Risk Latte - Fibonacci: The World’s First Financial Engineer

Fibonacci: The World’s First Financial Engineer

Team Latte
December 17, 2011

Who was the world’s first financial engineer? The same guy who gave us the Fibonacci numbers, Leonardo of Pisa.

We all know the importance of the "present value problem" in finance. The "present value analysis" of a stream of cash flows to determine the value of an investment is a well-established method in the theory of corporate finance and investment. The present value (PV) method is so ubiquitous, being used by MBA students, equity analysts, investment analysts and the CFOs of global corporations almost on a daily basis that hardly anybody pauses to think about the origin of this method.

Most financial historians think that it was Irving Fisher, the well-known American economist, who, in 1930, developed the present value method. And that is indeed true, as far as the concept stands today. However, the genesis of this idea can be traced back to early thirteenth century Europe, to a man named Leonardo, or Leonardo of Pisa, who is better known by another name, Fibonacci. This is the same man whose name is associated with the Fibonacci problem and the Fibonacci numbers.

Professor William Goetzmann of Yale School of Management in an NBER working paper titled Fibonacci and the Financial Revolution, in 2004 first pointed out this fact. In the paper, Professor Goetzmann highlights Leonardo of Pisa or Fibonacci’s contribution to mathematical finance. As Keith Devlin, in his recent book, The Man of Numbers, has argued Fibonacci was the first European to introduce the mathematical ideas and methods of the ancient and medieval Indians and Arab scholars, including the numeral system and the arithmetic associated with it. And he did this through his general purpose arithmetic book Liber Abaci that was published in 1202. This book laid the foundation for the development of modern symbolic algebra.

Professor Goetzmann has argued in the paper that Leonardo of Pisa developed an early form of present value analysis which incorporated the time value of money. He points out in the same paper that while Leonardo used many of the mathematical tools of the Indian and Arabic scholars, such as such as the use of interest rates, he invented "a new mathematical tool for financial decision making".

Here’s a quote from Professor Goetzmann’s paper: "Leonardo was arguably the first scholar in world history to develop a detailed and flexible mathematical approach to financial calculation. He was not only a brilliant analyst of the business problems of his day, but also a very early financial engineer whose work played a major role in Europe’s distinctive capital market development in the late Middle Ages and the renaissance."

If financial engineering entails solving – approximating a solution – complex problems numerically then Fibonacci was certainly the world’s first financial engineer. Keith Devlin’s book details how Leonardo was able to solve a cubic equation numerically; and, the algorithm that he followed had some similarity to the one used by many quants these days to approximate implied volatility of an option, namely the bisection method.

Leonardo was asked by Johannes of Palermo, one of the court mathematicians of Frederick, the Emperor of Germany, Italy, Sicily and Burgundy, to solve the following cubic equation:

Leonardo solved the above equation by approximation. Here is the algorithm that he followed (we follow the same notation used by Devlin in his book as well as his explanation of how Leonardo solved the above equation; Leonardo found out the solution using sexagesimal fractions that’s quite cumbersome to comprehend):

  1. Assume that solves the above cubic equation; then, can be written as,
  2. If is an approximate solution of the equation that is too high (or too low) then would be an approximate solution that would too low (or too high);
  3. Therefore, the average of the above two approximations would be a better solution of the equation, i.e.
  4. Starting with a seed value of say, 1.5 (which is ), we can perform an iteration that very quickly – within 19 steps – converges to a correct solution of 1.368808 (up to six decimal places). If we start with an initial value of say, 10, then within 26 steps of iteration it converges to the correct solution of 1.36880810 (up to eight decimal places). One can easily perform the above iterations using ExcelTM.

Leonardo, using sexagesimal fractions (fractions expressed in base 60), estimated the correct answer at 1.3688081075. Remember, Leonardo solved the above equation when there were no ExcelTM spreadsheets, no computers, no calculators and no notion of what algebraic equations were. In the early thirteenth century the common man in Europe had not even heard of "algebra". And Leonardo was talking to the ordinary men of commerce, traders and merchants of Pisa; his objective was to develop detailed and flexible mathematical methods for financial calculations.

Liber Abaci, in more ways than one, was in essence a manual of instructions for the traders and merchants of Pisa to transact and do business. It was a book of commercial arithmetic which used algorithms that would soon form the foundations of symbolic algebra. For example, in Chapter 12 of the book, he lists out a wide variety of arithmetic problems, including interest rate and present value examples. Chapter 13 of the book outlines a method of solution through linear interpolation. Of course, there were other chapters which present more abstract mathematical results.

In the same book, there is a problem which has now become known as "Fibonacci’s Problem of the Birds". Here is a problem in modern algebra where one has to solve a system of two equations in three unknown variables. In almost all cases, such problems cannot be solved because if there are three unknown variables then we require three algebraic equations. However, Leonardo’s specific problem was one where such a solution was possible if one looked at it carefully. And even though the problem was one of algebra, Leonardo formulated it as a commercial problem Once again, quoting straight from Devlin’s book, here’s Fibonacci’s bird problem:

  1. A man buys 30 birds which are partridges, pigeons and sparrows for 30 dinari. A partridge he buys for 3 dinari, a pigeon for 2 dinari, and 2 sparrows for 1 denaro, namely one sparrow for half denaro. It is sought how many birds he buys of each kind.

If , and are the number of partridges, pigeons and sparrows respectively then in terms of a system of algebraic equations, the above problem can be written as:

One can not solve for , and because there are only 2 equations. We need one more equation in the above system. However, if one looks carefully at the problem there is a crucial piece of information provided in the problem. All values of , and must be positive whole numbers. The value of these variables cannot be in fractions (because the man cannot buy a fraction of a bird) and has to be greater than zero (because the man buys all three kinds of birds). With this piece of information a solution can be found to the above problem, using some kind of iterative logic, which Devlin explains in his book.

In Finance, there is a branch of study called the Technical Analysis. Here, the chartists – the other name for technical analysts – study patterns and historical price moves of stocks and other financial assets and try to predict their future paths. Fibonacci numbers and the Golden Ratio figure prominently in these analyses. Fibonacci numbers and the Fibonacci series are not only popular amongst the high school math students, cryptographers and number theorists but they have found a place of recognition within quantitative finance as well.

However, over the centuries, Fibonacci, the man has been virtually forgotten. He was a great mathematician – and certainly, a great financial mathematician – who has been relegated to the footnotes of history. The Reading Professor Goetzmann’s paper and Devlin’s book is a pure delight and we strongly recommend all our readers to find time to read them.


  1. The Man of Numbers, Keith Devlin, Bloomsbury Publishing Ltd., 2011.
  2. Fibonacci and the Financial Revolution, William N. Goetzmann, NBER Working Paper Series, March 2004.

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