Computational Finance

The following post was written and contributed by Cornelis ("Kees") A. Los, PhD, Professor of Finance. Faculty of Management, The University of Lethbridge (Alberta, Canada).

An equation is what it is: it equates. in other words, what is at the left side of the equal (=) sign equals what is at the right side of that sign and it does not matter what is left or what is right, as long as they are equated. Therefore it is also called a balance equation.

The traditional Accounting equation can be written as A = L + E (= traditional accounting balance at one point in time, since A - L - E = 0). Modern finance, in particular financial engineering (e.g., options theory) has used this equation to compute a market value for equity E, when the assets (A) and liabilities are both valued at market prices: E = A - L.. Modern bankruptcy valuation, after the equity has "melted away" (E = 0) computes the value of the remaining liabilities by valuing the assets, so that L = A.

In a dynamic fashion, we can take differences of the entities involved: dA = dL + dE, which we can recognize as the net income (profit) equation: profits = dE = dA - dL = revenues - expenses.

With a simple manipulation, we can look at the (modified) duration equation: (dA/A) x A = (dL/L) x L + (dE/E) x E, which states that the relative sensitivity of the assets x total assets = relative sensitivity of the liabilities x liabilities + relative sensitivity of equity x equity. This useful equation becomes now a tool for financial risk analysis, since (dE/E) = (dA/A) x A - (dL/L) x L = (dA/A) x (E + L) - (dL/L) x L, so that the profit dE = (dA/A) + [(dA/A) - (dL/L)] x (L/E). Thus the profit depends on the relative sensitivities of the assets and liabilities and the leverage factor (L/E).

What sensitivities are we talking about? For example, banks are very sensitive to the level of the interest rate (r), so that we have (dA/dr) = (dL/dr) + dE/dr, i.e. the sensitivity of profits wrt. the interest rate dE/dr = (dA/dr)/A + [(dA/dr)/A - (dL/dr)/L] x (L/E). [Note: Bear Sterns' problem of a wipe-out was mostly its extremely high leverage factor of L/E close to 30 times, when its asset value A = its portfolio of loaned mortgages, was diminishing in value].

But another example would be to find the sensitivity of an institution wrt. the oil price (P): dE/dP = (dA/dP)/A + [(dA/dP)/A - (dL/dP)/L] x (L/E). When I was a Chief Economist of ING Bank for the US, in New York, in the early 1990s, I already computed the sensitivity of ING Bank's balance sheet wrt. the world oil price, as part of my duties as a financial risk analyst for ING Bank in New York. It is easy to imagine that many transportation companies are VERY sensitive to the oil price, so are agribusinesses and many manufacturing companies.

I have a discussion of this particular issue in my book Computational Finance, going all the way back to the Franciscan monk Luca Pacioli's original (1494) mathematical treatise on double-entry bookkeeping (called in Latin: Summa de Arithmetica, Geometria Proportioni et Proportionalita = Summary of Arithmeticvs, Geometry of Proportions and the Abacus, in English. Thus Pacioli clearly meant double-entry bookkeeping to be a mathematical construct, in particular an equation). So let nobody tell us what was meant by double-entry bookkeeping! The mathematician Pacioli earned his keep by tutoring mathematics to the sons and daughters of wealthy merchants or Florence, so he was very familiar with the double-entry bookkeeping used by those merchants of Florence and Venice and he generalized it into the now familiar balance equation A = L + E, even though he did it in Latin words and used no algebraic notation.

My book is Computational Finance, World Scientific Publishing Co, Singapore, 2001 (there is a small misprint in equation 1.5 on page 24, where a sign is reversed). I now even daresay that this balancing equation, which forms the bedrock of modern capitalism and finance, formed the inspiration for Isaac newton's balancing equation of forces: sum of all forces = sum of all (masses x accelerations) = 0), which is the bedrock for physics. Accounting preceded (Newtonian) Physics by ca 150 years. I will make the error correction in the second edition of my book, to be published at the end of this year.