Unit VII J

Journal of Accounting, Auditing & Finance

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sagepub.com/journals-permissions DOI: 10.1177/0148558X231165557

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Fair Value Measurement in Inactive Crypto Asset Markets

Eyal Beigman1, Gerard Brennan2, Sheng-Feng Hsieh3 , and Alexander J. Sannella4

Abstract

This article proposes a new dynamic method, the Principal Path Method (PPM), for pricing crypto asset against a primary or functional (fiat) currency in situations where these assets do not trade directly against the functional currency or trade at volumes that prevent resulting pricing information to qualify as Level 1 (ASC 820) for financial reporting. We base our method on the guidance provided in ASC 820, IFRS 13, and IAS 21. Our method is designed to extract prices from ‘‘compliant’’ markets that result in reliable inputs to the valuation process. We believe that our methodology improves the current techniques used to value thinly traded crypto assets such as using the last observable transaction price, cre- ating a weighted-average price across multiple markets, or using data on comparable tokens, if available. Furthermore, we present empirical evidence that suggests pricing infor- mation generated by our method for non-exchangeable, thinly traded, or illiquid crypto assets better reflects the fundamental qualitative characteristics of useful information, rele- vance and faithful representation, and results in more reliable inputs used in the valuation process. Unlike methods currently used in practice, our method ensures the integrity of the valuation data employed by selecting prices from compliant markets.

Keywords

crypto asset, fair value measurement, thinly traded, inactive markets

Introduction

The evolution of digital assets and blockchain technology since 2009 and the advent of

non-fungible tokens (NFTs) and Web 3.0 technology since 2020, have redefined the mean-

ing of assets, markets, and economies. Moving well beyond the status of payment tokens or

1Ramzgate Capital, USA 2Lukka, Inc., New York, NY, USA 3Department and Graduate Institute of Accounting, College of Management, National Taiwan University, Taipei,

Taiwan 4Rutgers Business School, Newark and New Brunswick, USA

Corresponding Author:

Alexander J. Sannella, Professor of Accounting and Information Systems, Rutgers Business School, Rutgers, the

State University of New Jersey, One Washington Park, Room 948, Newark, NJ 07102, USA.

Email: [email protected]

Article

2025, Vol. 40(1) 241–269

242 Journal of Accounting, Auditing & Finance 40(1)

stores of value, digital assets can now be works of art, virtual real estate, or entertainment

assets. Early issuance of access tokens such as MATIC,1 UNI,2 and GRT,3 was not only

instrumental in developing new economic ecosystems but also created unique possibilities

for raising investment capital such as crowd sales, initial coin offerings (ICO), and initial

exchange offerings (IEO). However, these new assets present ever-increasing challenges

for financial reporting, both conceptual, such as the classification of virtual real estate, and

methodological, such as auditing and reporting transactions on decentralized exchanges.

Many of the fundamental concepts must be reevaluated, reinterpreted, and expanded to the

realities of digital asset ecosystems.

In this article, we propose an expansion of the notion of fair value pricing for financial

reporting. Fair value is a market-based estimate of the price of an asset, namely, an estimate

of the price an asset could attain on the market in an orderly transaction. It is primarily

geared toward valuation of fungible assets traded on public exchanges against a fiat currency.

ASC 820 (Financial Accounting Standards Board [FASB], 2011) and IFRS 13

(International Accounting Standards Board [IASB], 2011) provide guidance for fair value

measurement and disclosure, outlining a three-level fair value hierarchy used to rank the

reliability of the inputs used in the fair value measurement. The levels, from high to low,

are as follows:

Level 1: Quoted prices in active markets for identical assets or liabilities that the

entity can access at the measurement date.

Level 2: Pricing inputs other than Level 1, observable directly or indirectly, including

quoted prices for similar assets and prices derived from or corroborated by obser-

vable markets.

Level 3: Pricing using unobservable inputs.

The classification in the hierarchy is driven by the lowest level of input reliability.

To expand fair value to digital assets, we revisit the notions of markets and transaction

and expand them to digital assets within the framework of IFRS 13 and ASC 820. First, we

show, as result of a world view focusing on decentralization to which many of the block-

chain pioneers adhere, these markets are inherently fragmented. Consequently, the notion

of principal market, central to the fair value measurement, must be subsumed by a new

construct that could be applied to digital assets. Second, we show that as result of geogra-

phical disparity, globalization, technological legacy and the nature of financing and value

appropriation, the market has become partitioned into multiple enclaves using different

digital assets as vehicles for value. Consequentially, there is no single unit of measure uni-

versally applicable to all transactions in digital assets, and most transactions involve

exchanges of digital assets for other digital assets. This situation closely resembles global

portfolio across assets denominated in different currencies. However, foreign currency con-

versions fall under a different set of guidelines in which they are not considered separate

transactions. This implies that exchange rates used to translate fair value measurements to

a functional currency are not considered additional measurements and therefore do not

impact level in the fair value hierarchy. Crypto asset conversions, however, are considered

transactions and fall under fair value guidelines, and as such a conversion of digital assets

going through one or more vehicles would be considered a chain of transactions. An esti-

mation of value obtained through such a chain would therefore require multiple measure-

ments. Currently, IFRS 13 and ASC 820 offer no guidelines for situations where value

estimation requires multiple measurements.

Beigman et al. 243

Market fragmentation has been previously addressed in the literature, in particular a

paper by Beigman et al. (2021) (BBHS, hereafter) extensively address the issue of fragmen-

tation in crypto markets, offering a methodology for selecting principal market in a

dynamic setting. Multiple measurements have so far received only limited attention within

the fair value framework; however, similar issues had been addressed in discussions

included in amendments to IAS 21 (IASB, 2001, 2021) focusing on situations where there

is a lack of exchangeability in foreign currency. In this article, we adapt key concepts from

these two lines of literature and apply them to the case of digital asset, offering a compre-

hensive methodology for pricing digital asset by fiat currency within the framework of

IFRS 13 and ASC 820.

This article is organized as follows. In the next section, we present a brief review of cur-

rent practices and recommendations for pricing crypto assets that do not directly trade

against fiat. We then discuss fragmented markets and provide an overview of the funda-

mental pricing assumptions used in Beigman et al. (2021) in section ‘‘Fragmented

Markets.’’ In section ‘‘Inactive Markets and Lack of Exchangeability,’’ we discuss cur-

rency conversion (the discussion around lack of exchangeability) and how it relates to inac-

tive markets. We then employ the notion of a ‘‘path of assets’’ as a new construct used for

pricing. In section ‘‘The Principal Path Method (PPM),’’ we expand these ideas and

develop a methodology for comprehensive evaluation of digital assets. In section ‘‘PPM:

An Empirical Demonstration,’’ we will demonstrate the methodology numerically and

empirically. Finally, we present our conclusions, limitations, and suggestions for future

research in section ‘‘Conclusions.’’

A Review of Current Practice

This paper contributes a novel approach to the valuation of crypto assets. Few papers

related to this issue have been published in the academic literature to date (2023).

However, there is much debate regarding the use of fair value to measure crypto assets in

practice.

In its publication, Accounting for and Auditing of Digital Assets (Association of

International Certified Professional Accountants [AICPA], 2022), the AICPA takes the

position that digital assets are intangible assets and should be valued at cost. Fair value

should be used only the case of impairment. Nevertheless, it does emphasize the need to

use as much observable data as available. However, more recently standard setters have

supported the use of fair value in measuring crypto assets. Specifically, in March 2022, the

SEC issued Staff Accounting Bulletin (SAB) 121 which indicates that it would be appropri-

ate for an entity that has an obligation to protect crypto assets held on its platform to

record and asset and related obligation at fair value (U.S. Securities and Exchange

Commission [SEC], 2022). In addition, recent decision by the FASB recommends the use

of fair value through earnings to measure cryptocurrency (FASB 2022a, 2022b). As a

result, the need to value crypto assets at fair value has expanded and assumes a central role

in accounting for such assets, whether actively or thinly traded.

Current techniques used to value thinly traded crypto assets include methods such as

using the last observable transaction price, creating a weighted-average price across multi-

ple markets, or using data on comparable tokens, if available. In general, firms are using a

simple management judgment-based approach identifying assets that are not actively traded

via an analysis of observable transactions and then using price discovery from the market

the entity transacted on most frequently. These techniques generally result in a Level 2

244 Journal of Accounting, Auditing & Finance 40(1)

classification.4 Although not specifically included in the current guidance, the use of multi-

ple markets may be justified from a risk management standpoint. In addition, a 2019 publi-

cation from PwC discussed the accounting considerations for cryptographic asset

transactions and included situations where an active market for a cryptographic asset is not

likely to exist (PricewaterhouseCoopers [PwC], 2019). It used the term ‘‘trading pairs,’’

which uses some of the elements found in our method. Specifically, the PwC monograph

provides an example that considers a cryptographic asset (CA ABC) that cannot be directly

converted into fiat currencies but can convert into other cryptographic assets (CA DEF).

PwC concludes that the price of the CA ABC would likely be the product of the exchange

rate of [Fiat/(CA DEF)] by the exchange rate of [(CA DEF)/(CA ABC)]. Moreover, PwC

indicates that

in practice, there are exchanges that do not offer the possibility for crypto to fiat trades at all.

In this instance, an entity might exchange a cryptographic asset for another cryptographic

asset, and then exchange the second cryptographic asset into fiat at another exchange. This

means that it is possible for a market to exist for a cryptographic asset in which there is fre-

quency and volume of trades, but that this market is not an active market under IFRS 13.

Our methodology improves upon the techniques used in current practice by proposing a

dynamic approach that relies on compliant markets for price discovery. This method will

result in more reliable and faithfully represented valuations for thinly traded crypto assets,

even in an ecosystem that includes fragmented markets. We discuss fragmented markets

further in the following section of this article.

Fragmented Markets

Although in some cases assets may be traded over multiple exchanges or exchange alterna-

tives (market makers, dark pools, or OTC), classical markets are designed to promote a

single centralized exchange as the primary source of pricing information, termed the princi-

pal market. This designation may be difficult to apply when dealing with certain crypto

asset markets. There are a multitude of centralized exchanges operating in this space (e.g.,

Coinbase, Kraken, and Binance) spanning diverse geographical regions and sovereignties.

Beyond that there are different types of decentralized market institutions, such as decentra-

lized exchanges (Uniswap, IDEX, and Sushiswap), automated market makers (Bancor and

Pancakeswap), and other trading platforms (dXdY, Metcha, and Balancer to name just a

few). The fragmentation this creates in the market, as well as the lack of centralization

within many of these entities, is not an arbitrary side effect, but part of an ideology of

decentralization that many of the pioneers in this space follow. At the heart of this ideology

is the concept of ‘‘code is law,’’ namely, a belief that code and protocol can replace inter-

mediary third-party institution as a source of trust in peer-to-peer transactions. This ideol-

ogy is in direct contrast to fair value as a price observation on a designated centralized

market which in this context serves as a third-party intermediary.

In their paper, Beigman et al. (2021) addressed the issue of fragmentation in crypto mar-

kets. In their setting, digital assets are traded over multiple public exchanges in volume suf-

ficient to qualify as Level 1; however, none of these exchanges have a designation that

would qualify them as principal market. Moreover, key relevance indicators, such as

volume and price discovery, tend to fluctuate drastically throughout the day such that there

is no one exchange that would be a natural focal point. The BBHS method enhances the

Beigman et al. 245

classical notion of principal market with a corresponding ephemeral construct dependent on

multiple factors, including exchange compliance, quality, volume, and freshness of the

data, used for pricing.5

This article employs elements of the BBHS method, including a new ephemeral con-

struct used to develop a methodology for pricing digital assets for which the asset to fiat

market is inactive, either do not exist or trade at volumes that are insufficient to qualify as

Level 1, but trade against other assets over public exchanges in sufficient volume.

Inactive Markets and Lack of Exchangeability

When Bitcoin first launched in 2009, it operated as a single-asset platform dedicated

almost exclusively to financial transfers. The underlying scripting language was too limited

for most other uses. This fundamentally changed with the introduction of Ethereum and the

Ethereum Virtual Machine (EVM), a Turing complete on-chain computing device sup-

ported by Solidity and Viper, full-fledged programming languages, opening the way for

smart contracts and the ERC-20 standard for fungible tokens. This in turn led to the prolif-

eration of new class of digital assets on top of the Ethereum blockchain and later, on other

EVM compatible blockchains like Polygon and Binance. Each of these platforms requires

access to a native token which is used to pay for gas, the resource required for transactions

on the blockchain. Moreover, many of the new ERC-20 tokens are access tokens to a whole

ecosystem built on top of these blockchains, such as APE,6 MANA,7 CAKE,8 and dXdY.9 In

many cases, these digital assets may not be directly exchangeable into a fiat currency. To

illustrate, we take a closer look at the ApeCoin,10 the governance and utility token used

within the APE ecosystem. This ecosystem includes a wide range of products and services

using the token, including the Bored Ape Yacht Club NFT series by Yuga Labs, as well as

additional NFT series and a metaverse. ApeCoin tokens are used for accessing different

NFTs, as governance tokens and as currency. NFT assets are traded on ApeCoin marketplace,

quoted in either ETH or APE. Transactions are either in wrapped ETH (an ERC-20 token

pegged to ETH, for technical reasons ETH cannot be used in the Ape ecosystem) or APE

and will come with a fee denoted in APE. Thus, to obtain a fair value measurement in terms

of a fiat currency such as the USD, will require a path of at least two ‘‘hops’’ (NFT-APE-

USD or NFT-WETH-USD) across at least two exchanges, and in practice more likely will be

at least three ‘‘hops’’ (NFT-APE-ETH-USD, NFT-APE-USDT-USD, etc.) across three

exchanges. Some centralized exchanges such as Binance (the largest exchange by volume),

support multiple stablecoins pegged to USD and non-USD fiat, as well as non-USD curren-

cies, offering many different paths through which to estimate price versus functional.

The immediate question is, of course, how should fair value be estimated in such set-

tings, and the level assigned to these estimates within the fair value hierarchy? As noted

earlier, conversion of one digital asset to another is considered a transaction and is not the

equivalent of foreign currency conversions from the accounting perspective. However, pric-

ing digital assets through valuations of other digital assets is not fundamentally different

than pricing foreign currency denominated assets through currency conversions. We there-

fore use analogous concepts found in the amendments to IAS 21 (IASB, 2014, 2021) which

focus on the lack of exchangeability.

IFRS 13 (IASB, 2011) states that assets that are quoted in a currency different from the

functional currency, should translate the quote to the functional currency, and will qualify

as Level 1 if the exchange rate is observable. IAS 21 addresses situations where an asset is

priced in a currency that is not directly convertible to the functional currency. Initially, the

246 Journal of Accounting, Auditing & Finance 40(1)

focus was on temporary non-exchangeability and was later expanded to settings where cur-

rencies are permanently non-exchangeable. The reality in the FX market is that 85% of the

volume involves USD pairs (Gourinchas et al., 2019), even though the U.S. economy

accounts for less than a quarter of the global economy. If we add to USD the EUR and

JPY volume, this accounts for over 95% of the volume. Moreover, the two established prin-

cipal markets for FX, EBS and Refinitiv both support only pairs that include at least one of

the vehicle currencies: USD, EUR, or JPY. Given there are over 150 currencies worldwide,

lack of direct conversion is norm rather than the exception. Research (Goldberg & Tille,

2008; Somogyi, 2022) indicates that up to 40% of the USD volume results from indirect

transactions. Both US GAAP and IFRS treat FX as a separate asset class, and as such, ASC

820 and IFRS 13 do not apply. IFRS 13 does not differentiate between direct and indirect

observable data. Nevertheless, in the accounting guidance that considers the lack of exchan-

geability, the standards indicate that a path, (i.e., a chain of multiple assets that are traded

over an active market), can be used to price the assets at the ends of the path. Agenda pro-

posal for foreign currency accounting issued by the Korean Accounting Standards Board

and submitted to the IASB discussed whether such evaluations qualify as single or multiple

measures and what exchange rate constraints on the chain must be satisfied (IASB, 2014).

In addition, the issue of whether low volumes and high volatility could disqualify a market

from consideration is discussed. These issues were partly addressed in a recent Exposure

Draft ED/2021/4 Lack of Exchangeability (IASB, 2021). The proposed guidelines specify

the requirements for reporting exchange rates when a pair is lacking exchangeability.

Specifically, it provides a procedure through which it is determined when the exchange rate

should be priced through direct or indirect observations and when model-based pricing

should be preferred. The proposal emphasizes estimating exchange rates through an obser-

vable exchange rate as the estimated spot exchange rate when that observable exchange

rate would have applied to an orderly transaction between market participants and a rate

that faithfully reflects the prevailing economic conditions (IASB, 2021).

Accountants face a similar situation where there is lack of exchangeability or thinly

traded crypto assets. Our objective is to develop a pricing method that improves upon current

practice by extracting the most reliable data from compliant exchanges and allows for the

valuation of crypto assets that do not directly trade or are thinly traded against fiat. We adapt

the IAS 21 implicit idea of pricing an asset through a path or chain of assets, if any link in

this chain is a pair of assets that are supported by an active market and transactions are obser-

vable, public, and orderly. We explicitly take the position that each conversion estimate is a

measurement, and thus, the overall price is not a direct quote but a derivation of multiple

observed inputs and should therefore fall primarily into Level 2 in the fair value hierarchy.

IAS 21 does not include any recommendations regarding the path used for FX valuation,

but given the volume involving USD pairs, virtually any asset can be converted directly to

USD or at most via EUR, thus virtually all paths include one vehicle with a small fraction

including two vehicles.

The situation is quite different for digital assets, while Bitcoin and Ethereum are sup-

ported almost universally, the choice of fiat currencies, stable coins, and exchange tokens

available on each exchange differ significantly from one exchange to the other.11 In some

cases, stable coins and exchange tokens command a significant portion of the volume. As

such, these crypto-to-crypto markets are critical in the price discovery process and should

not be ignored. We propose the Principal Path Method (PPM, hereafter) as a method to

systematically choose a path for pricing crypto assets. By this method, price is determined

by the exchange rates of assets on a path, as in FX pricing; however, due to the

Beigman et al. 247

fragmentation in crypto markets, each link is priced independently through the BBHS

method.12 Moreover, the chain used for pricing is not fixed as it typically is in FX, but

dynamically chosen based on market conditions. We designate this ephemeral construct the

principal path. Using data from both centralized and decentralized exchanges, we will

demonstrate that most thinly traded crypto assets have short chains connecting them to

functional, fiat currencies, with sufficient volume and activity to qualify as a principal

path, rarely with length greater than three links or ‘‘hops.’’ However, our method accounts

for chains of all lengths when determining a dynamic principal path. Chains with a signifi-

cant number of ‘‘hops’’ is in line with the new realities on the ground where in many cen-

tralized and decentralized exchanges technology-enabled trading bots are used to discover

possible superior liquidity provisions through longer chains of assets.

The determination of whether a crypto asset pair is trading in inactive market will be

made based on a volume threshold13 as well as additional characteristics of the market. The

use of a volume threshold for thinly traded, inactive, or illiquid assets has precedent as it is

recommended by the SEC in its analysis of the national market system (NMS) for equity

shares (SEC, 2018). Specifically, the SEC (2018) used the average daily share volume

(ADV) as the criterion to differentiate liquid from illiquid stocks. This approach was also

supported by the U.S. Department of the Treasury (SEC, 2019; the U.S. Department of the

Treasury, 2017). Although there is no specific accounting guidance provided to differenti-

ate between the liquid and illiquid crypto assets, our method can be employed under a des-

ignated volume threshold. Our proposed PPM intentionally does not propose explicit

thresholds; however, once sufficient experience has been gained, with input from practi-

tioners, market data can be used to establish a threshold.

The Principal Path Method (PPM)

Once a pair of crypto assets is determined to be thinly traded or lacking exchangeability, our

proposed PPM will conceptually follow the elements of the BBHS method by scoring each

possible path or chain of indirect conversion based on compliance, volume, and freshness of

data, and designating the exchange with the highest score as the principal path. It should be

noted that although we use the key elements of the BBHS method in our valuation, we are

not attempting to identify a principal market with the use of the PPM. Based on the current

standards, the resulting valuation cannot be considered as extracted from a principal market.

The guidance (e.g., IFRS 13) indicates that while the market predominantly used to trade

crypto currency should be considered the principal market, a market that does not exchange

crypto to fiat cannot be considered the principal market. Similarly, PwC (2019) advises that a

crypto asset that does not convert directly into fiat in an active market does not fulfill the cri-

teria for a Level 1 asset. Although the definition of an active market does not refer to fiat

currency, the presumption of this interpretation is that, to qualify as a Level 1 fair value mea-

surement, the transaction should be measured in a fiat currency. The fair value of a crypto-

currency that is convertible into a fiat currency through another cryptocurrency is likely to

qualify for a Level 2 asset in the fair value hierarchy. A Level 3 classification may result if

the asset is not readily convertible into a fiat currency (PwC, 2019).

PPM: Method Development

Specifically, the BBHS method dynamically identifies compliant potential markets with a

volume-based approach consistent with the accounting standards (ASC-820-10-35-5A-6A

248 Journal of Accounting, Auditing & Finance 40(1)

& IFRS 13, 17-19), for actively traded crypto asset pairs across all available exchanges.

The method determines its fair value from the exchange that is most compliant and exhibit-

ing the largest volume with the least decay. Specifically, the BBHS method applies a scor-

ing mechanism (Base Exchange Score or BES) that considers several exchange

characteristics, including oversight, microstructure, and technology which assure only com-

pliant markets are used, and then adjusts for volume and frequency of trades. Selecting

only compliant markets when any fair value modeling is critical in the crypto eco-system

as research has established that wash and fake trading, misreporting, price manipulation,

and other misconduct are widely pervasive in the crypto ecosystem in both centralized and

decentralized exchanges, and in particular in smaller and exchanges in jurisdictions that do

not have KYC/AML requirements (Aloosh & Li, 2019; Amiram et al., 2020; Chen et al.,

2022; Cui & Gao, 2022; von Wachter et al., 2022).

We make several assumptions in the development of our method. First, we assume that

markets are highly fragmented and decentralized, a pair of assets like BTC/USDT, ETH/

USD or even USDT/USD could trade over multiple exchanges in parallel, dispersed geo-

graphically and across sovereignties. Next, vast majority of active pairs are not supported

by active markets, and in particular, there are many assets that do not trade against USD or

other fiat in active markets, thus there is a need to price assets to USD (or other fiat) based

on trade against other crypto assets. Finally, we assume that liquid markets with higher

volume are more likely to support estimated prices in real-life transaction than markets

with lower volume. By applying the bottleneck methodology, we are able to identify the

path that is more likely to support the estimated price in real-life transactions than other

paths.

The method is making some trade-offs. For example, using a path, we are reconstructing

a price rather than using a direct trade, thus dropping to a lower level of reliability (to

Level 2). In addition, we are adding a significant computational burden of reporting.

Identifying principal path requires a significant amount of computations and knowledge of

algorithms.

The BBHS method is used in this article to dynamically identify compliant markets

(credibility and quality) that would provide more reliable information, for a certain pair,

than other markets for the same pair, but not to determine a principal market.

To rank the credibility and quality of each exchange, a score that incorporates the key

characteristics for each exchange is assigned through the following steps:

Step 1: Assign a Base Exchange Score (BES) for each exchange for the targeted

crypto asset pair based on the static exchange characteristics.

Step 2: Adjust the score by the relative monthly volume of the exchange Volume-

Adjusted Path Score (VAPS).

Step 3: Decay the adjusted score based on the time passed since the last trade on

exchange Decay Volume-Adjusted Path Score (DVAPS).

We take a similar approach for the indirect pricing of chains for non-exchangeable

pairs. For each chain, we look at the static characteristics, volume, and freshness for each

link, then, for each attribute separately, in we take the minimal score across all links as the

score of the chain for the given attribute. We then compute an adjusted score for the chain

and select the chain with the highest score as the principal path. Note that this a min-max

score and not a simple max volume.

Beigman et al. 249

The mathematical specification of the method is as follows, let A = {a0, . . ., ak} be the

set all of assets (XBT, ETH, USDT, USDC, BAT, MCO, DOGE, CRO, etc.). We say that a

pair (ai, aj) exchangeable if the pair ai/aj or aj/ai is traded on at least one exchange and

priced by the BBHS model let E be the set of exchangeable pairs in A. Thus, (XBT,

USDT) is exchangeable since this pair is traded on multiple exchanges, while (DOGE,

APE) is not exchangeable as of the time of this writing. We shall call G = (A, E) the asset

graph. A path sa0, ..., ak in the asset graph is a sequence a0, . . ., ak such that ai 2 A for 0 � i � k and (ai, ai+ 1) 2 E for 1 � i � k – 1 this corresponds to a chain of pairs ai/ai+ 1

where each link is an exchangeable pair ai, aj 2 A.

For any link ai, aj 2 A, let SBES ai, aj

be the base exchange score (BES) of the BBHS market

for the pair, we define the path base score (PBS) to be

SPBS sa0, ..., ak

=minfSBES ai, ai+1

j0 � i � k� 1g:

The link (ai, ai+1) on which this minimum is attained is the path score bottleneck. In

the same manner, we define the path volume to be

Volsa0, ..., ak =minfVolai, ai+1

j0 � i � k� 1g,

where, as before, Volai, aj is the volume on the BBHS path for (ai, a j) as measured in some

common numeraire, typically either USD or XBT, using the BBHS method. Let Vola0ak be

the volume passing through the graph with a0 as source and ak as sink,14 then we define

the volume-adjusted path score (VAPS) to be

SVAPS sa0, ..., ak

=SPBS sa0, ..., ak

3 Volsa0, ..., ak

Vola0ak :

Similarly, we define

tsa0, ..., ak =maxftai, ai+1

j0 � i � k� 1g,

where tai, aj is the time elapsed since the last trade of the pair on the BBHS market. Note

that the greater the time lapse, the lower the score. Finally, we define the decayed volume-

adjusted path score (DVAPS)

SDVAPS sa0, ..., ak

= e�k�tsa0, ..., ak 3SVAPS sa0, ..., ak

:

We then define the principal path as the path sa� 0 , ..., a�

k such that sa�

0 , ..., a�

k = argmax

{Bsa0, ..., ak � SDVAPS

sa0, ..., ak —for every path a0, . . ., ak in G},15 where Bsa0, ..., ak

depends on and is

descending on k, the latter reflects a higher score for Level 1 over Level 2 and for shorter

paths over longer ones. The fair value exchange rate of ak to a0 is then

ak=a0 = a�k=a � k�13a�k�1=a

� k�23 � � �3a�1=a

� 0:

One limitation of this method is that reconstructing the trades on each link may imply

transfer of assets between exchanges, transfers that may be costly in both time and gas.

Crypto assets traded in centralized exchanges are typically held in the exchange wallet and

250 Journal of Accounting, Auditing & Finance 40(1)

credited to client accounts, thus buying, and selling crypto assets on such exchanges is not

registered on the blockchain and does not require block confirmation. Transferring assets

between exchanges requires transfer of assets from one wallet to another; these transfers

are registered on the blockchain and require block confirmation. Blocks are added at fixed

time intervals depending on the specific blockchain and entail corresponding latency,16 as

well as a fee, typically referred to as gas price, to the miner for including a transaction in

the block. For modern blockchain platforms such as Avalanche and Solana, the latency is a

fraction of a second, but with older platforms such as the pre-merge PoW Ethereum,

latency was around 10 s, and for Bitcoin, at time of writing, is about 10 min. Another sig-

nificant factor is the volatility of gas prices, the process through which a slot on a block is

allocated to a transaction is essentially an auction, thus in times when there is a high

volume of on-chain transactions, the gas prices can go up significantly. All this suggests, of

course, that PPM is often not easy to attain, however this is not unlike many other asset

classes such as equities where there may be significant transaction costs that are accounted

for in ASC 820 guidelines for fair value.17 Regarding latency, the IAS 21 includes a provi-

sion by which ‘‘normal administrative delay in obtaining the other currency does not pre-

clude a currency from being exchangeable into that other currency’’ (IASB, 2021).

Moreover, we should recall that many of the more exotic assets, including many NFT’s

and specialized tokens, are traded on exchanges supporting only assets contained in a par-

ticular ecosystem. In these cases, traders use local exchanges for on/off ramping their fiat-

to-crypto and execute crypto-to-crypto trades on more liquid exchanges, thus multi-

exchange paths are quite common in practice.

PPM: An Illustration

In this section, we provide a basic example to illustrate the PPM. The illustration assumes

that we would need to value MCO in terms of the USD for financial reporting purposes.

However, MCO is not directly paired and traded with the USD but transacted with other

crypto assets. To measure MCO at fair value for financial reporting purposes, we will use

PPM to find the pairs of transactions with MCO that ultimately lead to the USD valuation.

In applying PPM, the first step is to identify ‘‘links’’ (crypto-crypto or crypto-fiat pairs)

in all observed chains/paths from the targeted crypto asset (MCO) to the destination fiat

currency18 (i.e., USD, for financial reporting purposes). Figure 1 shows the complete

‘‘asset graph’’ for MCO-USD. As indicated in Figure 1,19 MCO was mainly traded with

other crypto assets including XBT, ETH, and USDT on major exchanges worldwide.

Hence, three observed paths with one intermediary crypto asset would be MCO-XBT-USD,

MCO-ETH-USD, and MCO-USDT-USD. In addition, some observed paths have two inter-

mediary crypto assets such as MCO-XBT-USDT-USD, MCO-ETH-USDT-USD, MCO-

USDT-XBT-USD, and MCO-USDT-ETH-USD. It is also possible to identify observed

paths with more than two intermediary crypto assets. The more crypto assets or fiat curren-

cies that the targeted crypto asset could be traded against, the more complex the map of

observed paths would become—our method can address paths of any length. We know

from data observations and input from experts that most paths used for both centralized

and decentralized exchanges are short and consist of only two to three hops—this generally

reflects the behavior of a typical crypto market participant. For illustration purposes, we

will employ only the observed paths with one and two intermediary crypto assets.

Table 1 summarizes the simulated information for the illustration, including the princi-

pal path, the base exchange score, the fair value, the minute transaction volume (in USD),

Beigman et al. 251

and the time of the last trade for each link (crypto-to-crypto or crypto-to-fiat pairs). The

principal paths and base exchange scores of the links in all observed chains are individually

determined and obtained using the BBHS method. For this part of the illustration, we

implemented PPM at 16:06:00 of the day.20 PPM will determine the PBS, which is the min-

imum base exchange score among links in each observed chain, for each observed chain.

The result of the determination of the PBS for seven selected observed chains is presented

in Panel A of Table 2.

Panel B of Table 2 shows the determination process of the path volume, which is the

minimum transaction volume among links in each observed chain. The volume-adjusted

path score (VAPS) will be determined after obtaining the volume weights for each chain,

indicated in the Panel C of Table 2. Finally, the decayed volume-adjusted path score

(DVAPS) will be calculated after identifying the maximum time elapsed since the last trade

among all links for each observed chain. The observed chain with the highest DVAPS

would be designated as the principal path at that moment in time. As indicated in the

Figure 1. Illustration of Selected Observed Path Candidates for MCO-USD.

Table 1. Simulated Information for the Illustration of the Principal Path Method (PPM).

Pairs/Links Market Base exchange

score Fair value Minute transaction volume (USD)

Time of the last trade

MCO-ETH Exchange C 74.9889 0.0295 0.33 16:05:31.273 MCO-USDT Exchange A 92.1846 4.689 597.17 16:05:52.207 MCO-XBT Exchange A 94.9548 0.00054 606.64 16:05:52.143 ETH-USDT Exchange A 96.2627 161 355,471.40 16:05:59.870 XBT-USDT Exchange A 93.2820 8,698.97 7,409,847.00 16:05:59.988 ETH-XBT Exchange A 96.3611 0.0185 64,553.04 16:05:59.620 ETH-USD Exchange B 91.7013 160.7 79,671.79 16:05:57.405 USDT-USD Exchange D 75.9979 0.99775 1,901.71 16:05:59.268 XBT-USD Exchange B 86.3616 8,684.33 873,888.00 16:05:59.841

Note. The illustration is supposed to implement PPM at 16:06:00.000 of the day. For each pair, there is a BBHS

principal market. The base exchange score for each pair is derived from the BBHS method. The fair value for each

pair is derived from the BBHS method and presented on a basis of the first currency of each pair. For example, the

fair value for XBT-USD is 8,684.33, meaning 8,684.33 USD/Bitcoin. The minute transaction volume (USD) for each

link is the sum of transaction volume from 16:05:00.000 to 16:05:59.999 on the BBHS principal market. The time

of the last trade for each link is derived from the BBHS principal market. PPM = Principal Path Method.

252 Journal of Accounting, Auditing & Finance 40(1)

T a b le

2 .

D et er m in at io n o f th e P at h B as e Sc o re

(P B S) , th e P at h V o lu m e,

an d th e V o lu m e- A d ju st ed

P at h Sc o re

(V A P S)

U n d er

th e P ri n ci p al

P at h M et h o d

(P P M ). P an el

A . P at h B as e Sc o re

(P B S)

in P P M ; P an el

B . P at h V o lu m e in

P P M ; P an el

C . D et er m in at io n o f th e V o lu m e- A d ju st ed

P at h Sc o re

(V A P S)

u n d er

th e

P ri n ci p al P at h M et h o d (P P M ); P an el D .D

et er m in at io n o f th e D ec ay ed

V o lu m e- A d ju st ed

P at h Sc o re

(D V A P S)

u n d er

th e P ri n ci p al P at h M et h o d (P P M ).

P an el A

T h e B B H S m o d el (2 0 2 1 )—

B as e E x ch an ge

Sc o re

P at h B as e Sc o re

(P B S)

O b se rv ed

p at h

M C O -E T H

M C O -U

SD T

M C O -X

B T

E T H -U

SD T

X B T- U SD

T E T H -X

B T

E T H -U

SD U SD

T- U SD

X B T- U SD

M C O -X

B T- U SD

9 4 .9 54 8

8 6 .3 6 1 6

8 6 .3 6 1 6

M C O -E T H -U

SD 7 4 .9 8 8 9

9 1 .7 01 3

7 4 .9 8 8 9

M C O -U

SD T- U SD

9 2 .1 84 6

7 5 .9 9 7 9

7 5 .9 9 7 9

M C O -X

B T- ET

H -U

SD 9 4 .9 54 8

9 6. 3 6 1 1

9 1 .7 0 1 3

9 1 .7 0 1 3

M C O -X

B T- U SD

T- U SD

9 4 .9 54 8

9 3. 2 8 2 0

7 5 .9 9 7 9

7 5 .9 9 7 9

M C O -E T H -X

B T- U SD

7 4 .9 8 8 9

9 6. 3 6 1 1

8 6. 3 6 1 6

7 4 .9 8 8 9

M C O -E T H -U

SD T- U SD

7 4 .9 8 8 9

9 6 .2 62 7

7 5 .9 9 7 9

7 4 .9 8 8 9

M C O -U

SD T- ET

H -U

SD 9 2 .1 84 6

9 6 .2 62 7

9 1 .7 0 1 3

9 1 .7 0 1 3

M C O -U

SD T- X B T- U SD

9 2 .1 84 6

9 3. 2 8 2 0

8 6 .3 6 1 6

8 6 .3 6 1 6

N ot e.

T h e pa th

ba se

sc or e (P B S)

u n d er

P P M

is th e m in im u m

o f b as e ex ch an ge

sc o re s fo r lin ks

in ea ch

o b se rv ed

ch ai n , in d ic at ed

w it h th e b o ld

n u m be rs . T he

co n ce p t an d

ca lc ul at io n o f th e B as e E x ch an ge

Sc o re

ar e d er iv ed

fr o m

B ei gm

an , B re n n an , H si eh , an d Sa n ne lla

(B B H S, 2 0 2 1) .

P an el

B

O b se rv ed

p at h

M in u te

tr an sa ct io n vo lu m e (U

SD ) o n th e o b se rv ed

p at h d et er m in ed

b y th e B B H S m o d el (2 0 2 1 )

P at h V o lu m e

M C O -E T H

M C O -U

SD T

M C O -X

B T

E T H -U

SD T

X B T- U SD

T E T H -X

B T

E T H -U

SD U SD

T- U SD

X B T- U SD

M C O -X

B T- U SD

6 0 6 .6 4

8 7 3 ,8 8 8

6 0 6 .6 4

M C O -E T H -U

SD 0 .3 3

7 9 ,6 7 1 .7 9

0 .3 3

M C O -U

SD T- U SD

5 9 7 .1 7

1 ,9 0 1 .7 1

5 9 7 .1 7

M C O -X

B T- E T H -U

SD 6 0 6 .6 4

6 4 ,5 5 3 .0 4

7 9 ,6 7 1 .7 9

6 0 6 .6 4

M C O -X

B T- U SD

T- U SD

6 0 6 .6 4

7 ,4 0 9 ,8 4 7

1 ,9 0 1 .7 1

6 0 6 .6 4

M C O -E T H -X

B T- U SD

0 .3 3

6 4 ,5 5 3 .0 4

8 7 3 ,8 8 8

0 .3 3

M C O -E T H -U

SD T- U SD

0 .3 3

3 5 5 ,4 7 1 .4

1 ,9 0 1 .7 1

0 .3 3

M C O -U

SD T- E T H -U

SD 5 9 7 .1 7

3 5 5 ,4 7 1 .4

7 9 ,6 7 1 .7 9

5 9 7 .1 7

M C O -U

SD T- X B T- U SD

5 9 7 .1 7

7 ,4 0 9 ,8 4 7

8 7 3 ,8 8 8

5 9 7 .1 7

N ot e. Pa th

vo lu m e u n d er

P P M

is th e m in im um

o f tr an sa ct io n vo lu m e (U

SD ) o n th e B B H S m ar ke t fo r th e lin ks

in ea ch

o b se rv ed

ch ai n , in d ic at ed

w it h th e b o ld

n u m be rs .

Beigman et al. 253

Panel D of Table 2, the path MCO-XBT-ETH-USD is designated as the principal path

because it generated the highest DVAPS (45.8373) among all observed chains.

After determining the principal path, the fair value price or exchange rate for MCO-

USD would be the product of the BBHS exchange rates (ER) for the links in the chain.

Therefore, as the designated principal path is MCO-XBT-ETH-USD, the fair value price

for MCO-USD would be

FVMCO�USD =ERMCO�XBT 3 ERXBT�ETH 3 ERETH�USD =0:00054 3

1

0:0185

� � 3160:7=4:6907:

Table 2. (continued)

Panel C

Observed path Path Base Score (PBS) Volume weight Volume-Adjusted Path Score (VAPS)

MCO-XBT-USD 86.3616 606.64/1,204.14a 43.5087 MCO-ETH-USD 74.9889 0.33/1,204.14 0.0202518 MCO-USDT-USD 75.9979 597.17/1,204.14 37.6899 MCO-XBT-ETH-USD 91.7013 606.64/1,204.14 46.1989 MCO-XBT-USDT-USD 75.9979 606.64/1,204.14 38.2875 MCO-ETH-XBT-USD 74.9889 0.33/1,204.14 0.0202518 MCO-ETH-USDT-USD 74.9889 0.33/1,204.14 0.0202518 MCO-USDT-ETH-USD 91.7013 597.17/1,204.14 45.4777 MCO-USDT-XBT-USD 86.3616 597.17/1,204.14 42.8296

a1,204.14 (= 606.64 + 597.17 + 0.33) is ‘‘the overall volume passing through the graph.’’ As we explained in

section ‘‘Fragmented Markets,’’ the path bottleneck volume would be identified for each path and the VAPS would be

calculated based on the volume weight in which the denominator is the sum of all unique path bottleneck volume.

That’s the reason that 606.64, 597.17, and 0.33 are included only once when calculating the denominator (sum).

Panel D

Observed path Volume-Adjusted Path Score (VAPS)

Time of the last trade

Decayed Volume-Adjusted Path Score (DVAPS)

Computed price

MCO-XBT-USD 43.5087 16:05:52.143 43.1682 4.68954 MCO-ETH-USD 0.0202518 16:05:31.273 0.0196783 4.74065 MCO-USDT-USD 37.6899 16:05:52.207 37.3973 4.67845 MCO-XBT-ETH-USD 46.1989 16:05:52.143 45.8373 4.69070 MCO-XBT-USDT-USD 38.2875 16:05:52.143 37.9878 4.68687 MCO-ETH-XBT-USD 0.0202518 16:05:31.273 0.0196783 4.73947 MCO-ETH-USDT-USD 0.0202518 16:05:31.273 0.0196783 4.73881 MCO-USDT-ETH-USD 45.4777 16:05:52.207 45.1247 4.68026 MCO-USDT-XBT-USD 42.8296 16:05:52.207 42.4971 4.68111

Note. The time of the last trade for each observed chain is derived from the trade with the longest time lag among

all links of each observed chain.

254 Journal of Accounting, Auditing & Finance 40(1)

PPM: An Empirical Demonstration

In this section, we demonstrate the proposed PPM, determining the principal path and fair

value measures for digital assets that are thinly traded or inactive against USD. The empiri-

cal demonstration uses actual market data. In addition, we will compare the fair value mea-

sures generated by the method with the actual prices of trades of the pairs, if they are

available.

Specifically, we test eight different thinly traded crypto assets, including Basic Attention

Token (BAT), Dogecoin (DOGE), STASIS EURO (EURS), Ankr (ANKR), Bytecoin

(BCN), Crypto.com Coin (CRO), MCO (MCO), and NEM (XEM). The first three crypto

assets had direct trades to the USD available on at least one exchange, while the remaining

five crypto assets did not have any direct USD trades during the test period covering

January 1, 2019, to June 30, 2020 (totaling 787,680 minutes).

Figure 221 shows the minute transaction volume from thinly traded crypto assets to other

mainstream crypto assets and USD (if available) during the test period. Volumes from

BAT, DOGE, EURS, ANKR, BCN, CRO, MCO, and XEM to other crypto assets and the

USD are displayed in Panels A, B, C, D, E, F, G, and H, respectively. For BAT (Panel A)

and DOGE (Panel B), the transaction volume to the USD is much lower compared with the

volume for XBT, ETH, USDT, or USDC. Interestingly, the transaction volume for the

USD is the same level as the volume for USDT for EURS (Panel C). Although the remain-

ing crypto assets (Panel D-H) had no direct trades against the USD, they had frequent

trades and sufficient transaction volume with XBT, ETH, USDT, USDC, or BNB.

In the empirical demonstration, PPM is tested at the end of each minute for the period

from January 1, 2019, to June 30, 2020 (totaling 787,680 min). Table 3 presents the

number (percentage) of minutes available for the fair value measures from PPM and the

prices of direct trades to the USD for the eight selected thinly traded crypto assets. For

instance, the BAT-USD trades were accessible on four exchanges; however, they are infre-

quently traded. Specifically, only 28,079 (3.56%), 19,219 (2.44%), 25,117 (3.19%), and

5,698 (0.72%) min had trades on the four exchanges for BAT-USD during the test period,

while PPM could generate fair value measures for BAT-USD in 757,923 (96.22%) min.

Furthermore, DOGE-USD could only be transacted on Exchange A in 6,735 (0.86%) min,

but there were 783,579 (99.48%) min having fair value measures from PPM during the test

period. For ANKR, BCN, CRO, MCO, and XEM—crypto assets with no direct trades

against the USD—PPM could still determine fair value measures in terms of USD in

339,567 (43.11%), 254,746 (32.34%), 623,970 (79.22%), 667,862 (84.79%), and 777,042

(98.65%) min, respectively. PPM can provide timely, reliable, and frequent fair value mea-

sures for those thinly traded crypto assets with few or even no direct trades against the

USD.

Figure 3 shows the minute fair value measures of PPM for thinly traded crypto assets

and prices of trades from thinly traded crypto assets to the USD (if available) during the

test period. The fair value measures and prices (if available) from BAT, DOGE, EURS,

ANKR, BCN, CRO, MCO, and XEM to other crypto assets and the USD (if available) are

displayed in Panels A through H, respectively.

The fair value measures and the actual prices for BAT-USD were well matched with

each other for most minutes. However, PPM had one fair value outlier at 7:35 AM on

January 15, 2020. This outlier might result from the real-time, atypical transaction behavior

in the determined principal path or a possible data integrity issue on exchanges. Moreover,

there were some price outliers (large abnormal price jumps comparing the adjacent

Beigman et al. 255

Figure 2.

(continued)

256 Journal of Accounting, Auditing & Finance 40(1)

(continued)

Figure 2.

Beigman et al. 257

(continued)

Figure 2.

258 Journal of Accounting, Auditing & Finance 40(1)

Figure 2. Minute Transaction Volume from Thinly Traded Crypto Assets to Other Mainstream Crypto Assets and the USD (If Available). Note. This figure presents the minute transaction volume from thinly traded crypto assets to other mainstream

crypto assets and the USD (if available) from January 1, 2019, to June 30, 2020. Volumes from Basic Attention Token

(BAT), Dogecoin (DOGE), STASIS EURO (EURS), Ankr (ANKR), Bytecoin (BCN), Crypto.com Coin (CRO), MCO

(MCO), and NEM (XEM) to other crypto assets and the USD (if available) are displayed in Panels A, B, C, D, E, F, G,

and H, respectively. Panel A: Basic Attention Token (BAT), with available trades for BAT-USD. Panel B: Dogecoin

(DOGE), with available trades for DOGE-USD. Panel C: STASIS EURO (EURS), with available trades for EURS-USD.

Panel D: Ankr (ANKR), without available trades for ANKR-USD. Panel E: Bytecoin (BCN), without available trades

for BCN-USD. Panel F: Crypto.com Coin (CRO), without available trades for CRO-USD. Panel G: MCO (MCO),

without available trades for MCO-USD. Panel H: NEM (XEM), without available trades for XEM-USD.

Beigman et al. 259

Figure 3.

(continued)

260 Journal of Accounting, Auditing & Finance 40(1)

Figure 3.

(continued)

Beigman et al. 261

Figure 3.

(continued)

262 Journal of Accounting, Auditing & Finance 40(1)

Figure 3. Minute Fair Value Measures From the Principal Path Method (PPM) for Thinly Traded Crypto Assets and Prices From Thinly Traded Crypto Assets-USD Trades (If Available). This figure presents the minute fair value measures of PPM for thinly traded crypto assets and prices of trades from

thinly traded crypto assets to the USD (if available) from January 1, 2019, to June 30, 2020. The fair value measures

and prices (if available) from Basic Attention Token (BAT), Dogecoin (DOGE), STASIS EURO (EURS), Ankr (ANKR),

Bytecoin (BCN), Crypto.com Coin (CRO), MCO (MCO), and NEM (XEM) are displayed in Panels A, B, C, D, E, F, G,

and H, respectively. Panel A: Basic Attention Token (BAT), with available trades for BAT-USD. Panel B: Dogecoin

(DOGE), with available trades for DOGE-USD. Panel C: STASIS EURO (EURS), with available trades for EURS-USD.

Panel D: Ankr (ANKR), without available trades for ANKR-USD. Panel E: Bytecoin (BCN), without available trades

for BCN-USD. Panel F: Crypto.com Coin (CRO), without available trades for CRO-USD. Panel G: MCO (MCO),

without available trades for MCO-USD. Panel H: NEM (XEM), without available trades for XEM-USD.

Beigman et al. 263

minutes) of real trades for BAT-USD at 8:10 AM on April 8, 2020, and from 5:00 to 6:00

PM on May 5, 2020. As presented in Figure 4, the real trades of BAT-USD experienced

abnormal price jumps during the one-hour period; however, PPM constantly generated rela-

tively stable fair value measures in those 60 minutes.

Some thinly traded crypto assets, such as DOGE and EURS, would be initially traded

with other mainstream crypto assets rather than fiat currencies. As displayed in Panels B

and C of Figure 3, PPM could identify the principal path and determine the fair values for

DOGE-USD and EURS-USD even if there were no direct trades from DOGE and EURS

against the USD at the beginning of the test period. Furthermore, Panel C of Figure 3

revealed that the real trades of EURS-USD experienced many significant abnormal price

jumps, but the fair value measures from PPM were relatively stable, and therefore, more

reliable.

Panels D to H of Figure 3 show the fair value measures from ANKR,22 BCN, CRO,23

MCO, and XEM to the USD. At time of writing, none of these assets are traded against

USD on any exchange and can only be priced through PPM or some other methodology.

Conclusions

Innovative valuation methodologies for crypto assets are a critical component of the

ongoing acceptance and adoption of these emerging economic phenomena built on block-

chain/distributed ledger technology. Commercial market-to-market methodologies for

actively traded crypto assets in orderly markets exist and are offered in commercially avail-

able software products, but dynamic standards–aligned valuation models for thinly traded,

or Level 2, crypto assets are needed to provide more comprehensive pricing and asset

valuation. In addition, we believe that our methodology improves on current techniques

used to value thinly traded crypto assets such as using the last observable transaction price,

creating a weighted-average price across multiple markets, or using data on comparable

tokens, if available. Unlike methods currently used in practice, our method ensures the

Table 3. The Number of Minutes With Available Fair Value Measure Under the Principal Path Method (PPM) With Available Prices of Trades to the USD From January 1, 2019, to June 30, 2020.

Crypto assets Principal Path

Method

Direct trade to USD

Exchange A Exchange B Exchange C Exchange D

Basic Attention Token (BAT)

757,923 (96.22%) 28,079 (3.56%) 19,219 (2.44%) 25,117 (3.19%) 5,698 (0.72%)

Dogecoin (DOGE) 783,579 (99.48%) 6,7235 (0.86%) N/A N/A N/A STASIS EURO (EURS) 16,373 (2.08%) N/A 2,267 (0.29%) N/A N/A Ankr (ANKR) 339,567 (43.11%) N/A N/A N/A N/A Bytecoin (BCN) 254,746 (32.34%) N/A N/A N/A N/A Crypto.com Coin (CRO)

623,970 (79.22%) N/A N/A N/A N/A

MCO (MCO) 667,862 (84.79%) N/A N/A N/A N/A NEM (XEM) 777,042 (98.65%) N/A N/A N/A N/A

Note. We perform the proposed PPM in each minute from January 1, 2019, to June 30, 2020, totaling 787,680 min.

The names of available exchanges were anonymized as Exchanges A to D. Exchanges A to D in this table are

different from the ones in Table 1.

264 Journal of Accounting, Auditing & Finance 40(1)

integrity of the valuation data by selecting prices from compliant markets to ensure reliabil-

ity and faithful representation.

This article addresses these needs by proposing the PPM, as an approach to determining

fair value for non-exchangeable, thinly traded, or illiquid asset pairs. The PPM enables the

dynamic identification of the principal path for the pairs by scoring the static characteris-

tics, transaction volume, and the time elapsed from the last trade for all observed paths.

Only compliant exchanges are considered in establishing fair value for thinly traded crypto

pairs and transaction volume-informed fair value measures are derived by multiplying the

crypto asset prices of links in the determined principal path. Empirical evidence is also pro-

vided, supporting the assertion that PPM can derive more timely and reliable fair value

measures for crypto to fiat currency than standard model-based and other methods used in

practice. While non-exchangeable or thinly traded crypto assets represent a small part of

the total trading volume, they account for a significant part of the total crypto transaction

activity in absolute number of trades and many crypto assets may move between valuation

levels as they emerge, mature, or decline. Our exchange-compliant method will ensure

broader valuation coverage in the rapidly expanding crypto asset space and may be applied

to a wide variety of crypto asset types and pairs.

There are some limitations in our approach. As noted earlier in our paper, one limitation

of this method is that reconstructing the trades on each link may imply the transfer of

assets between exchanges, transfers that could potentially be costly in time, fee, and gas.

We discuss penalizing longer paths through a path-dependent coefficient Bsa0, ..., ak , but do

not provide any guidance regarding how this penalty should be implemented if appropriate.

In addition, market factors have changed over time and may impact the assumptions used

Figure 4. Minute Fair Value Measures From the Principal Path Method (PPM) and Prices From Real Trades for BAT-USD From 17:00 to 18:00 on May 5, 2020. This figure presents the minute fair value measures from the Principal Path Method (PPM) and prices from real

trades for BAT-USD from 17:00 to 18:00 on May 5, 2020. The real trades of BAT-USD experienced abnormal price

jumps during the 1-hr period. However, PPM constantly generated stable fair value measures during the period.

Beigman et al. 265

in our method. Several years ago, the best way to attain the PPM conversion rate would be

to use multiple centralized exchanges, transferring assets directly from one exchange wallet

to another. In this setting, we incur additional costs for gas required for the on-chain trans-

action, as well as latency for confirmation. Currently, a significant amount of trading is

done on decentralized exchanges. These are markets not structured as limit order book mar-

kets, applying their own logic for matching through smart contracts. While transactions and

all relevant information are recorded on the blockchain, there are currently no guidelines

regarding the use of on-chain information. Moreover, these exchanges live on a particular

blockchain platform, moving between exchanges requires moving between blockchain plat-

forms, typically done through special bridges, which have their own costs and risks. This

technology is still in its infancy, and it might take some time before it is clear how it

should be used but these issues merit revisiting in future research.

Further research is also needed to explore the development of models for valuing assets

or asset pairs where there are no observable transactions resulting in a ‘‘Level 3’’ classifi-

cation in the fair value hierarchy.24 There is also a need to determine how to define ‘‘thinly

traded markets.’’ As discussed in this article, a level like the ADV of 100,000 for NMS

stocks (SEC, 2018) could be employed. The threshold established would be based on the

evaluation of available market data with input and recommendations from a cross-section

of valuation experts in industry, accounting firms, and academic researchers’ familiar with

crypto assets and the supporting ecosystem.

Acknowledgment

The authors thank the data team at Lukka, Inc., for their support and help accessing the data needed

to make this study possible.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/

or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this

article.

ORCID iDs

Sheng-Feng Hsieh https://orcid.org/0000-0003-0185-4107

Alexander J. Sannella https://orcid.org/0000-0002-3102-4959

Data Availability

Contact the correspondence author.

Notes

1. Native token for the Polygon layer 2 sidechain, ranked #10, market cap US$10.9B.

2. Utility token for Uniswap decentralized exchange (DEX), ranked #18, market cap US$5.5B.

3. Governance and utility token for The Graph ecosystem, ranked #52 with market cap

US$981.9M.

266 Journal of Accounting, Auditing & Finance 40(1)

4. Information about the pricing techniques currently used in practice was obtained from extensive

conversations with accounting and other valuation professionals. As noted, the resulting valua-

tions would yield a Level 2 valuation. An exception would be in those cases where discounted

cash flow is used or if an observed transaction price is discounted for significant decay. A Level

3 valuation would probably be used is such cases.

5. The BBHS method refers to the fair value measurement method developed by Beigman et al.

(2021).

6. Utility token used in the APE ecosystem developed from the Bored Ape Yacht Club project,

ranked #30, market cap 2.2B, 5% transactions vs. fiat.

7. Utility token used for purchasing virtual real estate on in Decenterland, ranked #41, market cap

1.44B, 6.5% transactions vs. fiat.

8. Utility and governance token, used to reward liquidity provision for automated market makers in

the PancakeSwap platform, ranked #61, market cap 768M, does not trade against fiat.

9. Governance token, used for staking and trading on the dXdY Layer 2 decentralized exchange,

ranked #81, market cap 495M, only 0.44% vs. fiat.

10. More information is available at: https://apecoin.com/.

11. There is significant diversity within the class of stablecoins. While all are pegged to one fiat or

another, the pegging mechanism can differ substantially, ranging from fully backed, partially

backed or algorithmic stable coins. This diversity was in full display in the recent events around

TerraUSD (UST). The rapid pace at which a stable coin such as UST can move away from peg

target is one of the motivations for our layered approach for path scoring accounting for both

monthly volumes and minute to minute decay.

12. Note that the BBHS method is only used to identify compliant markets at each link of the chain.

The exchanges identified at each link are not considered principal markets. This approach not

only ensures the inclusion of compliant markets, but it also enhances the reliability of the inputs

used to value the crypto asset held. We are using elements of the BBHS method to address the

need for price discovery for Level 2 assets only and are using a ‘‘mark to model’’ approach. We

are not using this method to identify a principal market. FASB ASC-820 and IFRS-13 does not

require the use of a principal or most advantageous market to price the crypto assets considered

in this article.

13. The threshold established would be based on the evaluation of available market data with input

and recommendations from a cross-section of valuation experts in industry, accounting firms,

and academic researchers’ familiar with crypto assets and the supporting ecosystem.

14. Vol is a solution to the max flow problem which can be solved efficiently (namely, in polyno-

mial time) with various algorithms. For details, see Matousek and Gärtner (2007) and

Papadimitriou and Steiglitz (1998).

15. If there are more than two observed paths with the same highest DVAPS, then the PPM will take

the path with the least number of links to be the principal path.

16. Proposed paragraph A5 in the Exposure Draft ED/2021/4 Lack of Exchangeability (IASB, 2021)

reflects the Board’s conclusion that a normal administrative delay in obtaining the other currency

does not preclude a currency from being exchangeable into that other currency. Ignoring normal

administrative delays would, in the Board’s view, lead to entities inappropriately concluding that

exchangeability is lacking when a currency would, in effect, be exchangeable into that other cur-

rency. The Board decided not to propose application guidance on what would constitute a

‘normal administrative delay’—this assessment would depend on facts and circumstances (for

example, the jurisdiction in which an exchange transaction occurs and the type of exchange

mechanism) (IASB, 2021).

17. It should be noted that transactions costs would only play a role in the fair value process if we

were considering the most advantageous market. The most advantageous market is used when a

principal market cannot be identified. Because the principal path subsumes the principal market

when the latter is not relevant, the most advantageous market or path and consideration of

Beigman et al. 267

transactions costs are not pertinent to our method as in classical market situations where this

could allow for possible to cherry picking and methodological inconsistency. In addition, prices

observed in illiquid markets tend to be discounted (i.e., an implied fee). Nonetheless, the B

factor/coefficient penalizes length of chains, it is somewhat arbitrary as of time of writing, but as

with the threshold for thinly traded, when sufficient experience on crypto trading is attained this

should be revisited. In addition to penalizing longer paths, we are also decaying the price for

latency. We realize that most transactions have short paths simply because that reflects the beha-

vior of a typical market participant trying to convert crypto to a fiat currency or other crypto.

The short path would imply immaterial transactions costs. There may be many possible reasons

for a longer path, and it may only occur with the use of automated trading bots. In addition,

there may not a fiat offramp on the exchange the participant is trading on. A market participant

would generally only need one hop to find an exchange with a fiat off-ramp or another crypto

they want to trade. A review of current practice indicates that multiple markets are often consid-

ered when computing a weighted-average price for thinly traded cryptocurrencies. The use of

multiple markets might be justified from a risk management standpoint although not specifically

included in the current guidance.

18. The destination currency is not limited to fiat currencies and can be other crypto assets based on

the customized needs.

19. The MCO may be transacted with additional crypto assets or fiat currencies (represented by the

dotted lines in Figure 1), making the observed paths more various. Additional observed paths

may be possible in the future when MCO becomes more popular.

20. Users can implement PPM in any time to fit their specific purposes. We randomly selected

16:06:00 of the day because the thinly traded pairs (MCO-XBT, MCO-USDT, and MCO-ETH)

had trades reported in the previous minute (from 16:05:00.000 to 16:05:59.999).

21. The names of exchanges used in the empirical demonstration section were anonymized as

Exchange A to D. The names of the actual exchanges were used in an earlier version of our

paper. However, one of the exchanges asked us not to disclose the name of the exchanges, and

we complied.

22. The ANKR trade data were not available until March 5, 2019.

23. The CRO trade data were not available until March 7, 2019.

24. It should be noted that even if crypto assets continue to be classified as indefinite-lived intangi-

ble assets, a current FASB decision supports the valuation of cryptocurrency at fair value through

earnings. In addition, the SEC in SAB 121 indicates that crypto assets held in a custodial capac-

ity, along with the related liability be measured at fair value. The valuation technique developed

in our paper is needed to measure the fair value of thinly traded crypto currency for these

purposes.

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Author Biographies

Eyal Beigman received his PhD in mathematics from the Hebrew University, completed a

postdoctoral fellowship in the Center for Mathematical Studies in Economics and

Beigman et al. 269

Management Science at Kellogg Business School at Northwestern University, and spent

two years as faculty at the Olin Business School, in Washington University in St Louis,

studying applications of game theory to mechanism design and political science. He later

spent several years developing HFT algorithms for Deutsche Bank before joining Lukka as

director of research. His current interests are in blockchain technology, digital assets and

market design in this space. He has recently been working on a new startup for the next

generation of financial reporting.

Gerard Brennan CFE, PhD, is a practitioner, educator, frequent speaker, and published

researcher helping develop innovative accounting/audit data and reporting solutions for the

blockchain/DLT ecosystem. Formerly the Audit Director and NA Risk & Internal Control

Officer for Siemens Corp. now a researcher and advisor for Lukka, a US-based software

company that automates and optimizes financial business processes for professionals who

interact with distributed and decentralized technologies. Gerard coordinates joint block-

chain-related research with The Rutgers Universities Continuous Auditing/Reporting

Laboratory (CarLab). He is passionate about delivering practical research to address real-

world accounting and audit problems in the blockchain space.

Sheng-Feng Hsieh, PhD, is an assistant professor of accounting at College of

Management, National Taiwan University, Taiwan. He earned his doctoral degree at

Rutgers, the State University of New Jersey. He was one of the recipients of the AAA SET

Outstanding Dissertation Award in 2022. His research interests include accounting for and

auditing of digital assets, emerging technologies in accounting and auditing, and text

mining. His works have been published in Journal of Accounting, Auditing and Finance,

International Journal of Accounting Information Systems, International Journal of Digital

Accounting Research, and so on.

Alexander J. Sannella is a Professor of Accounting and Information Systems at the

Rutgers Business School. He earned a BBA in Finance and an MBA in Accounting from

Iona University’s LaPenta School of Business. He received his PhD in Accounting and

Finance from New York University and is a New York State Certified Public Accountant.

He has public accounting experience as an auditor for PricewaterhouseCoopers, LLP and

KPMG, LLP. He also served as a consultant to the Line of Business Program at the Federal

Trade Commission in Washington. He is the author of many scholarly journal articles and

three books.

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