on September 15 of a $2000 payment on July 4, if money is worth 6% pa? Step 1 July 4 September 15 Table Draw a Timeline $2000 69 Days Future Value Step 2 Sum = P(1+rt) Calculate Sum = 2000[1+.06(69/365)] = $

P = Sum/(1+rt) Equivalent Value Table Calculate
What is the equivalent value on May 18th of a $2000 payment due on the following December 15th if money can earn 5.2%? Step 1 May 18 December 15 Table Draw a Timeline $2000 Present Value 211 Days Calculate Step 2 P = Sum/(1+rt) = 2000/[1+.052(211/365)] = $

Equivalent Value Heather owes Mark $3000 payable on April 27.
If money can earn 4%, what amount should Mark accept in settlement of the debt: A) 30 days before the scheduled payment? B) 90 days before the scheduled payment?

P = Sum/(1+rt) Equivalent Value Solve for Calculate P Draw a Timeline
April 27 Present Value Step 1 Draw a Timeline $3000 30 days 90 days Calculate Step 2 Solve for P P = Sum/(1+rt)

P = Sum/(1+rt) P = $2990.17 P = $2970.17 Equivalent Value Solve for P
Answer B P = 3000/[1+.04(90/365)] P = $

If you can earn 6% on your money, which option should you choose?
Equivalent Value You can prepay $1234 tuition for a course or delay payment for 3 months and pay $1432. If you can earn 6% on your money, which option should you choose?

Which formula should you use? Find the PV of the future payment!
Equivalent Value $1234 = Present Value = P1 $1432 = Future Value = S2 Data P = Sum/(1+rt) Which formula should you use? Tip: Find the PV of the future payment! 3 months = 1/4 P2 = 1432/[1+.06(.25)] P2 = $ $ Money saved ($176.84) by paying now!

This completes Chapter 6

I S imple nterest Chapter 6 McGraw-Hill Ryerson©

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