Financial mathematics

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1 Fnancal mathematcs Jean-Luc Bouchot February 19, 2013 Warnng Ths s a work n progress. I can not ensure t to be mstake free at the moment. It s also lackng some nformaton. But at least, for those of you who could not attend a class (never attended?... remember classes are mandatory normally) you get a coarse dea of what s beng done. However nothng works better than dong t; practcng and askng questons when somethng s not clear. You wll not learn playng the drums by watchng random guys playng, don t expect t to be dfferent wth math. Last but not least... Don t be selfsh nor shy, f you see any mstakes (shall they be n the grammar or n the math) n these notes, please let me know (ether personnaly or per emal) so that your fellow students can also learn from correct materal. 1 Smple nterest In case of a smple nterest scheme, the fnal amount takes nto account only the ntal depost or loan and does not consder ntermedate nterests. The smple nterest can be calculated wth the followng formula: I = P rt where I represents the nterest (n dollar), P s called the prncpal and corresponds to the orgnal amount (your nvestemnt or what has been borrowed, n dollars), r corresponds to the yearly rate (n percent per year) and t s the duraton of the nvestement/loan (n year). Let us consder a concrete example to make t easer to understand. Assume you borrow $5000 to furnsh a new flat at a yearly rate of $4 and plan to remburse n 30 months. In that case, the prncpal s P =$5000, the tme t = 30/12 (remember, t s expressed n years and a years corresponds to 12 months) and the rate s Hence the total nterests wll be I =$500. Hence you would have to remburse $5500 after 30 months. The fnal amount you would have to remburse s called the maturty value or future value. It s composed of the orgnal prncpal and the nterest calculated from them, hence A = P + I = P + P rt = P (1 + rt) Example 1 (Fndng nterest and maturty values). Fnd the nterest and the maturty value on a $2000 loan for 6 month wth a 6% smple nterest rate. We apply the prevous formulas: I = P rt wth P = 2000, r = 0.06 and t = 6/12 = 0.5. Hence I =$60. The maturty value s the sum of the prncpal P and the nterest I hence A = P + I =$2060. Example 2 (Fndng rates). A loan charges $30 n smple nterest on a prncpal of $1600 whch were borrowed for 9 months. What s the rate of that loan? We wll once agan use ths formula: I = P rt but n ths case we know P = 1600, I = 30 and t = 0.75 and we need to solve for r. Hence we get r = I P t = = = 2.5% 1

2 MATH-101: Intro Analyss 1 Interests-Annutes Example 3 (Fndng the duraton). What s the term of a loan of $4000 at 5% smple nterest f the total due s $4100? We use A = P (1 + rt) where we know A = 4100, P = 4000 and r = 0.05 and solve for t: t = A P 1 = r = Exercse 1. Fnd the nterest f you borrow $3000 for 36 months at a smple yearly rate of 5% Exercse 2. If you borrowed a prncpal of $15000 and remburse a total of $20000 after 5 years at smple nterest. Fnd the nterest and the rate. Exercse 3. Fnd the nterest f $3000 are borrowed and f the nterest after 10 months s $200. How much should be pad back after 10 months? 2 Compound nterests Actually, nterests should generate themselves more nterests through the years; ths s why the noton of compound nterest s developped here. 2.1 Motvaton As an example, consder the case when you depost $1000 on an account that generates an nterest at a rate of $4 yearly. By the end of year 1, f you don t do anythng, you would have = 1040 on your savng account. And ths amount wll be consdered for the followng year and hence, by the end of year 2 you would have = The questons that arse here are: 1. If you don t do anythng how much would you have on your account after say 10 years? 2. How long should you wat to double your money? Of course we could sequentally calculate everythng to fnd the answer... but we can try (and wll succeed) to do smarter than that. 2.2 Compound nterests If we consder the example above, we have that the future value after 2 years s A = = 1040 (1+0.04) = ( ) (1+0.04) = 1000 (1+0.04) (1+0.04) = 1000 (1+0.04) 2 In other terms, f we consder the prncpal value P =$1000, the yearly rate r = 0.04 and the number of years t = 2 the followng formula holds true: A = P (1 + r) t Now t s common (at least n France) that the savng accounts are compounded twce a month at a yearly rate of around 2%. In ths case, the yearly rate r s dstrbuted unformly nto an nterest rate per perod. If m denotes the number of perod (= 24 n our case), we get by dvdng the yearly rate by the number of perods per year, hence = r m = and the number of compoundng perods becomes n and can be calculated as the number of perod per year multpled by the number of years the nvestment runs: n = mt. In that case, the formula reads, as a functon of the tme t (choose whchever you prefer) A(t) = P (1 + ) n = P (1 + r m )mt note that the fnal value s wrtten as a functon of the tme but the tme s n general fxed hence the usual notaton of A (wthout free varable). 2

3 MATH-101: Intro Analyss 1 Interests-Annutes Example 4 (Calculatng nterests and maturty values). A small busness borrows $50000 at 8% compound bmonthly. The loan s due n 5 years. How much would ths company have to repay? How much nterest do they pay? We apply the prevous formula wth P = 50000, r = 0.08, m = 24 and t = 5. Hence the fnal value s A =$ The nterests correspond to the dfference between what s repad and what was borrow: I = A P =$ Exercse 4. Assume you want to nvest $2000 you have for 3 years. A bank makes you two offers: ether an nterest compounded annually at a rate of 4.25% or an nterest of 4% compounded twce a month. Whch opton should you choose to maxmze your benefts? what s the amount of the nterests earned n both schemes? Assume the bank would gve you another choce: they offer you to nvest your money for three years at a smple nterest of 4%. How much would you have by the end of the 3 years? 2.3 Effectve rates If we consder an nvestment of $1000 over a year compounded quaterly at a yearly rate of $4%. By the end of year one, we have accumulated (n = 4, = 0.01, P = 1000) $ (note that t does not make any sense to go after two decmals n practcal applcatons...). Now we can actually compare the actual rate wth the one we had prevously: r E = = = 4.06% Ths rate s called the effectve rate and usually wrtten wth a subscrpt E to dfferentate: r E. Its computaton s done usng the followng formula for an nterest compounded m tmes a year: r E = ( 1 + r ) m A(1) 1 = m P 1 The effectve rate can be seen as the rate that a smple nterest should have to be as frutful as the compound one. 2.4 Playng wth the varables Remember the formula A = P (1 + ) n There are four varables n there. Whenever we are gven three of them, we can recover the fourth one. Hence, f we are local for the present value P, or the number of perods or the rate per perod, we can equvalently use one of these formulas: P = A (1 + ) n n = log b(a) log b (P ) log b (1 + ) ( ) 1/n A = 1 P note that you are free to choose whatever logarthm you prefer, but once you have chosen one, you have to keep the same for all members. The two last ones do not appear often, but the frst one s of prme nterest! Example 5 (Calculatng terms). What s the term of a $2100 loan at 6% nterest compounded monthly wth $2400 due at repayment? we appy the prevous formula wth P = 2100, A = 2400, r = 0.06 and m = 12. And we get t = 2.23 or around 2 years and trhee months. 3

4 MATH-101: Intro Analyss 1 Interests-Annutes Exercse 5. Assume you nvest $2000 for 5 years n a prvate company where the nterests are pad quaterly. After these 5 years, they pay back $2865. What s the annual nterest rate and the nterest rate over a perod? Exercse 6. After 3 years of an nvestments compounded quaterly at a year rate of 5% the nvestments pays back $3567. What s the present value? How much s the effectve rate? How long should you wat to double your actual value? 2.5 Contnuous nterests As a motvatng example consder the case where you can choose the number of perod per year but the yearly rate s fxed. For nstance, consder r = 4%. We consder the present value P =$1000 and an nvestment over a full year. Compoundng type m A r E Yearly Byearly Quaterly Monthly Bmonthly Weekly Daly Every hour Every mnute 525, Notce that as the number of perod becomes larger and larger the value tends to stablze. It turns out that ths value tends to be get closer and closer to P e r (= n our case). Ths s a general fact and we can show that, for contnuous compoundng we have the followng formula A(t) = P e rt wth r the yearly rate and t the number of years. In case of a contnuous nterest, the effectve rate can be calculated by comparng the status at the end of year 1 wth the status at the begnnng; here t yelds r E = e r 1 Exercse 7. Suppose an orgnal amount of $5000 s deposted at a contnuous rate of 6% yearly and stays n the bank for 12.5 years. What s the compound amount and the nterest earned? What s the effectve rate? How long should you have to wat to get to a fnal value of $15000 (or n other terms, multply by three your nvestment)? 3 Annutes 3.1 Motvaton Assume you want to buy a new car but don t want to borrow from the bank. You would need to nvest you money pror to buyng your car. You decde to nvest $100 every month on a savng account payng $4 % compounded monthly. After 2 years of savng, do you have enough money to buy a $5000 car? And after 4 years? Ths problem could be solve easly n that case by calculatng the nterests one year after another, but ths could get cumbersome whenever the rates are compounded monthly over 30 years, say. That would make you calculate 3600 values... 4

5 MATH-101: Intro Analyss 1 Interests-Annutes 3.2 Geometrc sequence Defnton 1 (Geometrc sequence). A geometrc sequnce s a sequence of numbers n whch each number can be calculated knowng the prvous one by multplyng by a certan constant. Example 6. The sequence of number 1, 2, 4, 8, 16, 32, s an example of (potentally nfnte) geometrc sere. In ths case 2 (the number to pass from a number to the followng one) s called the common rato. Property 1 (Sum of geometrc sequences). Assume we are gven a fnte number of consecutve elements from a geometrc sequence. The sum of these elements can be computed as S = 1 + X + X 2 + X X n = 1 Xn+1 1 X (1) Exercse 8. Calculate the sum Exercse 9. Calculate the sum Ordnary annuty Ths theory s used when we do regular depost on a savng account. In comparson to the prevous secton where we would depost a certan amount at the begnng of the contract and see the status of the account at the end of the contract, we wll here contnue every sngle perod to depost a certan predefned amount and add the nterest at the end of each perod. Suppose we nvest $D every perod (say month) on a account wth an earnng rate per perod. So every month (or perod...) we have: 1. R 2. R(1 + ) (what we had plus nterest) +R (as a new depost ths month) 3. R(1 + ) 2 (ntal depost and two nterest perods) +R(1 + ) (depost and nterest of the prevous month) +R (new depost) n R(1 + ) n 1 + R(1 + ) n R(1 + ) + R = R ( (1 + ) n 1 + (1 + ) n (1 + ) + 1 ) And hence the total amount by the end of the n th perod can be calculated as the sum of a geometrc sequence: R 1 (1+)n 1 (1+) = R (1+)n 1 wth = r/m the rate per perod and n = mt the total number of perods. Example 7. We want to solve the ntroductory example now. $100 are saved everey month whch corresponds to the R. = r/m = 0.04/12 and t = 2 years n the frst case and t = 4 n the second. Hence, for the frst case, the future value s S =$ and n the second case S =$5196. Therefore t s possble to buy the car after 4 years of savng but not after 2. Exercse 10. A 45 year old woman puts $1000 on a retrement plan at the end of each month untl her 60 th annversary and then let the plan by tself (.e. she doesn t depost anymore). Assume the account pays 6% compounded monthly. How much wll be n the account when she turns 65? Snkng funds s a term dsgnatng a savng account desgned to provde a gven amount after a certan tme. In other words, n ths case, we know when the end of a savng account occurs but have no dea when how much we should depost every perod to reach a gven amount on that account. Another way to see t (from a mathematcal perspectve) means that we know the end value and are lookng to solve everythng for R: R = S (1 + ) n (2) 1 5

6 MATH-101: Intro Analyss 1 Interests-Annutes Example 8. A man needs $10000 n 8 years. How much should he depost at the end of each quarter nto a snkng fund whch pays 8% nterest compounded quaterly to reach ths goal? we apply the prevous formula wth S = 10000, = 0.08/4 = 0.02 and n = 8 4 = 32 and hence R =$ per quarter. 4 Present value of an ordnary annuty 4.1 Introducton - Motvaton We are here nterested n knowng what the depost would be f we were to consder only a compounded nterest nstead of an annuty. In other words, f two people, one makng a sngle depost, and another one makng regular deposts, want to have the same amount at a gven tme, how hgh should be the deposts. If we assume that both account are compounded n tmes at a rate per perod of, than the maturty value n the frst case s P (1 + ) n and n the second case, we have R (1+)n 1 and we want both values to be equal to each other hence : P (1 + ) n = R (1 + )n 1 P = R(1 + ) n (1 + )n 1 P = R (1 + )n (1 + ) n 1 (1 + ) n 1 (1 + ) n P = R multply by (1 + ) n Or equvalently we have R = P 1 (1 + ) n The reason why t s nterestng to know these values s because ths relatonshp reflects the case of a loan one needs to amortze. 4.2 Amortzng a loan Consder the followng stuaton. You need to borrow $P for a loan to by a house and decde to remburse after n payments (for nstance n = 120 for 10 years payng every month). If we consder that the loan has an nterest per perod compounded perodcally (ether monthly, weekly, etc... dependng on the stuatons), we have the followng stuaton: 0 You borrow $P from the bank 1 At the end of the frst perod, you make a payment of $R. However, on the other sde, the bank charges the nterests for the prevous month. Hence you owe the bank $P (1 + ) R 2 You now make a new payment and have the nterests from the prevous perod to pay; you stll owe (P (1 + ) R) (1 + ) R = P (1 + ) 2 R(1 + ) R 3 By the end of perod number 3, you stll owe P (1 + ) 3 R ( (1 + ) 2 + (1 + ) + 1 )... And so on for a couple of perods n By the end of the last perod you stll owe the bank P (1 + ) n R ( (1 + ) n 1 + (1 + ) n (1 + ) + 1 ) 6

7 MATH-101: Intro Analyss 1 Interests-Annutes and f we have a look at the last perod, we should be done payng off the loan. Hence the amount due should equal 0. Moreover, we can notce that the factor next to R s the sum of a geometrc sere and hence we get P (1 + ) n R ( (1 + ) n 1 + (1 + ) n (1 + ) + 1 ) = 0 P (1 + ) n = R ( (1 + ) n 1 + (1 + ) n (1 + ) + 1 ) P (1 + ) n = R (1 + )n 1 and so we get the relatonshp between P and R as ntroduced n the prevous secton. 5 Examples and exercses 7