Asset Price Stability Analysis in Sparse Portfolios Under the Transactional Asset Pricing Approach

Abstract

The Transactional Asset Pricing Approach (TAPA) is able to handle asset valuations on a portfolio level of size constrained markets against the backdrop of low liquidity hindering the estimation of the variance of returns. Prompted by a numerical simulation of the TAPA algorithm, we develop stability conditions associated with the valuation convergence for any maximal positive time period and positive number of assets. We present stability conditions at the local level, both in continuous and discrete algorithmic time, and we develop them by means of log-linearisation about the steady state of its equations’ variables. We conclude on the analytical existence of stability conditions at the local level up to four assets and any positive time period. We adduce analytical applications within such a region and present a solution for a benchmark calibration of the steady state parameters given two time periods and a single asset.

1. Prologue

In the vast literature researching the issues associated with sparse portfolios and sparse markets under the Discounted Cash Flow (DCF) framework almost no attention has been paid to the usage of sparsely composed portfolios as benchmarks for the valuation of the very assets constituting such portfolios.

Such market referencing niche indices developed by analytics firms as niche Standard and Poor’s sub-indices are typically broad enough to be considered other than sparse as well as bereft of self-referential valuation properties. There could however arise cases in which such self-referential valuations based on limited benchmark portfolios induce not a mere theoretical interest but an altogether practical one in answering questions as to the stability of joint valuations of assets tied up in sparse valuation benchmark portfolios.

In a first such attempt from the transactional standpoint we specifically deploy the Transactional Asset Pricing Approach (TAPA, as a standalone noun) DCF framework so as to study conditions under which TAPA DCF valuations of income producing assets constituting self-referential portfolios against such portfolios converge to a settled value.

Before introducing our work it behoves us to elucidate upon self-referentiality as presently construed. Self-referential in the sparse portfolio or market context refers to the fact that the value of the subject income producing asset being valued depends on the value and performance of the valuation benchmark portfolio against which it is being valued.

The value and ex ante performance of the valuation benchmark portfolio itself depends in turn on the value and performance of the subject asset, as it non-marginally contributes to the portfolio benchmark. Such a loop property introduces instability into the joint valuations of subject assets constituting the sparse portfolio benchmark, as herein explored.

Let us ulteriorly offer to the reader, prior to commencement, a backdrop against which our efforts rise. A sparse market perspective of asset pricing in recent years has been explored through models of diverse complexity, in terms of agent interactive effects and dynamic properties.

Earlier and foundational works, such as [1,2], study the influence on intermediation and asset prices in over the counter markets on the part of illiquidity associated with search and bargaining, focussing on market makers’ bid and ask prices and market structure.

Ref. [3] appraise the impact of market imperfections on expected asset returns, being an asset-pricing exercise whose finding is that on illiquid or sparse markets exhibiting slower incorporation of information prices can become self-referential and reflective of trading conditions, rather than fundamentals alone.

A little further away from asset pricing on sparse markets proper, but still germanely to the field, is an influential strain of academic literature dedicated to the analysis of financial shocks and agent interconnexions, such as [4], who demonstrate that sparse interconnexions can either stabilise or destabilise prices depending on the shock structure.

From a level more slanted towards asset pricing and transactions, rather than from a market level perspective, refs. [5,6] lay down a broader historical outlook and develop the makings of a model which we employ herein.

Ref. [5]’s formulation of the transactional asset pricing model reminiscent of a DCF analysis is more dynamic than that found in [6], as it innovatively ties the terminal values of subject income producing assets in a loop with their present values, but at the cost of assuming exogenously imputed growth dynamics for asset values.

We locate the key particularity of both models in their rare foundational focus on inter-temporal transactional equity, rather than that investor specificity with which the conventional DCF approach to asset pricing is usually associated. A more refined version of such a model, which forms the basis of the present work, can be found in [7,8], who rely on a dynamic multi-period discount rate derivation framework extending the single period discount rate formulations of [9].

Of such two works that of [7] presents an operational TAPA model, suggesting its applicability to the pricing of illiquid assets making use of property valuation related examples; that of [8] by contrast perfects the model in order to discuss and simulate the anti-cyclical properties of TAPA DCF valuations as applied to broader markets.

The present work travels the opposite direction by exploring the applicability of the same TAPA model to a case in which the valuation benchmark portfolio employed is a sparse one, that is to say, one composed of a limited number of assets to which the subject assets being valued contribute in a non-marginal fashion.

While it may indeed be contextually possible to approach the modelling of asset pricing on sparse markets through the lens of market level dynamics, by assuming particular behavioural properties of market agents endowed with heterogeneous expectations, as is operated by [10], we pursue the transactional level pricing behaviour of agents to the problem, whose sole explicit linkages to the sparse market level aggregate are encapsulated in the usage of specific discount rates originating from the TAPA DCF framework.

Even though such other related aspects of market modelling as spreads and expectations adaptation may not be emergent or salient in the approach we take this work is to bear out the approach as fruitful enough to draw the rich asset pricing dynamics emerging from the DCF pricing conventions into relief.

2. Introduction

DCF valuation of financial assets on sparse markets enjoys numerous self-referential properties, endowing them with unpredictable dynamics. The vast body of literature on sparse portfolios relates to selecting optimally performing, parsimonious portfolios out of a bigger set in the mean variance optimisation framework accounting for transaction costs.

Sparse portfolios are occasionally pre-conditioned by constraints on the actual size of financial markets, rather out of concern for transaction costs. Such can be the case of shallow, illiquid financial markets operating under the conditions of capital controls. For example, the Tashkent Republican Stock Exchange in Uzbekistan, in which the authors worked throughout the composition of this work, trades in slightly more than a dozen liquid stocks.

In such an instance the valuation of assets against such thin markets or locked up in such sparse portfolios poses a self-referential challenge which can be handled in the mean variance Modern Portfolio Theory (MPT) context of [11] only ineptly, especially if the equity market is found to be poorly integrated with other asset classes [12].

An alternative DCF based first moment framework is represented by TAPA, presented by [8], which is able to handle asset valuations on a portfolio level of size constrained markets against the backdrop of low liquidity hindering the estimation of the variance of returns; such demarcates TAPA as a complementary framework for sparse mean variance optimisation particularly suited to illiquid markets in which valuation, rather than allocation, is the primary concern.

Indeed, TAPA represents a fruitful framework for research in such a context, and its applications to the study of sparse portfolios are still relatively novel and under-explored. Prompted by a numerical simulation of the TAPA algorithm applied to a self-referential portfolio nested valuation of four assets, we develop stability conditions associated with the valuation convergence for any maximal positive time period and positive number of assets.

Owing to its non-linear nature, we concentrate our efforts on the study of its local stability. We specifically present stability conditions at the local level, both in continuous and discrete algorithmic time, and we develop them by means of log-linearisation about the steady state of its equations’ variables.

We conclude on the analytical existence of stability conditions at the local level up to four assets and any positive time period. We adduce analytical applications within such a region and present a solution for a benchmark calibration of the steady state parameters given two time periods and a single asset; thus, the present study contributes to the academic literature by bridging sparse portfolio frameworks and transactional asset pricing as a nexus researching asset price stability.

In Section 3 we review TAPA, both in terms of academic literature and algorithmic equations. In Section 4 we present the said numerical simulation of the TAPA algorithm having prompted our analysis. In Section 5 we undertake the log-linearisation of the TAPA algorithm equations and in Section 6 and Section 7 we study their local stability. Section 8 concludes.

3. TAPA: Literature Review and Algorithm

3.1. TAPA in Literary Perspective

Following the seminal works by [1,5,6,7,8,13,14,15] comprehensively advance TAPA, whose particularity is the derivation of income approach valuation techniques through a focus on inter-temporal transactional equity, rather than investor specificity.

TAPA is presented as a viable alternative to valuation thinking by analogy, distinctive of situations in which the second moment of return distributions is unavailable to the end of applying more conventional asset valuation frameworks associated with MPT.

Such situations specifically originate with regard to illiquid assets traded on inefficient markets in which the Law of One Price (LOP) is liable to failure [7]. In the context of such situations the transaction itself, coupled with the interplay of the bargaining interests implicit in it, can indeed represent an elementary unit of analysis and a departure point for the reconstitution of asset valuation theory under the income approach on transactional grounds.

The transactional perspective on income producing asset pricing complements the individual investor specific perspective on capital valuation developed by [16,17] along with subsequent modifications proceeding from behavioural finance; it accordingly complements the efficient market perspective developed by [9] as well as such related subsequent works on MPT as the Capital Asset Pricing Model (CAPM [11]).

TAPA thereby effectively occupies a middle ground, straddling situations in which subjective valuations are no longer sufficient to explain income producing asset pricing, but in which the statistical approach based on first and second moment returns is as yet unavailable owing to the paucity of external pricing data; modelling the dynamics of an individual transaction in time steps is to fill the void, ensuring applications to situations in which market inefficiencies assert themselves patently.

TAPA’s line of reasoning for the justification of the DCF asset pricing format adduces certain transactional concepts to the analysis which are conspicuously absent in the more conventional DCF justifications based on a recursive single period individual investor perspective, pioneered by [16] (see [18,19]). In essence, TAPA establishes the DCF framework as an overlooked corollary originating from the dynamics of transactional equity explored many centuries ago in Aristotelian catallactics (Book 5, paragraphs 2–5; see Richard McKeon (1941, 1966) “The basic works of Aristotle”, Random House, pp. 1005–1010 [20]).

Apart from the inter-temporal principle of transactional equity, in technical terms the TAPA approach displays a number of novelties compared to the conventional DCF analysis. It employs time varying discount rates based on a direct multi-period extension of the [17] framework and associates the terminal value of the subject asset to its present value through the use of subject specific asset price or capital value growth rates.

The combination of such features, as is claimed in [8], endows TAPA valuations with anti-cyclical valuation properties. Specifically, TAPA emphasises the fact that any income approach valuation is also a comparative valuation by virtue of being undertaken against some reference portfolio benchmark. It is the rate adopted in a valuation to discount cashflows from the subject asset which allows for such a comparative bridge to the benchmark.

In some applications, such as the case of MPT and its pricing models (e.g., CAPM [11,21]), the portfolio benchmark is assumed to be a very broad and well diversified portfolio shaped upon the normative principles of risk return optimisation. By contrast, TAPA is free of any assumption of portfolio composition rules.

Of crucial importance in our study context, in contrast with such normative models of asset pricing as CAPM, is TAPA’s admission of flexible portfolio benchmarks for DCF valuation analyses, which can be represented by broad or narrow portfolio aggregates without any prescription as to portfolio composition rules.

Such is an appealing feature of the approach, as it prompts one to study the properties of asset valuation models in which reference portfolio benchmarks are represented by narrow investment portfolios, the composition of which is non-marginally affected by the performance of the subject asset itself.

3.2. TAPA Algorithm: Rationale

The financial rationale of the TAPA equations is to create a portfolio of assets which are to be valued and against which the same assets are to be valued in the context of a first revaluation iteration, so that upon having rebalanced the initial value conditions of the portfolio the said assets may be revalued over the subsequent iterations. Such is to result in the convergence or divergence of the asset valuations over the subsequent iterations from their initial values and in our analysis we aim to study the properties of such a convergence or divergence.

Much of the transformations outlined in the TAPA equations which are to follow represent inter-temporal and inter-asset aggregations required for constructing the sparse portfolio benchmark and for the attendant study of its inter-temporal growth properties, which in turn are to be translated into discount rates and utilised in the inter-asset revaluation iterations, being indeed undertaken under TAPA’s DCF version.

The reason for which we treat of inter-temporal and inter-asset aggregations is that our inputs of prospective financial information are period specific by assumption; additionally, in order to analyse growth and discount rates we must employ cumulative inputs aggregated from the start of the analysis. Ulterior complexity is introduced by the necessity to aggregate income producing assets into the portfolio by price and by projected income.

3.3. TAPA Algorithm: Indices, Variables, Parameters and Equations

Consider the following indices, variables and parameters. Indices are time periods and the numbers of assets. Time periods: $t\in [1,k]\subseteq {\mathbb{N}}_{+}.$ Number of assets: $i\in [1,h]\subseteq {\mathbb{N}}_{+}.$ Parameters are the initial income and income growth rate. Initial income: $Y_{i0}=Y_{i1}\in {\mathbb{R}}_{+}.$ Income growth rate: $u_{it}\in (0,\infty )={\mathbb{R}}_{++},$ since $u_{it}=1+{\overline{u}}_{it}$ for ${\overline{u}}_{it}\in (-1,\infty )\subset \mathbb{R},$ and $1=u_{i1}=1+{\overline{u}}_{i1}=1+0.$

Variables are the initial price and the price growth rate. Initial price: $P_{i0}\in {\mathbb{R}}_{++}.$ Price growth rate: $v_{it}\in (0,\infty )={\mathbb{R}}_{++},$ since $v_{it}=1+{\overline{v}}_{it}$ for ${\overline{v}}_{it}\in (-1,\infty )\subset \mathbb{R}.$ Notice that initial price $P_{i0}$ and price growth rate $v_{it}$ shall be treated as variables, rather than parameters. Consider the following system of TAPA equations in which assets are vested in a portfolio, against which they are to be valued in accord with its last equation.

Equation (1) models the price growth rate compounded over time, given assets:

$$v_{ik}!=\prod _{t=1}^k{v}_{it}=v_{i1}·\dots ·v_{ik}$$

(1)

Such an equation is inter-temporally aggregative and positioned at the asset level, modelling price growth rate $v_{it}$ compounded over time t up to and including terminal period of analysis $k;$ it applies to each individual asset i entering the reference valuation portfolio.

Equation (2) models the price growth rate compounded over sub-time, given assets:

$$v_{it}!=\prod _{j=1}^t{v}_{ij}=v_{i1}·\dots ·v_{it}$$

(2)

Such an equation is inter-temporally aggregative and positioned at the asset level, modelling price growth rate $v_{ij}$ compounded over sub-time j up to and including each sub-period of the forecast period $t;$ it applies to each individual asset i entering the reference valuation portfolio.

Equation (3) models the income growth rate compounded over sub-time, given assets:

$$u_{it}!=\prod _{j=1}^t{u}_{ij}=u_{i1}·\dots ·u_{it}$$

(3)

Such an equation is inter-temporally aggregative and positioned at the asset level, modelling income growth rate $u_{ij}$ compounded over sub-time j up to and including each sub-period of the forecast period $t;$ it applies to each individual asset i entering the reference valuation portfolio.

Equation (4) models the price growth rate compounded over sub-sub-time, given assets:

$$v_{ij}!=\prod _{l=1}^j{v}_{il}=v_{i1}·\dots ·v_{ij}$$

(4)

Such an equation is inter-temporally aggregative and positioned at the asset level, modelling price growth rate $v_{il}$ compounded over sub-sub-time l up to and including each sub-sub-period of the forecast sub-period $j\in [1,t]\subseteq {\mathbb{N}}_{+};$ it applies to each individual asset i entering the reference valuation portfolio.

Equation (5) models the income growth rate compounded over sub-sub-time, given assets:

$$u_{ij}!=\prod _{l=1}^j{u}_{il}=u_{i1}·\dots ·u_{ij}$$

(5)

Such an equation is inter-temporally aggregative and positioned at the asset level, modelling income growth rate $u_{il}$ compounded over sub-sub-time l up to and including each sub-sub-period of the forecast sub-period $j;$ it applies to each individual asset i entering the reference valuation portfolio. For a representation of Equations (1)–(5) see Figure 1.

Equation (6) models price growth over assets (portfolio level), given sub-sub-time. (For elements $a,b\in \left\{0,1\right\}$ exclusive disjunction $a⊻b=1$ if and only if $a\ne b,$ whence $a⊻b=a+b-2ab=1$):

$$\begin{array}{cc}\hfill & {\tilde{v}}_j=\sum _{i=1}^h{P}_{i0}\left(1⊻v_{ij}!\right)=\left(P_{10}+\dots +P_{h0}\right)⊻\left(P_{10}{v}_{1j}!+\dots +P_{h0}{v}_{hj}!\right)=\hfill \\ \hfill & =\sum _{i=1}^h{P}_{i0}\left[1+v_{ij}!-2\left(1\right)\left(v_{ij}!\right)\right]=\sum _{i=1}^h{P}_{i0}\left(1-v_{ij}!\right)=\hfill \\ \hfill & =\left(P_{10}+\dots +P_{h0}\right)-\left(P_{10}{v}_{1j}!+\dots +P_{h0}{v}_{hj}!\right)\hfill \end{array}$$

(6)

Such an equation models portfolio-level aggregation over individual assets i with regard to initial price $P_{i0}$ at forecast sub-period 0 and specific reference portfolio price growth rate $v_{ij}!$ weighted at initial price $P_{i0}$ at forecast sub-period $j.$

Equation (7) models income growth over assets (portfolio level), given sub-sub-time:

$${\tilde{u}}_j=\sum _{i=1}^h{Y}_{i1}{u}_{ij}!=\left(Y_{11}{u}_{1j}!+\dots +Y_{h1}{u}_{hj}!\right)$$

(7)

Such an equation models portfolio level aggregation over individual assets i with regard to specific reference portfolio income growth rate $u_{ij}!$ weighted at initial income $Y_{i1}$ at forecast sub-period $j.$

Equation (8) models the price growth ratio compounded over time and assets (portfolio level):

$$\begin{array}{c}{\tilde{v}}_{t-1}!=\prod _{j=1}^{t-1}1+{\displaystyle \frac{{\tilde{v}}_j-{\tilde{v}}_{j-1}}{{\tilde{v}}_{j-1}}}=\prod _{j=1}^{t-1}{\displaystyle \frac{{\tilde{v}}_{j-1}+{\tilde{v}}_j-{\tilde{v}}_{j-1}}{{\tilde{v}}_{j-1}}}=\hfill \\ =\prod _{j=1}^{t-1}{\displaystyle \frac{{\tilde{v}}_j}{{\tilde{v}}_{j-1}}}={\displaystyle \frac{{\tilde{v}}_1}{{\tilde{v}}_0}}·\dots ·{\displaystyle \frac{{\tilde{v}}_{t-1}}{{\tilde{v}}_{t-2}}}\hfill \end{array}$$

(8)

Such an equation models portfolio level cumulative dynamics with regard to reference portfolio price growth ratio ${\displaystyle \frac{{\tilde{v}}_j}{{\tilde{v}}_{j-1}}}$ over sub-time j up to forecast period $t-1.$

Equation (9) models the income growth ratio compounded over time and assets (portfolio level):

$$\begin{array}{c}{\tilde{u}}_t!=\prod _{j=1}^{t}1+{\displaystyle \frac{{\tilde{u}}_j-{\tilde{u}}_{j-1}}{{\tilde{u}}_{j-1}}}=\prod _{j=1}^t{\displaystyle \frac{{\tilde{u}}_{j-1}+{\tilde{u}}_j-{\tilde{u}}_{j-1}}{{\tilde{u}}_{j-1}}}=\hfill \\ =\prod _{j=1}^t{\displaystyle \frac{{\tilde{u}}_j}{{\tilde{u}}_{j-1}}}={\displaystyle \frac{{\tilde{u}}_1}{{\tilde{u}}_0}}·\dots ·{\displaystyle \frac{{\tilde{u}}_t}{{\tilde{u}}_{t-1}}}\mathrm{for}{\displaystyle \frac{{\tilde{u}}_1}{{\tilde{u}}_0}}=1\hfill \end{array}$$

(9)

Such an equation models portfolio level cumulative dynamics with regard to reference portfolio income growth ratio ${\displaystyle \frac{{\tilde{u}}_j}{{\tilde{u}}_{j-1}}}$ over sub-time j up to forecast period $t.$

Equation (10) models the price growth change over terminal sub-time, given assets (portfolio level):

$${\check{v}}_t={\displaystyle \frac{{\sum }_{i=1}^h{P}_{i0}{v}_{it}!-{\sum }_{i=1}^h{P}_{i0}{v}_{it-1}!}{{\sum }_{i=1}^h{P}_{i0}{v}_{it-1}!}}={\displaystyle \frac{\left(P_{10}{v}_{1t}!+\dots +P_{h0}{v}_{ht}!\right)-\left(P_{10}{v}_{1t-1}!+\dots +P_{h0}{v}_{ht-1}!\right)}{P_{10}{v}_{1t-1}!+\dots +P_{h0}{v}_{ht-1}!}}$$

(10)

Such an equation models the portfolio level change of reference portfolio price growth $v_{it}!$ weighted at initial price $P_{i0}$ over individual assets i at forecast period $t.$ For a representation of Equations (6)–(10) see Figure 2.

Equation (11) models the initial income to price ratio (portfolio level yield):

$$R={\displaystyle \frac{{\sum }_{i=1}^h{Y}_{i1}}{{\sum }_{i=1}^h{P}_{i0}}}={\displaystyle \frac{Y_{11}+\dots +Y_{h1}}{P_{10}+\dots +P_{h0}}}$$

(11)

Such an equation models the portfolio level ratio of initial income $Y_{i1}$ to initial price $P_{i0}$ over individual assets $i.$

Equation (12) models the rate of return over terminal sub-time:

$$r_t=R\left({\displaystyle \frac{{\tilde{u}}_t!}{{\tilde{v}}_{t-1}!}}\right)+{\check{v}}_t$$

(12)

Such an equation models the portfolio level ratio of income growth change compounded over time and assets ${\tilde{u}}_t!$ to price growth change compounded over time and assets ${\tilde{v}}_{t-1}!$ weighted by portfolio level yield R and added to the price growth rate over terminal sub-time, given assets, ${\check{v}}_t;$ such variables stem from Equations (8)–(11).

It embodies the rate of return over terminal sub-time $r_t$ as a reference portfolio attribute for individual asset i revaluations. It is a portfolio level property used as a time varying discount rate to revalue individual assets against the reference portfolio under the DCF of TAPA’s Basic Pricing Equation (BPE, Equation (15)).

The discount rate consequently typifies the synthesised portfolio level rate of return based on the initial portfolio yield adjusted for growth under the frameworks of [9,17], which introduce the additive decomposition of equity discount rates into the dividend yield as well as capital gain components in a single period framework (The single period framework was a celebrated analytical approach before being eclipsed by [22] and related decomposition approaches [23,24]); the multi-period extension hereby employed together with the derivation of the TAPA BPE can be found in [7]. For a representation of Equations (11) and (12) see Figure 3.

Equation (13) models the rate of return compounded over sub-time:

$$r_t!=\prod _{j=1}^{t}1+r_j$$

(13)

Such an equation models the rate of return over sub-time $r_j$ added to one compounded over sub-time j up to and including each sub-period of the forecast period $t.$ It embodies the rate of return compounded over sub-time $r_t!$ as a reference portfolio attribute for individual asset i revaluations. It is a compounded rate of return for the reference portfolio in that it is the chain product of all prior rates of return up to and including forecast period $t.$

Equation (14) models the rate of return compounded over time:

$$r_k!=\prod _{t=1}^{k}1+r_t$$

(14)

Such an equation models the rate of return over terminal sub-time $r_t$ added to one compounded over time t up to and including terminal period of analysis $k.$ It embodies the rate of return compounded over time $r_k!$ as a reference portfolio attribute for individual asset i revaluations. It is a compounded rate of return for the reference portfolio in that it is the chain product of all prior rates of return up to and including terminal period of analysis $k.$

Equation (15) models the change in the initial price (TAPA BPE for asset i):

$$W_i\equiv {\dot{P}}_{i0}={\displaystyle \frac{Y_{i1}{\sum }_{t=1}^k\frac{u_{it}!}{r_t!}}{1-\frac{v_{ik}!}{r_k!}}}$$

(15)

Such an equation models the ratio of income growth rate compounded over sub-time, given assets, $u_{it}!$ to the rate of return compounded over sub-time $r_t!$ accumulated over time t up to and including terminal period of analysis k weighted by initial income $Y_{i1}$ divided by one minus the ratio of price growth rate compounded over time, given assets, $v_{ik}!$ to the rate of return compounded over time $r_k!.$

It embodies the change in initial price ${\dot{P}}_{i0}\equiv W_i$ as a revaluation of individual asset i against the reference portfolio constituted by h assets. It is the TAPA BPE for individual asset i revaluation iterations. It is a DCF framework endowed with time varying discount rates $r_k!$ and $r_t!$ based on the attributes of the reference portfolio, in which asset i’s terminal compounded price growth rate $v_{ik}!$ in period of analysis k is linked to the asset’s present value $W_i$ as being conditioned by the fixed input pattern of Equation (1) and in which asset i’s income growth rate $u_{it}!$ over time t is linked to initial income $Y_{i1}$ as conditioned by the fixed input pattern of Equation (3); see [7] for its derivation as the restatement of the following equation in which asset present value $W_i$ is to be isolated:

$$W_i=Y_{i1}\sum _{t=1}^k{\displaystyle \frac{u_{it}!}{r_t!}}+{\displaystyle \frac{W_i{v}_{ik}!}{r_k!}}.$$

For a representation of Equations (13) and (15) see Figure 4. We additionally remark that in an attempt to smooth out the TAPA algorithm and thereby better gauge its stability conditions, although time t may be discrete, changes in initial price ${\dot{P}}_{i0}$ are initially regarded as continuous; eventually and ultimately, while accommodating the study of TAPA’s algorithm’s stability across assets $h>1$ and for completeness, they are albeit regarded as discrete.

Notice too that the equation for income growth rate over time, given assets, is not used: $\nexists u_{ik}!={\prod }_{t=1}^k{u}_{it}=u_{i1}·\dots ·u_{ik}.$ There is further no equation for the change in initial income ${\dot{Y}}_{i1},$ suggesting that income growth ${\tilde{u}}_j,$ income growth rates $u_{it}!$ and $u_{ij}!$ and income growth change ${\tilde{u}}_t!$ equations may not be used either in the analysis of local stability.

4. Asset Valuation Against Small Portfolios: TAPA Example

Besides being a self-referential theoretical case, it can stand for a positivistic scenario encountered in inefficient markets in which asset investors are lightly diversified. For instance, in some emerging economies, national stock markets are constituted by a few stocks in the presence of restrictions on the capital account to invest abroad. A case in point is again that of Uzbekistan, in which only about 20 stocks from select industries are represented in active trading on the stock market.

We take up such a case as the baseline for our research. In it the subject asset contributes non-marginally to the composition of the benchmark portfolio, that is, it constitutes a sizeable portion of it, against which it is also valued under TAPA. The case is inaugurated by a theoretical example in which the valuation performance of four assets is analysed over the forecast horizon of five years. The assets are to be valued under DCF with reference to the overall portfolio characterised thereby.

The present example is conducted in Microsoft Excel and is freely available for download at https://disk.yandex.ru/i/cuH4eltHnOSnZQ (accessed on 1 September 2025) (On having downloaded the file, in order to solve the TAPA BPE for asset valuation by numerical substitution, please activate the Microsoft Excel solver for circular references, under “File”, “Options”, “Formulas” and “Enable iterative calculations”). As mentioned, four income producing assets are valued under the TAPA BPE over a forecast horizon of five years. The reference portfolio for the discount rate derivation purposes is fully characterised by the assets themselves. Portfolio asset growth rates $v_{it}$ and $u_{it},$ which are model parameters, are presented in

Table 1. $v_{it}$ and $P_{i0}$.

	Asset Price Growth Rate ${\mathit{v}}_{\mathit{i}\mathit{t}}$ (Percentage)					Initial Asset Price ${\mathit{P}}_{\mathit{i}0}$ (Currency Units)
Asset, Year	1	2	3	4	5	0
1	$1.02$	$1.03$	$1.04$	$1.02$	$1.07$	1000
1 (compound)	$1.02$	$1.05$	$1.09$	$1.11$	$1.19$
2	$1.04$	$1.03$	$1.2$	$0.94$	$1.05$	1300
2 (compound)	$1.04$	$1.07$	$1.28$	$1.21$	$1.27$
3	$1.04$	$1.04$	1	$0.9$	$1.05$	900
3 (compound)	$1.04$	$1.08$	$1.08$	$0.97$	$1.022$
4	1	$1.05$	$0.85$	$0.95$	$1.2$	1400
4 (compound)	1	$1.05$	$0.89$	$0.85$	$1.02$
						4600

Note. Input data on the initial expected growth parameters for the subject assets, that is, asset price or capital value growth rate $v_{it}$ and initial asset price or capital value $P_{i0}.$ The same can be viewed on lines 3 to 14 of the first tab of the referenced Microsoft Excel file.

and

Table 2. $u_{it}$ and $Y_{i1}$.

	Net Operating Income Growth Rate ${\mathit{u}}_{\mathbf{it}}$ (Percentage)					Initial Net Operating Income ${\mathit{Y}}_{\mathit{i}1}$ (Currency Units)
Asset, Year	1	2	3	4	5	1
1	1	$1.01$	$0.98$	$1.1$	$0.8$	100
1 (compound)	1	$1.01$	$0.99$	$1.089$	$0.87$
2	1	$0.95$	$0.95$	$1.04$	1	95
2 (compound)	1	$0.95$	$0.9$	$0.94$	$0.94$
3	1	$1.04$	1	$0.97$	$1.05$	110
3 (compound)	1	$1.04$	$1.04$	$1.01$	$1.06$
4	1	$1.07$	$1.021$	$1.01$	$0.9$	120
4 (compound)	1	$1.07$	$1.09$	$1.1$	$0.99$
						425

Note. Input data on the initial expected growth parameters for the subject assets, that is, net operating income growth rate $u_{it}$ and initial net operating income $Y_{i1}.$ The same can be viewed on lines 17 to 28 of the first tab of the referenced Microsoft Excel file.

.

By executing Equation (12) we derive discount rate $r_t$ at the level of the portfolio aggregate (four assets) as reported in

Table 3. $r_t$.

Year	1	2	3	4	5
Discount rate $r_t$ (percentage)	$1.116$	$1.13$	$1.108$	$1.041$	$1.179$

Note. Such an estimate is set forth on line 74 of the first tab of the referenced Microsoft Excel file; its consistency is ensured by applying a direct bottom up, period by period approach to portfolio return analysis on line 70 of the said first tab.

. Furthermore, by utilising the said discount rates we are able to provide a first time DCF based valuation $W_i$ for each of the four assets constituting the portfolio at time $t=0,$ being the start of the first period, relative to the portfolio’s overall constitution. Such a valuation is set forth in accord with the TAPA BPE (Equation (15)) for which discount rate estimates $r_t$ are those obtained in Table 3.

Upon execution of such DCF valuations on the basis of Equation (15) for each subject asset i characterising the four asset portfolio (See tabs “BPE asset 1” to “BPE asset 4” in the referenced Microsoft Excel file) we remark that at the algorithm’s first pass (See array $I4:I14$ on the first tab of the referenced Microsoft Excel file) not only will the completed valuations of individual assets in the portfolio $W_i$ differ from their assumed initial prices $P_{i0},$ as indicated in the rightmost column of Table 1, but that the sum of such estimates $W_i$ will also differ from the original initial total amount (See the final valuation of 4780 currency units juxtaposed to that of 4600 currency units for the portfolio in the rightmost column of Table 1), as discernible in

Table 4. $P_{i0}$ and $W_i$.

Asset	Initial Asset Price ${\mathit{P}}_{\mathit{i}0}$	Asset Valuation (Iteration 1, Currency Units) ${\mathit{W}}_{\mathit{i}}$	Asset Valuation (Iteration 2, Currency Units) ${\mathit{W}}_{\mathit{i}}$	Valuation Change (Percentage)
1	1000	1201	1202	$20.2$
2	1300	1272	1274	$-2$
3	900	1028	1027	$14.1$
4	1400	1140	1139	$-18.6$
	4600	4641	4642	$0.9$

Note. Initial asset price inputs $P_{i0}$ and valuation results $W_i,$ following two iterations, for each of the four assets rendered against their portfolio under TAPA DCF.

.

Equally noteworthy is that the result of valuations $W_i$ thus obtained at the first iteration, both for each individual asset in the portfolio and for the value of the entire portfolio, remains stable under each subsequent iteration of the valuations (See tabs “Portfolio aver after 1 iteration” and beyond in the referenced Microsoft Excel file), that is, whenever the obtained vector of valuations W substitutes for initial prices $P_0$ as inputs (See array $G5:G14$ in tab “Portfolio aver after 1 iteration”) and the analysis loops over into the second valuation iteration (See the return of the new valuation vector $K5:K14$ collated on the same tab).

Such results illustrate the case of stability in asset valuations against small portfolios converging to a set valuations W vector. It is clear that if the example assumptions are changed (For instance, in array $D2:F2$, being the first tab of the referenced Microsoft Excel file, we could introduce sizeable (greater than 1) period wise shocks to capital value growth rates with regard to the assets constituting the portfolio) such a stability cannot then remain unconditional (An example in which there is no convergence following two iterations of the discussed valuation algorithm is presented at https://disk.yandex.ru/i/C1wTndt_SNpGzw (accessed on 1 September 2025)), but is to remain subject to certain relations, which the present work aims at further exploring analytically.

Consequently, the research question treated in the present work is the inquiry into the properties of stability conditions for valuations of assets obtained by means of the TAPA BPE, whereby such valuations are rendered against small portfolios. In order to attain to the market level of valuations, whereby the contribution of individual assets to the market portfolio is infinitesimal, it is generally interesting to study the properties of valuation benchmarks (investment portfolios), which are characterised by non-marginal contributions of the subject assets, as well as the associated valuations of the subject assets against the portfolio to which they contribute.

For reasons of interest in reduced transactional costs or parsimony, as well as in the high non-linear sensitivity of well diversified portfolio weights to small changes in initial mean variance parameters [25,26,27,28], modern investment research in a Markovitz mean or variance environment focusses on small portfolios through the introduction of weighting penalty functions into portfolio optimisation frameworks or the employment of sophisticated regression models (e.g., Least Absolute Shrinkage and Selection Operators).

Anew, following [21], we refer to such portfolios as small, sparse or non-marginal and to such valuations as self-referential, on which to date research has been scarce. Aside from optimal portfolio selection issues, ref. [29] come close to adumbrating the practical impact of the problem of small portfolios on private value asset managers, however, they eschew asset valuation aspects.

The research which comes closest to the subject area is the analysis of the composition of small portfolios endowed with the desired characteristics in a mean variance space. For instance, ref. [30] adopt the paradigm of decoupling shrinkage and selection in the orbit of a penalised utility approach; similarly, ref. [31] exploit other penalised likelihood optimisation techniques in order to achieve mean reverting portfolios.

Yet, the mean variance framework deals with asset pricing issues in terms of absolute monetary value only obliquely; we additionally maintain that the exploration of stability of portfolio-level valuations for income producing capital assets placed in small portfolios may be a challenging exercise with practical implications for thin and illiquid capital markets. Albeit Appendix A.1 Section A0 may be consulted for a backdrop thereof, we consequently submit to have established sufficient familiarity with TAPA and now proceed to the development of our contributions.

5. Log-Linearisation of TAPA Equations

We reiterate that in a numerical simulation of the TAPA algorithm, presented in the previous section, over five time periods, across four assets and through $n>1$ iterations, we observe that the change in initial price brought about by the TAPA algorithm as of the second iteration does not differ from that of the first iteration, thereby discerning particular or inductive stability, whose general or deductive study we undertake in this work.

Since TAPA is a non-linear model we concentrate our efforts on the study of its local stability by means of a log-linearisation of its equations about their variables’ steady state, leaving the study of its global stability in its non-linear form for future work.

In this work we specifically develop stability conditions for any maximal time period and number of assets on the positive natural number line. Our contribution is therefore both notional and methodological: we notionally set forth stability conditions for TAPA at the local level and we methodologically develop them by means of the log-linearisation about the steady state of its equations’ variables.

Consider the following log-linearisation of such equations around their variables’ steady state (i.e., balanced growth path) by means of a first order Taylor expansion:

$$f\left(x\right)=f\left(x^{*}\right)+{\displaystyle \frac{f^{\prime }\left(x^{*}\right)}{1!}}(x-x^{*}),$$

in which $x^{*}\in \mathbb{R}$ is variable x’s steady state for function $f:{\mathbb{R}}^m\to {\mathbb{R}}^m$ and dimension $m\in \mathbb{R}$ and $\widehat{x}\equiv {\displaystyle \frac{x-x^{*}}{x^{*}}}$ is variable x’s growth rate relative to its steady state $x^{*}.$

For elaborative generality initial income ${\dot{Y}}_{i1},$ income growth ${\tilde{u}}_j,$ income growth rates $u_{it}!$ and $u_{ij}!$ and income growth change ${\tilde{u}}_t!$ shall be treated as variables until the end of the log-linearisation, whereupon they shall return to being treated as parameters, thereby disappearing. The details of the below derivations are collected in the Appendix A.2 and Appendix A.3 Sections A1 and A2.

5.1. Multiplicative Equations

Equation (1) models the price growth rate compounded over time, given assets:

$$v_{ik}!=\prod _{t=1}^k{v}_{it}=v_{i1}·\dots ·v_{ik}⟶{\widehat{v}}_{ik}!=\sum _{t=1}^k{\widehat{v}}_{it}$$

Equation (2) models the price growth rate compounded over sub-time, given assets:

$$v_{it}!=\prod _{j=1}^t{v}_{ij}=v_{i1}·\dots ·v_{it}⟶{\widehat{v}}_{it}!=\sum _{j=1}^t{\widehat{v}}_{ij}$$

Equation (3) models the income growth rate compounded over sub-time, given assets:

$$u_{it}!=\prod _{j=1}^t{u}_{ij}=u_{i1}·\dots ·u_{it}⟶{\widehat{u}}_{it}!=\sum _{j=1}^t{\widehat{u}}_{ij}$$

Equation (4) models the price growth rate compounded over sub-sub-time, given assets:

$$v_{ij}!=\prod _{l=1}^j{v}_{il}=v_{i1}·\dots ·v_{ij}⟶{\widehat{v}}_{ij}!=\sum _{l=1}^j{\widehat{v}}_{il}$$

Equation (5) models the income growth rate compounded over sub-sub-time, given assets:

$$u_{ij}!=\prod _{l=1}^j{u}_{il}=u_{i1}·\dots ·u_{ij}⟶{\widehat{u}}_{ij}!=\sum _{l=1}^j{\widehat{u}}_{il}$$

Equation (8) models the price growth ratio compounded over time and assets (portfolio level):

$${\tilde{v}}_{t-1}!=\prod _{j=1}^{t-1}{\displaystyle \frac{{\tilde{v}}_j}{{\tilde{v}}_{j-1}}}⟶{\widehat{\tilde{v}}}_{t-1}!=\sum _{j=1}^{t-1}{\widehat{\tilde{v}}}_j-{\widehat{\tilde{v}}}_{j-1}$$

Equation (9) models the income growth ratio compounded over time and assets (portfolio level):

$${\tilde{u}}_t!=\prod _{j=1}^t{\displaystyle \frac{{\tilde{u}}_j}{{\tilde{u}}_{j-1}}}⟶{\widehat{\tilde{u}}}_t!=\sum _{j=1}^t{\widehat{\tilde{u}}}_j-{\widehat{\tilde{u}}}_{j-1}$$

Equation (13) models the rate of return compounded over sub-time:

$$r_t!=\prod _{j=1}^{t}1+r_j⟶{\widehat{r}}_t!=\sum _{j=1}^t{\displaystyle \frac{{\widehat{r}}_j{r}_j^{*}}{\left(1+r_j^{*}\right)}}$$

Equation (14) models the rate of return compounded over time:

$$r_k!=\prod _{t=1}^{k}1+r_t⟶{\widehat{r}}_k!=\sum _{t=1}^k{\displaystyle \frac{{\widehat{r}}_t{r}_t^{*}}{\left(1+r_t^{*}\right)}}$$

5.2. Additive Equations

Equation (6) models price growth over assets (portfolio level), given sub-sub-time:

$${\tilde{v}}_j=\sum _{i=1}^h{P}_{i0}\left(1-v_{ij}!\right)⟶{\widehat{\tilde{v}}}_j=\sum _{i=1}^h\left[{\displaystyle \frac{P_{i0}^{*}\left(1-v_{ij}^{*}!\right){\widehat{P}}_{i0}}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}}-{\displaystyle \frac{P_{i0}^{*}{v}_{ij}^{*}!{\widehat{v}}_{ij}!}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}}\right]$$

Equation (7) models income growth over assets (portfolio level), given sub-sub-time:

$${\tilde{u}}_j=\sum _{i=1}^h{Y}_{i1}{u}_{ij}!⟶{\widehat{\tilde{u}}}_j=\sum _{i=1}^h\left({\displaystyle \frac{Y_{i1}^{*}{u}_{ij}^{*}!}{{\sum }_{i=1}^h{Y}_{i1}^{*}{u}_{ij}^{*}!}}\right)\left({\widehat{Y}}_{i1}+{\widehat{u}}_{ij}!\right)$$

Notice that income growth over assets (portfolio level), given sub-sub-time,

$${\widehat{\tilde{u}}}_j=\sum _{i=1}^h{\displaystyle \frac{Y_{i1}^{*}{u}_{ij}^{*}!{\widehat{u}}_{ij}!}{{\sum }_{i=1}^h{Y}_{i1}^{*}{u}_{ij}^{*}!}},$$

if initial income $Y_{i1}$ is a parameter.

Equation (10) models the price growth change over terminal sub-time, given assets (portfolio level):

$$\begin{array}{cc}\hfill & {\check{v}}_t={\displaystyle \frac{{\sum }_{i=1}^h{P}_{i0}{v}_{it}!-{\sum }_{i=1}^h{P}_{i0}{v}_{it-1}!}{{\sum }_{i=1}^h{P}_{i0}{v}_{it-1}!}}⟶\hfill \\ \hfill & {\widehat{\check{v}}}_t=\sum _{i=1}^h\left\{{\displaystyle \frac{{\widehat{v}}_{it}!v_{it}^{*}!}{{\sum }_{i=1}^h\left(v_{it}^{*}!-v_{it-1}^{*}!\right)}}+{\widehat{v}}_{it-1}!v_{it-1}^{*}!\left[{\displaystyle \frac{{\sum }_{i=1}^{h}2v_{it-1}^{*}!-{\sum }_{i=1}^h{v}_{it}^{*}!}{{\sum }_{i=1}^h\left(v_{it}^{*}!-v_{it-1}^{*}!\right){\sum }_{i=1}^h{v}_{it-1}^{*}!}}\right]\right\}\hfill \end{array}$$

Equation (11) models the initial income to price ratio (portfolio level yield):

$$R={\displaystyle \frac{{\sum }_{i=1}^h{Y}_{i1}}{{\sum }_{i=1}^h{P}_{i0}}}⟶\widehat{R}=\sum _{i=1}^h\left({\displaystyle \frac{{\widehat{Y}}_{i1}{Y}_{i1}^{*}}{{\sum }_{i=1}^h{Y}_{i1}^{*}}}-{\displaystyle \frac{{\widehat{P}}_{i0}{P}_{i0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}}\right)$$

Notice that initial income to price ratio (portfolio level yield)

$$\widehat{R}=-\sum _{i=1}^h{\displaystyle \frac{{\widehat{P}}_{i0}{P}_{i0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}},$$

if initial income $Y_{i1}$ is a parameter.

Equation (12) models the rate of return over terminal sub-time:

$$r_t=R\left({\displaystyle \frac{{\tilde{u}}_t!}{{\tilde{v}}_{t-1}!}}\right)+{\check{v}}_t⟶{\widehat{r}}_t=R^{*}\left({\displaystyle \frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}}\right)\left(\widehat{R}+{\widehat{\tilde{u}}}_t!-{\widehat{\tilde{v}}}_{t-1}!\right)+{\widehat{\check{v}}}_t{\check{v}}_t^{*}$$

Equation (15) models the change in the initial price (TAPA BPE for asset i):

$$W_i\equiv {\dot{P}}_{i0}={\displaystyle \frac{Y_{i1}{\sum }_{t=1}^k\frac{u_{it}!}{r_t!}}{1-\frac{v_{ik}!}{r_k!}}}⟶{\widehat{\dot{P}}}_{i0}={\widehat{Y}}_{i1}+\sum _{t=1}^k\left({\widehat{u}}_{it}!+{\widehat{r}}_t!\right)+\left[{\displaystyle \frac{v_{ik}^{*}!}{r_k^{*}!\left(1-\frac{v_{ik}^{*}!}{r_k^{*}!}\right)}}\right]\left({\widehat{v}}_{ik}!+{\widehat{r}}_k!\right)$$

Notice that change in initial price

$${\widehat{\dot{P}}}_{i0}=\sum _{t=1}^k\left({\widehat{u}}_{it}!+{\widehat{r}}_t!\right)+\left[{\displaystyle \frac{v_{ik}^{*}!}{r_k^{*}!\left(1-\frac{v_{ik}^{*}!}{r_k^{*}!}\right)}}\right]\left({\widehat{v}}_{ik}!+{\widehat{r}}_k!\right),$$

if initial income $Y_{i1}$ is a parameter.

6. TAPA State Equation

6.1. System

The resulting system of log-linearised equations is the following.

Equation (15) models the change in the initial price (TAPA BPE for asset i):

$${\widehat{\dot{P}}}_{i0}=\sum _{t=1}^k\left({\widehat{u}}_{it}!+{\widehat{r}}_t!\right)+\left[{\displaystyle \frac{v_{ik}^{*}!}{r_k^{*}!\left(1-\frac{v_{ik}^{*}!}{r_k^{*}!}\right)}}\right]\left({\widehat{v}}_{ik}!+{\widehat{r}}_k!\right)$$

It invokes log-linearised TAPA Equations (1), (3), (13) and (14), of which log-linearised TAPA Equations (1) and (3) are parametric.

Equation (3) models the income growth rate compounded over sub-time, given assets (parametric):

$${\widehat{u}}_{it}!=\sum _{j=1}^t{\widehat{u}}_{ij}$$

Equation (13) models the rate of return compounded over sub-time:

$${\widehat{r}}_t!=\sum _{j=1}^t{\displaystyle \frac{{\widehat{r}}_j{r}_j^{*}}{\left(1+r_j^{*}\right)}}$$

Equation (1) models the price growth rate compounded over time, given assets (parametric):

$${\widehat{v}}_{ik}!=\sum _{t=1}^k{\widehat{v}}_{it}$$

Equation (14) models the rate of return compounded over time:

$${\widehat{r}}_k!=\sum _{t=1}^k{\displaystyle \frac{{\widehat{r}}_t{r}_t^{*}}{\left(1+r_t^{*}\right)}}$$

It invokes log-linearised TAPA Equation (12).

Equation (12) models the rate of return over terminal sub-time:

$${\widehat{r}}_t=R^{*}\left({\displaystyle \frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}}\right)\left(\widehat{R}+{\widehat{\tilde{u}}}_t!-{\widehat{\tilde{v}}}_{t-1}!\right)+{\widehat{\check{v}}}_t{\check{v}}_t^{*}$$

It invokes log-linearised TAPA Equations (8)–(11), of which log-linearised TAPA Equations (9) and (10) are derivatively parametric.

Equation (11) models the initial income to price ratio (portfolio level yield):

$$\widehat{R}=-\sum _{i=1}^h{\displaystyle \frac{{\widehat{P}}_{i0}{P}_{i0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}}$$

Equation (9) models the income growth change compounded over time and assets (portfolio level) (derivatively parametric):

$${\widehat{\tilde{u}}}_t!=\sum _{j=1}^t{\widehat{\tilde{u}}}_j-{\widehat{\tilde{u}}}_{j-1}$$

It invokes log-linearised TAPA Equation (7), which is derivatively parametric.

Equation (7) models income growth over assets (portfolio level), given sub-sub-time (derivatively parametric):

$${\widehat{\tilde{u}}}_j=\sum _{i=1}^h{\displaystyle \frac{Y_{i1}^{*}{u}_{ij}^{*}!{\widehat{u}}_{ij}!}{{\sum }_{i=1}^h{Y}_{i1}^{*}{u}_{ij}^{*}!}}$$

It invokes log-linearised TAPA Equation (5), which is parametric.

Equation (5) models the income growth rate compounded over sub-sub-time, given assets (parametric):

$${\widehat{u}}_{ij}!=\sum _{l=1}^j{\widehat{u}}_{il}$$

Equation (8) models the price growth change compounded over time and assets (portfolio level):

$${\widehat{\tilde{v}}}_{t-1}!=\sum _{j=1}^{t-1}{\widehat{\tilde{v}}}_j-{\widehat{\tilde{v}}}_{j-1}$$

It invokes log-linearised TAPA Equation (6).

Equation (6) models price growth over assets (portfolio level), given sub-sub-time:

$${\widehat{\tilde{v}}}_j=\sum _{i=1}^h\left[{\displaystyle \frac{P_{i0}^{*}\left(1-v_{ij}^{*}!\right){\widehat{P}}_{i0}}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}}-{\displaystyle \frac{P_{i0}^{*}{v}_{ij}^{*}!{\widehat{v}}_{ij}!}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}}\right]$$

It invokes log-linearised TAPA Equation (4), which is parametric.

Equation (4) models the price growth rate compounded over sub-sub-time, given assets (parametric):

$${\widehat{v}}_{ij}!=\sum _{l=1}^j{\widehat{v}}_{il}$$

Equation (10) models the price growth change over terminal sub-time, given assets (portfolio level) (derivatively parametric):

$${\widehat{\check{v}}}_t=\sum _{i=1}^h\left\{{\displaystyle \frac{{\widehat{v}}_{it}!v_{it}^{*}!}{{\sum }_{i=1}^h\left(v_{it}^{*}!-v_{it-1}^{*}!\right)}}+{\widehat{v}}_{it-1}!v_{it-1}^{*}!\left[{\displaystyle \frac{{\sum }_{i=1}^{h}2v_{it-1}^{*}!-{\sum }_{i=1}^h{v}_{it}^{*}!}{{\sum }_{i=1}^h\left(v_{it}^{*}!-v_{it-1}^{*}!\right){\sum }_{i=1}^h{v}_{it-1}^{*}!}}\right]\right\}$$

It invokes log-linearised TAPA Equation (2), which is parametric.

Equation (2) models the price growth rate compounded over sub-time, given assets (parametric):

$${\widehat{v}}_{it}!=\sum _{j=1}^t{\widehat{v}}_{ij}$$

6.2. Reduced System

Parametric log-linearised TAPA equations reduce the system to the following, updated log-linearised TAPA equations.

Equation (15) models the change in the initial price (TAPA BPE for asset i):

$${\widehat{\dot{P}}}_{i0}=\sum _{t=1}^k{\widehat{r}}_t!+\left[{\displaystyle \frac{v_{ik}^{*}!}{r_k^{*}!\left(1-\frac{v_{ik}^{*}!}{r_k^{*}!}\right)}}\right]{\widehat{r}}_k!⟶\sum _{t=2}^k{\widehat{r}}_t!+\left[{\displaystyle \frac{v_{ik}^{*}!}{r_k^{*}!\left(1-\frac{v_{ik}^{*}!}{r_k^{*}!}\right)}}\right]{\widehat{r}}_k!,$$

as $\nexists {\widehat{r}}_1!,$ because of log-linearised TAPA Equation (14).

Equation (13) models the rate of return compounded over sub-time:

$${\widehat{r}}_t!=\sum _{j=1}^t{\displaystyle \frac{{\widehat{r}}_j{r}_j^{*}}{\left(1+r_j^{*}\right)}}⟶\sum _{j=2}^t{\displaystyle \frac{{\widehat{r}}_j{r}_j^{*}}{\left(1+r_j^{*}\right)}},$$

as $\nexists {\widehat{r}}_1⟶\nexists {\widehat{r}}_1!,$ because of log-linearised TAPA Equation (12); subsumed by log-linearised TAPA Equation (14).

Equation (14) models the rate of return compounded over time:

$${\widehat{r}}_k!=\sum _{t=1}^k{\displaystyle \frac{{\widehat{r}}_t{r}_t^{*}}{\left(1+r_t^{*}\right)}}⟶\sum _{t=2}^k{\displaystyle \frac{{\widehat{r}}_t{r}_t^{*}}{\left(1+r_t^{*}\right)}},$$

as $\nexists {\widehat{r}}_1⟶\nexists {\widehat{r}}_1!,$ because log-linearised TAPA Equation (12); repeated $k-1$ times.

Equation (12) models the rate of return over terminal sub-time:

$${\widehat{r}}_t=R^{*}\left({\displaystyle \frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}}\right)\left(\widehat{R}-{\widehat{\tilde{v}}}_{t-1}!\right)$$

such that $\nexists {\widehat{\tilde{v}}}_0!⟶\nexists {\widehat{r}}_1,$ because log-linearised TAPA Equation (8); repeated $k-1$ times.

Equation (11) models the initial income to price ratio (portfolio level yield):

$$\widehat{R}=-\sum _{i=1}^h{\displaystyle \frac{{\widehat{P}}_{i0}{P}_{i0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}}⟶\widehat{R}=-\sum _{i=1}^1{\displaystyle \frac{{\widehat{P}}_{i0}{P}_{i0}^{*}}{{\sum }_{i=1}^1{P}_{i0}^{*}}}$$

for maximal assets $h=1.$

Equation (8) models the price growth change compounded over time and assets (portfolio level):

$${\widehat{\tilde{v}}}_{t-1}!=\sum _{j=1}^{t-1}{\widehat{\tilde{v}}}_j-{\widehat{\tilde{v}}}_{j-1}$$

such that

$$\nexists {\widehat{\tilde{v}}}_0!=\sum _{j=1}^0{\widehat{\tilde{v}}}_j-{\widehat{\tilde{v}}}_{j-1},\nexists {\tilde{v}}_0^{*}!=\prod _{j=1}^0{\displaystyle \frac{{\tilde{v}}_j^{*}}{{\tilde{v}}_{j-1}^{*}}}\mathrm{and}{\widehat{\tilde{v}}}_1!={\widehat{\tilde{v}}}_1-{\widehat{\tilde{v}}}_{1-1}={\widehat{\tilde{v}}}_1;$$

repeated $k-2$ times.

Equation (6) models price growth over assets (portfolio level), given sub-sub-time:

$${\widehat{\tilde{v}}}_j=\sum _{i=1}^h{\displaystyle \frac{P_{i0}^{*}\left(1-v_{ij}^{*}!\right) {\widehat{P}}_{i0}}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}}⟶\sum _{i=1}^1{\displaystyle \frac{P_{i0}^{*}\left(1-v_{ij}^{*}!\right) {\widehat{P}}_{i0}}{{\sum }_{i=1}^1{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}}$$

for maximal assets $h=1;$ repeated $k-1$ times.

6.3. State Space Form

Variables are ${\widehat{P}}_{i0},{\left\{{\widehat{r}}_t!\right\}}_{t=2}^k,{\left\{{\widehat{r}}_t\right\}}_{t=2}^k,\widehat{R},{\left\{{\widehat{\tilde{v}}}_{t-1}!\right\}}_{t-1=2-1}^{k-1}$ and ${\left\{{\widehat{\tilde{v}}}_j\right\}}_{j=1}^{k-1}$ and parameters are ${\left\{r_t^{*}!\right\}}_{t=2}^k{\left\{{\tilde{u}}_t^{*}!\right\}}_{t=2}^k,{\left\{{\tilde{v}}_{t-1}^{*}!\right\}}_{t-1=2-1}^{k-1}$ and ${\left\{v_{ij}^{*}!\right\}}_{j=1}^k.$ The system can therefore be written in state space form (see Appendix A.4 Section A3):

$$A\dot{x}=Bx.$$

7. TAPA Stability

7.1. Stability Conditions

The stability of state equation $A\dot{x}=Bx⟶\dot{x}=A^{-1}Bx=Cx$ is verified if and only if Euclidean distance

$$d(x,x^{*})<\delta ⟶d\left[\dot{x}\left(C,x\right),x^{*}\right]<\epsilon ,$$

$\forall \delta ,\epsilon >0,$ such that limit

$$\underset{t\to \infty }{lim}\dot{x}=\underset{t\to \infty }{lim}Cx=0,$$

in which Euclidean distances $d(x,x^{*})$ and $d\left[\dot{x}\left(C,x\right),x^{*}\right]$ depend on time $t.$ If there existed an inverse companion matrix $A^{-1}$ then the stability of state equation $\dot{x}=A^{-1}Bx=Cx$ would be studied as follows: $\forall n\in [6,\infty )\subset {\mathbb{N}}_{+},$

$$\begin{array}{cc}\hfill & I_n{\displaystyle \frac{dx}{dt}}=Cx⟶I_n\left({\displaystyle \frac{1}{x}}\right) dx=Cdt⟶\hfill \\ \hfill & I_n\int {\displaystyle \frac{1}{x}}dx=C\int dt⟷I_n\left(lnx+c_L\right)=C\left(t+c_R\right)⟶\hfill \\ \hfill & lnx=Ct+C{c}_R-I_n{c}_L\equiv Ct+D_C⟶\hfill \\ \hfill & x=e^{Ct+D_C}=x_0{e}^{Ct}=x_0{e}^{PJ{P}^{-1}t},\hfill \end{array}$$

in which J is a Jordan matrix of eigenvalues ${\lambda }_{C\left(\lambda \right)}$ and P is a Jordan matrix of eigenvectors $v_{C\left(\lambda \right)}$ in eigenvalue problem

$$Cv=\lambda v⟶\left(C-\lambda I_n\right) v=0⟶det\left(C-\lambda I_n\right)=det\left[C\left(\lambda \right)\right]=0$$

if and only if inverse characteristic polynomial matrix $C^{-1}\left(\lambda \right)$ does not exist if and only if rank $rk\left[C\left(\lambda \right)\right]<n$ if and only if solution $v=0$ is not unique, whence limit

$$\underset{t\to \infty }{lim}\dot{x}=\underset{t\to \infty }{lim}Cx=\underset{t\to \infty }{lim}C{x}_0{e}^{PJ{P}^{-1}t}=0$$

if and only if eigenvalues ${\lambda }_{C\left(\lambda \right)}<0.$ However, because determinant

$$det\left(A\right)=0⟶\nexists A^{-1},$$

the stability of state equation $A\dot{x}=Bx$ must be studied as follows:

$$\begin{array}{cc}\hfill & A{\displaystyle \frac{dx}{dt}}=Bx⟶A\left({\displaystyle \frac{1}{x}}\right) dx=Bdt⟶\hfill \\ \hfill & A\int {\displaystyle \frac{1}{x}}dx=B\int dt⟷A\left(lnx+c_L\right)=B\left(t+c_R\right)⟶\hfill \\ \hfill & Alnx=Bt+B{c}_R-A{c}_L\equiv Bt+D_B⟶\hfill \\ \hfill & e^{A}x=e^{Bt+D_B}=x_0{e}^{Bt}⟷e^{QS{Z}^{\top }}x=x_0{e}^{QT{Z}^{\top }t},\hfill \end{array}$$

in which companion matrices $A=QS{Z}^{\top }$ and $B=QT{Z}^{\top }$ such that matrix product $Q{Q}^{\top }=Z{Z}^{\top }=I_n$ and matrices $S,T$ are upper triangular, according to the generalised Schur decomposition for generalised eigenvalue problem $Bv=\lambda Av$ with eigenvalues ${\lambda }_i={\displaystyle \frac{T_{ii}}{S_{ii}}}$ for index $i\in [1,n]\subset {\mathbb{N}}_{+},$ whence

$$\underset{t\to \infty }{lim}A{e}^{QS{Z}^{\top }}\dot{x}=\underset{t\to \infty }{lim}B{e}^{QS{Z}^{\top }}x=\underset{t\to \infty }{lim}B{x}_0{e}^{QT{Z}^{\top }t}=0$$

if and only if ${\lambda }_i<0.$

7.2. Application

For simplicity, let the maximal time period $k=2$ and the maximal number of assets $h=1:$

$$\underset{A}{\underbrace{\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 0\end{array}\right]}}\underset{\dot{x}}{\underbrace{\left[\begin{array}{c}{\widehat{\dot{P}}}_{10}\\ {\widehat{\dot{r}}}_2!\\ {\widehat{\dot{r}}}_2\\ \widehat{\dot{R}}\\ {\widehat{\dot{\tilde{v}}}}_1!\\ {\widehat{\dot{\tilde{v}}}}_1\end{array}\right]}}=\underset{B}{\underbrace{\left[\begin{array}{cccccc}0& 1+{\displaystyle \frac{v_{12}^{*}!}{r_2^{*}!\left(1-\frac{v_{12}^{*}!}{r_2^{*}!}\right)}}& 0& 0& 0& 0\\ 0& -1& 0& {\displaystyle \frac{r_2^{*}}{\left(1+r_2^{*}\right)}}& 0& 0\\ 0& 0& -1& R^{*}\left({\displaystyle \frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}}\right)& -R^{*}\left({\displaystyle \frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}}\right)& 0\\ -1& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& -1& 1\\ 1& 0& 0& 0& 0& -1\end{array}\right]}}\underset{x}{\underbrace{\left[\begin{array}{c}{\widehat{P}}_{10}\\ {\widehat{r}}_2!\\ {\widehat{r}}_2\\ \widehat{R}\\ {\widehat{\tilde{v}}}_1!\\ {\widehat{\tilde{v}}}_1\end{array}\right]}};$$

such a system is stable if and only if ${\lambda }_i={\displaystyle \frac{S_{ii}}{T_{ii}}}<0$ for $i\in [1,6]\subset {\mathbb{N}}_{+}.$ Yet, given the relative simplicity of such a system its stability can be directly studied in equation form (see Appendix A.5 Section A4):

$${\widehat{\dot{P}}}_{10}=\left[1+{\displaystyle \frac{v_{12}^{*}!}{r_2^{*}!\left(1-\frac{v_{12}^{*}!}{r_2^{*}!}\right)}}\right]\left[{\displaystyle \frac{-2R^{*}\left(\frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}\right)r_2^{*}}{\left(1+r_2^{*}\right)}}\right]{\widehat{P}}_{10},$$

which can be solved such that

$$\begin{array}{cc}\hfill & \dot{x}={\displaystyle \frac{dx}{dt}}=bx⟶{\displaystyle \frac{1}{x}}dx=bdt⟶\hfill \\ \hfill & \int {\displaystyle \frac{1}{x}}dx=b\int dt⟷lnx+c_L=b\left(t+c_R\right)⟶\hfill \\ \hfill & lnx=bt+b{c}_R-c_L\equiv bt+c⟶\hfill \\ \hfill & x=e^{bt+c}=x_0{e}^{bt},\hfill \end{array}$$

whence limit

$$\underset{t\to \infty }{lim}\dot{x}=\underset{t\to \infty }{lim}b{x}_0{e}^{bt}=0$$

if and only if parameter

$$b=\left[1+{\displaystyle \frac{v_{12}^{*}!}{r_2^{*}!\left(1-\frac{v_{12}^{*}!}{r_2^{*}!}\right)}}\right]\left[{\displaystyle \frac{-2R^{*}\left(\frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}\right)r_2^{*}}{\left(1+r_2^{*}\right)}}\right]<0⟶\left[1+{\displaystyle \frac{v_{12}^{*}!}{r_2^{*}!\left(1-\frac{v_{12}^{*}!}{r_2^{*}!}\right)}}\right]\left[{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}\right)r_2^{*}}{\left(1+r_2^{*}\right)}}\right]>0.$$

An exemplifying parametrisation involves the following calibration such that both stability and instability are contemplated. Parameter $r_2^{*}!=0.05$ is indicative of a $5\%$ return. $R^{*}=0.02$ represents such a small positive portfolio level yield as that of a risk free asset. ${\tilde{u}}_2^{*}!=1$ and ${\tilde{v}}_1^{*}!=1$ delineate normalised relative growth rates, as does initial condition $x_0=1.v_{12}^{*}=0.01,0.5$ finally typifies a twofold transition calibration ensuring stability and instability, respectively. See Figure 5.

7.3. Equation Form

Indeed, stability can be directly studied in equation form for any maximal time period k and maximal number of assets h on positive natural number line ${\mathbb{N}}_{+}:$

$$\begin{array}{cc}\hfill & {\widehat{\dot{P}}}_{i0}={\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}\right)\left\{-\left(\frac{{\widehat{P}}_{10}{P}_{10}^{*}}{P_{10}^{*}+ \dots +P_{h0}^{*}}+\dots +\frac{{\widehat{P}}_{h0}{P}_{h0}^{*}}{P_{10}^{*}+ \dots +P_{h0}^{*}}\right)+\right.}{\left(1+r_2^{*}\right)}}\hfill \\ \hfill & {\displaystyle \frac{\left.-\left[\frac{P_{10}^{*}\left(1-v_{11}^{*}!\right) {\widehat{P}}_{10}}{P_{10}^{*}\left(1-v_{11}^{*}!\right)+ \dots +P_{h0}^{*}\left(1-v_{h1}^{*}!\right)}+\dots +\frac{P_{h0}^{*}\left(1-v_{11}^{*}!\right) {\widehat{P}}_{h0}}{P_{10}^{*}\left(1-v_{11}^{*}!\right)+ \dots +P_{h0}^{*}\left(1-v_{h1}^{*}!\right)}\right]\right\}{r}_2^{*}}{\left(1+r_2^{*}\right)}}+\dots +\left[1+{\displaystyle \frac{v_{ik}^{*}!}{r_k^{*}!\left(1-\frac{v_{ik}^{*}!}{r_k^{*}!}\right)}}\right]·\hfill \\ \hfill & ·\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}\right)\left\{-\left(\frac{{\widehat{P}}_{10}{P}_{10}^{*}}{P_{10}^{*}+ \dots +P_{h0}^{*}}+\dots +\frac{{\widehat{P}}_{h0}{P}_{h0}^{*}}{P_{10}^{*}+ \dots +P_{h0}^{*}}\right)-\left[\frac{P_{10}^{*}\left(1-v_{11}^{*}!\right) {\widehat{P}}_{10}}{P_{10}^{*}\left(1-v_{11}^{*}!\right)+ \dots +P_{h0}^{*}\left(1-v_{h1}^{*}!\right)}+\dots +\frac{P_{h0}^{*}\left(1-v_{11}^{*}!\right) {\widehat{P}}_{h0}}{P_{10}^{*}\left(1-v_{11}^{*}!\right)+ \dots +P_{h0}^{*}\left(1-v_{h1}^{*}!\right)}\right]\right\}{r}_2^{*}}{\left(1+r_2^{*}\right)}}+\right.\hfill \\ \hfill & +\dots +{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_k^{*}!}{{\tilde{v}}_{k-1}^{*}!}\right)\left\{-\left(\frac{{\widehat{P}}_{10}{P}_{10}^{*}}{P_{10}^{*}+ \dots +P_{h0}^{*}}+\dots +\frac{{\widehat{P}}_{h0}{P}_{h0}^{*}}{P_{10}^{*}+ \dots +P_{h0}^{*}}\right)+\right.}{\left(1+r_k^{*}\right)}}\hfill \\ \hfill & {\displaystyle \frac{-\left\{\left[\frac{P_{10}^{*}\left(1-v_{11}^{*}!\right) {\widehat{P}}_{10}}{P_{10}^{*}\left(1-v_{11}^{*}!\right)+ \dots +P_{h0}^{*}\left(1-v_{h1}^{*}!\right)}+\dots +\frac{P_{h0}^{*}\left(1-v_{11}^{*}!\right) {\widehat{P}}_{h0}}{P_{10}^{*}\left(1-v_{11}^{*}!\right)+ \dots +P_{h0}^{*}\left(1-v_{h1}^{*}!\right)}\right]+\dots +\right.}{\left(1+r_k^{*}\right)}}\hfill \\ \hfill & {\displaystyle \frac{+\left\{\left[\frac{P_{10}^{*}\left(1-v_{1k-1}^{*}!\right) {\widehat{P}}_{10}}{P_{10}^{*}\left(1-v_{1k-1}^{*}!\right)+ \dots +P_{h0}^{*}\left(1-v_{hk-1}^{*}!\right)}+\dots +\frac{P_{h0}^{*}\left(1-v_{1k-1}^{*}!\right) {\widehat{P}}_{h0}}{P_{10}^{*}\left(1-v_{1k-1}^{*}!\right)+ \dots +P_{h0}^{*}\left(1-v_{hk-1}^{*}!\right)}\right]+\right.}{\left(1+r_k^{*}\right)}}\hfill \\ \hfill & \left.{\displaystyle \frac{\left.\left.\left.-\left[\frac{P_{10}^{*}\left(1-v_{1k-2}^{*}!\right) {\widehat{P}}_{10}}{P_{10}^{*}\left(1-v_{1k-2}^{*}!\right)+ \dots +P_{h0}^{*}\left(1-v_{hk-2}^{*}!\right)}+\dots +\frac{P_{h0}^{*}\left(1-v_{1k-2}^{*}!\right) {\widehat{P}}_{h0}}{P_{10}^{*}\left(1-v_{1k-2}^{*}!\right)+ \dots +P_{h0}^{*}\left(1-v_{hk-2}^{*}!\right)}\right]\right\}\right\}\right\}{r}_k^{*}}{\left(1+r_k^{*}\right)}}\right\},\hfill \end{array}$$

the full derivation of which is collected in the Appendix A.6 Section A5. Thus, if the maximal number of assets $h=1$ then change in the initial price equation

$$\begin{array}{cc}\hfill & {\widehat{\dot{P}}}_{10}=\left\{{\displaystyle \frac{-2R^{*}\left(\frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}\right)r_2^{*}}{\left(1+r_2^{*}\right)}}+\dots +\left[1+{\displaystyle \frac{v_{1k}^{*}!}{r_k^{*}!\left(1-\frac{v_{1k}^{*}!}{r_k^{*}!}\right)}}\right]·\right.\hfill \\ \hfill & \left.·\left\{{\displaystyle \frac{-2R^{*}\left(\frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}\right)r_2^{*}}{\left(1+r_2^{*}\right)}}+\dots +{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_k^{*}!}{{\tilde{v}}_{k-1}^{*}!}\right)\left\{-1-\left\{1+\dots +\left(1+1\right)\right\}\right\}{r}_k^{*}}{\left(1+r_k^{*}\right)}}\right\}\right\}{\widehat{P}}_{10}\hfill \\ \hfill & \dot{x}=bx,\hfill \end{array}$$

which is stable if and only if

$$\begin{array}{cc}\hfill & \left\{{\displaystyle \frac{-2R^{*}\left(\frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}\right)r_2^{*}}{\left(1+r_2^{*}\right)}}+\dots +\left[1+{\displaystyle \frac{v_{1k}^{*}!}{r_k^{*}!\left(1-\frac{v_{1k}^{*}!}{r_k^{*}!}\right)}}\right]·\right.\hfill \\ \hfill & \left.·\left\{{\displaystyle \frac{-2R^{*}\left(\frac{{\tilde{u}}_2^{*}!}{{\tilde{v}}_1^{*}!}\right)r_2^{*}}{\left(1+r_2^{*}\right)}}+\dots +{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_k^{*}!}{{\tilde{v}}_{k-1}^{*}!}\right)\left\{-1-\left\{1+\dots +\left(1+1\right)\right\}\right\}{r}_k^{*}}{\left(1+r_k^{*}\right)}}\right\}\right\}<0.\hfill \end{array}$$

7.4. System Form

Though all equations may be collapsed into that for change in initial price ${\widehat{\dot{P}}}_{i0},$ if maximal number of assets $h>1$ we must then return to studying the stability of the TAPA algorithm in terms of a system (see Appendix A.7 Section A6), so that change in initial price equation for asset 1

$$\begin{array}{cc}\hfill & {\widehat{\dot{P}}}_{10}=\sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-{\sum }_{i=2}^h\frac{{\widehat{P}}_{i0}{P}_{i0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{{\sum }_{i=2}^h\frac{P_{i0}^{*}\left(1-v_{ij}^{*}!\right){\widehat{P}}_{i0}}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[{\sum }_{i=2}^h\frac{P_{i0}^{*}\left(1-v_{ij-1}^{*}!\right){\widehat{P}}_{i0}}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{1k}^{*}!}{r_k^{*}!\left(1-\frac{v_{1k}^{*}!}{r_k^{*}!}\right)}}\right]+\hfill \\ \hfill & +\sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{10}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{10}^{*}\left(1-v_{1j}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{10}^{*}\left(1-v_{1j-1}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{1k}^{*}!}{r_k^{*}!\left(1-\frac{v_{1k}^{*}!}{r_k^{*}!}\right)}}\right]{\widehat{P}}_{10}\hfill \end{array}$$

until change in initial price equation for asset h

$$\begin{array}{cc}\hfill & {\widehat{\dot{P}}}_{h0}=\sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-{\sum }_{i=1}^{h-1}\frac{{\widehat{P}}_{i0}{P}_{i0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{{\sum }_{i=1}^{h-1}\frac{P_{i0}^{*}\left(1-v_{ij}^{*}!\right){\widehat{P}}_{i0}}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[{\sum }_{i=1}^{h-1}\frac{P_{i0}^{*}\left(1-v_{ij-1}^{*}!\right){\widehat{P}}_{i0}}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{hk}^{*}!}{r_k^{*}!\left(1-\frac{v_{hk}^{*}!}{r_k^{*}!}\right)}}\right]+\hfill \\ \hfill & +\sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{h0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{h0}^{*}\left(1-v_{hj}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{h0}^{*}\left(1-v_{hj-1}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{hk}^{*}!}{r_k^{*}!\left(1-\frac{v_{hk}^{*}!}{r_k^{*}!}\right)}}\right]{\widehat{P}}_{h0},\hfill \end{array}$$

whence the system

$$\begin{array}{cc}\hfill & \left[\begin{array}{c}{\widehat{\dot{P}}}_{10}\\ ⋮\\ {\widehat{\dot{P}}}_{h0}\end{array}\right]=\left[\begin{array}{cc}\sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{10}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{10}^{*}\left(1-v_{1j}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{10}^{*}\left(1-v_{1j-1}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{1k}^{*}!}{r_k^{*}!\left(1-\frac{v_{1k}^{*}!}{r_k^{*}!}\right)}}\right]& \dots \\ ⋮& \ddots \\ \sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{h0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{10}^{*}\left(1-v_{1j}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{10}^{*}\left(1-v_{1j-1}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{hk}^{*}!}{r_k^{*}!\left(1-\frac{v_{hk}^{*}!}{r_k^{*}!}\right)}}\right]& \dots \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{cc}\dots & \sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{h0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{h0}^{*}\left(1-v_{hj}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{h0}^{*}\left(1-v_{hj-1}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{1k}^{*}!}{r_k^{*}!\left(1-\frac{v_{1k}^{*}!}{r_k^{*}!}\right)}}\right]\\ \ddots & ⋮\\ \dots & \sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{h0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{h0}^{*}\left(1-v_{hj}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{h0}^{*}\left(1-v_{hj-1}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{hk}^{*}!}{r_k^{*}!\left(1-\frac{v_{hk}^{*}!}{r_k^{*}!}\right)}}\right]\end{array}\right]\left[\begin{array}{c}{\widehat{P}}_{10}\\ ⋮\\ {\widehat{P}}_{h0}\end{array}\right],\hfill \end{array}$$

which as $\dot{x}=Ax$ is stable if and only if eigenvalues ${\lambda }_{A\left(\lambda \right)}<0$ in characteristic polynomial matrix $A\left(\lambda \right)=A-\lambda I_h$ for determinant $det\left[A\left(\lambda \right)\right]=0,$ meaning

$$\begin{array}{cc}\hfill & det\left[\begin{array}{cc}\sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{10}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{10}^{*}\left(1-v_{1j}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{10}^{*}\left(1-v_{1j-1}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{1k}^{*}!}{r_k^{*}!\left(1-\frac{v_{1k}^{*}!}{r_k^{*}!}\right)}}\right]-\lambda & \dots \\ ⋮& \ddots \\ \sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{h0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{10}^{*}\left(1-v_{1j}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{10}^{*}\left(1-v_{1j-1}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{hk}^{*}!}{r_k^{*}!\left(1-\frac{v_{hk}^{*}!}{r_k^{*}!}\right)}}\right]& \dots \end{array}\right.\hfill \\ \hfill & \left.\begin{array}{cc}\dots & \sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{h0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{h0}^{*}\left(1-v_{hj}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{h0}^{*}\left(1-v_{hj-1}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{1k}^{*}!}{r_k^{*}!\left(1-\frac{v_{1k}^{*}!}{r_k^{*}!}\right)}}\right]\\ \ddots & ⋮\\ \dots & \sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{h0}^{*}}{{\sum }_{i=1}^h{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{h0}^{*}\left(1-v_{hj}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{h0}^{*}\left(1-v_{hj-1}^{*}!\right)}{{\sum }_{i=1}^h{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{hk}^{*}!}{r_k^{*}!\left(1-\frac{v_{hk}^{*}!}{r_k^{*}!}\right)}}\right]-\lambda \end{array}\right]=0.\hfill \end{array}$$

Consequently, for the maximal number of assets $h=2$ there arises that system

$$\left[\begin{array}{c}{\widehat{\dot{P}}}_{10}\\ {\widehat{\dot{P}}}_{20}\end{array}\right]=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\left[\begin{array}{c}{\widehat{P}}_{10}\\ {\widehat{P}}_{20}\end{array}\right],$$

in which parameters

$$\begin{array}{cc}\hfill & a=\sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{10}^{*}}{{\sum }_{i=1}^2{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{10}^{*}\left(1-v_{1j}^{*}!\right)}{{\sum }_{i=1}^2{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{10}^{*}\left(1-v_{1j-1}^{*}!\right)}{{\sum }_{i=1}^2{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{1k}^{*}!}{r_k^{*}!\left(1-\frac{v_{1k}^{*}!}{r_k^{*}!}\right)}}\right],\hfill \\ \hfill & b=\sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{20}^{*}}{{\sum }_{i=1}^2{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{20}^{*}\left(1-v_{2j}^{*}!\right)}{{\sum }_{i=1}^2{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{20}^{*}\left(1-v_{2j-1}^{*}!\right)}{{\sum }_{i=1}^2{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{1k}^{*}!}{r_k^{*}!\left(1-\frac{v_{1k}^{*}!}{r_k^{*}!}\right)}}\right],\hfill \\ \hfill & c=\sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{10}^{*}}{{\sum }_{i=1}^2{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{10}^{*}\left(1-v_{1j}^{*}!\right)}{{\sum }_{i=1}^2{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{10}^{*}\left(1-v_{1j-1}^{*}!\right)}{{\sum }_{i=1}^2{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{2k}^{*}!}{r_k^{*}!\left(1-\frac{v_{2k}^{*}!}{r_k^{*}!}\right)}}\right],\hfill \\ \hfill & d=\sum _{t=2}^k\left\{{\displaystyle \frac{R^{*}\left(\frac{{\tilde{u}}_t^{*}!}{{\tilde{v}}_{t-1}^{*}!}\right)\left\{-\frac{P_{20}^{*}}{{\sum }_{i=1}^2{P}_{i0}^{*}}-{\sum }_{j=1}^{t-1}\left\{\frac{P_{20}^{*}\left(1-v_{2j}^{*}!\right)}{{\sum }_{i=1}^2{P}_{i0}^{*}\left(1-v_{ij}^{*}!\right)}-\left[\frac{P_{20}^{*}\left(1-v_{2j-1}^{*}!\right)}{{\sum }_{i=1}^2{P}_{i0}^{*}\left(1-v_{ij-1}^{*}!\right)}\right]\right\}\right\}{r}_t^{*}}{\left(1+r_t^{*}\right)}}\right\}\left[1+{\displaystyle \frac{v_{2k}^{*}!}{r_k^{*}!\left(1-\frac{v_{2k}^{*}!}{r_k^{*}!}\right)}}\right],\hfill \end{array}$$

whence determinant

$$\begin{array}{cc}\hfill & det\left[A\left(\lambda \right)\right]=\left[\begin{array}{cc}a-\lambda & b\\ c& d-\lambda \end{array}\right]=0\hfill \\ \hfill & \left(a-\lambda \right)\left(d-\lambda \right)-bc=0⟶\hfill \\ \hfill & {\lambda }^2-\left(a+d\right)\lambda +\left(ad-bc\right)=0⟶\hfill \\ \hfill & {\lambda }_{1,2}={\displaystyle \frac{\left(a+d\right)\pm \sqrt{{\left(a+d\right)}^2-4\left(1\right)\left(ad-bc\right)}}{2\left(1\right)}}={\displaystyle \frac{\left(a+d\right)\pm \sqrt{{\left(a-d\right)}^2+4bc}}{2}},\hfill \end{array}$$

$\dot{x}=Ax$ being stable if and only if

$${\lambda }_{1,2}={\displaystyle \frac{\left(a+d\right)\pm \sqrt{{\left(a-d\right)}^2+4bc}}{2}}<0.$$

Finally, by regarding changes in initial price ${\widehat{P}}_{i0}$ as discrete, for system $x_{t+1}=A{x}_t$, the stability of the TAPA algorithm is studied as follows:

$$\begin{array}{cc}\hfill & x_{t+2}=A{x}_{t+1}=A^2{x}_t\hfill \\ \hfill & ⋮\hfill \\ \hfill & x_{t+n}=A{x}_{t+n-1}=A^n{x}_t,\hfill \end{array}$$

whence solution

$$x_t=A^t{x}_0$$

for initial condition $x_0\in {\mathbb{R}}^h$ is such that the solution check $x_{t+1}=A^{t+1}{x}_0=A\left(A^t{x}_0\right)=A{x}_t;$ indeed, for Jordan normal form $x_t={\left(PJ{P}^{-1}\right)}^t{x}_0$ limit

$$\underset{t\to \infty }{lim}{x}_{t+1}=\underset{t\to \infty }{lim}{A}^{t+1}{x}_0=\underset{t\to \infty }{lim}{\left(PJ{P}^{-1}\right)}^t{x}_0=0$$

if and only if modulus eigenvalues $|{\lambda }_{A\left(\lambda \right)}|<1.$ Therefore, stability of the TAPA algorithm in discrete time is guaranteed for any maximal time period k and maximal number of assets h on natural number line ${\mathbb{N}}_{+}$ if and only if modulus eigenvalues

$$|{\lambda }_{1,2}|<1,$$

which for maximal assets $h=2$ lays down inequality

$$-1<{\displaystyle \frac{\left(a+d\right)\pm \sqrt{{\left(a-d\right)}^2+4bc}}{2}}<1.$$

In closing, we note that the stability of the TAPA algorithm, studied through the equation for change in initial price ${\widehat{\dot{P}}}_{i0},$ for any maximal time period $k\in {\mathbb{N}}_{+}$ and maximal number of assets $h\in {\mathbb{N}}_{+}\setminus \left\{1\right\}$ is analytically examinable until maximal number of assets $h=4,$ as by the Abel–Ruffini Theorem and Galois Theory there exists no general solution in radicals for polynomial equations beyond the fourth degree, whence determinant $det\left[A\left(\lambda \right)\right]=0$ for maximal number of assets $h\ge 5$ cannot be generally solved in radicals.

8. Conclusions

This work has advanced the study of DCF valuations of income producing assets against sparse benchmark valuation portfolios by utilising TAPA in the investigation of conditions under which such valuations could converge to settled values; it is the first such application of the TAPA DCF framework to a problem which is exquisitely under-explored.

Prompted by a numerical simulation of the TAPA algorithm, in which the change in initial price brought about by its algorithm as of the second iteration does not differ from that of the first iteration, we have developed stability conditions for any maximal positive time period and positive number of assets.

Since TAPA is a non-linear model we have concentrated our efforts on the study of its local stability. We have specifically presented stability conditions for TAPA at the local level, both in continuous and discrete algorithmic time, and we have developed them by means of log-linearisation about the steady state of its equations’ variables.

Such is indeed the standard and immediate approach in dynamic economic analysis, albeit one abstracting from higher order non-linearities potentially central to the behaviour of such a self-referential valuation algorithm as TAPA’s, as erstwhile mentioned; in particular, convergence or divergence properties far from the steady state, multiple equilibria and cycles are to be hardly assessed within the proposed framework.

Provided that thin markets and sparse portfolios may be the exact environments in which non-linear feedback effects could be strongest, the absence of global stability analysis leaves important questions open about the robustness of TAPA valuations under large shocks or misspecified initial conditions.

Further rarefaction of which our work admits, to be adduced in subsequent TAPA elaborations, speaks to the fact that income growth rates for portfolio assets are largely treated as exogenous or parametric such that feedback from valuation outcomes to income dynamics is ruled out by construction; while such an assumption may simplify the log-linearised system and allow for the isolation of the price adjustment mechanism it does restrict the economic richness of the model.

Be that as it may, we have especially remarked the analytical existence of stability conditions for TAPA at the local level, through a first order steady state log linearisation, up to four assets and any positive time period. We have adduced analytical applications within such a region and have presented a solution for a benchmark calibration of the steady state parameters given two time periods and one asset. In future work we plan on further studying TAPA’s local parametrisation and exploring global stability in TAPA’s original non-linear form.