A smooth used-car salesman is offering you a great deal on a “preowned” car. He

Question

A smooth used-car salesman is offering you a great deal on a “preowned” car. He says, “For only five annual end-of-the year payments of $1,980, the beautiful 2011 Honda Civic can be yours.” If you can borrow money a 4.56%, what is the price of the car? If you are to make the payment at the beginning of the year, what is the price of the car?

You borrow a 15-year $289,000 mortgage loan at annual interest rate of 3.50%.

What is the monthly mortgage payment for your mortgage (payment is made at the end of each month)?

Find the amount of principal paid, interest paid, and the ending balance for the first month.

Find the amount of principal paid, interest paid, and the ending balance for the last month (i.e., month 180)

d.

e.

Today is Denise’s 30th birthday and she determines to become a millionaire by age sixty-five. She plant to start saving annually at her 30th birthday for 35 years. She anticipates an annual return of 4.5% on her investments. How much does she need to save annually to be a millionaire at her sixty- five birthday?   If she saves monthly for 35 years, how much does she need to save monthly to achieve her goal? (draw a simple time line for this question)

4. A group of college students won a record Powerball jackpot of $423 million. The group had 2 choices:

      A) Receive 30 beginning of the year payments of 14.1 million each

      B) Receive a lump sum payment today of $211.5 million immediately

Draw the time lines for the two choices.

Find the FV (at the end of year 30) of the two choices assuming interest rate is 11%? Which choice should the group select?

Find the PV of the two choices assuming interest rate is 11%. Which choice should the group select?

Find the FV (at the end of year 30) for the two choices assuming interest is 4%. Which choice should the group select?

Find the PV of the two choices assuming interest rate is 4%. Which choice should the group select?

Explanation / Answer

1) We can find price of car using annuities

FV(annuity) = A[(1+r)^n-1 / r]

A=1980$ , n = 5 , r =4.56%

FV(annuity) = 1980[(1+4.56%)^5-1/ 0.0456]

=1980[(1.0456)^5-1 / 0.0456]

=1980[1.2497-1 / 0.0456]

=1980[5.47727]

=10845$

If payments were made at begining of year, cost of car would be

FV(annuity due) = A[(1+r)^n-1 / r]*(1+r)

=10845(1.0456)

11339.53$

2) here annual interest rate = 3.5%

Monthly rate = 3.5/12 = 0.29167%

total no. of months = 15*12 = 180 months

EMI = loan/PVIFA(0.29167%,180 months)

=289000/139.8827

=2066.016$

Repayment schedule

3) here we need 1 million after 35 years at rate of 4.5%

FV(annuity) = A[(1+r)^n-1/r]

1000,000 = A[(1+0.045)^35-1 / 0.045]

=1000000 = A[4.6673-1 / 0.045]]

=1000,000 = A[81.49662]

=12270.45$

If one saves monthly, r = 4.5/12 = 0.375% , n = 35*12 = 420 months

1000,000 = A[(1+0.00375)^420 - 1 / 0.00375]

=1000,000 = A[4.816532-1/0.00375]

=1000,000 = A[1017.742]

A = 982.5673$

4) First of all lets find FV of bott option at 11%

FV(annuity due) = A[(1+r)n - 1 / r )(1+r)

=14.1[(1+0.11)^30 - 1 / 0.11](1+0.11)

=14.1[199.02](1.11)

=3114.876

If it were to receive at lumsome rate

FV =PV(1+r)^n

211.5(1+0.11)^30

=211.5(22.8923)

=4841.72$

Now lets find FV of both option at 4%

FV(annuity due) = A[(1+r)n - 1 / r )(1+r)

=14.1[(1+0.04)^30 - 1 / 0.04](1+0.04)

=14.1[56.08494](1.04)

=822.4295$

If it were to receive at lumsome rate

FV =PV(1+r)^n

211.5(1+0.04)^30

=211.5(3.243398)

=685.9786$

Now lets find PV for annuity at 11%

PV(annuity due) = A[1- 1/(1+r)^ / r](1+r)

=14.1[1-1/(1.11)^30 / 0.11](1+0.11)

=14.1[1-0.043/0.11](1.11)

=14.1[8.6937](1.11)

=136.0665

Now lets find PV for annuity at 4%

PV(annuity due) = A[1- 1/(1+r)^ / r](1+r)

=14.1[1-1/(1.04)^30 / 0.04](1+0.04)

14.1[1-0.308319 / 0.04](1.04)

=14.1[ 0.691681/0.04](1.04)

=14.1[17.29203](1.04)

=253.5704

Months opening balance EMI Principal Interest @0.29167% Closing balance 1 289000 2066 1223 843 287777 2 287777 2066 1227 839 286550 3 286550 2066 1230 836 285320 4 285320 2066 1234 832 284086 5 284086 2066 1237 829 282849 6 282849 2066 1241 825 281608 7 281608 2066 1245 821 280363 8 280363 2066 1248 818 279115 9 279115 2066 1252 814 277863 10 277863 2066 1256 810 276607 11 276607 2066 1259 807 275348 12 275348 2066 1263 803 274085 13 274085 2066 1267 799 272819 14 272819 2066 1270 796 271548 15 271548 2066 1274 792 270274 16 270274 2066 1278 788 268997 17 268997 2066 1281 785 267715 18 267715 2066 1285 781 266430 19 266430 2066 1289 777 265141 20 265141 2066 1293 773 263848 21 263848 2066 1296 770 262552 22 262552 2066 1300 766 261252 23 261252 2066 1304 762 259948 24 259948 2066 1308 758 258640 25 258640 2066 1312 754 257328 26 257328 2066 1315 751 256013 27 256013 2066 1319 747 254693 28 254693 2066 1323 743 253370 29 253370 2066 1327 739 252043 30 252043 2066 1331 735 250712 31 250712 2066 1335 731 249378 32 249378 2066 1339 727 248039 33 248039 2066 1343 723 246696 34 246696 2066 1346 720 245350 35 245350 2066 1350 716 244000 36 244000 2066 1354 712 242645 37 242645 2066 1358 708 241287 38 241287 2066 1362 704 239925 39 239925 2066 1366 700 238558 40 238558 2066 1370 696 237188 41 237188 2066 1374 692 235814 42 235814 2066 1378 688 234436 43 234436 2066 1382 684 233054 44 233054 2066 1386 680 231667 45 231667 2066 1390 676 230277 46 230277 2066 1394 672 228883 47 228883 2066 1398 668 227484 48 227484 2066 1403 664 226082 49 226082 2066 1407 659 224675 50 224675 2066 1411 655 223264 51 223264 2066 1415 651 221850 52 221850 2066 1419 647 220431 53 220431 2066 1423 643 219007 54 219007 2066 1427 639 217580 55 217580 2066 1431 635 216149 56 216149 2066 1436 630 214713 57 214713 2066 1440 626 213274 58 213274 2066 1444 622 211830 59 211830 2066 1448 618 210381 60 210381 2066 1452 614 208929 61 208929 2066 1457 609 207472 62 207472 2066 1461 605 206011 63 206011 2066 1465 601 204546 64 204546 2066 1469 597 203077 65 203077 2066 1474 592 201603 66 201603 2066 1478 588 200125 67 200125 2066 1482 584 198643 68 198643 2066 1487 579 197156 69 197156 2066 1491 575 195665 70 195665 2066 1495 571 194170 71 194170 2066 1500 566 192670 72 192670 2066 1504 562 191166 73 191166 2066 1508 558 189658 74 189658 2066 1513 553 188145 75 188145 2066 1517 549 186628 76 186628 2066 1522 544 185106 77 185106 2066 1526 540 183580 78 183580 2066 1531 535 182049 79 182049 2066 1535 531 180514 80 180514 2066 1540 527 178975 81 178975 2066 1544 522 177431 82 177431 2066 1549 518 175882 83 175882 2066 1553 513 174329 84 174329 2066 1558 508 172772 85 172772 2066 1562 504 171210 86 171210 2066 1567 499 169643 87 169643 2066 1571 495 168072 88 168072 2066 1576 490 166496 89 166496 2066 1580 486 164916 90 164916 2066 1585 481 163331 91 163331 2066 1590 476 161741 92 161741 2066 1594 472 160147 93 160147 2066 1599 467 158548 94 158548 2066 1604 462 156944 95 156944 2066 1608 458 155336 96 155336 2066 1613 453 153723 97 153723 2066 1618 448 152105 98 152105 2066 1622 444 150483 99 150483 2066 1627 439 148856 100 148856 2066 1632 434 147224 101 147224 2066 1637 429 145587 102 145587 2066 1641 425 143946 103 143946 2066 1646 420 142300 104 142300 2066 1651 415 140649 105 140649 2066 1656 410 138993 106 138993 2066 1661 405 137332 107 137332 2066 1665 401 135667 108 135667 2066 1670 396 133997 109 133997 2066 1675 391 132322 110 132322 2066 1680 386 130641 111 130641 2066 1685 381 128956 112 128956 2066 1690 376 127267 113 127267 2066 1695 371 125572 114 125572 2066 1700 366 123872 115 123872 2066 1705 361 122167 116 122167 2066 1710 356 120458 117 120458 2066 1715 351 118743 118 118743 2066 1720 346 117023 119 117023 2066 1725 341 115299 120 115299 2066 1730 336 113569 121 113569 2066 1735 331 111834 122 111834 2066 1740 326 110094 123 110094 2066 1745 321 108349 124 108349 2066 1750 316 106599 125 106599 2066 1755 311 104844 126 104844 2066 1760 306 103084 127 103084 2066 1765 301 101319 128 101319 2066 1770 296 99548 129 99548 2066 1776 290 97772 130 97772 2066 1781 285 95992 131 95992 2066 1786 280 94206 132 94206 2066 1791 275 92414 133 92414 2066 1796 270 90618 134 90618 2066 1802 264 88816 135 88816 2066 1807 259 87009 136 87009 2066 1812 254 85197 137 85197 2066 1818 248 83379 138 83379 2066 1823 243 81557 139 81557 2066 1828 238 79728 140 79728 2066 1833 233 77895 141 77895 2066 1839 227 76056 142 76056 2066 1844 222 74212 143 74212 2066 1850 216 72362 144 72362 2066 1855 211 70507 145 70507 2066 1860 206 68647 146 68647 2066 1866 200 66781 147 66781 2066 1871 195 64910 148 64910 2066 1877 189 63033 149 63033 2066 1882 184 61151 150 61151 2066 1888 178 59264 151 59264 2066 1893 173 57370 152 57370 2066 1899 167 55472 153 55472 2066 1904 162 53568 154 53568 2066 1910 156 51658 155 51658 2066 1915 151 49742 156 49742 2066 1921 145 47821 157 47821 2066 1927 139 45895 158 45895 2066 1932 134 43963 159 43963 2066 1938 128 42025 160 42025 2066 1943 123 40082 161 40082 2066 1949 117 38132 162 38132 2066 1955 111 36178 163 36178 2066 1960 106 34217 164 34217 2066 1966 100 32251 165 32251 2066 1972 94 30279 166 30279 2066 1978 88 28301 167 28301 2066 1983 83 26318 168 26318 2066 1989 77 24329 169 24329 2066 1995 71 22333 170 22333 2066 2001 65 20333 171 20333 2066 2007 59 18326 172 18326 2066 2013 53 16313 173 16313 2066 2018 48 14295 174 14295 2066 2024 42 12271 175 12271 2066 2030 36 10240 176 10240 2066 2036 30 8204 177 8204 2066 2042 24 6162 178 6162 2066 2048 18 4114 179 4114 2066 2054 12 2060 180 2060 2066 2060 6 0

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