Parallel Addition, Series-Parallel Duality, and Financial Mathematics

This is Chapter 12 in my book: Ellerman, David. 1995. Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics. Lanham MD: Rowman & Littlefield.

Boolean duality can be algebraically represented on Rota’s valuation rings, which have two ring multiplications and one addition, as the complementation anti-isomorphism on the ring that interchanges the two multiplications (see previous chapter).  The work in this last chapter arose by extending the same type of treatment to series-parallel duality.  The roles of addition and multiplication are reversed.  Series-parallel algebras are defined with two additions (the usual series addition and parallel addition) and one multiplication.  Duality on series-parallel algebras that allow division (such as the positive rationals) is represented by the reciprocity anti-isomorphism that interchanges the two additions.

Series-parallel duality has been previously studied in electrical circuit theory and combinatorial theory.  The parallel sum arose naturally when resistors are connected in parallel instead of series.  Given two resistors with the positive real resistances of a and b, their combined resistance is a+b when connected in series and 1/(1/a + 1/b) when connected in parallel.  The full colon (:) notation will be used for the parallel sum, a:b =1/(1/a + 1/b).

Any equation on the positive reals concerning the two sums and multiplication, can be “dualized” by applying the “take-reciprocals” map to obtain another equation.  Each number is replaced by its reciprocal and the two additions are interchanged.  The following equation

holds for any positive real x.  Add any x to one and add its reciprocal to one.  The results are two numbers larger than one and their parallel sum is exactly one.  Dualizing yields the equation

for all positive reals x.  Taking the parallel sum of any x and its reciprocal with one yields two numbers smaller than one that sum to one.  For instance, taking x = 2, we have that

(1+2):(1+(1/2)) = 3:(3/2) = 1 and (1:(1/2)) + (1:2) = (1/3) + (2/3) = 1.

The principal application considered in the essay is series-parallel duality in financial arithmetic.  The basic result is that the parallel sum of the one-shot “balloon” payments at different times that would pay off a given loan is the equal amortization payment that would pay off the loan if paid at each of those times.  This interpretation is not restricted to financial arithmetic.  For example, suppose a forest of initial size PV (in harvestable boardfeet) grows at the rate ri in the ith period.  Let

Then Pm would be the one-shot harvest that could be obtained at the end of the mth period.  For instance,  P3, P17, and P23 are the amounts that could be harvested if the whole forest was harvested at the end of the 3rd, 17th, or the 23rd period.  But what is the smooth or equal harvest PMT so that if PMT was harvested at the end of the 3rd, 17th, and the 23rd periods, then the forest would just be completely harvested at end of that last period?  That equal harvest amount is just the parallel sum of the one-time harvests:

PMT = P3 : P17 : P23.

In the standard application to financial arithmetic, PV is the principal value of a loan, ri is the interest rate for the ith period, and PMT is the equal amortization payment.  Ordinarily, the amortization payments are made at equal time intervals, but this example showed that equal intervals are not necessary.

Since the parallel sum has a natural interpretation, any equation in financial arithmetic can be dualized and interpreted in the field.  Suppose the constant interest rate is 20 percent per period.  Then the discounted present value of two amortization payments of 1 at the end of the first and second period is principal value of the loan paid off by those payments, i.e.,

The equation dualizes to:

The amounts (1.2)1 = 1.2 and (1.2)2 = 1.44 are the compounded principal values of a one dollar loan so they are the one-shot or balloon payments that would pay off the loan if paid, respectively, at the end of the first or the second period.  Their parallel sum, 36/55, is the equal amortization payment that would pay off the loan if paid at the end of both the first and second periods.

These facts can be arranged in the following dual format.


Primal Fact:

The series sum of the


amortization payments for a loan

is the principal of the loan.

Dual Fact:

The parallel sum of the


principals of a loan

is the amortization payment for the loan.


In this manner, each equation can be paired with a dual equation to reveal the structure of series-parallel duality within financial arithmetic.

In the appendix to the essay, commutative series-parallel algebras are defined that are to series-parallel duality as Rota’s valuation rings are to Boolean duality.  In a series-parallel algebra where every element has a multiplicative inverse, called a “series-parallel division algebra,” the take-reciprocals map is an anti-isomorphism that interchanges the two additions.  It is easy to find such algebras since every group generates a series-parallel division algebra.  For instance, the trivial group {1} (written multiplicatively) generates the positive rationals Q+.   This means that a electrical circuit of any given rational resistance can be constructed solely from series and parallel connections of one ohm resistances.

Non-commutative series-parallel algebras are also defined, and every non-commutative group also have a series-parallel completion.  The principal model for a non-commutative series-parallel algebra is the algebra of monotonic increasing real functions on a single real variable.  Series and parallel sums are respectively the vertical and horizontal sum of functions.  Multiplication is functional composition, so the reciprocal of an element is its functional inverse.

In the move from commutative to non-commutative series-parallel algebras, the standard models change from an algebra of numbers such as Q+ to an algebra of functions such as the monotonic increasing real functions.  That is important because it allows us to see the connection to convex duality.

In modern economics, duality on convex functions has an important role.  Duality in linear programming is a well-known special case of convex duality.  We argue informally that convex duality is related to series-parallel duality as a function is related to its derivative.  The derivatives of differentiable strictly convex functions of one variable are the monotonic increasing functions used in the standard model of a non-commutative series-parallel algebra.  The “series” and “parallel” sums of convex functions are, respectively, the usual sum and the infimal convolution.  The derivatives of these two sums of convex functions are, respectively, the vertical and horizontal sums of the derivatives.  Each convex function has a dual or convex conjugate.  The derivative of the convex conjugate of a convex function is the functional inverse of the derivative of the original convex function.

In this manner, the operations expressing duality on convex functions map via differentiation into the operations of the series-parallel algebra of increasing monotonic functions.  In that sense, series-parallel duality can be seen as the “derivative” of convex duality.

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