Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part of ths revew wll gve you varous sample problems to work on you should know how to do all these sample problems for Exam II. The sample exam s posted to the Math 113 webpage please come to the Math Resource Center f you need help! The sum of money you depost nto a savngs account or borrow from a bank s called the prncpal. The fee to borrow money s called nterest. When you borrow money you pay back the prncpal and nterest to your lender. When you depost money nto a savngs account or other nvestment, the bank pays you back the prncpal and the nterest earned. Interest s calculated as a percent of the money borrowed, a percent of the prncpal. Dependng on the type of loan, nterest can be calculated n a varety of ways. Smple Interest Loans (Secton 5.1) : Smple Interest nvolves a sngle payment and the nterest computed on the prncpal only. The equaton for smple nterest s a lnear functon. If the problem refers to a smple nterest rate, then you know you need to use smple nterest rate formulas. Equaton #1: I = Pr t where : I = the amount of nterest pad for borrowng the money P= the prncpal or the amount of money you borrowed from the bank r = s the smple nterest rate ths s a per annum rate (.e. yearly) t = the amount of tme the money s borrowed for ths needs to be n years f the problem gves you t n months then you need to dvde the number of months by to convert t nto years. Equaton #2: A = (P+Prt)=P*(1+rt) A = the future value - the total amount the borrower owes at the end of the loan perod ths s Prncpal plus Interest. I = the amount of nterest pad for borrowng the money P= the prncpal or the amount of money you borrowed from the bank r = s the smple nterest rate ths s a per annum rate (.e. yearly) t = the amount of tme the money s borrowed for ths needs to be n years f the problem gves you t n months then you need to dvde the number of months by to convert t nto years. Equaton #3: I = A P A = the total amount you owe at the end of the loan perod ths s Prncpal plus Interest. I = the amount of nterest pad for borrowng the money 1
P= the prncpal or the amount of money you borrowed from the bank Problems wth smple nterest n these problems you need to dentfy what you are gven and what you are solvng for: 1. Fnd the nterest due on each loan f $100 s borrowed for 6 months at an 8% smple nterest rate. For ths problem you need to solve for the nterest owed on the loan or I. Use Equaton #1: I = Prt P = Prncpal or the amount of the loan = $100 r = the smple nterest rate = 8% per year t= tme of the loan BUT ths must be n years so, t = (8/) I = Prt = (100)*(.08)*(8/) = $5.33 2. Fnd the smple nterest rate for a loan where $500 s borrowed and the amount owed after 8 months s $600. Ths s a smple nterest problem where you are solvng for the smple nterest rate or r. You can use Equaton #1 and solve for r. If I = Prt, then r = I Pt You are gven t = amount of tme the money s borrowed = 8 months BUT you need to convert ths to years by dvdng by so, t = (8/). Then you need to calculate I. You are gven P (the prncpal,.e. the amount borrowed) ths equals $500 and you are also gven A the amount owed at the end of the loan. A = $600. You can use equaton #3 to calculate I. I = A-P = 600 500 = 100 then, r I = Pt 100 = 8 500*( ) =.30 = 30% 3. What s the term of a loan where $600 was borrowed at a smple nterest of 8% and the nterest pad on the loan was $156. You know you need to use smple nterest formulas and n ths problem you need to solve for t the term of the loan or the amount of tme you borrow the money for. You can use Equaton #1 and solve for t. If I = Prt, then t = I Pr, t wll be n years. 2
Ths problem s pretty straght forward, you are gven I = nterest pad = $156, P= amount borrowed = $600, and r = smple nterest rate = 8%, so I 156 t = = = 3.25years. Pr 600*.08 Dscounted Loans: A Dscounted Loan s a loan where the lender deducts the nterest due on the loan from the amount of money borrowed pror to gvng the money to you. For nstance, you may want to borrow $1000 from a bank f the loan s dscounted the bank does not gve you $1000, they gve you $1000 mnus the nterest due, for example say the nterest on ths loan was $100 you would receve $900 from the bank. However you owe the bank $1000 at the end of the loan perod. Key words to dentfyng a loan as a dscounted loan are: dscounted loan and proceeds. Equaton #4: R=L-Lrt = L*(1-rt) R= the amount of money the bank gves you or the proceeds L = the amount you need to pay back to the bank r = the dscounted nterest rate, ths s a yearly or per annum rate t = the amount of tme you have to pay the back the bank ths s n years Lrt = the dscount or the amount deducted from the loan the nterest on the loan. You can solve Equaton #4 for L, for t or for r. Problems wth dscounted loans agan n these problems you need to dentfy what you are gven and what you are solvng for: 1. A borrower sgns a note for a dscounted loan and agrees to pay the lender $1000 n 9 months at a 10% rate of nterest. How much does the borrower receve? For ths problem you need to use equaton #4 to solve for R the proceeds or the amount of money the borrower receves. You are gven L = the amount the borrower owes the bank = $1000, r = 10%, and t = 9 months = (9/) years, so: R = L*(1-rt) = $1000*(1-(.10)*(9/)) = $925 2. You wsh to borrow $10,000 for 3 months. If the person you are borrowng from offers a dscounted loan at 8%, how much must you repay at the end of the loan? For ths problem you use equaton #4 but now you are solvng for L, the amount you need to pay back to the bank. You are gven R the proceeds or the amount you receve from the bank = $10,000, r = the nterest rate = 8%, and t = the tme you have to pay back the loan = 3 months you need to convert ths to years = (3/) years, so: 3
If R = L*(1-rt), then: R L = (1 rt) 10000 = (1 (.08) *( 3 )) = $10,204.08 3. How much must you repay the bank f the proceeds of your loan are $1500 for 18 months at 10%? You need to solve for L= the loan amount or the amount or money you need to repay the bank. You are gven the value of R the proceeds = the amount of money you get from the bank = $1500. You need to use equaton #4 and solve for L. R 1500 L = = = $1, 764.71 (1 rt) 18 (1 (.10)*( )) Compounded Interest (Secton 5.2) If at the end of a payment perod the nterest s renvested at the same rate, then over the next payment perod you wll earn nterest on the nterest from the frst payment perod plus the orgnal prncpal. Interest pad on nterest renvested s called compound nterest. The formula for compound nterest s an exponental functon, so f the rates and lengths of the loan are the same a compounded nvestment wll earn more money than a smple nterest nvestment. Compounded nterest s wth loans here you owe money so you look for a loan wth lower nterest rate and fewer compound perods per year. Compound nterest s also used wth savngs accounts and CDs here you are earnng money on the nterest from the last payment perod so you want a hgher nterest rate and multple compound perods. The followng payment perods (number tmes nterest s compounded per year) apply to compounded nterest problems: Annually Once per Year (m = 1) Semannually Twce per year (m = 2) Quarterly 4 tmes per year (m = 4) Monthly tmes per year (m = ) Daly 365 tmes per year (m = 365) 4
Equaton #5 Compound Interest Formula: A = P(1+) n A = the total amount owed (nterest plus prncpal) or the total amount earned on a savngs account or CD (prncpal plus nterest) after n payment perods. P= the prncpal or orgnal amount of the loan or nvestment r = the nterest rate for the compound perod s calculated by dvdng the annual nterest rate m by the number of compound perods n a year. n = the number of deposts made for the duraton of the annuty (m * t) m= the number of compound perods n a year t = length of the annuty n years Compound nterest problems usng ths formula nvolve a sngle payment and the amount of nterest earned over the length of the nvestment/loan. Problems wth Compounded Interest: 1. If $5000 s nvested at an annual rate of nterest of 10%, what s the amount after 5 years f the compoundng takes place (a) annually (b) monthly or (c) daly? For ths problem you need to use the Compounded Interest formula and solve for A n. The value of the prncpal P remans the same for each problem but the value of and n wll change dependng on the number of compound perods that take place If compounded annually = annual rate/1 and n = 1 * number of years, f compounded monthly = annual rate/ and n = * 5, and f compounded daly = annual rate/365 and n = 365*5. (5 1) n.10 a. A = P(1 = 5000 1 + = $8,052. 55 1 (5 ) n.10 b. A = P(1 = 5000 1 + = $8,226. 54 (5 365) n.10 c. A = P(1 = 5000 1 + = $8,243. 04 365 2. You need $5000 and you need to take out a loan. Whch loan would be the best loan for you to take? The tme perod for all the loans s 1 year. a. A dscounted loan wth an nterest rate of 10.5% b. A loan wth a rate of 10% compounded monthly c. A loan wth a rate of 9.5% compounded daly To solve ths problem you need to compute how much money you would need to payback to the bank wth each type of loan. 5
a. Ths s a dscounted loan and you need to use the dscounted loan equaton and solve for L the amount of money you need to payback to the bank. You need $5000 so ths s the amount of money you want from the bank or your proceeds R = $5000, r =.105, and t =1 R 5000 L = $5,586.59 (1 rt) = (1.105*1) = b. Ths s a compounded nterest loan and you need to use the compounded nterest formula and solve for A n the amount you owe the bank Snce the nterest s compounded monthly for a perod of one year = (.10/) and n =. A = P(1 n.10 = 5000 1 + (1 ) = $5,523.57 c. Ths s a compounded nterest loan and you need to use the compounded nterest formula and solve for A n the amount you owe the bank. Snce the nterest s compounded daly for a perod of one year = (.095/365) and n = 365. (1 365) n.095 A = P(1 = 5000 1 + = $5,498.23 365 Snce the loan that s compounded daly gves us the lowest value for the amount of money you owe the bank (prncpal plus nterest), the best loan for you s loan c. For more examples of problems wth compounded nterest you should revew secton 5.2 n your textbook and be able to do the homework problems n ths secton. Annutes & Snkng Funds (Secton 5.3) An Annuty s a sequence of equal perodc deposts. The perodc payments can be made annually, sem-annually, quarterly, monthly,etc. The amount of the annuty s the sum of all the deposts made PLUS the nterest earned on those deposts. The nterest earned on those deposts s compounded nterest. Wth annutes we are sometmes nterested n the future value of the annuty or sometmes we are nterested n the present value of the annuty. Secton 5.3 deals wth the future value of an annuty. Wth the future value of an annuty deposts are made and the account grows. The equaton used for future value of an annuty s: 6
Equaton #6: Future Value of an Annuty n (1 1 A= P* A = the present value of the annuty ths s the sum of the deposts PLUS the nterest earned on those deposts. P = the amount of the depost or payment for each payment perod r = the nterest rate for the compound perod s calculated by dvdng the annual nterest m rate by the number of compound perods n a year. n = the number of deposts made for the duraton of the annuty (m * t) m= the number of compound perods n a year t = length of the annuty n years Example: Ian pays $150 every month nto an annuty payng 6% compounded monthly. What s the value of hs annuty after 30 deposts? Use equaton #6, where we are solvng for A the amount of the annuty, P= the amount of the deposts = $150, =.06/, and n = 30..06 30 n (1 + ( )) 1 (1 1 A= P* = 150* = $4,842.00.06 ( ) The amount of nterest earned on the deposts can also be calculated. If the deposts dd not earn any nterest the amount n the account after 30 deposts would be 30 *150 or $4,500, so the nterest earned on the deposts = A-nP = 4,842.00-4,500 = $342 Snkng Funds If a person knows they wll owe a certan debt n the future they may decde to depost a certan amount of money nto an account ( monthly, sem-annually, or annually) that s accumulatng nterest (compounded nterest) on the payments n order to have the amount of money needed to settle ther debt by a certan tme perod. Problems wth snkng funds usually requre you to fnd the amount of the depost requred to meet a certan debt. Ths s the P value n the future value of an annuty equaton. You can use the future value of an annuty equaton plug-n what you know and then solve for P or you can rearrange the equaton for the future value of an annuty to solve t for P as shown n Equaton #7. 7
Equaton #7 Future Value of an Annuty solved for P-payment ( use wth Snkng Fund problems) P = A ((1 n 1) Example: The ABC company needs to provde for the payment of a $500,000 debt maturng n 7 years. Contrbutons to the fund are made quarterly and nterest on the account s compounded quarterly at 10%. Fnd the amount necessary to depost on a quarterly bass so the company can pay ther debt. For ths problem you can use the annuty equaton and solve for P the amount of the depost. A = $500,000 the amount of the annuty you need n seven years, = (.10/4) because nterest s compounded monthly, and n = 4*7 = 28. n (1 1 A= P* =.10 28 (1 + ( )) 1 500000 = P* 4 =.10 ( ) 4 500000 = P*39.86 P = $,543.90 For more examples of annutes and snkng funds look n Secton 5.3 of your book you should be able to do the homework problems on pages 288 290. Present Value of an Annuty Amortzaton (Secton 5.4) The present value of an annuty can be thought of as the amount of money needed today to nvest at percent nterest such that n number of wthdrawals can be made over a perod of tme at the end of whch there wll not be any money n the account 8
The equaton for present value of an annuty s: Equaton # 8: V 1 (1 = P* n V= the present value of an annuty = the amount of money you need now or owe now P = the amount of the depost or payment for each payment perod r = the nterest rate for the compound perod s calculated by dvdng the annual nterest m rate by the number of compound perods n a year. n = the number of deposts made for the duraton of the annuty (m * t) m= the number of compound perods n a year t = length of the annuty n years Mr. Smth at the age of 75, can expect to lve for 15 more years. If he can nvest at % per annum compounded monthly, how much does he need now to guarantee hmself $500 every month for the next 15 years? For ths problem you need to use the equaton for present value and solve for V. P = the amount wthdrawn perodcally = $500, I =./ (nterest s compounded monthly), and n = (15*) (monthly payments for 15 years) V. (15*) n 1 (1 + ( )) 1 (1 = P* = 500* = $41, 660.83. ( ) Now try problem # 15 on page 299. We can also have problems were we know the present value of the annuty and want to fnd P. These problems usually consst of a present value loan of V dollars and a perodc payment of P. After makng n payments your loan s pad-off, because the loan s an annuty your payments nclude payments towards the orgnal loan PLUS nterest the cost for borrowng money. Ths s also known as amortzaton. When you amortze a loan you make perodc payments on the loan untl the balance s zero or the debt has been retred. For nstance f you borrowed $20,000 to buy a car, you can amortze the loan over a perod of 4 years by makng monthly payments. The amount of those monthly payments s what you would solve for, to do ths, take the equaton for present value of an annuty and solve for P: 9
Equaton #9 Amortzaton Equaton P= V* (1 (1 ) n + Ths s known as the amortzaton equaton and you use ths equaton when tryng to calculate requred monthly payments to pay off loans such as car loans, mortgage payments, credt cards, determnng nhertance payments, and determnng retrement ncome. Example: Jacque and Jm want to buy ther dream house. If they make a down payment of $75,000 on a house that costs $650,000 and the mortgage rate s 7.5% wth a term of 30 years, what wll there monthly payment be? Use Equaton #8, the amortzaton equaton and solve for P. V = the amount of the loan = $650,000-75,000 = $575,000, =.075/, and n = 30* = 360..075 P= V* 575000* $4, 020.48 n 360 (1 (1 ) = = +.075 1 1+ What s the total amount of nterest pad? Ths s equal to what they pay over thrty years (n*p) mnus the amount of the orgnal loan (V): np V = (360* 4020.48) 575000 = $872,372.80 If Jacque and Jm decde to pay-off ther mortgage after 10 years what amount do they owe? The balance of the unpad loan s equal to the PRESENT VALUE of the remanng payments. Each payment equals $4,020.48 and after ten years of payments Jacque and Jm stll have to make 240 payments (20 years of payments). To fnd the present value we use Equaton #8: 240 1 1+ V = P ( ) n 1 1+.075 = 4020.48*.075 = $499,071 How much nterest wll Jacque and Jm pay f they make ten years of payments and then pay the house off? Frst fnd out how much they pay ten years of payments plus the present value of the remanng 240 payments = 10**4,020.48 + 499,071 = $981,529. The amount of nterest s what they pad mnus the orgnal amount of the loan or 981,529 575,000 = $406,529. So they save $872,372.80 406,529 = $465,843.80 n nterest by payng the loan off n ten years nstead of 30. 10