^{1}

^{*}

^{2}

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 mutliple-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, consisting 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.

Cryptocurrencies, led by bitcoin, are virtual currencies, used in transactions recorded 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 [

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 [

The academic literature is sparse, with few studies evaluating the addition of bitcoin to portfolios. A notable exception is the [

The remainder of this paper is organized as follows. Section 2 is a Review of Literature, Section 3 includes Single-Asset Cryptocurrency Portfolios, while Section 4, consists of Multiple-Asset Capital Market Line Portfolios. Section 5 describes the Conclusions.

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. [

By definition, informed traders capitalize on profit-making opportunities present in information events ( [

The only examination of informed trading in the bitcoin market was undertaken by [

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, [

Multiple-Asset Investments. [

Single-Asset Bitcoin Portfolios. Perceiving the risk-taking benefit of the blockchain, risk-taking informed traders invest in bitcoin, their lack of concern for risk being exemplified by the upward-sloping utility function, OS, in

( 1 − z 2 ) ⋅ d 2 w / d z 2 − 2 z d w / d z + [ v ( v + 1 ) − μ 2 / ( 1 − z 2 ) ] w = 0 (1)

where, μ and v are complex constants. The branch points of z are at F and P, where the direction of risk aversion changes.

We add a utility function that is equivalent to this compensation to Equation (1), as derived by [

( 1 − z 2 ) ⋅ d 2 w / d z 2 − 2 z d w / d z + [ v ( v + 1 ) − μ 2 / ( 1 − z 2 ) ] w + [ m 2 / 3 m 1 − 1 ] / [ θ 2 − 2 θ + ( θ 3 − 3 θ 2 + 3 θ ) m 2 / 3 m 1 ] (2)

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.

at A. Risk-averse traders purchase at bargain prices, at B, their compensation for additional risk being represented by XP.

With a completely unknown trajectory, bitcoin’s prices may follow the Geometric Levy process, which is used for highly uncertain jump processes (See [_{k} is the price of bitcoin at time period, k, the compound return on a bitcoin investment may be represented by ∆Z_{k} ( [

Δ Z k = log ( S k / S k − 1 ) (3)

[

Z t = σ W t + b + 0.5 σ 2 + t ∫ 0 t e x − 1 − x 1 ( | x | ≤ 1 ) x v d x + ∫ 0 t ∫ x < 1 ∞ ( e x − 1 ) N p d s d x (4)

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.

Z t = b + 0.5 σ 2 + t ∫ 0 t e x − 1 − x 1 ( | x | ≤ 1 ) x v d x + ∫ 0 t ∫ x < 1 ∞ ( e x − 1 ) N p d s d x (5)

Excessive volatility, measured by the variance, skewness, and kurtosis, of deviations of prices from mean bitcoin values, is overcome by adding a Laplace transform to Equation (5). The σ^{2} term measuring volatility is absorbed by the Laplace transform, as follows, (see the L term below),

Z t = b + 0.5 σ 2 + t ∫ 0 t ( e x − 1 − x 1 ( | x | ≤ 1 ) x ) v d x + ∫ 0 t ∫ x < 1 ∞ ( e x − 1 ) N p d s d x + L [ ( x − μ ) 2 / σ + ( x − μ ) 3 / σ + ( x − μ ) 4 ] / σ (6)

The necessary condition for the maximum price is the second derivative of Equation (6),

t ( e x − 1 − x 1 ( | x | ≤ 1 ) ( x ) ) v + ∫ 0 t ∫ x < 1 ∞ ( e x − 1 ) N p + L ′ [ ( x − μ ) 2 / σ + ( x − μ ) 3 / σ + ( x − μ ) 4 / σ ] = 0 (7)

The sufficient condition for the maximum price is the second derivative of Equation (6),

t ( e x − x 1 ( | x | ≤ 1 ) ( x ) ) v + ( e x ) N ′ p + L ″ [ ( x − μ ) 2 / σ + ( x − μ ) 3 / σ + ( x − μ ) 4 / σ ] = 0 (8)

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),

2 d 4 w / d z 4 − 2 − μ 2 ( 1 − 2 z ) d w / d z (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,

2 d 4 w / d x 4 − 2 − μ 2 ( 1 − 2 x ) d w / d x = t ( e x − x 1 ( | x | ≤ 1 ) ( x ) ) v + ( e x ) N ′ p + L ″ [ ( x − μ ) 2 / σ + ( x − μ ) 3 / σ + ( x − μ ) 4 / σ ] (10)

For the risk-averse trader, we add the [

2 d 4 w / d x 4 − 2 − μ 2 ( 1 − 2 x ) d w / d x + [ m 2 / 3 m 1 − 1 ] / [ θ 2 − 2 θ + ( θ 3 − 3 θ 2 + 3 θ ) m 2 / 3 m 1 ] = t ( e x − x 1 ( | x | ≤ 1 ) ( x ) ) v + ( e x ) N ′ p + L ″ [ ( x − μ ) 2 / σ + ( x − μ ) 3 / σ + ( x − μ ) 4 / σ ] (11)

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 ( [_{t} be the martingale measure for Geometric Levy processes, f(s, x) and g(s, x) ( [

L t ( f , g ) = exp { ∫ 0 t f s d W s − 0.5 f s d W s − 0.5 ∫ 0 t f 2 s d s + ∫ 0 t g ( s , x ) N d s d x − ∫ 0 t ∫ x < 1 ∞ [ e g ( s , x ) − 1 − g ( s , x ) ] v d x d s } (12)

We remove the Weiner process term, differentiating to yield the necessary condition for the maximum price,

L ′ t ( f , g ) = 0.5 f s d s + g ( s , x ) N d s d x − ∫ x < 1 ∞ [ e g ( s , x ) − 1 − g ( s , x ) ] v d x d s (13)

Differentiating Equation (13) to obtain the sufficient condition for price maximization,

L ″ t ( f , g ) = 0.5 f ′ s d s + g ′ ( s , x ) N d s d x − ∫ x < 1 ∞ [ e g ( s , x ) − 1 − g ( s , x ) ] v d x d s (14)

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,

2 d 4 w / d x 4 − 2 − μ 2 ( 1 − 2 x ) d w / d x = 0.5 f ′ s d s + g ′ ( s , x ) N d s d x − ∫ x < 1 ∞ [ e g ( s , x ) − 1 − g ( s , x ) ] v d x d s (15)

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,

2 d 4 w / d x 4 − 2 − μ 2 ( 1 − 2 x ) d w / d x + [ m 2 / 3 m 1 − 1 ] / [ θ 2 − 2 θ + ( θ 3 − 3 θ 2 + 3 θ ) m 2 / 3 m 1 ] + Y = 0.5 f ′ s d s + g ′ ( s , x ) N d s d x − [ e g ( s , x ) − 1 − g ( s , x ) ] v d x d s (16)

Single-Asset Ether Portfolio. Ether’s strength lies in its low transaction fee, which was $ 0.33, compared to $ 23.00, for bitcoin ( [_{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,

P ( x ) = m 3 ∗ [ ( 1 − x 2 ) d 2 w / d x 2 − 2 x d w / d x + [ v ( v + 1 ) − μ 2 / ( 1 − x 2 ) ] w ] (17)

P ( x ) = m 4 ∗ [ ( 1 − x 2 ) d 2 w / d x 2 − 2 x d w / d x + [ v ( v + 1 ) − μ 2 / ( 1 − x 2 ) ] w ] (18)

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,

( k + m 4 ) ∗ [ ( 1 − x 2 ) d 2 w / d x 2 − 2 x d w / d x + [ v ( v + 1 ) − μ 2 / ( 1 − x 2 ) ] w ] + m 4 ∗ [ ( 1 − x 2 ) d 2 w / d x 2 − 2 x d w / d x + [ v ( v + 1 ) − μ 2 / ( 1 − x 2 ) ] w ] (19)

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 [

P ( x ) = b t + ∫ 0 t x N d x d s + ∫ x > 1 ∞ x v ( d x ) ( d s ) (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),

( k + m 4 ) ∗ [ ( 1 − x 2 ) d 2 w / d x 2 − 2 x d w / d x + [ v ( v + 1 ) − μ 2 / ( 1 − x 2 ) ] w ] + m 4 ∗ [ ( 1 − x 2 ) d 2 w / d x 2 − 2 x d w / d x + [ v ( v + 1 ) − μ 2 / ( 1 − x 2 ) ] w ] = b t + ∫ 0 t ∫ x > 1 t x N ( d x , d s ) + ∫ 0 t ∫ x > 1 t x v ( d x ) ( d s ) (21)

The first differential of Equation (21) yields the necessary condition for maximization of risk-taking trader gain, (S − P), (sales price − purchase price of ether),

( S − P ) ∗ m ′ 4 ∗ [ ( − 2 x ) d 3 w / d x 3 − 2 x d 2 w / d x 2 + [ v ( v + 1 ) + μ 2 / ( 2 x ) ] w ] + m ′ 4 ∗ [ ( − 2 x ) d 3 w / d x 3 − 2 d 2 w / d x 2 + [ v ( v + 1 ) + μ 2 / ( 2 x ) ] w ] = x N + x v (22)

The second differential of Equation (21) yields the sufficient condition for the maximization of this trader’s gain,

( S − P ) ∗ m ″ 4 ∗ [ ( − 2 ) d 4 w / d x 4 − 2 d 3 w / d x 3 + [ v ( v + 1 ) + μ 2 / 2 ] w ] + m ″ 4 ∗ [ − 2 d 4 w / d x 4 − 2 d 3 w / d x 3 + [ v ( v + 1 ) + μ 2 / 2 ] w ] = N + v (23)

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,

P ( z ) = ( 1 − x 2 ) d 2 w / d x 2 − 2 x d w / d x + [ v ( v + 1 ) − μ 2 / ( 1 − x 2 ) ] w + [ m 4 / 3 m 3 − 1 ] / ( θ 1 / θ 2 ) ( θ 1 / θ 2 ) − 2 ( θ 1 / θ 2 ) + ( θ 1 / θ 2 ) / ( θ 1 / θ 2 ) ( θ 1 / θ 2 ) − 3 ( θ 1 / θ 2 ) / [ ( θ 1 / θ 2 ) + 3 ( θ 1 ⋅ θ 2 − 1 m 4 / 3 m 3 ) ] = N + v (24)

Differentiating Equation (24) twice, and equating to the right side of Equation (23),

( − 2 ) d 4 w / d x 4 − 2 d 3 w / d x 3 + [ v ( v + 1 ) − μ 2 / ( − 2 ) ] w + [ m ″ 4 / 3 m ″ 3 ] / ( θ 1 / θ 2 ) 2 − 2 ( θ 1 / θ 2 ) + ( θ 1 / θ 2 ) / ( θ 1 / θ 2 ) ( θ 1 / θ 2 ) − 3 ( θ 1 / θ 2 ) / [ ( θ 1 / θ 2 ) + 3 ( θ 1 ⋅ θ 2 − 1 m ″ 4 / 3 m ″ 3 ) ] = N + v (25)

The gain to the risk-averse trader is,

( S − P ) ∗ ( − 2 ) d 4 w / d x 4 − 2 d 3 w / d x 3 + [ v ( v + 1 ) − μ 2 / ( − 2 ) ] w + [ m ″ 4 / 3 m ″ 3 ] / ( θ 1 / θ 2 ) 2 − 2 ( θ 1 / θ 2 ) + ( θ 1 / θ 2 ) / ( θ 1 / θ 2 ) ( θ 1 / θ 2 ) − 3 ( θ 1 / θ 2 ) / [ ( θ 1 / θ 2 ) + 3 ( θ 1 ⋅ θ 2 − 1 m ″ 4 / 3 m ″ 3 ) ] = N + v (26)

[

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. [_{b}.

Minimize .5 w T Σ w (27)

Subject to,

m T w ≥ μ b and e T w = 1

where, e is a vector of 1 values. The necessary and sufficient conditions for optimality are,

0 = Σ w − ϕ m − γ e (28)

μ b ≤ m T w , e T w = 1 (29)

ϕ ( m T − μ b ) = 0 (30)

w = weights of assets,

m = return on assets,

Σ = covariance risk of the portfolio.

We increase the weight of the riskless asset, and include the cryptocurrency. The objective function becomes,

Minimize 1 / 2 ( w 1 + w 2 + w 3 + w 4 ) ∗ Σ ′ w (31)

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 covariance risk, Σ ′ > 0 . Assuming a sufficient quantity of riskless assets are added, the covariance risk could be minimized with Σ − 1 multiplying Σ ′ to yield Σ , the covariance risk before the addition of bitcoin. The new optimal objective function becomes,

Minimize 1 / 2 ( w 1 + w 2 + w 3 + w 4 ) ∗ Σ w (32)

Yet, as the reduced return of the new portfolio, fails to meet the threshold return.

μ_{b}, we cannot prove that m^{T}w is positive definite,

m T ∗ symmetricmatrix = a * a + a * b + b * a + b * b (33)

This result = 0, when a^{*} and b^{*} are 0, which occurs when ϕ ( m T − μ b ) = 0 or ϕ = 1 . Since m^{T} < μ_{b}, m^{T}w, cannot be positive definite, the Capital Market Line portfolio of risk-free asset, risky asset, and bitcoin, cannot achieve minimum risk with maximum return.

Short Selling the Risky Asset. The linear programming model to be minimized,

Minimize ( w 2 σ 2 x 2 + w 3 σ 2 x 3 + 2 w 1 w 2 σ 2 x 2 σ 2 x 3 ρ x 2 x 3 + S K x 3 + K x 3 − 0.4 σ 2 x 2 ) (34)

Subject to,

w 1 r 1 + w 2 r 2 + w 3 r 3 ≥ μ b (35)

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,

ρ x 1 x 2 = 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),

Minimize ( 0.6 w 2 σ 2 x 2 + w 3 σ 2 x 3 − 1.8 w 2 w 3 σ 2 x 2 σ 2 x 3 + S K x 3 + K x 3 ) − P ( w 1 r 1 x 1 + w 2 r 2 x 2 + w 3 r 3 x 3 − μ b ) (36)

0.6 w 2 σ 2 x 2 = 60 % of the risky asset that remains in the portfolio, 1.8 w 2 w 3 σ 2 x 2 σ 2 x 3 = twice the correlation coefficient of bitcoin with the risky asset, [

= 2 L t 2 − ∫ − ∞ ∞ ( e 2 g − 1 ) N ( d t , d x ) + L t 2 ( f t 2 + ∫ − ∞ ∞ ( e g ( t , x ) − 1 ) 2 v d x ) d t (37)

where,

L_{t} = a martingale process,

(g, s) and f(t, x) = predictable processes,

N = Poisson process,

v = a constant.

The necessary condition for minimization is the first derivative of Equation (36), and Equation (37),

1.2 w 2 σ x 2 + 2 w 3 σ x 3 − 7.2 w 2 w 3 σ x 2 σ x 3 + 3 σ 2 x 3 + 4 σ 3 x 3 − P ( w 1 r 1 + w 2 r 2 + w 3 r 3 ) = 4 L t − ( e 2 g ) ( N ′ ( d t , d t ) ) + 2 L t ( ( e g ( t , x ) ) 2 v ( d x ) ) d t (38)

The sufficient condition for minimization is the derivative of Equation (38),

1.2 w 2 ( 1 − μ 2 ) ⋅ n − 1 + 2 w 3 ( 1 − μ 3 ) ⋅ n − 1 − 7.2 w 2 w 3 ( 1 − μ 2 ) ⋅ n − 1 ( 1 − μ 3 ) ⋅ n − 1 + 6 σ x 3 + 12 σ x 3 − P ( w ′ 1 r 1 + w ′ 2 r 2 + w ′ 3 r 3 + w ′ 4 r 4 − μ b ) = 4 L ′ t − ( ( e 2 g N ″ ( d t , d x ) ) ) + 2 L ′ t ( ( e g ( t , x ) ) v d x ) d t (39)

Adding Foreign Currencies.

In

Minimize w 2 σ 2 x 2 + w 3 σ 2 x 3 + w 4 σ 2 x 4 + w 5 σ 2 x 5 + 2 w 2 w 3 σ 2 x 2 σ 2 x 3 ρ x 2 x 3 + 2 w 2 w 4 σ 2 x 2 σ 2 x 4 ρ x 2 x 4 + 2 w 2 w 5 σ 2 x 2 σ 2 x 5 ρ x 2 x 5 + 2 w 3 w 4 σ 2 x 3 σ 2 x 4 ρ x 3 x 4 + 2 w 3 w 5 σ 2 x 3 σ 2 x 5 ρ x 3 x 5 + 2 w 4 w 5 σ 2 x 4 σ 2 x 5 ρ x 4 x 5 + ∑ S k x = 3 to 5 + ∑ K x = 3 to 5 − P ( w 1 r 1 x 1 + w 2 r 2 x 2 + w 3 r 3 x 3 + w 4 r 4 x 4 + w 5 r 5 x 5 − μ b ) (40)

where,

x_{1} = the risk-free asset,

x_{2} = the risky asset,

x_{3} = the first foreign currency,

x_{4} = the second foreign currency,

x_{5} = bitcoin,

We assume positive correlation coefficients of +1, between 1) the risky asset, and the first foreign currency, 2) the risky asset, and the second foreign currency, and 3) the risky asset and bitcoin. Negative correlations of −1 are assumed between 1) the 2 foreign currencies, 2) the first foreign currency and bitcoin, and 3) the second foreign currency and bitcoin. The three positive measures of covariance risk exactly offset the three negative measures of covariance risk, resulting in the minimum variance portfolio, shown in Equation (41),

Minimize w 2 σ 2 x 2 + w 3 σ 2 x 3 + w 4 σ 2 x 4 + w 5 σ 2 x 5 + 2 w 2 w 3 σ 2 x 2 σ 2 x 3 + 2 w 2 w 4 σ 2 x 2 σ 2 x 4 + 2 w 2 w 5 σ 2 x 2 σ 2 x 5 − 2 w 3 w 4 σ 2 x 3 σ 2 x 4 − 2 w 3 w 5 σ 2 x 3 σ 2 x 5 − 2 w 4 w 5 σ 2 x 4 σ 2 x 5 + ∑ S k x = 3 t o 5 x 4 / n ∑ K x = 3 to 5 − P ( w 1 r 1 x 1 + w 2 r 2 x 2 + w 3 r 3 x 3 + w 4 r 4 x 4 + w 5 r 5 x 5 − μ b ) (41)

The necessary condition for optimization is obtained by differentiating Equation (41) and equating to the right side of Equation (38), given the assumption of a minimal distance martingale measure for the price process,

2 w 2 σ x 2 + 2 w 3 σ x 3 + 2 w 4 σ x 4 + 2 w 5 σ x 5 + 8 w 2 w 3 σ x 2 σ x 3 + 8 w 2 w 4 σ x 2 σ x 4 + 8 w 2 w 5 σ x 2 σ x 5 − 8 w 3 w 4 σ x 3 σ x 4 − 8 w 3 w 5 σ x 3 σ x 5 − 8 w 4 w 5 σ x 4 σ x 5 + 3 ( σ 2 x 3 + σ 2 x 4 + σ 2 x 5 ) + 4 ( σ 3 x 3 + σ 3 x 4 + σ 3 x 5 ) − P ( w 1 r 1 + w 2 r 2 + w 3 r 3 + w 4 r 4 + w 5 r 5 ) = 4 L t − ( ( e 2 g ) N d t d x ) + 2 L t ( ( e g ( t , x ) ) 2 v d x ) d t (42)

The sufficient condition is obtained by differentiating Equation (42),

2 w 2 ( 1 − μ 2 ) ⋅ n − 1 + 2 w 3 ( 1 − μ 3 ) ⋅ n − 1 + 2 w 4 ( 1 − μ 4 ) ⋅ n − 1 + 2 w 5 ( 1 − μ 5 ) ⋅ n − 1 + 8 w 2 w 3 ( 1 − μ 2 ) ⋅ n − 1 ( 1 − μ 3 ) + 8 w 2 w 4 n − 1 ( 1 − μ 2 ) ( 1 − μ 4 ) n − 1 + 8 w 2 w 5 ⋅ n − 1 ( 1 − μ 2 ) ( 1 − μ 5 ) n − 1 − 8 w 3 w 4 ( 1 − μ 3 ) ⋅ n − 1 ( 1 − μ 4 ) ⋅ n − 1 − 8 w 3 w 5 ( 1 − μ 3 ) ⋅ n − 1 ( 1 − μ 5 ) n − 1 − 8 w 4 w 5 ( 1 − μ 4 ) n − 1 ( 1 − μ 5 ) n − 1 + 6 ( ∑ σ x = 3 to 5 ) + 12 ∑ σ 3 = 3 to 5 ) = 4 L ′ t − ( e 2 g N ″ ( d t , d x ) ) + 2 L ′ t ( ( e g ( t , x ) ) v d x ) d t (43)

Only the Capital Market Line (henceforth, CML) with foreign currencies, described by Equation (43), reduces the risk of bitcoin sufficiently, to achieve minimum variance, with maximum return. The CML portfolio with additional risk-free assets, has lower return, while the CML portfolio that short sells the risky asset, may have excessive risk. Will similar theoretical formulations result from CML portfolios with other cryptocurrencies? Litecoin and peercoin are so similar to bitcoin, that they may not be expected to differ from bitcoin. Yet, there is no literature that describes its correlations with securities, or foreign currencies. Therefore, ether has greater uncertainty of price expectations, with similar returns. To construct a minimum variance portfolio for ether, we add an additional stable foreign currency, such as the euro, which is constrained to a strict band by the regulations of the European Monetary System, or the Australian dollar, or the New Zealand dollar ( [

2 w 2 σ x 2 + 2 w 3 σ x 3 + 2 w 4 σ x 4 + 2 w 5 σ x 5 + 2 w 6 σ x 6 + 8 w 2 w 3 σ x 2 σ x 3 + 8 w 2 w 4 σ x 2 σ x 4 + 8 w 2 w 5 σ x 2 σ x 5 + 8 w 2 w 6 σ x 2 σ x 6 − 8 w 3 w 4 σ x 3 σ x 4 − 8 w 3 w 5 σ x 3 σ x 5 − 8 w 4 w 5 σ x 4 σ x 5 − 8 w 5 w 6 σ x 5 σ x 6 + 3 ( σ 2 x 3 + σ 2 x 4 + σ 2 x 5 + σ 2 x 6 + ∑ S K x = 3 to 6 ) − P ( w 1 r 1 + w 2 r 2 + w 3 r 3 + w 4 r 4 + w 5 r 5 + w 6 r 6 ) = 4 L t − ( ( e 2 g ) N ( d t , d x ) ) + 2 L t ( e g ( t , x ) ) 2 v d x d t (44)

The sufficient condition is obtained by differentiating Equation (44),

2 w 2 ( 1 − μ 2 ) ⋅ n − 1 + 2 w 3 ( 1 − μ 3 ) ⋅ n − 1 + 2 w 4 ( 1 − μ 4 ) ⋅ n − 1 + 2 w 5 ( 1 − μ 5 ) ⋅ n − 1 + 2 w 6 ( 1 − μ 6 ) ⋅ n − 1 + 8 w 2 w 3 ( 1 − μ 2 ) ⋅ n − 1 ( 1 − μ 3 ) ⋅ n − 1 + 8 w 2 w 4 ( 1 − μ 2 ) ⋅ n − 1 ( 1 − μ 4 ) ⋅ n − 1 + 8 w 2 w 5 ( 1 − μ 2 ) ⋅ n − 1 ( 1 − μ 5 ) ⋅ n − 1 + 8 w 2 w 6 ( 1 − μ 2 ) ⋅ n − 1 ( 1 − μ 6 ) n − 1 − 8 w 3 w 4 ( 1 − μ 3 ) ⋅ n − 1 ⋅ ( 1 − μ 4 ) ⋅ n − 1

+ 8 w 3 w 5 ( 1 − μ 3 ) ⋅ n − 1 ( 1 − μ 5 ) ⋅ n − 1 − 8 w 4 w 5 ( 1 − μ 4 ) ⋅ n − 1 ( 1 − μ 5 ) ⋅ n − 1 − 8 w 5 w 6 ( 1 − μ 5 ) ⋅ n − 1 ( 1 − μ 6 ) ⋅ n − 1 + 6 ( σ x 3 + σ x 4 + σ x 5 + σ x 6 + ( ∑ x = 3 to 6 ) ( 2 σ 2 x + 1.5 σ 3 x ) − P ( w 1 r ″ 1 + w 2 r ″ 2 + w 3 r ″ 3 + w 4 r ″ 4 + w 5 r ″ 5 + w 6 r ″ 6 ) = 4 L ′ t − ( e 2 g N ′ d t d x ) + 2 L ′ t ( e g ( t , x ) ) 2 v d x d t (45)

The drawback of the CML portfolio, is its employment of the riskless asset. 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) soybean futures, 5) Treasury bond futures, that rise with declines in bitcoin, or currency options on volatile Mexican pesos 6) risky technology and biotechnology stocks, 7) cyclical oil futures, 8) relatively stable currency call options on US dollars, and 9) and highly volatile currency call options on Mexican peso. This combination of assets has high returns above the CML. This is in accordance with [

( 1 − z 2 ) d 2 x / d z 2 z d w / d z + [ v ( v + 1 ) − μ 1 2 / ( 1 − z 2 ) ] w = ( 1 / x 1 σ 1 2 Π ) e ( ln x 1 − μ 1 ) 2 / 2 σ 1 2 + Z t b + t ∫ 0 t ( e x 2 − 1 − x 2 ( | x 2 | ≤ 1 ) ( x 2 ) ) v d x + ∫ 0 t ∫ x < 1 ∞ ( e x 2 − 1 ) N d s d x + L [ ( x 2 − μ 2 ) 2 / σ 2 + ( x 2 − μ 2 ) 3 / σ 2 + ( x 2 − μ 2 ) 4 / σ 2 ] + ∂ t p ( x 3 , t ) + Π θ x ∂ x 3 2 σ 3 ( x 3 , t ) 2 / 2 ⋅ p ( x 3 , t ) + Y 4 − a 4 b 4 + Y 5 − a 5 b 5 + ( 1 / x 6 σ 6 2 Π ) e − ( ln x 6 − μ 6 ) 2 ⋅ 5 σ 6 2 + 1 / 2 Π σ 7 2 exp − ( x 7 − μ 7 ) / 2 σ 7 2 + ∫ e i θ x 8 t I | x < 1 | Π d x + ∫ e i θ x 9 t I | x > 1 | Π d x (46)

where,

x_{1} = risky stock underlying equity options, following a lognormal distribution, with probability density function ( [

x_{2} = bitcoin value, whose distribution is represented by Equation (6) of this paper,

x_{3} = US dollar value, the Fokker-Planck equation with US dollar value,

with currency values based on changes in macroeconomic variables, including inflation, short-term interest rates, long-term interest rates, government debt, export prices, import prices, and political stability.

x_{4} = soybean futures, Y 4 = a 4 + b 4 x 4

x_{5} = Treasury bond futures, Y 5 = a 5 + b 5 x 5

x_{6} = technology/biotechnology stocks, following a lognormal distribution,

x_{7} = oil futures values, described by a continuous normal distribution to capture the cyclical nature of oil prices,

x_{8} = currency options on US dollars, Levy-Khintchine distribution of a currency call option with small jumps,

x_{9} = currency options on Mexican pesos, Levy-Khintchine distribution for a currency call option with large jumps,

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) is given below,

μ 2 w / 2 x = ( 1 / 2 Π ) e ∧ ( ln x 1 − μ 1 ) 2 σ 2 / 2 + Z t b + t ( e x 2 − 1 − x 2 ( | x 2 | ≤ 1 ) ( x 2 ) ) v + ∫ x < 1 ∞ ( e x 2 − 1 ) N p d s d x + L ′ [ ( x 2 − μ 2 ) 2 / σ 2 + ( x 2 − μ 2 ) 3 / σ 2 + ( x 2 − μ 2 ) 4 / σ 2 ] + ∂ ′ t p ( x 3 , t ) + d y 4 / d x 4 − a 4 d 2 y 4 / d x 4 2 + d y 5 / d x 5

+ a 5 d 2 y 5 / d x 5 2 + ( 1 / 2 Π ) e ∧ − ( ln x 6 − μ 6 ) 2 σ 6 2 / 2 − [ [ ( x 7 − μ 7 ) / 2 σ 7 2 ] exp − ( x 7 − μ 7 ) − 1 ] + ( e i θ x 8 t − 1 − i θ x 8 t I | x < 1 | ) Π + ( e i θ 8 t − 1 − i θ x 9 t I | x < 1 | ) Π (47)

The sufficient condition for optimization is the second derivative of Equation (46), given below. We add the gradient vector to reduce volatility, and the Laplace transform to reduce skewness and kurtosis.

μ 2 w / 2 = ( 1 / 2 Π ) e ∧ ( ln x 1 − μ 1 ) 2 σ 2 / 2 + ( e x 2 ( | x 2 | ≤ 1 ) x 2 ) v + ( e x 2 − 1 ) N p + L ″ [ ( x 2 − μ 2 ) 2 / σ 2 + ( x 2 − μ 2 ) 3 / σ 2 − ( x 2 − μ 2 ) 4 / σ 2 ] + ∂ ″ t p ( x 3 , t ) + 2 Π θ x ∂ 4 x 3 / d x 4 ( x 3 , t ) p ( x 3 , t ) + d 2 y 4 / d y 4 2 − a 4 d 3 y 4 / d x 4 3 + d 2 y 5 / d x 5 2

+ ( 1 / 2 Π ) e ∧ − ( ln x 6 − μ 6 ) 2 σ 6 2 / 2 − 2 [ 1 − μ 7 ] / 4 σ 7 ( Π σ 7 ) ( x 7 − μ 7 − 1 ) ( x 7 − μ 7 ) + ( e i θ x 8 t − i θ x 8 t I | x < 1 | ) Π + ( e i θ 8 t − i θ x 9 t I | x < 1 | ) Π + ∇ g 2 1 + L ( s , x ) ∑ S K x = 2 to 9 + L ∑ K x = 2 to 9 (48)

where,

Sk = skewness of bitcoin, foreign currency, technology stock, and currency options,

Kx = kurtosis of bitcoin, foreign currency, technology stock, and currency options,

∇ g q 2 = gradient vectors to suppress stock volatility,

L(s, x) = Laplace transforms to suppress skewness and kurtosis.

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, suggesting the need for the inclusion of negatively correlated, or uncorrelated, assets. The proposed portfolio consists of 9 assets, which are predominantly uncorrelated with each other, or are uncorrelated with the cryptocurrency, to reduce risk. As cryptocurrencies are deflationary, we recommend the inclusion of a hedge against deflation. One such inflationary asset, is the Treasury bond future. This suggestion contrasts with the [

Both the [

The next stage of the theoretical analysis, is to ground cryptocurrency investments within the framework of the Capital Asset Pricing Model, (henceforth, CAPM, [

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Abraham, R. and Tao, Z. (2019) The Valuation of Cryptocurrencies in Single-Asset and Multiple-Asset Models. Theoretical Economics Letters, 9, 1093-1113. https://doi.org/10.4236/tel.2019.94071