"Fibonacci, Fermat, and Finance"

I went to a seminar yesterday in the Physics Department to see the manager of a hedge fund, John Seo, talk about "Fibonacci, Fermat, and Finance: How a Biophysicist Built a Multi-Billion Dollar Catastrophe Bond Fund after Re-Reading the Foundations of Modern Finance" (NY Times magazine story). One thing I learned at the seminar and from asking questions at dinner afterward was about the origins of present value analysis. It goes back to the mathematician Leonardo Fibonacci and chapter 12 of his book "Liber Abaci" written in 1202. Here's a working paper on the topic:

Fibonacci and the Financial Revolution William N. Goetzmann NBER Working Paper No. 10352: ...Traveling Merchant Problems The second type of financial problem is a set of “traveling merchant” examples, akin to accounting calculations for profits obtained in a series of trips to trading cities.

The first example is:

A certain man proceeded to Lucca on business to make a profit doubled his money, and he spent there 12 denari. He then left and went through Florence; he there doubled his money, and spent 12 denari. Then he returned to Pisa, doubled his money and it is proposed that he had nothing left. It is sought how much he had at the beginning.

[Update: the problem isn't clear about this, but 12 denari are spent at each of the three stops, including Pisa] Leonardo proposes an ingenious solution method. Since capital doubles at each stop, the discount factor for the third cash flow (in Pisa) is ½ ½ ½ . He multiplies the periodic cash flow of 12 denari times a discount factor that is the sum of the individual discount factors for each trip i.e. (1/2) + (1/4) + (1/8). The solution is 10½ denari. The discount factor effectively reduces the individual cash flows back to the point before the man reached Lucca.

Notice that this approach can be generalized to allow for different cash flows at different stages of the trip, a longer sequence of trips, different rates of return at each stop, or a terminal cash flow. In the twenty examples that follow the Lucca-Florence- Pisa problem, Leonardo presents and solves increasingly complex versions with various unknown elements. For example, one version of the problem specifies the beginning value and requires that the number of trips to be found – e.g. “A certain man had 13 bezants, and with it made trips, I know not how many, and in each trip he made double and he spent 14 bezants. It is sought how many were his trips.” This and other problems demonstrate the versatility of his discounting method. They also provide a framework for the explicit introduction of the dimension of time, and the foundation for what we now consider finance.

In case you don't see this, though he didn't explicitly use this formula, he realized that the "present value" of the cash flow is:

PV = [R₁/(1+i)¹] + [R₂/(1+i)²] + [R₃/(1+i)³]

where R_j is the cash spent at each point j on the trip, and i is the profit rate. In this case,  R_j = 12 for all j, and i=100%, or 1.0. Thus:

PV = [12/(1+1)¹] + [12/(1+1)²] + [12/(1+1)³]

      = [12/2] + [12/4] + [12/8]

      = 12[(1/2) + (1/4) + (1/8)] (as above)

      = 10.5

But does Fibonacci realize this applies not just to traveling merchants, but also to discounting financial cash flows over time? Yes. All you have to do is convert the problem back to the traveling merchant example. Going back to the NBER paper:

Immediately following the trip problems, Fibonacci poses and solves a series of banking problems. Each of these follows the pattern established by the trips example – the capital increases by some percentage at each stage, and some amount is deducted. For example:

A man placed 100 pounds at a certain [banking] house for 4 denari per pound per month interest and he took back each year a payment of 30 pounds. One must compute in each year the 30 pounds reduction of capital and the profit on the said 30 pounds. It is sought how many years, months, days and hours he will hold money in the house....

Fibonacci explains that the solution is found by using the same techniques developed in the trips section. Intervals of time replace the sequence of towns visited and thus a time-series of returns and cash draw-downs can be evaluated. Once the method of trips has been mastered, then it is straightforward to construct a multiperiod discount factor and apply it to the periodic payment of 30 pounds – although in this problem the trick is to determine the number of time periods used to construct the factor. Now we might use logarithms to address the problem of the nth root for an unknown n, but Fibonacci lived long before the invention of logarithms. Instead, he solves it by brute force over the space of three pages, working forward from one period to two periods etc. until he finds the answer of 6 years, 8 days and [5 and 1/2] hours. The level of sophistication represented by this problem alone is unmatched in the history of financial analysis. Although the mathematics of interest rates had a 3,000 year history before Fibonacci, his remarkable exposition and development of multi-period discounting is a quantum leap above his predecessors.

And, one final example:

Present Value Analysis The most sophisticated of Fibonacci’s interest rate problems is “On a Soldier Receiving Three Hundred Bezants for his Fief.” In it, a soldier is granted an annuity by the king of 300 bezants per year, paid in quarterly installments of 75 bezants. The king alters the payment schedule to an annual year-end payment of 300. The soldier is able to earn 2 bezants on one hundred per month (over each quarter) on his investment. How much is his effective compensation after the terms of the annuity have changed? ...

As before, Fibonacci explains how to construct a multi-period discount factor from the product of the reciprocals of the periodic growth rate of an investment, using the model developed from mercantile trips in which a percentage profit is realized at each city. In this problem, he explicitly quantifies the difference in the value of two contracts due to the timing of the cash flows alone. As such, this particular example marks the discovery of one of the most important tools in the mathematics of Finance – an analysis explicitly ranking different cash flow streams based upon their present value.