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  1. Appendix A--Learning Objectives • 1. Differentiate between simple and compound interest

  2. Interest • The charge for the use of money for a specified period of time

  3. The basic interest formula is • I = P x r x n • where • I = the amount of interest • P = the principal • r = the rate • n = the number of periods or time

  4. Another useful formula is • A = (P x r x n) + P • where • A = is the final amount or maturity value • P = the principal • r = the rate • n = the number of periods or time

  5. Simple interest • Interest accrues on the principal only • Suppose we have $10,000 • We can earn 12 percent • and we can wait 5 years: • How much money will we have • at the end of that time ?

  6. Simple interest • A = (P x r x n) + P • A = ($10,000 x .12 x 5) + $10,000 • A = $6,000 + $10,000 • A = $16,000 • At the end of the five years, • we will have $16,000

  7. Compound interest • Is nothing more than simple interest • over and over again • with interest on the interest • as well as the principal • Let’s check it out

  8. $10,000 in 5 years at 12 %compounded annually • The first year • A = ( P x r x n ) + P • A = ($10,000 x .12 x 1) + $10,000 • A = $1,200 + $10,000 • A = $11,200

  9. $10,000 in 5 years at 12 %compounded annually • It gets better in the second year • (because we have more money) • A = ( P x r x n ) + P • A = ($11,200 x .12 x 1) + $11,200 • A = $1,344 + $11,200 • A = $12,544

  10. $10,000 in 5 years at 12 %compounded annually • The third year is even better • A = ( P x r x n ) + P • A = ($12,544 x .12 x 1) + $12,544 • A = $1,505 + $12,544 • A = $14,049

  11. $10,000 in 5 years at 12 %compounded annually • The fourth year is better yet • A = ( P x r x n ) + P • A = ($14,049 x .12 x 1) + $14,049 • A = $1,686 + $14,049 • A = $15,735

  12. $10,000 in 5 years at 12 %compounded annually • And the fifth year is best • A = ( P x r x n ) + P • A = ($15,735 x .12 x 1) + $15,735 • A = $1,888 + $15,735 • A = $17,623

  13. Note the difference • With compound interest we got • $17,623 • With simple interest we got • $16,000 • The difference of $1,623 is not bad • compensation for getting the words • “compounded annually” • into the agreement

  14. The “over and over” method worked,but it was a lot of trouble • Another approach is to use the formula • A = P x ( 1 + r ) n • where • A = Amount • P = Principal • 1 = The loneliest number • r = Rate • n = number of periods

  15. $10,000 in 5 years at 12 %compounded annually • A = P x ( 1 + r ) n • A = $10,000 x ( 1.12 ) 5 • A = $10,000 x 1.7623 • A = $17,623 • This bears an awesome resemblance • to what we got a minute ago

  16. Another way is with the table( Table A-1 in our book ) • Interest rates are across the top • And number of periods down the side • Just find the intersection • n/r 11% 12% • 1 1.1100 1.1200 • 2 1.2321 1.2544 • 3 1.3676 1.4049 • 4 1.5181 1.5735 • 5 1;6851 1.7623

  17. The table is faster ! • Multiply the number from the table • 1.7623 • times the principal • $10,000 • and we have the answer • $17,623

  18. Future value • $17,623 • could be referred to as the • future value • of $10,000 at 12 percent for 5 years • compounded annually • That is what we will usually call it

  19. A note about financial calculators • A number of calculators have built-in financial functions and can solve problems of this type very quickly • Your instructor will advise you as to what the calculator policies are for your course and your school • But remember, a fancy calculator will not solve all of your problems for you

  20. FANCY CALCULATORSARE LIKE FOUR-WHEEL DRIVE • THEY WILL NOT KEEP YOU FROM GETTING STUCK

  21. BUT THEY WILL LET YOU GET STUCK • IN MORE REMOTE PLACES

  22. Now for a change • Instead of having $10,000 now • let’s say we have to wait 5 years • to get the $10,000 • the interest rate is still 12% • compounded annually • What is that worth to us now ?

  23. In other words • What is the present value • of $10,000 to be received in 5 years • if the interest rate is 12 percent • compounded annually ?

  24. A reciprocal • The future value interest formula was • ( 1 + r ) n • and the basic present value formula is • 1 / [ ( 1 + r ) n ] • the future value example was • ( 1.12 ) 5 or 1.7623 • and the reciprocal is • 1 / 1.7623 or .5674

  25. Factors for the present value of 1are found in Table A-2 • The present value factor for $1 to be received in five years at 12 percent compounded annually is .5674 • We are looking for the present value of $10,000 • All we need to do is multiply the factor by the amount to obtain the answer of $5,674 • In other words, the present value of $10,000 to be received five years from now is $5,674 if the interest rate is 12% compounded annually

  26. Appendix A--Learning Objectives • 2. Distinguish a single sum from an annuity

  27. Annuity • A series of equal payments • at equal intervals • at a constant interest rate

  28. Types of annuities • Ordinary annuity--payments at the ends of the periods • Annuity due--payments at the beginnings of the periods • Deferred annuity--one or more periods pass before payments start

  29. Ordinary annuity assumptions • Today is January 1, 2001 • We will receive five annual payments of $1,000 each starting on December 31, 2001 • Money is worth 12 percent per year compounded annually • What will the payments be worth on December 31, 2005 ?

  30. Future value of an ordinary annuity 2001 2002 2003 2004 2005 • The five payments are equal amounts at equal intervals at a constant interest rate • They come at the ends of the periods, so this is an ordinary annuity • We are looking for the future value $1,000 $1,000 $1,000 $1,000 $1,000 ?

  31. A slow solution approach--finding the FV of each payment 2001 2002 2003 2004 2005 • 1st. 1,000 1,574 • 2nd. 1,000 1,405 • 3rd. 1,000 1,254 • 4th. 1,000 1,120 • 5th. 1,000 • Total 6,353 • First payment earns 4 years of interest. Last earns none. $1,000 $1,000 $1,000 $1,000 $1,000

  32. The faster approach is to use Table A-3 2001 2002 2003 2004 2005 • Table A-3 gives us a factor of 6.3528 for 12% interest and five payments (periods) • For annuities, we multiply the factor by the amount of each payment--$1,000 in this case • The result is the same answer--$6,353 (rounded to the nearest dollar) $1,000 $1,000 $1,000 $1,000 $1,000

  33. Another ordinary annuity situation • Today is January 1, 2001 • We will receive five annual payments of $1,000 each starting on December 31, 2001 • Money is worth 12 percent per year compounded annually • What are the payments worth to us today ?

  34. Present value of an ordinary annuity 2001 2002 2003 2004 2005 • This is an ordinary annuity with the payments at the ends of the periods • We want to know what the 5 payments are worth to us NOW $1,000 $1,000 $1,000 $1,000 $1,000 ?

  35. We could discount each payment 2001 2002 2003 2004 2005 • 893 1,000 • 797 1,000 • 712 1,000 • 636 1,000 • 567 1,000 • 3,605 • First payment discounted for one year, last for five years ? $1,000 $1,000 $1,000 $1,000 $1,000

  36. But using Table A-5 is much faster 2001 2002 2003 2004 2005 • Table A-5 gives us a factor of 3.6048 for 12% interest and five payments (periods) • Multiply by the payment amount--$1,000 • The result is the same answer--$3,605 (rounded to the nearest dollar) ? $1,000 $1,000 $1,000 $1,000 $1,000

  37. Appendix A--Learning Objectives • 3. Differentiate between an ordinary annuity and an annuity due

  38. Another type of annuity is the annuity due • The ordinary annuity has the payments at the ends of the periods • But the annuity due has the payments at the beginnings of the periods

  39. An annuity due situation • Today is January 1, 2001 • We will receive five annual payments of $1,000 each starting today • Money is worth 12 percent per year compounded annually • What will the payments be worth on December 31, 2005 ?

  40. $1,000 $1,000 $1,000 $1,000 $1,000 Future value of an annuity due 2001 2002 2003 2004 2005 • The five payments come at the beginning of the periods, so this is an annuity due • We are looking for the future value ?

  41. $1,000 $1,000 $1,000 $1,000 $1,000 A slow solution approach--finding the FV of each payment 2001 2002 2003 2004 2005 • 1,000 (1st.) 1,762 • 2nd. 1,000 1,574 • 3rd. 1,000 1,405 • 4th. 1,000 1,254 • 5th. 1,000 1,120 • Total 7,115 • Even the last payment earns interest for one year. ?

  42. $1,000 $1,000 $1,000 $1,000 $1,000 Table A-4 solves the problem fast 2001 2002 2003 2004 2005 • The table factor is 7.1152 • Once again, we multiply by the amount of each payment--$1,000 in this example • The result is the same number--$7,115 (rounded to the nearest dollar) ?

  43. Another annuity due situation • Today is January 1, 2001 • We will receive five annual payments of $1,000 each starting today • Money is worth 12 percent per year compounded annually • What is the series of payments worth to us today ?

  44. $1,000 $1,000 $1,000 $1,000 $1,000 Present value of an annuity due 2001 2002 2003 2004 2005 • The five payments come at the beginning of the periods, so this is an annuity due • We are looking for the present value ?

  45. $1,000 $1,000 $1,000 $1,000 $1,000 Once again, we could discount each payment ? 2001 2002 2003 2004 2005 • 1,000 ( First payment needs no discounting) • 893 1,000 • 797 1,000 • 712 1,000 • 635 1,000 • 4,037

  46. $1,000 $1,000 $1,000 $1,000 $1,000 Table A-6 is the fast way ? 2001 2002 2003 2004 2005 • The table factor is 4.0373 • Once again, we multiply by the amount of each payment--$1,000 in this example • The result is the same number--$4,037 (rounded to the nearest dollar)

  47. The last type of annuity we will look at is the deferred annuity • A deferred annuity is also a series of equal payments at equal intervals at a constant interest rate • but • two or more periods elapse before the first payment is made

  48. Deferred annuity example • Today is January 1, 2001 • We are going to receive three annual payments of $1,000 each • We get the first payment on December 31, 2003, the second on December 31, 2004, and the third on December 31, 2005 • The interest rate is 12% compounded annually • What is the series of payments worth to us today ?

  49. Here is the fact situation: • Each of the three payments is $1,000 • We want to know the value as of January 1, 2001 • The first payment does not occur until the end of the third year We are here 1st payment 2nd payment 3rd payment 2001 2002 2003 2004 2005

  50. We are here 1st payment 2nd payment 3rd payment • We could discount the payments individually: • 712 1,000 • 636 1,000 • 567 1,000 • 1,915 • This is OK if there are only a few payments 2001 2002 2003 2004 2005