An Overview of Financial Mathematics

Transcription

1 An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take exam FM/2 jontly offered by the Socety of Actuares and Casualty Actuaral Socety. As such, ths gude wll follow the course objectves for the August 2012 sttng. That beng sad, I hope t wll be of some use to anyone studyng fnancal mathematcs. It s best used n conjuncton wth another more rgorous text. I recommend Mathematcal Interest Theory by Vaaler & Danel. Both the Actex and ASM manuals are very helpful n preparng for exam FM thanks to ther thousands of practce problems. Ths gude covers the tme value of money, annutes, loans, bonds, stocks, nterest rates, and an ntroducton to dervatves, forwards and futures. 1

2 1 Introducton to the Tme Value of Money 1.1 Interest Explaned The central dea of nterest theory s that the same amount of money s worth dfferent amounts at dfferent tmes. In other words, a ratonal person would rather have fve thousand dollars rght now than the promse of fve thousand dollars n three years the ratonal person has the opton to save the money for three years or to do somethng else wth t. Ths or adds value to havng the money now. Because of ths dfference n values a lender who loans money to a borrower can reasonably charge hm for t. The money that the borrower pays to the lender s called nterest. Most of us, much to Polonus dsmay, are both borrowers and lenders. We ve borrowed money to pay for our cars and therefore have to pay some amount of nterest on that loan. At the same tme we ve lent money to the bank (by depostng our money there) and they pay us nterest. There s a reason that the nterest rate, the rate that determnes the amount of nterest owed, s dfferent n our arrangement wth the car dealershp and the bank s arrangement wth us: we are rsker. The dealershp has far reason to worry that I mght splt wthout payng for all of the car or fnd myself bankrupt and unable to pay. The dealershp s tryng to put a prce on the rsk they are acceptng. Therefore, f I am able to convnce the dealer I m a sure thng to pay t all back, I mght be able to get a lower rate. On the other hand, I m not at all concerned that the bank s gong to run away wth my money so I can t reasonably charge them a hgh rate for borrowng my money. However, I am stll assumng some rsk, the bank mght go under, so I do charge some nterest. These are just two examples of hundreds of dfferent ways that people and organzatons can lend money to each other. Ths revew gude s more concerned wth valung the nterest and determnng the value of money at dfferent tmes than workng out what a far nterest rate s or why nterest exsts. Those topcs wll be explored n exams MFE/3F and MLC/3L. For now, just assume a non-negotable nterest rate and don t worry too much about why exactly t exsts. 1.2 Smple and Compound Interest Let s assume that you depost some money nto a savngs account at a bank and plan to leave t there for a whle wthout depostng nto t or wthdrawng from t. The amount of money n the account wll therefore ncrease over tme as the bank adds the nterest payments to the savngs account. Ths phenomenon s called the growth of money or tme value of money. The money can grow n one of two man ways. 2

3 The frst s that each year the bank mght add some percent of the orgnal amount deposted. For example, you mght depost $100 nto the account. Then one year later the bank adds an nterest payment of amount $5 so that the savngs account contans $105. Then one year after that, two years after the orgnal depost was made, the bank adds another $5 nto the account so that the account contans $110. Ths model of growth s called smple nterest and can be represented by the followng equaton: A(t) = K(1 + t) (1) where A, the amount functon, s the value of the account at tme t; K, the prncpal, s the amount orgnally deposted; s the nterest rate governng the account; and t s the tme snce the orgnal depost. In ths example, A(0) = $100, A(1) = $105, A(2) = $110, K = $100, and =.05 (remember, the nterest rate s just that, a rate). Sometmes t s more convenent to deal wth the case n whch just one dollar was orgnally nvested. a(t) = 1 + t (2) Here a s called the accumulaton functon. Note that A(t) = Ka(t). Smple nterest s not used very much n the real world for an obvous reason. Look back over the example and see f you can fgure out why. A savvy nvestor, nstead of dong what the nvestor n the example dd, would take hs $105 and open a brand new account. That way at the end of the second year the bank would add = 5.25 to hs account. Ths nvestor was able to take advantage of the system and get an A(2) equal to $ The banks forseeng ths problem usually offer a savngs account that grows accordng to compound nterest. A(t) = Ka(t) = K(1 + ) t (3) To explore smple and compound nterest n more depth t s necessary that I ntroduce some new vocabulary. The effectve nterest rate, [t1,t 2], tells us the rate that corresponds to the amount of nterest pad n the nterval [t 1, t 2 ]. [t1,t 2] = a(t 2) a(t 1 ) a(t 1 ) = A(t 2) A(t 1 ) A(t 1 ) Note that the nterval [n 1, n] s called the n th perod. Wth these defntons n hand, t s clear that for smple nterest the effectve nterest rate s constantly decreasng whle for compound nterest t s the same regardless of the perod (look back to the example and verfy ths for yourself). 1.3 Nomnal Interest Rates In our example the nterest was pad at the end of each year. But there s no reason that the nterest must be pad yearly. Indeed, t s very often pad monthly, quarterly, or sem-annually. Ideally, we would always be gven the effectve rate for the nterval and the length of the nterval. For example, an (4) 3

4 nvestment mght pay 1% per month (f t doesn t say nterest s smple or compound, t s assumed to be compound), and we would know that at the end of each month our nvestment would be worth 1.01 tmes what t was worth at the begnnng of the month. Unfortunately, ths s not always the case. The nomnal nterest rate s the nterest rate per perod tmes the number of those perods n a year. In our example we would say the nvestment pays nterest at a nomnal rate of 12% convertble/credted/payable monthly. Ths nomnal rate s wrtten (12) = 12% (note the paranthess around the 12, t s not an exponent) and we say upper twelve. Note that f we deposted $100 n ths account we would not have $112 at the end of the year but rather $(100)(1.01)(1.01)...(1.01) = (100)(1.01) 12 = $ We can use formula (4) to determne that ths account pays nterest at an annual EIR of = 12.68%. There s a specfc formula for convertng a nomnal nterest rate to an effectve nterest rate, but I prefer to thnk about what s actually happenng to the money. a(t) = (1 + ) t = ) tm (1 + (m) (5) m You can then rearrange whatever you want to solve for what the problem requres. 1.4 Dscount Rates So far, I ve talked about what happens when an amount of money changes hands at the begnnng of a tme perod and nterest s pad on that money. But sometmes, two partes make an arrangement where the money changes hands at the end of a tme perod nstead. Snce the money s worth more now than n the future, the person gettng the money at the end of the year ought to get a cheap prce on t. In other words, he gets t at a dscounted prce. A smple example best demonstrates the dfference between dscount and nterest rates. The nterest rate s the amount the money grows. The nvestor lends $100 to the bank, the bank pays hm 5% nterest, and at the end of the year the nvestor gets $105. The dscount rate s the rate at whch the money s dscounted. The nvestor lends the bank $95 and at the end of the year the bank pays hm $100. In ths case the dscount rate was 5%. We calculate the effectve dscount rate n a smlar fashon to how we calculated the effectve nterest rate. d [t1,t 2] = a(t 2) a(t 1 ) a(t 2 ) = A(t 2) A(t 1 ) A(t 2 ) In our ntroductory example we had just one dscount tme perod, but there can, of course, be multple perods. Both smple and compound dscountng (6) 4

5 exst. a(t) = 1 1 dt (7) a(t) = (1 d) t (8) Also note that dscount rates can be be wrtten as nomnal rates. The concepts are all equvalent to the nterest rate concepts. The followng equalty wll be useful. ) pt a(t) = (1 d) t = (1 d(p) (9) p The prevous example also shows how nterest and dscount rates are related. A dfferent nvestor at a dfferent bank could have lent the bank $95 and watched that money grow, accordng to some nterest rate, to $100. Be aware that the $95 dd not grow at an nterest rate of 5% but =.0526 or 5.26%. A lttle algebrac manpulaton of equatons (4) and (6) reveals that (1 + )(1 d) = 1 (10) Wth the concept of dscountng we are now well prepared to truly understand the noton of the tme value of money. 1.5 Present and Future Values After readng the prevous example comparng nterest and dscount rates a natural thng to wonder s how much money dd the nvestor have to nvest at the frst bank where the nterest rate was 5% to end the perod wth $100. To fgure ths out, smply use equaton (3) and solve for K. In fact, ths sort of calculaton s done so often that a new functon, called the dscount functon, s used. The dscount functon s, of course, the nverse of the accumulaton functon. v(t) = 1 a(t) (11) To make very clear what values we are referrng to, we use the terms present value and future value. The present value s the value of the money rght now. The future value s the value of that same money plus the accumulated nterest. F V = P V a(t) (12) It should be becomng clear how the dscount rate plays a role n ths. Let s say an nvestor wants hs nvestment to grow to $1 over the course of the year. The 1 present value he has to nvest s 1+. Recall that the dscount rate s the rate that tells us how much the future value s on sale or marked down. Therefore, d = = 1 + (13) 5

6 Ths s smply another way of wrtng what we already algebracally showed n equaton (7). Fnally, we see that 1.6 The Force of Interest d = 1 v (14) Whle generally accounts pay nterest n dscrete ntervals, t s theoretcally possble that an account could pay nterest contnuously. To calculate what the accumulaton functon s under contnuous nterest you can do a couple thngs. One, you can take a lmt of a nomnal rate. Frst, we wll defne the force of nterest, δ, as follows. lm m (m) = ln(1 + ) = δ (15) It s not necessary that you be able to prove t for exam FM, but t s a good proof to know. Wth that defnton and a recollecton of the defnton of e n hand we can fnally calculate the accumulaton functon. lm m ) m (1 + (m) = lm e(m) = e δ (16) m m A completely dfferent approach s to start wth what contnuous nterest actually means. It means that at every pont n tme your nstantaneous change n your account s δ tmes the amount of money currently n the account, snce that s the nterest amount payable. Mathematcally we can wrte: d a(t) = δa(t) (17) dt Ths s just an elementary dfferental equaton wth soluton: a(t) = e δt (18) I ll reassure you now that you don t need to be able to derve the accumulaton functon under contnuously compounded nterest. Smply memorzng the accumulaton functon s suffcent. The last explanaton ntroduced the dea of the nstantaneous change n the account value. Whle solvng the dfferental equaton we see that δ = a (t) a(t). In that case, the force of nterest, δ, was constant. However, n accounts earnng dscrete nterest there s also a force of nterest at all tmes. And, as luck would have t, the formula s the same. The only dfference wll be that δ wll depend on tme,.e., t s not a constant. In general, gven a δ functon n terms of t we can determne what a(t) s. t a(t) = e δ(u)du 0 (19) 6

7 1.7 Bg Equaton The prevous sectons have ntroduced the most common ways that money can grow. All of the nterest theory presented so far can be represented by one bg equalty. a(t) = (1 + ) t = (1 + (m) m ) mt ) pt = (1 d) t = (1 d(p) = e δt (20) p And f, nstead of fndng the future value of some amount of money, you need to fnd a present value use v(t) = 1 a(t). 1.8 Inflaton Inflaton s defned as the natural ncrease n the cost of goods. Inflaton as a concept s complcated and beyond the scope of the overvew, but ts mpact on the value of money s mportant. As a smple example, n 1980 $100 could buy 278 wdgets whle today t can only buy 100 wdgets. It s reasonable to assume that n the future $100 wll buy even fewer wdgets. In other words, nflaton eats away at the value of money. Because of ths, there s an mportant rate, the real nterest rate, that descrbes how the purchasng power of money grows over tme. 1 + j = 1 + (21) 1 + r where j s the real nterest rate, s the nterest rate, and r s the rate of nflaton. Be forewarned that the notaton wll change n the later exams. It s standard exam FM notaton, though. 1.9 Equatons of Value Up to ths pont I have only presented stuatons that nvolve a sngle present value and a sngle future value that can be related by usng a sngle dscount or accumulaton factor. However, the real meat of fnancal mathematcs s puttng these concepts to use n more complcated cash flows. A classc example mght ask us to fnd the present value of three payments assumng a compound nterest rate of 6%. A payment of $1000 made one year from now, a payment of $5000 made two years from now, and a payment of $3000 made three years from now. The present value can smply be thought of as the sums of the present values of four dfferent future values. P V = (1000)(1.06) 1 + (5000)(1.06) 2 + (3000)(1.06) 3 = $ (22) Another problem mght assume that the nterest rate s 4% n the frst year and 7% n the second year. Fnd the present value of $2000 pad at the end of the 7

8 frst year and $3000 pad at the end of the second year. For problems lke ths t s helpful to draw a tmelne. P V = (2000)(1.04) 1 + (3000)(1.07) 1 (1.04) 1 = $ (23) To fnd the future value of ths set of cash flows at the end of year 2, you can ether brng forward the present value amount we solved for or the ndvdual cash flows. Draw a tmelne. F V = ( )(1.04)(1.07) = $ (24) F V = (2000)(1.07) = $ (25) There are several dfferent ways to solve these sorts of problems. It helps to always break t down nto lttle bts money moves from one perod to the next perod at some rate. It s always possble to solve equatons of value ths way. There are many, many examples of them and I encourage you to do enough of them untl you feel comfortable movng money around accordng to the theory lad down n ths frst secton. 2 Annutes An annuty s a knd of cash flow that nvolves payng the owner a fxed sum of money each year for some number of years. The key thng to remember s that the amount of money pad s fxed. Ths wll allow us to reduce PV and FV calculatons for annutes to short formulas nstead of strngs of values summed together as n secton Annutes-Immedate An mmedate annuty s an annuty that makes payments at the end of specfed ntervals. The payments are always the same amount, the tme ntervals are always the same length, and the nterest rate s constant. Score! Assumng an end-of-nterval payment amount 1, an effectve nterest rate per perod, and n ntervals the present value of the annuty-mmedate s as follows. P V = (1 + ) 1 + (1 + ) (1 + ) n (26) Recall ths very mportant equaton for the sum of the terms of a geometrc sequence. Sum = frst term next after last term 1 rato (27) From equaton (26), we can see that the frst term s (1 + ) 1, the next term after the last s (1+) (n+1), and the rato s (1+) 1. We let the present value be represented by the symbol a n pronounced a angle n. 8

9 1 (1 + ) n P V = a n = = 1 vn (28) The future value, s n, s angle n, can be determned usng ether the method we used n (24) or the method used n (25). As a remnder the present value, a n, s the value one full nterval before the frst payment. The future value s the value the nstant after the n th payment. F V = s n = (a n )(1 + ) n = (1 + )n 1 Another useful relatonshp exsts between a n and s n. (29) 1 a n = 1 s n + (30) We have been assumng that the level payment s of amount 1. If t s some other amount, say P (remember that the payment amount must be constant), then the present value s smply P (a n ) and the future value s P (s n ) 2.2 Annutes-Due Ths s exactly the same dea as an mmedate annuty, except nstead of the payments beng made at the end of each nterval they re made at the begnnng of each nterval. We wrte the present value as (ä n ) and say a double dot angle n. Notce that the present value s the value of the cash flows the nstant before the frst payment. ä n = (1 + )(a n ) = 1 vn (31) d It s worth takng a few mnutes to make sure you understand why t s true. Remember that the payments of a n are farther away so ther present value must be lower. The future value formulas are analogous. F V = s n = (1 + )(s n ) = (1 + )n 1 (32) d There are a number of relatonshps between present and future values and between annutes due and mmedate. In realty they re all knd of slly. But n terms of learnng t s good to thnk through them and n terms of passng exam FM t s good to know them as they can really speed up a soluton. a n + 1 = ä n+1 (33) s n + 1 = s n+1 (34) 1 ä n = 1 s n + d (35) (a n )(1 + ) n (1 + ) = s n (36) 9

10 s n (1 + ) n (1 + ) 1 = a n (37) As before, for payment amounts other than 1, multply the annuty symbol by the amount of the payment. 2.3 Annutes wth Arthmetc Progresson The actuaral exam wll requre that you be able to fnd the present value of some specal non-level-payment annutes. Annutes whose payments are geometrcally ncreasng can be dealt wth usng equaton (27). There also exst annutes whose payments ncrease by some fxed amount each nterval. Consder an annuty that pays P at the end of the frst perod, P + Q at the end of the second perod, and P + (n 1)Q and the end of the n th perod. Then the present value of the ncreasng annuty at tme 0 can be represented as follows. (I P,Q a) n = P a n + Q (a n n(1 + ) n ) (38) Ths equaton makes a lot of sense and you should derve t yourself. Ths wll help you understand what all the parts mean whch wll n turn help you adapt the equaton to many stuatons. And usng the same logc as before: (I P,Q s) n = (I P,Q a) n (1 + ) n (39) (I P,Q ä) n = (I P,Q a) n (1 + ) (40) (I P,Q s) n = (I P,Q a) n (1 + ) n (1 + ) (41) For completeness I ll nclude the rest of ths secton, but I really urge you to understand what s gong on. If you can t derve the followng equatons t doesn t mean you haven t memorzed enough to be ready for the test, t means you don t understand enough to be ready for the test. (I 1,1 a) n = (Ia) n = (1 + )(a n) n(1 + ) n (42) In the event that P = n and Q = 1 we have what s called a decreasng annuty. 2.4 Perpetutes (Da) n = n a n (43) (Ds) n = (Da) n (1 + ) n (44) A perpetuty s exactly what t sounds lke - an annuty that goes on perpetually. It should be clear rght away that a perpetuty can t have a future value snce we ve been defnng the future value as the value of the annuty after the last payment, whch n ths case doesn t exst. To calculate the present value of the perpetuty, a, wrte out (26) for ths stream of cash flows and then realze that for (27) the next term after the last must be (1 + ) ( +1) = 0. 10

11 a = 1 ä = a + 1 = 1 + = 1 d (45) (46) (I P,Q a) = P + Q 2 (47) Remember that n all three of these cases annuty mmedate, annuty due, and perpetuty s the nterest rate per perod. So f you re gven (4) make sure to convert to = (4) 4 and to use the number of years tmes 4 as the number of payments. 2.5 The Werd Annuty Almost all of the annutes and perpetutes that the actuaral exam wll throw at you can be dealt wth usng a tmelne, the formulas from the prevous fve sectons, and a lttle common sense. However, t s worth your tme to memorze the formula for the werd annuty. It pays an amount 1/m at the end of each m th of the frst year, amount 2/m at the end of each m th of the second year, up to amount n/m at the end of each m th of the n th year. The symbol for ths knd of annuty s (Ia) (m) n. (Ia) (m) n 2.6 Contnuously Payng Annutes = än n(1 + ) n (m) (48) The dea of these s that nstead of gettng pad at the end of every year, or every month, the annutant gets pad at the end of every nstant. The present value of ths contnuously payng annuty s symbolzed by ā n. You can derve the followng formula by ether takng the lmt of equaton (48) as m goes to nfnty or ntegratng over the approprate tme span. ā n = n 0 v t dt = 1 (1 + ) n δ = 1 e nδ δ (49) The general formula for a contnuously payng annuty s really all you need to memorze. P V = n 0 f(t)v t dt (50) Where f(t) s the rate of payment. Very often t s 1 per year n whch case the formula for the present value s what we already saw n equaton (49). Other tmes, as n an ncreasng contnuously payng annuty that pays an amount t at tme t, the present value s P V = n 0 tvt dt. And by now you should know how 11

12 to move a lump sum from the present to the future, smply multply by (1 +) n. The perpetuty (n ) formulas pop rght out of equatons (49) and (50). ā = 1 δ (51) (Īā) = 1 δ 2 (52) where f(t) = 1 for (51) and f(t) = t for (52). 3 Loans Patence! 12

13 References [1] Beckley, Jeffrey A. Math/Stat 373 Class Notes [2] Cherry, Harold and Rck Gorvett. ASM Study Manual for SOA FM / CAS 2. 11th ed. ASM, [3] Hasset, Matthew J, Ton Coombs Carca and Amy C. Steeby. ACTEX Study Manual SOA Exam FM. Actex Publcatons, [4] McDonald, Robert L. Dervatves Markets. 2nd ed. Pearson Educaton, Inc., [5] Smon, Carl P. and Lawrence Blume. Mathematcs for Economsts. W. W. Norton & Company, Inc., [6] Vaaler, Lesle Jane Federer and James W. Danel. Mathematcal Interest Theory. 2nd ed. Pearson Educaton, Inc.,