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## Appendix A--Learning Objectives

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**Appendix A--Learning Objectives**• 1. Differentiate between simple and compound interest**Interest**• The charge for the use of money for a specified period of time**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**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**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 ?**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**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**$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**$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**$10,000 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**$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**$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**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**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**$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**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**The table is faster !**• Multiply the number from the table • 1.7623 • times the principal • $10,000 • and we have the answer • $17,623**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**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**FANCY CALCULATORSARE LIKE FOUR-WHEEL DRIVE**• THEY WILL NOT KEEP YOU FROM GETTING STUCK**BUT THEY WILL LET YOU GET STUCK**• IN MORE REMOTE PLACES**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 ?**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 ?**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**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**Appendix A--Learning Objectives**• 2. Distinguish a single sum from an annuity**Annuity**• A series of equal payments • at equal intervals • at a constant interest rate**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**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 ?**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 ?**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**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**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 ?**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 ?**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**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**Appendix A--Learning Objectives**• 3. Differentiate between an ordinary annuity and an annuity due**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**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 ?**$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 ?**$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. ?**$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) ?**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 ?**$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 ?**$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**$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)**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**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 ?**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**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