Series-Parallel Duality: Part II: Financial arithmetic

Reciprocal formulas in financial arithmetic

In financial arithmetic and in the appraisal literature, it has been noticed that the basic formulas occur in pairs, one being the reciprocal of the other. For instance, one popular text on real estate appraisal presents the “Basic Functions of Compound Interest and Their Reciprocals” [Friedman, Jack P. and Nicholas Ordway 1988. Income Property Appraisal and Analysis. Englewood Cliffs: Prentice Hall, p. 70]. The functions could be presented as follows to bring out the underlying symmetry.

Principal Retired by Payment of One

$(1+r)^{-n}$

Payment to Retire Principal of One

$(1+r)^{n}$

Principal Amortized by Payments of One $a(n,r)=\frac{1}{(1+r)^{1}}+\ldots+\frac{1}{(1+r)^{n}}$	Payments to Amortize a Principal of One $\frac{1}{a(n,r)}=(1+r)^{1}:\ldots:(1+r)^{n}$
Fund Accumulated by One per Period $s(n,r)=(1+r)^{n-1}+\ldots+(1+r)+1$	Payments to Accumulate a Fund of One $\frac{1}{s(n,r)}=\frac{1}{(1+r)^{n-1}}:\ldots:\frac{1}{(1+r)}:1$

The Six Functions of One

This Part II of the series-parallel duality post shows that these reciprocal formulas are an example of the SP duality normally associated with electrical circuit theory.

Parallel Sums in Financial Arithmetic

The parallel sum has a natural interpretation in finance so that each equation and formula in financial arithmetic can be paired with a dual equation or formula. The parallel sum “smooths” balloon payments to yield the constant amortization payment to pay off a loan. If r is the interest rate per period, then PV(1+r)ⁿ is the one-shot balloon payment at time n that would pay off a loan with the principal value of PV. The similar balloon payments that could be paid at times t =1, 2,…, n, any one of which would pay off the loan, are

$PV(1+r)^{1},PV(1+r)^{2},\ldots,PV(1+r)^{n}$ .

But what is the equal amortization payment PMT that would pay off the same loan when paid at each of the times t =1, 2, …, n? It is simply the parallel sum of the one-shot balloon payments:

$PMT=PV(1+r)^{1}:PV(1+r)^{2}:\ldots:PV(1+r)^{n}$

$= P_{i=1}^{n}PV(1+r)^{i}$ .

Amortization Payment is Parallel Sum of Balloon Payments

This use of the parallel sum is not restricted to financial arithmetic. For example, suppose a forest of initial size PV (in harvestable board feet) grows at the rate r_i in the i-th period. Then

$P_{m}=PV\Pi_{i=1}^{m}(1+r_{i})$

would be the one-shot harvest that could be obtained at the end of the m-th 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 3^rd, 17^th, or the 23^rd period. But what is the smooth or equal harvest PMT so that if PMT was harvested at the end of the 3^rd, 17^th, and the 23^rd periods, then the forest would just be completely harvested at end of that last period? That smooth harvest amount is just the parallel sum of the one-time harvests:

$PMT=P_{3}:P_{17}:P_{23}$ .

Returning to financial arithmetic, the discounted present value at time zero of n one dollar payments at the end of periods 1, 2,…, n is a(n,r), the present value of an annuity of one.

$a(n,r)=\frac{1}{(1+r)^{1}}+\frac{1}{(1+r)^{2}}+\ldots+\frac{1}{(1+r)^{n}}$

Present Value of Payments of One

Dualizing [i.e., applying the reciprocity map from Part I and using the fact that r(1/1+r) = 1+r] yields:

$\frac{1}{a(n,r)}=(1+r)^{1}:(1+r)^{2}:\ldots:(1+r)^{n}$

Payments to Amortize a Principal of One

For the principal value of one dollar at time zero, the one-shot payments at times 1, 2,…, n that would each pay off the principal are the compounded principals (1+r)¹, (1+r)²,…, (1+r)ⁿ. The parallel sum (1+r)¹: (1+r)² paid at times 1 and 2 would pay off the $1 principal. Similarly, the parallel sum of the first three one-shot payments paid at times 1, 2, and 3 would pay off the $1 principal, and so forth.

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., 55/36:

$a(2,.20)=\frac{55}{36}=\frac{1}{(1.20)^{1}}+\frac{1}{(1.20)^{2}}$ .

The equation dualizes to:

$\frac{1}{a(2,.20)}=\frac{36}{55}=(1.20)^{1}:(1.20)^{2}$ .

The amounts (1.2)¹ = 6/5 and (1.2)² = 36/25 are the compounded principal values of a $1 loan so they are the one-shot or balloon payments that would pay off a loan of principal value $1 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 that loan of $1 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 discounted amortization payments for a loan is the principal of the loan.

Dual Fact:

The parallel sum of the compounded principals of a loan is the amortization payment for the loan.The example illustrates some of the substitutions involved in dualizing the interpretation.

series sum $\leftrightarrow$ parallel sum

discounting $\leftrightarrow$ compounding

principals $\leftrightarrow$ payments.

Future Values and Sinking Fund Deposits

Another staple of financial arithmetic is the computation of sinking fund deposits. The compounded future value at time n of n one dollar deposits at times 1,2,…, n is s(n,r), the accumulation of one per period.

$s(n,r)=(1+r)^{n-1}+(1+r)^{n-2}+\ldots+(1+r)+1=a(n,r)(1+r)^{n}$

Fund Accumulated by One per Period

The discounted values 1/(1+r)^n-1,…, 1/(1+r), 1 of a one-dollar fund are the one-shot deposits at times 1,…, n‑1, n that would each by itself yield a one-dollar future value for the sinking fund at time n. The parallel sum of these one-shot deposits is the (equal) sinking fund deposit at times 1,…, n-1, n that would yield a one-dollar fund at time n:

$\frac{1}{s(n,r)}=\frac{1}{(1+r)^{n-1}}:\frac{1}{(1+r)^{n-2}}:\ldots:\frac{1}{(1+r)}:1$

Sinking Fund Factor: Payments to Accumulate a Fund of One

The dual interpretations might be stated as follows.

The series sum of the n compounded one-dollar deposits is the sinking fund that is accumulated by the one-dollar deposits.

The parallel sum of the n discounted one-dollar funds is the deposit that accumulates to a one-dollar sinking fund.

We now have reproduced the six basic functions of the valuation literature as three pairs of series-parallel duals.

Function	Reciprocal
Principal Retired by Payment of One $(1+r)^{-n}$	Payment to Retire Principal of One $(1+r)^{n}$
Principal Amortized by Payments of One $a(n,r)=\frac{1}{(1+r)^{1}}+\ldots+\frac{1}{(1+r)^{n}}$	Payments to Amortize a Principal of One $\frac{1}{a(n,r)}=(1+r)^{1}:\ldots:(1+r)^{n}$
Fund Accumulated by One per Period $s(n,r)=(1+r)^{n-1}+\ldots+(1+r)+1$	Payments to Accumulate a Fund of One $\frac{1}{s(n,r)}=\frac{1}{(1+r)^{n-1}}:\ldots:\frac{1}{(1+r)}:1$

The Six Functions of One

Infinite Streams of Payments

The formulas for amortization payments can be extended to an infinite time horizon. This involves a financial interpretation for the dual geometric series from Part I with indices beginning at i = 1:

$\sum_{i=1}^{\infty}(1:x)^{i}=x=P_{i=1}^{\infty}(1+x)^{i}$ .

Taking x = 1/r so that 1:x = 1:1/r = 1/(1+r) in the series summation yields the fact that the discounted present value of the constant stream of one-dollar payments at times 1, 2,… is reciprocal of the interest rate x = 1/r.

$\sum_{i=1}^{\infty}(1+r)^{-i}=\frac{1}{r}$ .

Perpetuity Capitalization Formula

Taking x = r in the parallel summation yields the fact that the parallel sum of compounded values of one dollar is the interest rate r, the constant payment at t = 1, 2,… that pays off a principal value of one dollar.

$P_{i=1}^{\infty}(1+r)^{i}=r$

Dual of Perpetuity Capitalization Formula

Thus the dual to the annuity capitalization formula (1/r is the principal whose payments are 1) is the fact that the constant income stream of r is the equivalent of the capital of $1 (r is the payments whose principal is 1).

The series sum of the stream of discounted $1 amortization payments (which is the principal amortized by a $1 amortization payment) is the reciprocal of the interest rate, $(1+r)^{-1}+(1+r)^{-2}+\ldots=r^{-1}$ .

The parallel sum of the stream of compounded $1 principals (which is the payment that amortizes a $1 principal) is the interest rate, $(1+r)^{1}:(1+r)^{ 2}:\ldots=r$ .

More material on series-parallel duality can be found in Chapter 12 of my 1995 book, Intellectual Trespassing as a Way of Life: Essays in Philosophy, Economics, and Mathematics, (Rowman & Littlefield) available to be downloaded on this site.

Series-Parallel Duality: Part II: Financial arithmetic

Reciprocal formulas in financial arithmetic

Parallel Sums in Financial Arithmetic

Future Values and Sinking Fund Deposits

Infinite Streams of Payments

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