# Example: Suppose that we deposit \$1000 in a bank account offering 3% interest, compounded monthly. How will our money grow?

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1 Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent, or need a loan, we need ore atheatics. Siple Interest Discussing interest starts with the principal, or aount your account starts with. This could be a starting investent, or the starting aount of a loan. Interest, in its ost siple for, is calculated as a percent of the principal. For exaple, if you borrowed \$1 fro a friend and agree to repay it with 5% interest, then the aount of interest you would pay would just be 5% of 1: \$1(.5) = \$5. The total aount you would repay would be \$15, the original principal plus the interest. Siple One-tie Interest I = Pr A= P (1 ) + I = P + Pr = P + r I is the interest A is the end aount: principal plus interest P is the principal (starting aount) r is the interest rate (in decial for. Exaple: 5% =.5) Exaple: A friend asks to borrow \$3 and agrees to repay it in 3 days with 3% interest. How uch interest will you earn? P = \$3 (the principal) r =.3 (3% rate) I = \$3(.3) = \$9. You will earn \$9 interest. One-tie siple interest is only coon for extreely short-ter loans. For longer ter loans, it is coon for interest to be paid on a daily, onthly, quarterly, or annual basis. In that case, interest would be earned regularly. For exaple, bonds are essentially a loan ade to the bond issuer (a copany or governent) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a aturity date, at which tie the issuer pays back the original bond value. Exaple: Suppose your city is building a new park, and issues bonds to raise the oney to build it. You obtain a \$1, bond that pays 5% interest annually that atures in 5 years. Each year, you would earn 5% interest: \$1(.5) = \$5 in interest. So over the course of five years, you would earn a total of \$25 in interest. When the bond atures, you would receive back the \$1, you originally paid, leaving you with a total of \$1,25. David Lippan Creative Coons BY-SA

2 112 Siple Interest over Tie I = Prt A= P (1 ) + I = P + Prt = P + rt I is the interest A is the end aount: principal plus interest P is the principal (starting aount) r is the interest rate (in decial for. Exaple: 5% =.5) t is tie Exaple: Treasury Notes (T-notes) are bonds issued by the federal governent to cover its expenses. Suppose you obtain a \$1, T-note with a 4% annual rate, paid sei-annually, with a aturity in 4 years. How uch interest will you earn? First, it is iportant to know that interest rates are usually given as an annual percentage rate (APR) the total interest that will be paid in the year. Since interest is being paid seiannually (twice a year), the 4% interest will be divided into two 2% payents. P = \$1 (the principal) r =.2 (2% rate per half-year) t = 8 (4 years = 8 half-years) I = \$1(.2)(8) = \$16. You will earn \$16 interest total over the four years. Copound Interest With siple interest, we were assuing that we pocketed the interest when we received it. In a standard bank account, any interest we earn is autoatically added to our balance. This reinvestent of interest is called copounding. Exaple: Suppose that we deposit \$1 in a bank account offering 3% interest, copounded onthly. How will our oney grow? The 3% interest is an annual percentage rate (APR) the total interest to be paid during the year. Since interest is being paid onthly, each onth, we will earn 3%/12 =.25% per onth. So in the first onth, P = \$1 r =.25 (.25%) I = \$1(.25) = \$2.5 A = \$1 + \$2.5 = \$12.5 So in the first onth, we will earn \$2.5 in interest, raising our account balance to \$12.5. In the second onth,

3 Finance 113 P = \$12.5 r =.25 (.25%) I = \$12.5 (.25) = \$2.51 (rounded) A = \$ \$2.51 = \$15.1 Notice that in the second onth we earned ore interest than we did in the first onth. This is because we earned interest not only on the original \$1 we deposited, but we also earned interest on the \$2.5 of interest we earned the first onth. This is the key advantage that copounding of interest gives us. Calculating out a few ore onths: Month Starting balance Interest earned Ending Balance To find an equation to represent this, if P represents the aount of oney after onths, then P = \$1 P = (1+.25)P -1 To built an explicit equation for growth, P = \$1 P 1 = 1.25P = 1.25 (1) P 2 = 1.25P 1 = 1.25 (1.25 (1)) = (1) P 3 = 1.25P 2 = 1.25 ( (1)) = (1) P 4 = 1.25P 3 = 1.25 ( (1)) = (1) Observing a pattern, we could conclude P = (1.25) (\$1) Notice that the \$1 in the equation was P, the starting aount. We found 1.25 by adding one to the growth rate divided by 12, since we were copounding 12 ties per year. Generalizing our result, we could write

4 114 r P = P 1+ k In this forula: is the nuber of copounding periods (onths in our exaple) r is the annual interest rate k is the nuber of copounds per year. While this forula works fine, it is ore coon to use a forula that involves the nuber of years, rather than the nuber of copounding periods. If N is the nuber of years, then = N k. Making this change gives us the standard forula for copound interest: Copound Interest N k r PN = P 1+ k In this forula: P N is the balance in the account after N years. P is the starting balance of the account (also called initial deposit, or principal) r is the annual interest rate (in decial for. Exaple: 5% =.5) k is the nuber of copounding periods in one year. If the copounding is done annually (once a year), k = 1. If the copounding is done quarterly, k = 4. If the copounding is done onthly, k = 12. If the copounding is done daily, k = 365. The ost iportant thing to reeber about using this forula is that it assues that we put oney in the account once and let it sit there earning interest. If you re unsure how to raise nubers to large powers, see the Using Your Calculator section at the end of the chapter. Exaple 1. A certificate of deposit (CD) is savings instruent that any banks offer. It usually gives a higher interest rate, but you cannot access your investent for a specified length of tie. Suppose you deposit \$3 in a CD paying 6% interest, copounded onthly. How uch will you have in the account after 2 years? In this exaple, P = \$3 (the initial deposit) r =.6 (6%) k = 12 (12 onths in 1 year) N = 2, since we re looking for P 2 So P = = \$ (round your answer to the nearest penny)

5 Finance 115 Let us copare the aount of oney you will have fro copounding against the aount you would have just fro siple interest Years Siple Interest (\$15 per onth).5% interest copounded each onth. 5 \$39 \$ \$48 \$ \$57 \$ \$66 \$ \$75 \$ \$84 \$ \$93 \$ Account Balance (\$) Years As you can see, over a long period of tie, copounding akes a large difference in the account balance. Exaple 2. You know that you will need \$4, for your child s education in 18 years. If your account earns 4% copounded quarterly, how uch would you need to deposit now to reach your goal? In this exaple, We re looking for P. r =.4 (4%) k = 4 (4 quarters in 1 year) N = 18 P 18 = \$4, In this case, we re going to have to set up the equation, and solve for P = P = P (2.471) 4 P = = \$ So you would need to deposit \$19, now to have \$4, in 18 years. Note on rounding: You ay have to round nubers soeties while working these probles. Try to keep 3 significant digits after the decial place (keep 3 non-zero nubers). For exaple: Round to Round to For ore inforation on rounding, see the Note on Rounding at the end of the chapter.

6 116 Annuities For ost of us, we aren t able to just put a large su of oney in the bank today. Instead, we save by depositing a saller aount of oney fro each paycheck into the bank. This idea is called a savings annuity. This is what ost retireent plans are. An annuity can be described recursively in a fair siple way. Recall that basic copound interest follows fro the relationship r P = 1+ P 1 k For a savings annuity, we siply need to add a deposit, d, to the account with each copounding period: r P = 1+ P 1 + d k Taking this equation fro recursive for to explicit for is a bit trickier than with copound interest. It will be easiest to see by working with an exaple rather than working in general. Exaple: Suppose we will deposit \$1 each onth into an account paying 6% interest. We assue that the account is copounded with the sae frequency as we ake deposits unless stated otherwise. So in this exaple: r =.6 (6%) k = 12 (12 copounds/deposits per year) d = \$1 (our deposit per onth) So.6 P = 1+ P 1+ 1 = ( 1.5) P Assuing we start with an epty account, we can begin using this relationship: P = P1 = + = P = = = ( 1.5) P 1 1 ( ) P ( )( ) ( ) ( ) P ( ) ( ) ( ) ( ) P = = = (1.5) + 1 Continuing this pattern, 2 P ( ) ( ) 1 2 = (1.5) + 1

7 Finance 117 In other words, after onths, the first deposit will have earned copound interest for -1 onths. The second deposit will have earned interest for -2 onths. Last onths deposit would have earned only one onth worth of interest. The ost recent deposit will have earned no interest yet. This equation leaves a lot to be desired, though it doesn t ake calculating the ending balance any easier! To siplify things, ultiply both sides of the equation by 1.5: ( ( ) 1 ( ) 2 ) 1.5 = (1.5) + 1 P Distributing on the right side of the equation gives ( ) ( ) = (1.5) + 1(1.5) P Now we ll line this up with like ters fro our original equation, and subtract each side ( ) ( ) ( ) 1 1.5P = (1.5) P 1 = (1.5) + 1 Alost all the ters cancel on the right hand side, leaving ( ) 1.5P P = Solving for P (( ) ) ( ).5P = P ( ) =.5 Replacing onths with 12N, where N is years gives P N (( ) 12 N ) =.5 Recall.5 was r/k and 1 was the deposit d. 12 was k, the nuber of deposit each year. Generalizing this, we get:

8 118 Annuity Forula N k r d k P = N r k In this forula: P N is the balance in the account after N years. d is the regular deposit (the aount you deposit each year, each onth, etc.) r is the annual interest rate (in decial for. Exaple: 5% =.5) k is the nuber of copounding periods in one year. The copounding frequency is not always explicitly given. But: If you ake your deposits every onth, use onthly copounding, k = 12. If you ake your deposits every year, use yearly copounding, k = 1. If you ake your deposits every quarter, use quarterly copounding, k = 4. Etc. The ost iportant thing to reeber about using this forula is that it assues that you put oney in the account on a regular schedule (every onth, year, quarter, etc.) and let it sit there earning interest. Copound interest: One deposit Annuity: Many deposits. Exaple 1. A traditional individual retireent account (IRA) is a special type of retireent account in which the oney you invest is exept fro incoe taxes until you withdraw it. If you deposit \$1 each onth into an IRA earning 6% interest. How uch will you have in the account after 2 years? In this exaple, d = \$1 (the onthly deposit) r =.6 (6%) k = 12 (since we re doing onthly deposits, we ll copound onthly) N = 2, since we re looking for P 2 Putting this into the equation:

9 Finance 119 P P P P (12) = = (( 24 ) ) (.5) ( ) (.5) ( ) (.5) = = = \$462 So you will have \$46,2 after 2 years. Notice that you deposited into the account a total of \$24, (\$1 a onth for 24 onths). The difference between what you end up with and how uch you put in is the interest earned. In this case it is \$46,2 - \$24, = \$22,2. Exaple 2. You want to have \$2, in your account when you retire in 3 years. Your retireent account earns 8% interest. How uch do you need to deposit each onth to eet your retireent goal? In this exaple, We re looking for d. r =.8 (8%) k = 12 (since we re depositing onthly) N = 3 (3 years) P 3 = \$2, In this case, we re going to have to set up the equation, and solve for d. 3(12).8 d , = d (( 1.667) 1) 2, = (.667) 2, = d( ) 2, d = = \$ So you would need to deposit \$134.9 each onth to have \$2, in 3 years.

10 12 Payout Annuities In the last section you learned about annuities. In an annuity, you start with nothing, put oney into an account on a regular basis, and end up with oney in your account. In this section, we will learn about a variation called a Payout Annuity. With a payout annuity, you start with oney in the account, and pull oney out of the account on a regular basis. Any reaining oney in the account earns interest. After a fixed aount of tie, the account will end up epty. Payout annuities are typically used after retireent. Perhaps you have saved \$5, for retireent, and want to take oney out of the account each onth to live on. You want the oney to last you 2 years. This is a payout annuity. The forula is derived in a siilar way as we did for savings annuities. The details are oitted here. Payout Annuity Forula N k r d k P = r k P is the balance in the account at the beginning (starting aount, or principal). d is the regular withdrawal (the aount you take out each year, each onth, etc.) r is the annual interest rate (in decial for. Exaple: 5% =.5) k is the nuber of copounding periods in one year. N is the nuber of years we plan to take withdrawals Like with annuities, the copounding frequency is not always explicitly given, but is deterined by how often you take the withdrawals. The ost iportant thing to reeber about using this forula is that it assues that you take oney fro the account on a regular schedule (every onth, year, quarter, etc.) and let the rest sit there earning interest. Copound interest: One deposit Annuity: Many deposits. Payout Annuity: Many withdrawals Exaple 1. After retiring, you want to be able to take \$1 every onth for a total of 2 years fro your retireent account. The account earns 6% interest. How uch will you need in your account when you retire? In this exaple, d = \$1 (the onthly withdrawal) r =.6 (6%) k = 12 (since we re doing onthly withdrawals, we ll copound onthly) N = 2, since were taking withdrawals for 2 years

11 Finance 121 We re looking for P ; how uch oney needs to be in the account at the beginning. Putting this into the equation: P = P = 2(12) 24 ( ( ) ) (.5) ( ) (.5) P = = \$139,6 So you will need to have \$139,6 in your account when you retire. Notice that you withdrew a total of \$24, (\$1 a onth for 24 onths). The difference between what you pulled out and what you started with is the interest earned. In this case it is \$24, - \$139,6 = \$1,4 in interest. Exaple 2. You know you will have \$5, in your account when you retire. You want to be able to take onthly withdrawals fro the account for a total of 3 years. Your retireent account earns 8% interest. How uch will you be able to withdraw each onth? In this exaple, We re looking for d. r =.8 (8%) k = 12 (since we re withdrawing onthly) N = 3 (3 years) P = \$5, In this case, we re going to have to set up the equation, and solve for d. 3(12).8 d , = d ( 1 ( 1.667) ) 5, = (.667) 5, = d( ) 5, d = = \$ So you would be able to withdraw \$3,67.21 each onth for 3 years.

12 122 Loans In the last section, you learned about payout annuities. In this section, you will learn about conventional loans (also called aortized loans or installent loans). Exaples include auto loans and hoe ortgages. These techniques do not apply to payday loans, add-on loans, or other loan types where the interest is calculated up front. One great thing about loans is that they use exactly the sae forula as a payout annuity. To see why, iagine that you had \$1, invested at a bank, and started taking out payents while earning interest as part of a payout annuity, and after 5 years your balance was zero. Flip that around, and iagine that you are acting as the bank, and a car lender is acting as you. The car lender invests \$1, in you. Since you re acting as the bank, you pay interest. The car lender takes payents until the balance is zero. Loans Forula r d k P = r k N k Like before, the copounding frequency is not always explicitly given, but is deterined by how often you ake payents. The ost iportant thing to reeber about using this forula is that it assues that you ake loan payents on a regular schedule (every onth, year, quarter, etc.) and are paying interest on the loan. Copound interest: One deposit Annuity: Many deposits. Payout Annuity: Many withdrawals Loans: Many payents In this forula: P is the balance in the account at the beginning (aount of the loan). d is your loan payent (your onthly payent, annual payent, etc) r is the annual interest rate (in decial for. Exaple: 5% =.5) k is the nuber of copounding periods in one year. N is the length of the loan, in years Exaple 1. You can afford \$2 per onth as a car payent. If you can get an auto loan at 3% interest for 6 onths (5 years), how expensive of a car can you afford (in other words, what aount loan can you pay off with \$2 per onth)?

13 Finance 123 In this exaple, d = \$2 (the onthly payent) r =.3 (3%) k = 12 (since we re doing onthly payents, we ll copound onthly) N = 5, since we re aking onthly payents for 5 years We re looking for P, the starting aount of the loan P = P = ( ( 6 ) ) (.25) ( ) (.25) 5(12) P = = \$11,12 So you can afford a \$11,12 loan. You will pay a total of \$12, (\$2 per onth for 6 onths) to the loan copany. The difference between the aount you pay and the aount of the loan is the interest paid. In this case, you re paying \$12,-\$11,12 = \$88 interest total. Exaple 2. You want to take out a \$14, ortgage (hoe loan). The interest rate on the loan is 6%, and the loan is for 3 years. How uch will your onthly payents be? In this exaple, We re looking for d. r =.6 (6%) k = 12 (since we re paying onthly) N = 3 (3 years) P = \$14, (the loan aount) In this case, we re going to have to set up the equation, and solve for d. 3(12).6 d , = d ( 1 ( 1.5) ) 14, = (.5) 14, = d( ) 14, d = = \$

14 124 So you will ake payents of \$ per onth for 3 years. You re paying a total of \$32,173.2 to the loan copany (\$ per onth for 36 onths). You are paying a total of \$32, \$14, = \$162,173.2 in interest over the life of the loan. Which equation to use? When presented with a finance proble (on an exa or in real life), you're usually not told what type of proble it is or which equation to use. Here are soe hints on deciding which equation to use based on the wording of the proble. The easiest types of proble to identify are loans. Loan probles alost always include words like: "loan", "aortize" (the fancy word for loans), "finance (a car)", or "ortgage" (a hoe loan). Look for these words. If they're there, you're probably looking at a loan proble. To ake sure, see if you're given what your onthly (or annual) payent is, or if you're trying to find a onthly payent. If the proble is not a loan, the next question you want to ask is: "A I putting oney in an account and letting it sit, or a I aking regular (onthly/annually/quarterly) payents or withdrawals?" If you're letting the oney sit in the account with nothing but interest changing the balance, then you're looking at a copound interest proble. The exception would be bonds and other investents where the interest is not reinvested; in those cases you re looking at siple interest. If you're aking regular payents or withdrawals, the next questions is: "A I putting oney into the account, or a I pulling oney out?" If you're putting oney into the account on a regular basis (onthly/annually/quarterly) they're you're looking at a basic Annuity proble. Basic annuities are when you are saving oney. Usually in an annuity proble, your account starts epty, and has oney in the future. If you're pulling oney out of the account on a regular basis, then you're looking at a Payout Annuity proble. Payout annuities are used for things like retireent incoe, where you start with oney in your account, pull oney out on a regular basis, and your account ends up epty in the future. Reeber, the ost iportant part of answering any kind of question, oney or otherwise, is first to correctly identify what the question is really asking, and to deterine what approach will best allow you to solve the proble.

15 Finance 125 Using your Calculator When we need to calculate soething like 5 3 it is easy enough to just ultiply 5 5 5=125. But when we need to calculate soething like , it would be very tedious to calculate this by ultiplying 1.5 by itself 36 ties! So to ake things easier, we can harness the power of our scientific calculators. Most scientific calculators have a button for exponents. It is typically either labeled like: ^, y x, or x y. To evaluate we'd type 1.5 ^ 36, or 1.5 y x 36. Try it out - you should get soething around With soe probles, you need to raise nubers to negative powers. Most calculators have a separate button for negating a nuber that is different than the subtraction button. Soe calculators label this (-), soe with +/-. The button is often near the = key or the decial point. If your calculator displays operations on it (typically a calculator with ultiline display), to calculate you'd type soething like: 1.3 ^ (-) 1 If your calculator only shows one value at a tie, then usually you hit the (-) key after a nuber to negate it, so you'd hit: 1.3 y x 1 (-) = Give it a try - you should get = A note on rounding It is iportant to be very careful about rounding when calculating things with exponents. In general, you want to keep as any decials during calculations as you can. Be sure to keep at least 3 significant digits (nubers after any leading zeros). So rounding to.123 will usually give you a close enough answer, but keeping ore digits is always better. To see why, suppose you were investing \$1 at 5% interest copounded onthly for 3 years. P = \$1 (the initial deposit) r =.5 (5%) k = 12 (12 onths in 1 year) N = 3, since we re looking for P 3 If we first copute r/k, we find.5/12 =

16 126 Here is the effect of rounding this to different values: r/k rounded to: Gives P 3 to be: Error.4 \$ \$ \$ \$ \$ \$ \$ \$ \$ \$.6 no rounding \$ If you re working in a bank, of course you wouldn t round at all. But for our purposes, the answer we got by rounding to three significant digits is close enough - \$5 off of \$45 isn t too bad. Certainly keeping that fourth decial place wouldn t have hurt.

17 Finance 127 Exercises Skills 1. A friend lends you \$2 for a week, which you agree to repay with 5% interest. How uch will you have to repay? 2. Suppose you obtain a \$3, T-note with a 3% annual rate, paid quarterly, with aturity in 5 years. How uch interest will you earn? 3. A T-bill is a type of bond that is sold at a discount over the face value. For exaple, suppose you buy a 13-week T-bill with a face value of \$1, for \$9,8. This eans that in 13 weeks, the governent will give you the face value, earning you \$2. What interest rate have you earned? 4. Suppose you are looking to buy a \$5 face value 26-week T-bill. If you want to earn at least 1% interest, what is the ost you should pay for the T-bill? 5. You deposit \$3 in an account earning 5% interest copounded annually. How uch will you have in the account in 1 years? 6. How uch will \$1 deposited in an account earning 7% interest copounded annually be worth in 2 years? 7. You deposit \$2 in an account earning 3% interest copounded onthly. a. How uch will you have in the account in 2 years? b. How uch interest will you earn? 8. You deposit \$1, in an account earning 4% interest copounded onthly. a. How uch will you have in the account in 25 years? b. How uch interest will you earn? 9. How uch would you need to deposit in an account now in order to have \$6, in the account in 8 years? Assue the account earns 6% interest copounded onthly. 1. How uch would you need to deposit in an account now in order to have \$2, in the account in 4 years? Assue the account earns 5% interest. 11. You deposit \$2 each onth into an account earning 3% interest copounded onthly. a. How uch will you have in the account in 3 years? b. How uch total oney will you put into the account? c. How uch total interest will you earn? 12. You deposit \$1 each year into an account earning 8% copounded annually. a. How uch will you have in the account in 1 years? b. How uch total oney will you put into the account? c. How uch total interest will you earn?

18 Jose has deterined he needs to have \$8, for retireent in 3 years. His account earns 6% interest. a. How uch would you need to deposit in the account each onth? b. How uch total oney will you put into the account? c. How uch total interest will you earn? 14. You wish to have \$3 in 2 years to buy a fancy new stereo syste. How uch should you deposit each quarter into an account paying 8% copounded quarterly? 15. You want to be able to withdraw \$3, each year for 25 years. Your account earns 8% interest. a. How uch do you need in your account at the beginning b. How uch total oney will you pull out of the account? c. How uch of that oney is interest? 16. How uch oney will I need to have at retireent so I can withdraw \$6, a year for 2 years fro an account earning 8% copounded annually? a. How uch do you need in your account at the beginning b. How uch total oney will you pull out of the account? c. How uch of that oney is interest? 17. You have \$5, saved for retireent. Your account earns 6% interest. How uch will you be able to pull out each onth, if you want to be able to take withdrawals for 2 years? 18. Loren already knows that he will have \$5, when he retires. If he sets up a payout annuity for 3 years in an account paying 1% interest, how uch could the annuity provide each onth? 19. You can afford a \$7 per onth ortgage payent. You ve found a 3 year loan at 5% interest. a. How big of a loan can you afford? b. How uch total oney will you pay the loan copany? c. How uch of that oney is interest? 2. Marie can affort a \$25 per onth car payent. She s found a 5 year loan at 7% interest. a. How expensive of a car can she afford? b. How uch total oney will she pay the loan copany? c. How uch of that oney is interest? 21. You want to buy a \$25, car. The copany is offering a 2% interest rate for 48 onths (4 years). What will your onthly payents be? 22. You decide finance a \$12, car at 3% copounded onthly for 4 years. What will your onthly payents be? How uch interest will you pay over the life of the loan?

19 Finance You want to buy a \$2, hoe. You plan to pay 1% as a down payent, and take out a 3 year loan for the rest. a. How uch is the loan aount going to be? b. What will your onthly payents be if the interest rate is 5%? c. What will your onthly payents be if the interest rate is 6%? 24. Lynn bought a \$3, house, paying 1% down, and financing the rest at 6% interest for 3 years. a. Find her onthly payents. b. How uch interest will she pay over the life of the loan? Concepts 25. Suppose you invest \$5 a onth for 5 years into an account earning 8% copounded onthly. After 5 years, you leave the oney, without aking additional deposits, in the account for another 25 years. How uch will you have in the end? 26. Suppose you put off aking investents for the first 5 years, and instead ade deposits of \$5 a onth for 25 years into an account earning 8% copounded onthly. How uch will you have in the end? 27. Mike plans to ake contributions to his retireent account for 15 years. After the last contribution, he will start withdrawing \$1, a quarter for 1 years. Assuing Mike's account earns 8% copounded quarterly, how large ust his quarterly contributions be during the first 15 years, in order to accoplish his goal? 28. Kendra wants to be able to ake withdrawals of \$6, a year for 3 years after retiring in 35 years. How uch will she have to save each year up until retireent if her account earns 7% interest? Exploration 29. Pay day loans are short ter loans that you take out against future paychecks: The copany advances you oney against a future paycheck. Either visit a pay day loan copany, or look one up online. Be forewarned that any copanies do not ake their fees obvious, so you ight need to do soe digging or look at several copanies. a. Explain the general ethod by which the loan works. b. We will assue that we need to borrow \$5 and that we will pay back the loan in 14 days. Deterine the total aount that you would need to pay back and the effective loan rate. The effective loan rate is the percentage of the original loan aount that you pay back. It is not the sae as the APR (annual rate) that is probably published. c. If you cannot pay back the loan after 14 days, you will need to get an extension for another 14 days. Deterine the fees for an extension, deterine the total aount you will be paying for the now 28 day loan, and copute the effective loan rate.

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