The Valuation of Cryptocurrencies in Single-Asset and Multiple-Asset Models

Cryptocurrencies are virtual currencies employed in blockchain transactions. They are particularly worthy of theoretical examination, given the limited academic literature on the subject. This paper constructs valuation models of bitcoin and altcoins, both as single investments and components of mut-liple-asset portfolios. As single investments, cryptocurrencies are valued at the confluence of Legendre utility functions, with Esscher transformed Geometric Levy pricing processes. As part of portfolios, cryptocurrencies are contained in traditional Markowitz portfolios which are varied by increasing the proportion of the riskless asset, shorting the risky asset, or adding currency options. Theoretical formulations show that Markowitz models combined with bitcoin, located on the Capital Market Line (which we term CML portfolios), have low returns, mainly due to the presence of the riskless asset. Such portfolios are appropriately suited to the investment goals of risk-averse traders, while overlooking the preferences of risk-takers. To satisfy less risk-averse investors, we propose a high-return portfolio with 9 asset choices, con-sisting of risky assets, cryptocurrencies, US dollars, soybean futures, Treasury bond futures, oil futures, currency options on the US dollar, currency options on the Mexican peso, and technology, or biotechnology stocks. Laplace transforms are employed to suppress volatility, skewness, or kurtosis of returns, which empirical studies have found to contribute to tail risk contained in outliers in fat-tailed distributions.

Theoretical Economics Letters orded on the blockchain, a global public ledger. The sentiment that the blockchain will have far-reaching disruptions, from global supply chains to shipping, has led to a surge in bitcoin prices. With a 168% increase in bitcoin prices from 2016-2017, reaching a peak of $2700 in May, 2017, the market capitalization of funds flowing into cryptocurrencies rose from $ 25 billion in April 2016-$1000 billion in June, 2017 [1].
The blockchain is available to all computers with the bitcoin protocol, granting every user access to each transaction. A seller verifies that a buyer has the bitcoins associated with his or her address, then completes the sale, recording it in the public blockchain ledger. As an address cannot be linked to a user, the privacy of buyer and seller are maintained [2]. Unlike coins and paper currency, cryptocurrencies lack intrinsic value, meaning that they have no physical form.
Their value is not supported by a government. The benefits of decentralization of authority are that it avoids the concentration of power that would permit a single user to control the system, or loss of resilience from the failure of the computer system at a central point [3]. Alternate cryptocurrencies to bitcoin, such as litecoin, with a faster hashtag function, peercoin, with superior verification, ether, with lower cost, and appcoins, which can be used as currencies have emerged, though bitcoin still dominates, with 93% market share [4].
The academic literature is sparse, with few studies evaluating the addition of bitcoin to portfolios. A notable exception is the [5] addition of bitcoin to a portfolio of stocks, bonds, foreign currencies, commodities, and real estate, using weekly 2010-2013 data. The addition of bitcoin significantly improved the mean return, while reducing the risk of the portfolio [5]. [4] supported this finding with their observation that bitcoin depreciated against the US dollar, while other currencies appreciated against it. They coined the term, reinforcement effect, whereby certain initiators of bitcoin trading are joined by others, resulting in new users flooding the market. There may be a substitution effect, with [4] observing some replacement of bitcoin by alternate currencies, after bitcoin's sharp decrease in prices, in October, 2013. This paper attempts to fill three gaps in the academic literature, 1) There is a need for valuation models for the leading cryptocurrencies, such as there are for stocks, bonds, and derivatives ( [6] [7] [8]). 2) [9] observed informed trading in the bitcoin market in advance of large positive and negative events, suggesting that cryptocurrencies may be an investment choice for informed traders. We posit that such traders may differ in the extent of risk aversion. The microstructure literature suggests that, on the one hand, risk-taking informed traders initiate purchases of cryptocurrencies ( [10]), while on the other, risk-averse informed traders sell information ( [11]). We derive cryptocurrency valuation models for both types of traders, 3) There is a need for models that update traditional portfolio theory models of risk-free assets and risky assets ( [12]). What is the effect on risk and return of a portfolio of risk-free assets and risky assets, if cryptocurrencies are added? Alternatively, could risk reduction be achieved by short selling risky assets? What would be the impact of adding foreign currencies to such a portfolio? We address all of these questions R. Abraham

Investor Demand for Cryptocurrencies
The reinforcement effect consists of risk-taking informed traders, recognizing the future uses of cryptocurrencies, in reducing both a business's cost of goods sold, and its administrative expenses. Consider an importer. Currently, the firm will need to pay for foreign goods through bank loans, having to provide letters of credit to the foreign bank, along with verification documents. After the first transaction, all of these items may be stored on the blockchain, with cryptocurrencies being used to complete the transaction. All management of inventory and accounts receivable will occur on the blockchain, diminishing administrative expenses. Other businesses in the global supply chain, shipping, and retail, will benefit, leading to significant positive spillover economic effects. [4] predicted that this "winner-take-all," feature of bitcoin would lead to the network effect of attracting more and more investors. In the four quarters from May  [18]). [11] theorized that informed traders differed in their attitudes towards risk aversion. Risk-taking informed traders actively trade on the basis of privileged information. For example, they may purchase stocks forecasted to exceed earnings expectations, the day before the earnings announcement, in anticipation of price increases the next day. In contrast, risk-averse informed traders prefer to forego trading fearing losses from momentary price changes. They may sell information by creating a mutual fund. The load on the fund permits the fund manager to charge for information that he or she provides to investors, including the investments selected by the fund manager for the client, and the optimal time to purchase.
The only examination of informed trading in the bitcoin market was undertaken by [9], who observed informed trading on both positive and negative signals. On positive signals, order sizes of informed buyer-initiated bitcoin orders were significantly higher than uninformed seller-initiated bitcoin orders. On negative signals, order sizes of informed seller-initiated bitcoin orders were significantly higher than buyer-initiated bitcoin orders.

The Role of Cryptocurrencies in a Portfolio
Single-Asset Investments. Successive empirical examinations of the determinants of bitcoin prices have established that prices are being fueled by expectations of future gain. For instance, [19] found that initial interest in cryptocur- Multiple-Asset Investments. [12] introduced the concept of portfolio diversification, by specifying that an optimal portfolio, of risky corporate securities, and riskless government securities, will maximize returns with minimum risk.
His minimum variance portfolio frontier, with return-risk combinations of portfolios that lay on the frontier, yielded optimal returns with the least risk. The addition of cryptocurrencies to such portfolios, elevates returns, while increasing volatility. To achieve risk-return optimality, [5] created a 13-asset portfolio, including global stocks, bonds, currencies, commodities, hedge funds, and real es-m 1 = coefficient of absolute risk aversion, m 2 = change in coefficient of absolute risk aversion, θ function = profit function of investing in a risky gamble.
The risk-averse trader's acceptance of a risky bitcoin investment in a mutual fund is predicated upon his or her desire to avoid risk (risk aversion), decreasing more than the increase in profit potential, or −∆[m 2 /3m 1 ] > ∆ [the θ profit function].
The price function for bitcoin follows a martingale process, in that the next price is equal to the present observed value, which is unpredictable from past observations.  With a completely unknown trajectory, bitcoin's prices may follow the Geometric Levy process, which is used for highly uncertain jump processes (See [25] for a review). The Esscher transformation is frequently used to represent collective risk ( [26])), in which asset prices react to the investment by a number of investors whose actions are influenced by each other. Therefore, we present bitcoin's price distribution as an Esscher-Transformed Martingale Measure Geometric Levy process. If S k is the price of bitcoin at time period, k, the compound return on a bitcoin investment may be represented by ∆Z k ( [27]).
( ) [27] presents the return process as, We remove the W t term, which represents a Weiner process based on a predictable continuous normal distribution. We retain the N p , the discontinuous Poisson process with a function x < 1, that describes small jumps.
( ) ( ) The necessary condition for the maximum price is the second derivative of Equation (6), The sufficient condition for the maximum price is the second derivative of Equation (6), Equation (8) must be equated to the second derivative of Equation (1) to obtain the maximum price for the risk-taking informed trader. The second derivative of Equation (1) is presented in Equation (9), Setting z = x, and equating Equation (8) with Equation (9), yields an expression that may be used to obtain the optimal price, x, for the risk-taking informed trader, For the risk-averse trader, we add the [24] function, to the left side of Equation (10). This function describes the probability of a risk-averse trader earning a profit with a high-risk gamble. It compensates for the acceptance of risk, that is excessively higher than the risk-averse trader's normal threshold for risk, Single-Asset Litecoin Portfolios. Litecoin processes a block every 2.5 minutes, as opposed to bitcoin's slower 10-minute processing speed. However, it has never had bitcoin's popularity, presumably due to greater memory demands, and more expensive mining ( [28] We remove the Weiner process term, differentiating to yield the necessary condition for the maximum price, Differentiating Equation (13) to obtain the sufficient condition for price maximization, Equating the second derivative of the utility function in Equation (9) and Equation (14), the risk-taking trader's optimal price is given by x below, Given the lower volume of usage for litecoin, we assume that there is greater uncertainty associated with adding it to a portfolio. This increased risk may not affect the risk-taking trader, so that the utility function specified in Equation (1), will remain unchanged. However, the risk-averse trader will demand higher compensation. The compensation will differ from that obtained in Equation (2).
With greater risk, the risk-averse investor may demand a definite dollar amount of compensation, termed Y. Since litecoin is less well-known than bitcoin, investor expectations may be satisfied if the optimal least favorable Minimal Distance Martingale Measure for a Geometric Levy process is obtained. Esscher transformations cannot be used, as the low volume of investment in litecoin suggests that fewer risk-averse traders are engaging in collective action. For the risk-averse trader, the second derivative of the utility function in Equation (11) is equated to the price function in Equation (14), with the compensation, Y, included on the left side, Single-Asset Ether Portfolio. Ether's strength lies in its low transaction fee, which was $ 0.33, compared to $ 23.00, for bitcoin ( [29]). Both the risk-taking trader, and the risk-averse trader, would obtain significant cost savings, by substituting ether for bitcoin. In Figure 2, the risk-taking trader's utility function is based upon the relative risk aversion of bitcoin in relation to ether. The investor will desire ether more than bitcoin, if the slope of relative risk aversion to bitcoin, represented by OS, is steeper than the slope of relative risk aversion relative to ether, PQ. We define m 3 , as the coefficient of relative risk aversion for bitcoin due to preference for ether, and m 4 , the change in relative risk aversion for bitcoin due to preference for ether. If ether investments become more attractive than bitcoin,, m 3 and m 4 will be positive.. Conversely, if ether investments become less attractive with respect to bitcoin, m 3 and m 4 will be negative. Including m 3 and m 4 to Equation (1) with z = x, yields Equation (17) and Equation (18), the components of the utility function of the risk-taking trader, As m 3 > m 4 ,, the displacement of bitcoin by ether slows over time, we may recast m 3 = k (constant)+m 4 , and adding Equation (17) and Equation (18), as they are the utility functions of a single investor, Given that ether is a substitute, price expectations may be lower for ether than bitcoin, so the price function may yield the optimal least favorable solution. This price process [27], is given in Equation (20), Where N(ds, dx) is a Poisson process, while v is a constant. We designate x > 1, as the price of ether, subject to large jump discontinuities. The unknown nature of future ether paths suggests excessive uncertainty about future prices, or large jump discontinuities. We equate Equation (19) and Equation (20), The first differential of Equation (21) The second differential of Equation (21)  Risk-averse traders will invest in ether, if they feel that their investment in bitcoin is becoming unprofitable. Therefore, ether will act as a substitute for bitcoin, if the trader's perception of profit, as depicted in Equation (2) for bitcoin declines with respect to the utility of profit for ether, i.e. ether promises higher future gain. In Equation (24), the ratio of profit expectations for ether to bitcoin = θ 1 (profit expectations for ether/θ 2 (profit expectations for bitcoin) > 0, Differentiating Equation (24) twice, and equating to the right side of Equation The gain to the risk-averse trader is,    [12] combined risk-free securities with risky stocks, to create diversified portfolios that minimize variance, and increase return, on the Capital Market Line.

Multiple-Asset Capital Market Line Portfolios
The objective of the Capital Market Line is to minimize risk, with the optimal portfolio being the market portfolio of all risky assets. We assume that the diversification of the Capital Market Line reduces the risk of portfolios lying on the Line, to the extent that they meet the risk preferences of risk-averse investors.
We exclude risk-takers, who are unlikely to select these portfolios, given their practice of disregarding risk.
If we add bitcoin to a portfolio lying on the Capital Market Line, both the volatility and returns will rise substantially. While the increase in return is attractive, the excessive volatility violates the minimum variance objective of the portfolio. Possible solutions include: 1) Increasing the proportion of the risk-free asset, whose reduced volatility will partly offset the heightened volatility of bitcoin.
2) Short selling the risky asset to reduce volatility, will result in less reduction of returns, than increasing the risk-free asset. The drawback is that high borrowing costs may limit the volume of short selling. 3) Adding foreign currencies, such as the Australian dollar, the British pound, and the euro, reduces covariance risk, as bitcoin has low correlations with these currencies.
The Increased Risk-Free Asset Portfolio. [12] presented the minimum variance portfolio of the risk-free asset and the risky asset as follows, with the objective of minimizing risk, Σ , while maintaining returns, m, above a threshold return, μ b . We increase the weight of the riskless asset, and include the cryptocurrency.
The objective function becomes, Minimize 1 2 w w w w w ′ + + + * Σ where, w 1 , w 2 , w 3 , and w 4 are the weights of riskless asset, risky asset, bitcoin, and the additional riskless asset. ′ Σ is the first derivative of the covariance risk, so that the change in cova- where σ 2 x 1 , the variance of the risk-free asset = 0, w 1 = weight of the risk-free asset, w 2 = weight of the risky asset, w 3 = weight of bitcoin, μ b = the return of a portfolio of the risk-free asset, the risky asset, and bitcoin, σ 2 x 2 = the variance of the risky asset, σ 2 x 3 = the variance of bitcoin, 1 2 x x ρ = the correlation coefficient of the risky asset and bitcoin, SK 3 = skewness of bitcoin, Kx 3 = kurtosis of bitcoin, −0.4σ 2 x 2 = 40% of the risky asset that is short sold, We assume that skewness and kurtosis are only on bitcoin, and that there is a correlation coefficient of −0.9, between bitcoin and the risky asset. Applying a Lagrangian function with coefficient, P, to Equation (35), The sufficient condition for minimization is the derivative of Equation (38), Adding Foreign Currencies. Figure 3 depicts three Capital Market Line (CML) portfolios. The optimal market portfolio of risk-free assets, and risky assets intersects the CML at P. The original lending portfolio, S, becomes a borrowing portfolio with the addition of cryptocurrency. 1) With the increase in risk-free assets, the lowest return portfolio, U, results, 2) With the short selling of risky assets, the next highest return portfolio, F, results, and 3) With the addition of foreign currencies, the highest return portfolio, G, is obtained.
In Figure 3, the optimal market portfolio of risk-free assets, and risky assets intersects the CML at P. The original lending portfolio, S, becomes a borrowing portfolio at U. Among the borrowing portfolios, 1) The increase in risk-free assets yields the lowest-return portfolio, U, 2) The short selling of risky assets creates the next highest return portfolio, F, and 3) The addition of foreign currencies yields the highest return portfolio, G.
The reduction in risk may only be achieved by reducing returns. This reduction of returns is exacerbated with the strategy of short selling the risky asset. It may be concluded that an optimal cryptocurrency portfolio should be broadly diversified to reduce portfolio risk, without the reduction in returns of the riskless asset.
The Proposed Multiple-Asset Cryptocurrency Portfolio. We assume that risk-averse traders will pursue one of the aforementioned CML portfolio strategies. For the risk-taker, we take 1)risky stock underlying equity options, 2) bitcoins, which has volatile prices, 3) US dollars, with relatively stable prices, 4) combination of assets has high returns above the CML. This is in accordance with [5]'s 13-asset portfolio, though that portfolio was less successful in controlling risk. This paper's proposed portfolio contains assets with correlation coefficients of zero with other assets, which reduce covariance risk of the portfolio.
Equating the Legendre utility function for risk-takers, with the price function of the aforementioned 9 assets, where, x 1 = risky stock underlying equity options, following a lognormal distribution, with probability density function ( [31]), x 2 = bitcoin value, whose distribution is represented by Equation (6)  The necessary condition for the optimization of portfolio risk and return is obtained by differentiating Equation (46). If w = the coefficient of relative risk aversion, the relative risk aversion across 9 assets cancels out, resulting in dw/dx = 0, or the left side reducing to μ 2 w/2x. The first derivative of Equation (46) The sufficient condition for optimization is the second derivative of Equation

Conclusions
This paper is the first attempt to create a theoretical portfolio containing cryptocurrencies, either as a single asset, or as part of a multiple-asset portfolio, within the framework of modern portfolio theory. As a single asset, the cryptocurrency may be retained, by either risk-taking informed traders, or risk-averse informed traders, who base its value on the beneficial impact of recording transactions on the blockchain, without the services of intermediaries, such as banks.
Two of the CML portfolios of increasing the proportion of riskless assets, or short selling risky assets, failed to achieve minimum risk. Only the CML portfolio with foreign currencies, mitigated the excessive risk of the cryptocurrency, risk-avoiders will reduce value-at-risk to levels 1.5 times that of G10 currencies.
Does this suggest that traditional diversification should not be employed at all?
We do not support this position, maintaining that diversification, as in our proposed model may be deployed to reduce overall covariance risk. However, diversification will be ineffective in reducing tail risk, as it does not limit the outliers that cause tail risk. Therefore, diversification must be supplemented with the elimination of outliers. Another method is the addition of volatile currencies, that follow a Laplace distribution. Laplace transforms, may be employed to reduce kurtosis. We suggest that the following currencies be added to a cryptocurrency portfolio. All of them fluctuate considerably, as their economies are commodity-driven, so that the currencies rise and drop sharply, with changes in commodity prices. Political instability, and the removal of capital controls, resulting in capital flight, also contributes to currency destabilization. The currencies include, the Azerbaijani manat, the Kazakhastan tenge, tied to oil prices, the Zambian kwacha, linked to the price of copper, the Belarusian ruble, the Argen- The next stage of the theoretical analysis, is to ground cryptocurrency investments within the framework of the Capital Asset Pricing Model, (henceforth, CAPM, [8], and the Fama-French models of security valuation. CAPM views the return on a security in terms of market movements. Yet, cryptocurrencies correlate poorly with the market, so that the return on a cryptocurrency cannot be explained by CAPM. Consequently, cryptocurrencies are a CAPM anomaly. The [31] model adjusted CAPM for the aberrations in security returns, due to size (small firms were observed to have higher returns, than large firms), book-to-market (high book-to-market firms had higher returns, than low book-to-market firms), and momentum (stocks, with large price increases in the previous period, outperformed their counterparts, without such increases). As cryptocurrencies are not larger, have high book-to-market, or higher momentum, their prices cannot be explained by the Fama-French model. If cryptocurrencies are anomalous in traditional security return models, they may be speculative investments, such as derivatives. While derivatives are supported by the value of stocks for equity options, real estate for real options, or currency, for currency derivatives, cryptocurrencies are supported by belief in the power of the blockchain to reduce transactions costs for businesses. In essence, cryptocurrencies are supported by faith.
For less rational investors, who seek speedy profits, cryptocurrency valuations have taken the form of a bubble, with speculators driving up prices. The extent of rationality in cryptocurrency price-setting, as opposed to bubble-like speculation, may form the basis for future research. Another area of research, is the relationship of cryptocurrencies to gold. Both are hedges against inflation. What is the relationship between them? Should a portfolio contain one, or both? Theoretical and empirical relationships must be explored.