Comments on "Fibonacci, Fermat, and Finance"TypePad2008-11-08T08:24:00ZMark Thomahttps://economistsview.typepad.com/economistsview/tag:typepad.com,2003:https://economistsview.typepad.com/economistsview/2008/11/fibonacci-ferma/comments/atom.xml/Alexander Body commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2011571042f7f970c2009-07-12T15:29:31Z2009-07-12T15:29:31ZAlexander BodySorry, my formulas got out of whack. I'll upload a new copy when my ADOBE gets fixed. Alex B.<p>Sorry, <br />
my formulas got out of whack.<br />
I'll upload a new copy when <br />
my ADOBE gets fixed.<br />
Alex B.</p>Alexander Body commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2011571f8fcc3970b2009-07-12T15:23:20Z2009-07-12T15:23:20ZAlexander Body“Fibonacci, Fermat, and Finance” Article By: Mark Thoma 11/08/08 Comments By: Alex Body 07/12/09. No doubt, the story-line of the...<p>“Fibonacci, Fermat, and Finance” Article By: Mark Thoma 11/08/08<br />
Comments By: Alex Body 07/12/09.</p>
<p>No doubt, the story-line of the Leonardo/Fibonacci Travelling <br />
Merchant Problem has been partially lost in the translation.<br />
The same omission is present in the source book.<br />
Current mortgage formulas show the astonishing fact that his 1202 AD Present Value and balance calculations are right on the button,providing that in each of three cities the merchant doubled his balance and spent 12 dinari.<br />
For brevity, I use the condensed Present Value formula for annually paid and annually compounded standard American mortgage. </p>
<p> 1 - (1 + IF)^-N <br />
PV = Bo = PMT * ------------------<br />
IF </p>
<p> 1-(1+1)^-3 <br />
= 12 * -------------- = 12 * 7/8 = 10.5 <br />
1<br />
Where:<br />
PV = Present Value = Bo = the merchant’s Original Balance prior to travelling to the three subject cities. <br />
PMT = amount of money he spent at each of the three cities = 12 dinari.<br />
IF = Interest Factor for the payment period = 1, because he doubled his outstanding balance in each city.<br />
N = number of cities visited = number of payments made.<br />
Note:<br />
The discount factors corresponding to the three cities become:<br />
Lucca: 1/(1+IF)^1 = 1/(1+1) = 1/2<br />
Florence: 1/(1+IF)^2 = 1/(1+1)^2 = 1/4<br />
Pisa: 1/(1+IF)^3 = 1/(1+1)^3 = 1/8</p>
<p>The above verify Leonardo's problem definition and results:<br />
PV = Bo = 12 * (1/2 + (1/2)^2 +(1/2)^3)) = 10.5<br />
End Balance = 10.5 *2)-12)*2)-12)*2)-12) = 0 </p>
<p>In summary: <br />
The merchant started his sales trip with the original balance of 10.5 dinari.<br />
At each of the three cities, he doubled his balance, but spent 12 dinari. <br />
As a result, his Final Balance has diminished to zero.</p>Lafayette commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535f4dc8c970c2008-11-14T11:06:20Z2008-11-14T11:06:20ZLafayetteSi cerca (It is sought ...) cm: rent seeking and arbitrage are not recent inventions. It should surprise no one,...<p>Si cerca (It is sought ...)</p>
<p>cm: rent seeking and arbitrage are not recent inventions.</p>
<p>It should surprise no one, for this article, to learn that the first banks were established in ... Florence and not Switzerland. In fact, the Swiss developed their tiny, mountainous country by charging a "toll" to get across the Alps from Italy to Northern Europe, a trade route. Banking came much later.</p>
<p>I submit that what was sought was not the answer to the riddle but the money rent necessary to obtain certain sums, also a cornerstone of modern finance. Or, iow, the ROI of revenue streams. Si cerca l'utile sul capitale investito, or What's the ROI? </p>
<p>Even back then, the notion of return on investment was firmly embedded in Finance, and the Florentines were masters of the art. An interesting read of students of the history of finance can be found here. It all started long, long ago. Thanks, btw, to Google Book indexing.)</p>
<p>Who said the Italians only new how to make pizza ...?</p>reason commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535eea2a3970c2008-11-12T10:03:43Z2008-11-12T10:03:43Zreasonhttp://profile.typekey.com/jim-brady/That was me.<p>That was me.</p> commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535eea287970c2008-11-12T10:03:07Z2008-11-12T10:03:07ZThis just says to me what I have always known, that insurance (and issuing mortgages is a form of insurance...<p>This just says to me what I have always known, that insurance (and issuing mortgages is a form of insurance - a pricing of risk - if you think about it) is a difficult market, because naive newcomers can undercut the market, but leave lots of customers with an unknown counterparty risk. That is not talked about at all, but underlies the whole discussion. It is a fundamental problem with the market and this discussion offers no solution. As regards his anecdote about Florida and the state offering catastrophe insurance, the easy answer to the regressiveness is to place a (fairly mostly) upper limit on the insurance offered to any individual.</p>dlaw commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e07716970b2008-11-09T23:07:16Z2008-11-09T23:07:17Zdlawhttp://financialroadtosocialism.blogspot.comI thought I had the answer until I looked at the original text, which I found very confusing.<p>I thought I had the answer until I looked at the original text, which I found very confusing. </p>Asif Dowla commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e66c6a970c2008-11-09T22:45:08Z2008-11-09T22:45:10ZAsif DowlaJohn Seo's father is an economics professor at SMU, Ken Seo. In addition to his Econ Ph.D, Ken Seo also...<p>John Seo's father is an economics professor at SMU, Ken Seo. In addition to his Econ Ph.D, Ken Seo also holds a Ph.D in Math. I took Stochastic Calculus course with him in graduate school with Ken Seo.</p>dlaw commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e65f6b970c2008-11-09T22:17:38Z2008-11-09T22:17:38Zdlawhttp://financialroadtosocialism.blogspot.comOkay,but what's the answer to the final question? What is the difference between the yearly and quarterly contracts? Can you...<p><br />
Okay,but what's the answer to the final question?</p>
<p>What is the difference between the yearly and quarterly contracts?</p>
<p>Can you elucidate further?</p>
<p>Thanks so much. </p>Kaleberg commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e4682f970c2008-11-09T04:58:43Z2008-11-09T04:58:43ZKalebergThat was also the era in which double entry bookkeeping was developed. That provided a link between static and dynamic...<p>That was also the era in which double entry bookkeeping was developed. That provided a link between static and dynamic accounting analysis. That is, Renaissance merchants could track their instantaneous net worth and interval flow. If you have ever tried to figure this out without double entry bookkeeping you'd know this is surprisingly tricky. (In college I ran a soda cooperative and needed to know when I could declare a dividend, payable in cookies. I wound up half re-inventing double entry bookkeeping, though I didn't quite get there.)</p>Mark Thoma commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e40153970c2008-11-09T03:48:20Z2008-11-09T04:15:25ZMark ThomaHe starts with 10.5. It doubles to 21. He spends 12 leaving 9. The 9 doubles to 18. He spends...<p>He starts with 10.5. It doubles to 21. He spends 12 leaving 9. The 9 doubles to 18. He spends 12 leaving 6. The 6 doubles to 12. He spends 12 leaving 0.</p>
<p>I added an update to clarify that 12 is spent at each city, including Pisa (I took the solution as presented in the seminar and in the NBER paper, but rereading it, it doesn't talk about spending anything in the third city. I assume the original text, which includes the calculations, is clear on this point).</p>Julio commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535dd08ae970b2008-11-09T01:45:21Z2008-11-09T01:45:21ZJulioRichard E, I think the statement was imprecise; he also spent 12 dinars in Pisa. (If he didn't, then you're...<p>Richard E,</p>
<p>I think the statement was imprecise; he also spent 12 dinars in Pisa. </p>
<p>(If he didn't, then you're right but it's kind of an odd problem; the last step "doubled his money and had nothing" would always mean that "his money" which he doubled = 0).</p>Lynn commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e3a4ec970c2008-11-09T01:36:15Z2008-11-09T01:36:15ZLynnRichard E You are correct. The answer to the stated problem is 9. The problem failed to mention that the...<p>Richard E<br />
You are correct. The answer to the stated problem is 9. The problem failed to mention that the traveler spent 12 denari in Pisa. Then the answer would be 10.5</p>Bruce Wilder commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535dcfcf0970b2008-11-09T01:11:38Z2008-11-09T01:11:38ZBruce WilderWikipedia actually has a bit on his representation of fractions and the related system known as Egyptian fractions, and how...<p>Wikipedia actually has a bit on his representation of fractions and the related system known as Egyptian fractions, and how that facilitated currency conversion, as well as the use of customary measures.</p>
<p>Liber abacci also describes lattice multiplication. I had never quite made the connection, though, that computers multiply by a variation on lattice multiplication known as Egyptian multiplication. </p>
<p>The peripatetic model for calculation of present value served, I suspect, the purposes of bills of exchange, which not only financed the Crusades and the revival of international trade, but also provided convenient circumvention of the medieval Church's prohibition on usury as well a severe shortage of both law enforcement and coin. The "story" of the story problem isn't just quaint pedagogy, it is subversive political propaganda.<br />
</p>Richard E. commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e399b3970c2008-11-09T01:06:33Z2008-11-09T01:06:33ZRichard E.I feel obtuse - I must not be interpreting the problem statement correctly because: Suppose the man starts with 9...<p>I feel obtuse - I must not be interpreting the problem statement correctly because:</p>
<p>Suppose the man starts with 9 denari.</p>
<p>"A certain man proceeded to Lucca on business to make a profit doubled his money, and he spent there 12 denari."</p>
<p>(9 * 2) - 12 = 6</p>
<p>"He then left and went through Florence; he there doubled his money, and spent 12 denari."</p>
<p>(6 * 2) - 12 = 0</p>
<p>"Then he returned to Pisa, doubled his money and it is proposed that he had nothing left."</p>
<p>0 * 2 = 0</p>
<p>"It is sought how much he had at the beginning."</p>
<p>Seems to me that 9 denari is the correct answer - please illuminate my mistake. </p>
<p>Thanks, Richard.</p>Alex Tolley commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e38c9a970c2008-11-09T00:32:09Z2008-11-09T00:32:09ZAlex Tolleyhttp://www.mymeemz.comBW: "It makes me wonder if there's been an arbitrage operation fleecing an obsolete system of economizing on information and...<p>BW: "It makes me wonder if there's been an arbitrage operation fleecing an obsolete system of economizing on information and calculation."</p>
<p>That's a very interesting and possibly perceptive thought...I must think along those lines further.</p>Mark Thoma commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e33239970c2008-11-08T20:07:12Z2008-11-08T20:10:20ZMark ThomaBruce: If you have access (I couldn't find an open link), the parts of the NBER article on "the rule...<p>Bruce:</p>
<p>If you have access (I couldn't find an open link), the parts of the NBER article on "the rule of three" and "the rule of five", as well as the currency conversion calculations, etc., support exactly what you are saying. As noted in the article, math was used as a tool - much as math is used now as a tool in hedge funds - to gain an advantage in arbitrage calculations.</p>
<p>He had a very different way of expressing fractions, for example, but the method he chose made some of these issues such as currency conversion much more transparent than they are with our system of expressing fractions.</p>RW commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535dc7ed3970b2008-11-08T20:05:40Z2008-11-08T20:05:40ZRWAlfred Crosby follows the connection of math or, more broadly, quantification and representation for some distance in his text, "The...<p>Alfred Crosby follows the connection of math or, more broadly, quantification and representation for some distance in his text, "The measure of reality: Quantification and western society, 1250-1600"</p>
<p>The short answer for increasing Western dominance is "science and technology" but Crosby tries to follow outcomes of a complex, 350 year process whereby certain habits of mind, the growing 'obsession' with splitting the world and temporality into symmetric fragments that could then be counted, classified, measured (and communicated across language boundaries), developed.</p>
<p>Beginning with the Venerable Model whereby time was governed by daily religious observances (hours expanded or contracted to assure 12 per day regardless of season), Crosby describes a series of rather remarkable developments that become increasingly visible in the 14th century: the use of large town clocks with fixed hours (shared time), Gregorian rationalization of the Julian calendar, capitalization and punctuation in writing, musical notation (including rests), single perspective in art, Arabic numerals, cartographic grids and projections, double-entry bookkeeping (leading to a math student population explosion), and so forth.</p>
<p>The core of Crosby's argument is that once the necessary social conditions (not confined to an elite) came about -- a rise in commerce, the growth of nation-states and the revival of accessible learning -- what kindled and accelerated the trend toward a rather dramatic change in human outlook was the advent of techniques for visualizing information; quantification and visualization together were the key to the epochal shift in European mentality.</p>
<p>It's an interesting read.</p>Bruce Wilder commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535dc75bf970b2008-11-08T19:37:26Z2008-11-08T19:37:26ZBruce WilderThis post launched me onto a meandering thru Wikipedia, which was really fun. So much of foundational math is tied...<p>This post launched me onto a meandering thru Wikipedia, which was really fun.</p>
<p>So much of foundational math is tied to systems of notation, which facilitated calculation, and that ties back to the curious business of computers being used for formal proof -- I wonder what the availability of computers is doing to re-invent math.</p>
<p>In the current financial crisis, all kinds of ties are made to, say, the ratings agencies, or financial modeling gone awry, and most of the popular press accounts are, basically, moral fables, but I wonder if we are not in a process of re-inventing finance to run on the vast computing power we have available. There's really nothing to prevent the financial system from having available real-time databases that work up from atoms of individual household income or particular houses or, even, autos -- in such a world, the moral integrity of Moody's is less questionable than its usefulness in avoiding calculation. If Google Maps can take a photo of every house in America for a free service . . . If I have a database of every business and household in Ashtabula, Ohio, why do I need Moody's to rate the bonds of the Ashtabula Sewage Treatment authority? It makes me wonder if there's been an arbitrage operation fleecing an obsolete system of economizing on information and calculation.</p>
<p>Liber abacci was part of the invention of capitalism -- the application of math to merchant banking facilitated the re-invention of international trade in the high middle ages, which would drive, first, the age of discovery, and finally, the industrial revolution. Math, which, in ancient civilizations was usually the province of priests and astronomers, and, sometimes civil engineers, became, in Italian renaissance towns, the toolkit of artists and bookkeepers, and we got perspective in painting, and the Medici bank. Math would be taught in the abbaco schools, precursors to the grammar schools, which would become modern elementary schools -- Liber abacci and its pedagogy of "story problems" would be a foundation of abacco schools.</p>
<p>Just some random thoughts on too much caffeine on an idle Saturday . . .</p>cm commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e2b082970c2008-11-08T14:49:37Z2008-11-08T14:49:37Zcmken melvin, rent seeking and arbitrage are not recent inventions.<p>ken melvin, rent seeking and arbitrage are not recent inventions.<br />
</p>ken melvin commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535e27dea970c2008-11-08T12:16:37Z2008-11-08T12:16:37Zken melvinSeems to me there's an (inverted?, perhaps?) association with everyone trying to live of investment and the consequences of an...<p>Seems to me there's an (inverted?, perhaps?) association with everyone trying to live of investment and the consequences of an 8% return, and one with growing our way out of anything forever.</p>hari commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535db98b2970b2008-11-08T09:06:55Z2008-11-08T09:06:56ZhariThe best part of your story is my favourite cities - Lucca, Pisa, Florence - which may be 100kms or...<p>The best part of your story is my favourite cities - Lucca, Pisa, Florence - which may be 100kms or so on horseback at the time this financial calculation originated. For all its antiquity resemblace, I admire your keen interest in the origin of financial analysis. </p>chris commented on '"Fibonacci, Fermat, and Finance"'tag:typepad.com,2003:6a00d83451b33869e2010535db9878970b2008-11-08T09:05:53Z2008-11-08T09:05:53ZchrisWhat did Warren Buffett, channeling Virgil, say about this kind of stuff: "Beware geeks bearing formulas"?<p>What did Warren Buffett, channeling Virgil, say about this kind of stuff: "Beware geeks bearing formulas"?</p>