TANTON S TAKE ON CONTINUOUS COMPOUND INTEREST

Transcription

1 CURRICULUM ISPIRATIOS: IOVATIVE CURRICULUM OLIE EXPERIECES: TATO TIDBITS: TATO S TAKE O COTIUOUS COMPOUD ITEREST DECEMBER 208 How does one teach students the t compound inteest fomula Be fo the final balance afte investing B dollas at an inteest ate of % pe annum (expessed as a decimal numbe) fo a total of t yeas? Does one simply pesent a mysteious fomula and say, Hee you go!? How does one pove that computing simple inteest computed ove fine and fine time intevals conveges to the peculia expession B ( ) t? How did Jacob Benoulli in 683 come to conclude thee must be an expession of this type? Do pe-calculus students have the tools to follow a deivation? Comment: Benoulli ecognized thee is a constant e with a value between 2 and 3 that lies at the heat of undestanding compound inteest. Decades late Leonhad Eule computed the value of this constant to 8 decimal places and ecognized its impotance thoughout many banches of mathematics. He denoted the constant as e, not afte his own name, but simply because e is the next vowel afte a, a lette he had aleady used many times in othe papes to epesent a numeical value. The lette e has since emained the standad notation fo continuous compound inteest constant

2 O FIRST ITRODUCIG COTIUOUS COMPOUD ITEREST One typically intoduces this topic with a stoy akin to the following one. I wish to invest $000 fo a yea. The cuent going ate, fo all banks, is 4% pe annum. I look aound and find five banks with the following deals: BAK : Bank with us. We povide simple inteest at 4% pe annum. Clean and simple. BAK 2: Bank with us. We povide an inteest ate of 4% pe annum and we compute it monthly. BAK 3: Bank with us. We povide an inteest ate of 4% pe annum and we compute it weekly. BAK 4: Bank with us. We povide an inteest ate of 4% pe annum and we compute it daily. BAK 5: Bank with us. We povide an inteest ate of 4% pe annum and we compute it evey hou. How much money would I ean with each bank? Which bank offes the best deal fo me? Aside: Banks povide a sevice: they hold and potect customes money. Don t you find it cuious that banks PAY YOU fo thei sevice? (What othe institutions pay its customes?) Why is that? Reseach the histoy of banking and ty to find out when financies fist stated paying customes to povide them sevices. Let s calculate the total money eaned with each bank. BAK : The tem simple inteest means that one is given a stated faction of one s balance at one specified time. The detail at a ate of 4% pe annum infoms us that we shall ean the faction 4 / 00 of ou balance one time at the end of one yea. Aside: Roman empeo Augustus levied a fee of one pat pe hunded on all tansactions and exchanges that occued at auctions. The pecent sign % is deived fom the Lain tem pe cento, which was tanslated to pe cento in Italian, and witten in shothand with a p and an o. The symbol o/oo is used fo pe mille, one pat pe thousand. Thus afte one yea with bank we ll have in ou account the oiginal balance of $000 plus the faction 4 / 00 of that balance. We ean $ = 000( ) = $040 Comment: We see, in geneal, that awading inteest on a balance $ B, multiplies B by the facto ( + ), whee is the inteest ate expessed as a faction. ew Balance B B B( ) = + = +. BAK 2: This bank will divide the inteest ate into 2 pats, one fo each month, and povide that divided inteest on the balance at the end each month. Afte the fist month, my balance will thus gow to =

3 Afte a second month ou balance shall be this amount plus an additional 0.04/2 of that amount = Afte a thid month, the balance shall gow by that same facto again to And so on, fo 2 months. My final balance in the account shall be $ I ll ean $40.74 with bank 2. A bette deal! BAK 3: This bank will add the faction 0.04 to you balance at the end of each 52 week fo 52 weeks. (Fo ease, let s ignoe the final patial week of the yea.) Thus afte one yea my balance would be $ I ll ean $ Even bette! BAK 4: Woking with a non-leap yea we see my balance will be $ (It s actually just unde $040.8.) Bette still! 2 BAK 5: As thee ae 8760 hous in a yea we see my balance will be $ (It s actually just ove $040.8.) Question: What would the balance be if inteest was calculated eve minute? Evey second? Evey nano-second? It seems as inteest is computed and compounded ove smalle and smalle time intevals, the final balance inceases and, moeove, seems to convege to some ultimate idea value. (Is that ideal value the esult of computing inteest at each and evey instant? What could that mean?) How can we compute that ideal value? COMPUTIG COTIUOUS COMPOUD ITEREST We e looking at the quantity fo lage and lage values of. The facto of 000 is somewhat immateial in this wok and we need not be locked into the numbe 0.04, so we eally want to look at the values of + as gows. And to get us going we might as well be kind to ouselves and fist ty woking with a vey simple value fo, namely, with =. Can we say, and pove, anything about the values of +

4 as gows? Hee ae the values of ( / ) = 2,3, 4,5,6, and fo = = = = = = = It seems the values ae inceasing and, if you inset lage and lage values fo into ( + / ) we neve seem to get a value lage than 3. If these two assetions ae tue, then It is intuitively clea that the sequence of values we ae seeing must convege ( cunch up ) to some value, call it e, to the left of 3 on the numbe line, o maybe to the numbe 3 itself. (Univesity students taking a couse in eal analysis will ealize that thee is an assumed popety of the eal numbes at play hee.) But in ode to pove this numbe e exists, we still need to pove ou two assetions: The values of +. steadily incease as inceases, and 2. neve exceed the value 3. THE GEERAL COTIUOUS COMPOUD ITEREST FORMULA Befoe we pove these two assetions let s complete ou analysis of compound inteest. We have, allegedly, that + e as gows in value. (Hee, the aow is shothand fo gets close and close to. ) ow let s examine +. What value does this appoach as gows? Well, if we loosely phase ou esult as big + e big then we might think to ewite + as ( / ) + ( / ) = + ( / ) = + ( / ) ( / ) ow, if is big, then so is /. So we have hee something that looks like + big big Ou esult then says that

5 big + ( e) = e big fo lage and lage inputs. This establishes that + e as gows, and so that B + Be. Fo ou opening example, with B = 000 and = 0.04 we see that the ideal final balance of my account, computing inteest, continuously, ove eve possible moment of time (whateve that means) will be e $040.8 factionally highe than computing inteest houly. In geneal: We have just shown that one yea of continuous compound inteest has the effect of multiplying a balance B by a facto e. A second yea of continuous compound inteest will multiply by anothe facto of e, and so one s final balance 2 afte two yeas shall be ( Be ) e = Be. A thid yea of continuous compound inteest intoduces anothe facto of e and so will 3 yield the balance Be. And so on. Afte t yeas of continuous compound inteest, the final balance shall be t Be. This is the fomula pesented in textbooks. Question: How can one justify that the t fomula Be is still valid if t epesents a factional count of yeas? PROVIG THE TWO ASSERTIOS ow we come to the cunch of mattes. How do we pove that that the values of + steadily incease as inceases and neve exceed the value 3? This all elies on the combinatoics of expanding backets and its connection to Pascal s tiangle. ( x y) 0 + = x+ y = x+ y x + y = x + 2xy + y x + y = x + 3x y + 3xy + y x + y = x + 4x y + 6x y + 4xy + y To expand 4 ( x+ y) = ( x+ y)( x+ y)( x+ y)( x+ y), say, we must choose one tem fom each set of paentheses, multiply them togethe, and do this fo all possible selections of tems, and sum the esults. 2 2 Fo example, the tem x y can appea by selecting x and x and y and y, and again by selecting y and y and x and x ; 2 2 and so on. In fact, the tem x y will appea as many times as it is possible to aange two x s and two y s. Thee ae 6 such ways. xxyy yyxx xyyx xyxy yxyx yxxy

6 One leans in a counting couse (see my online notes hee) that this count is eally 4! = 6 the numbe of ways to aange 4 2!2! lettes: 2 x s and 2 y s. ( a+ b In geneal, thee ae )! ways to ab!! aange a x s and b y s, and this leads to the binomial theoem:!!!! x 2!2!! x 3!3! x+ y = x + x y + + ( ) ( ) ( ) y 2 2 y We can simplify these coefficients a little! = ( )!!! = ( ) ( 2 )!2! 2!! = ( 3 )!3! 3! so that when we expand + we obtain + = + ( 2) 2 + ( ) 2 2! 3 + ( )( 2) 3! + This equals ! 3! o equivalently ! 3! So we have just detemined + = + + 2! 2 + 3! ! + which is a sum with + tems. We can copy this fomula and deduce that + + = ! ! ! and since we have k k < + + < This establishes the fist assetion: the sequence of values is indeed stictly inceasing. (Sneaky!).

7 We can also see that + = + + 2! 2 + 3! ! + < ! 3! 4! = < = The final sum is a sum of a finite numbe of tems. But it is clea that any finite sum is less than Thus = 3. is sue to be less than Ou cleve way of witing the expansion of + has led to establishing the two claims. We have now poved that the numbe e exists! IS THIS APPROPRIATE FOR THE CURRICULUM? The wok hee to establish the textbook t fomula Be is hefty, to say the least. Most textbooks choose not to shae it. But that leaves the students with nothing to hold on to with egad to playing with continuous compound inteest. Some cuicula do have students exploe the values of + as gows, but they typically get muky in linking this with evaluating + and, in the end, fall back onto the message just tust us, thee is a fomula. I pesonally think it IS impotant to show the full mathematics of this wok and I have no qualms about giving sessions fo students that ae puely optional pay attention o not (with cetainly no assessment attached). Pesenting this mateial in peson on a whiteboad is not actually too difficult o ovewhelming. I ve had success doing this. But it does equie woking with a goup of students who have aleady had some expeience counting wods with epeated lettes and the binomial theoem and who have fully intenalized the lovely and inticate mathematics hidden within Pascal s tiangle. (Again, see my counting notes.) So one needs to set mattes up months befoe and sadly most pecalculus cuiculums don t. Can we ensue that some meaningful discete mathematics falls within the ealy high-school cuiculum? Challenge: Students late encounte in thei schooling a numbe, also called e, in calculus class. It is defined as the base of an exponential function whose deivative equals itself:

8 d e x e x =. dx Is thee any eason to believe that this numbe e is the same numbe that aises in the study of continuous compound inteest? (This is not a tivial question.) 208 James Tanton tanton.math@gmail.com