# Can Bitcoin Replace Gold in an Investment Portfolio?

^{*}

## Abstract

**:**

## 1. Introduction

## 2. What Is Bitcoin—Currency or Asset?

## 3. Empirical Model

_{TR}. The optimal portfolio weights are found by solving the following optimization problem:

_{t}are the portfolio weights. There are no restrictions on short sales. The solution to Equation (1) gives the expression for the optimal portfolio weights:

_{t}. An AR(1) process for the asset returns, r

_{t}, conditional on the information set I

_{t}

_{−1}is written as:

_{t}is the conditional covariance matrix of r

_{t}and z

_{t}is a n × 1 i.i.d. random vector of errors.

_{t}is a n × n conditional covariance matrix, R

_{t}is the conditional correlation matrix, and D

_{t}is a diagonal matrix with time-varying standard deviations on the diagonal.

_{t}can be written as:

_{t}is a symmetric positive definite matrix.

_{i,t}(z

_{i,t}= ε

_{i,t}/√h

_{i,t}). The parameters θ

_{1}and θ

_{2}are non-negative. These parameters are associated with the exponential smoothing process that is used to construct the dynamic conditional correlations. The DCC model is mean reverting as long as θ

_{1}+ θ

_{2}< 1. The correlation estimator is:

_{i,t}

_{−1}) is equal to one if ε

_{i,t}

_{−1}< 0 and 0 otherwise. A positive value for d means that negative residuals tend to increase the variance more than positive returns. The asymmetric effect, which is sometimes referred to as the “leverage effect”, is designed to capture an often-observed characteristic of financial assets that an unexpected drop in asset prices tends to increase volatility more than an unexpected increase in asset prices of the same magnitude. This can be interpreted as bad news increasing volatility more than good news.

_{t}when less than zero and zero otherwise. The matrices $\overline{Q}$ and ${\overline{Q}}^{-}$ are the unconditional matrices of z

_{t}and ${z}_{t}^{-}$, respectively.

_{t}, onto a set of uncorrelated components, z

_{t}, using a mapping Z.

_{t}, are normalized to have unit variance. Each component of y

_{t}can be described by a GARCH process. For example, consider a standard GARCH(1,1) process with a normal distribution.

_{t}is H

_{0}= I. The conditional covariance matrix of r

_{t}is:

_{t}to the observed returns r

_{t}. There exists an orthogonal matrix U such that:

_{t}. Recent work on GO-GARCH is concentrated on finding different ways to parameterize and estimate the matrix U. Boswijk and van der Weide (2006) provide a more detailed discussion of these efforts.

_{t}is a diagonal matrix. An orthogonal GARCH (OGARCH) model is the result when Z is restricted to be orthogonal (Alexander 2001). The OGARCH model can be estimated using principle components on the normalized data and GARCH models estimated on the principle components. This corresponds to U being an identity matrix. In the original formation of the GO-GARCH model, Van Der Weide (2002) uses a 1-step maximum likelihood approach to jointly estimate the rotation matrix and the dynamics. This method, however, is impractical for many assets because the maximum likelihood estimation procedure may fail to converge. The matrix U can also be estimated using nonlinear least squares (Boswijk and van der Weide 2006) and method of moments (Boswijk and van der Weide 2011), both of which involve two-step and three-step estimation procedures. More recently, it has been proposed that U can be estimated by independent component analysis (ICA) (Broda and Paolella 2009; Zhang and Chan 2009) and is the method employed in this paper3.

_{p}, the superscripts a and b denote the alternative portfolio and the benchmark portfolio, respectively, and γ denotes the degree of risk relative risk aversion.

## 4. Data

## 5. Results

## 6. Robust Analysis: Long Only Portfolios

## 7. Conclusions and Implications

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | Daily data are available at https://blockchain.info/charts. By convention, we use Bitcoin with a capital “B” to denote the Bitcoin network and “bitcoin” with a small “b” to denote the unit of account. |

2 | |

3 | The rotation matrix U needs to be estimated. For all but a few factors, maximum likelihood is not feasible. For a larger number of factors, alternative estimation methods must be used. ICA is a fast statistical technique for estimating hidden factors in relation to observable data. |

4 | Bitcoin price data was from 18 July 2010, but there was not much price variability over the first few months. |

5 | Summary statistics are computed using continuously compounded daily returns. Portfolio weights are estimated using discrete returns because discrete returns are additive across assets. The resulting portfolio returns are then converted to continuous returns for the calculation of portfolio summary statistics. |

6 | GARCH models are estimated using 1200 observations, and 519 one step forecasts are generated using rolling window estimation. The estimation window of 1200 observations is chosen based on a Monte Carlo comparison of RMSE. GARCH models are refitted every 60 observations. The portfolio results are robust to refits between 40 and 120 observations. |

VNQ | GLD | TLT | SPY | EFA | BIT | |
---|---|---|---|---|---|---|

median | 0.083 | 0.024 | 0.076 | 0.061 | 0.052 | 0.247 |

mean | 0.037 | −0.008 | 0.028 | 0.049 | 0.022 | 0.582 |

SE.mean | 0.026 | 0.025 | 0.022 | 0.022 | 0.028 | 0.154 |

CI.mean.0.95 | 0.052 | 0.050 | 0.042 | 0.042 | 0.054 | 0.301 |

var | 1.193 | 1.100 | 0.799 | 0.805 | 1.316 | 40.537 |

std.dev | 1.092 | 1.049 | 0.894 | 0.897 | 1.147 | 6.367 |

coef.var | 29.151 | −134.32 | 32.012 | 18.296 | 53.190 | 10.934 |

skewness | −0.364 | −0.610 | −0.121 | −0.572 | −0.778 | 0.148 |

skew.2SE | −3.080 | −5.165 | −1.023 | −4.843 | −6.591 | 1.251 |

kurtosis | 7.349 | 5.915 | 1.696 | 5.182 | 6.711 | 9.633 |

kurt.2SE | 31.142 | 25.066 | 7.188 | 21.961 | 28.439 | 40.822 |

normtest.W | 0.934 | 0.948 | 0.986 | 0.938 | 0.930 | 0.843 |

normtest.p | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

VNQ | GLD | TLT | SPY | EFA | BIT | |
---|---|---|---|---|---|---|

VNQ | 1 | 0.07 * | −0.19 * | 0.73 * | 0.66 * | 0.07 * |

GLD | 0.07 * | 1 | 0.2 * | −0.03 | 0.06 * | 0.02 |

TLT | −0.19 * | 0.2 * | 1 | −0.5 * | −0.47 * | −0.02 |

SPY | 0.73 * | −0.03 | −0.5 * | 1 | 0.88 * | 0.04 |

EFA | 0.66 * | 0.06 * | −0.47 * | 0.88 * | 1 | 0.03 |

BIT | 0.07 * | 0.02 | −0.02 | 0.04 | 0.03 | 1 |

BIT | Mean | Sd | ||||||||

VNQ | TLT | SPY | EFA | BIT | VNQ | TLT | SPY | EFA | BIT | |

DCC-13 | −0.073 | 0.457 | 0.602 | −0.003 | 0.017 | 0.067 | 0.074 | 0.204 | 0.119 | 0.009 |

DCC-15 | −0.073 | 0.441 | 0.653 | −0.050 | 0.028 | 0.068 | 0.079 | 0.224 | 0.133 | 0.010 |

DCC-17 | −0.072 | 0.425 | 0.704 | −0.096 | 0.039 | 0.069 | 0.084 | 0.248 | 0.152 | 0.011 |

DCC-GMV | −0.071 | 0.464 | 0.570 | 0.022 | 0.015 | 0.066 | 0.071 | 0.192 | 0.114 | 0.012 |

ADCC-13 | −0.061 | 0.439 | 0.629 | −0.022 | 0.015 | 0.084 | 0.081 | 0.238 | 0.132 | 0.010 |

ADCC-15 | −0.061 | 0.423 | 0.680 | −0.068 | 0.026 | 0.086 | 0.085 | 0.255 | 0.145 | 0.011 |

ADCC-17 | −0.060 | 0.407 | 0.731 | −0.115 | 0.037 | 0.088 | 0.090 | 0.277 | 0.162 | 0.012 |

ADCC-GMV | −0.059 | 0.446 | 0.595 | 0.006 | 0.012 | 0.084 | 0.081 | 0.235 | 0.134 | 0.012 |

GO-13 | −0.126 | 0.483 | 0.654 | −0.028 | 0.017 | 0.087 | 0.055 | 0.122 | 0.064 | 0.007 |

GO-15 | −0.127 | 0.468 | 0.702 | −0.072 | 0.028 | 0.086 | 0.063 | 0.141 | 0.087 | 0.008 |

GO-17 | −0.128 | 0.453 | 0.751 | −0.116 | 0.040 | 0.087 | 0.072 | 0.167 | 0.113 | 0.010 |

GO-GMV | −0.124 | 0.495 | 0.606 | 0.011 | 0.012 | 0.087 | 0.048 | 0.120 | 0.048 | 0.013 |

GOLD | Mean | Sd | ||||||||

VNQ | GLD | TLT | SPY | EFA | VNQ | GLD | TLT | SPY | EFA | |

DCC−13 | −0.058 | −0.021 | 0.432 | 0.890 | −0.244 | 0.089 | 0.066 | 0.100 | 0.187 | 0.078 |

DCC−15 | −0.047 | −0.089 | 0.447 | 1.049 | −0.360 | 0.103 | 0.081 | 0.116 | 0.208 | 0.088 |

DCC−17 | −0.036 | −0.157 | 0.461 | 1.208 | −0.476 | 0.119 | 0.097 | 0.132 | 0.231 | 0.105 |

DCC−GMV | −0.074 | 0.123 | 0.393 | 0.580 | −0.021 | 0.065 | 0.044 | 0.063 | 0.192 | 0.105 |

ADCC−13 | −0.045 | −0.024 | 0.422 | 0.875 | −0.228 | 0.107 | 0.082 | 0.120 | 0.214 | 0.091 |

ADCC−15 | −0.033 | −0.093 | 0.435 | 1.033 | −0.343 | 0.123 | 0.099 | 0.139 | 0.245 | 0.107 |

ADCC−17 | −0.020 | −0.161 | 0.449 | 1.190 | −0.458 | 0.141 | 0.116 | 0.158 | 0.277 | 0.129 |

ADCC−GMV | −0.064 | 0.115 | 0.381 | 0.597 | −0.030 | 0.073 | 0.047 | 0.068 | 0.228 | 0.120 |

GO−13 | −0.087 | −0.004 | 0.436 | 0.948 | −0.293 | 0.075 | 0.044 | 0.058 | 0.177 | 0.066 |

GO−15 | −0.076 | −0.076 | 0.456 | 1.104 | −0.408 | 0.082 | 0.055 | 0.063 | 0.191 | 0.072 |

GO−17 | −0.065 | −0.149 | 0.476 | 1.260 | −0.523 | 0.092 | 0.067 | 0.071 | 0.207 | 0.081 |

GO−GMV | −0.103 | 0.168 | 0.383 | 0.591 | −0.038 | 0.078 | 0.052 | 0.070 | 0.118 | 0.049 |

Bitcoin Portfolio | ||||||||||||

DCC-13 | DCC-15 | DCC-17 | DCC-GMV | ADCC-13 | ADCC-15 | ADCC-17 | ADCC-GMV | GO-13 | GO-15 | GO-17 | GO-GMV | |

Mean | 11.737 | 13.684 | 15.623 | 10.899 | 12.277 | 14.244 | 16.204 | 11.483 | 12.954 | 14.934 | 16.906 | 11.881 |

Sd | 6.340 | 6.462 | 6.694 | 6.325 | 6.139 | 6.261 | 6.497 | 6.115 | 6.463 | 6.589 | 6.830 | 6.427 |

Sharp | 1.823 | 2.089 | 2.307 | 1.694 | 1.970 | 2.246 | 2.466 | 1.848 | 1.976 | 2.239 | 2.449 | 1.820 |

Sharpe VaR | 1.194 | 1.384 | 1.542 | 1.104 | 1.298 | 1.497 | 1.659 | 1.212 | 1.303 | 1.492 | 1.646 | 1.192 |

Sharpe ES | 0.937 | 1.084 | 1.205 | 0.868 | 1.018 | 1.171 | 1.295 | 0.951 | 1.021 | 1.167 | 1.285 | 0.936 |

Sortino | 0.170 | 0.198 | 0.222 | 0.156 | 0.189 | 0.218 | 0.243 | 0.175 | 0.190 | 0.218 | 0.241 | 0.173 |

Omega | 0.365 | 0.427 | 0.475 | 0.338 | 0.400 | 0.464 | 0.514 | 0.372 | 0.393 | 0.452 | 0.499 | 0.359 |

Information | 1.849 | 2.153 | 2.411 | 1.705 | 2.010 | 2.328 | 2.591 | 1.872 | 2.024 | 2.329 | 2.582 | 1.847 |

Drawdown | 0.074 | 0.069 | 0.064 | 0.077 | 0.062 | 0.057 | 0.051 | 0.065 | 0.055 | 0.050 | 0.044 | 0.054 |

Gold Portfolio | ||||||||||||

DCC-13 | DCC-15 | DCC-17 | DCC-GMV | ADCC-13 | ADCC-15 | ADCC-17 | ADCC-GMV | GO-13 | GO-15 | GO-17 | GO-GMV | |

Mean | 11.589 | 12.594 | 13.578 | 8.774 | 11.827 | 12.941 | 14.035 | 9.888 | 11.565 | 12.368 | 13.151 | 9.644 |

Sd | 6.843 | 7.610 | 8.554 | 6.098 | 6.613 | 7.377 | 8.324 | 5.907 | 6.813 | 7.561 | 8.478 | 6.112 |

Sharp | 1.667 | 1.631 | 1.566 | 1.409 | 1.761 | 1.729 | 1.664 | 1.643 | 1.671 | 1.612 | 1.530 | 1.548 |

Sharpe VaR | 1.085 | 1.060 | 1.015 | 0.907 | 1.150 | 1.128 | 1.083 | 1.068 | 1.087 | 1.046 | 0.990 | 1.003 |

Sharpe ES | 0.853 | 0.834 | 0.799 | 0.715 | 0.904 | 0.887 | 0.851 | 0.840 | 0.855 | 0.823 | 0.779 | 0.789 |

Sortino | 0.156 | 0.152 | 0.146 | 0.130 | 0.166 | 0.162 | 0.156 | 0.156 | 0.158 | 0.151 | 0.142 | 0.149 |

Omega | 0.346 | 0.338 | 0.318 | 0.274 | 0.362 | 0.356 | 0.337 | 0.323 | 0.339 | 0.324 | 0.303 | 0.297 |

Information | 1.683 | 1.653 | 1.592 | 1.387 | 1.784 | 1.762 | 1.701 | 1.641 | 1.686 | 1.631 | 1.549 | 1.540 |

Drawdown | 0.063 | 0.058 | 0.063 | 0.086 | 0.059 | 0.062 | 0.069 | 0.073 | 0.046 | 0.050 | 0.054 | 0.065 |

DCC-13 | DCC-15 | DCC-17 | DCC-GMV | ADCC-13 | ADCC-15 | ADCC-17 | ADCC-GMV | GO-13 | GO-15 | GO-17 | GO-GMV | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Diff | 0.010 | 0.028 | 0.046 | 0.018 | 0.013 | 0.032 | 0.050 | 0.013 | 0.019 | 0.039 | 0.057 | 0.017 |

p value | 0.611 | 0.257 | 0.146 | 0.230 | 0.476 | 0.192 | 0.115 | 0.389 | 0.351 | 0.142 | 0.070 | 0.345 |

DCC-13 | DCC-15 | DCC-17 | DCC-GMV | ADCC-13 | ADCC-15 | ADCC-17 | ADCC-GMV | GO-13 | GO-15 | GO-17 | GO-GMV | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

γ = 1 | 14.873 | 109.067 | 204.591 | 212.566 | 45.007 | 130.366 | 217.066 | 159.585 | 139.024 | 256.772 | 375.691 | 223.790 |

γ = 5 | 28.161 | 141.374 | 261.263 | 206.981 | 57.096 | 160.762 | 271.158 | 154.603 | 148.323 | 284.251 | 426.049 | 215.929 |

γ = 10 | 44.840 | 181.937 | 332.429 | 199.976 | 72.271 | 198.929 | 339.095 | 148.353 | 159.995 | 318.748 | 489.280 | 206.067 |

DCC-13 | DCC-15 | DCC-17 | DCC-GMV | ADCC-13 | ADCC-15 | ADCC-17 | ADCC-GMV | GO-13 | GO-15 | GO-17 | GO-GMV | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

BIT | 0.125 | 0.136 | 0.148 | 0.129 | 0.129 | 0.140 | 0.153 | 0.133 | 0.078 | 0.093 | 0.109 | 0.074 |

Gold | 0.128 | 0.148 | 0.172 | 0.129 | 0.136 | 0.157 | 0.182 | 0.133 | 0.063 | 0.073 | 0.085 | 0.065 |

TC = $5 | ||||||||||||

BIT | 1.576 | 1.708 | 1.870 | 1.623 | 1.620 | 1.759 | 1.931 | 1.673 | 0.979 | 1.169 | 1.378 | 0.927 |

Gold | 1.618 | 1.870 | 2.163 | 1.624 | 1.716 | 1.977 | 2.288 | 1.678 | 0.788 | 0.916 | 1.069 | 0.821 |

TC = $10 | ||||||||||||

BIT | 3.151 | 3.417 | 3.741 | 3.247 | 3.239 | 3.518 | 3.862 | 3.347 | 1.959 | 2.338 | 2.756 | 1.854 |

Gold | 3.236 | 3.741 | 4.326 | 3.247 | 3.432 | 3.954 | 4.576 | 3.355 | 1.576 | 1.832 | 2.138 | 1.641 |

TC = $20 | ||||||||||||

BIT | 6.303 | 6.833 | 7.481 | 6.493 | 6.479 | 7.035 | 7.725 | 6.693 | 3.917 | 4.676 | 5.511 | 3.707 |

Gold | 6.472 | 7.481 | 8.652 | 6.495 | 6.864 | 7.908 | 9.151 | 6.710 | 3.152 | 3.663 | 4.276 | 3.282 |

BIT | GLD | |||||
---|---|---|---|---|---|---|

DCC-GMV | ADCC-GMV | GO-GMV | DCC-GMV | ADCC-GMV | GO-GMV | |

Mean | 10.557 | 10.475 | 11.088 | 8.316 | 8.778 | 8.785 |

Sd | 6.405 | 6.231 | 6.440 | 6.191 | 6.056 | 6.146 |

Sharp | 1.620 | 1.652 | 1.693 | 1.314 | 1.419 | 1.400 |

SharpeVaR | 1.052 | 1.074 | 1.103 | 0.843 | 0.914 | 0.901 |

SharpeES | 0.828 | 0.845 | 0.867 | 0.665 | 0.721 | 0.710 |

Sortino | 0.146 | 0.152 | 0.156 | 0.119 | 0.131 | 0.131 |

Omega | 0.324 | 0.329 | 0.337 | 0.257 | 0.277 | 0.269 |

Information | 1.623 | 1.656 | 1.706 | 1.285 | 1.398 | 1.378 |

Drawdown | 0.081 | 0.071 | 0.065 | 0.093 | 0.082 | 0.077 |

DCC-GMV | ADCC-GMV | GO-GMV | |
---|---|---|---|

γ = 1 | 224.177 | 169.771 | 230.425 |

γ = 5 | 218.885 | 165.505 | 223.109 |

γ = 10 | 212.250 | 160.155 | 213.935 |

DCC-GMV | ADCC-GMV | GO-GMV | |
---|---|---|---|

BIT | 0.088 | 0.086 | 0.037 |

Gold | 0.081 | 0.084 | 0.033 |

TC = $5 | |||

BIT | 1.104 | 1.088 | 0.465 |

Gold | 1.019 | 1.055 | 0.410 |

TC = $10 | |||

BIT | 2.208 | 2.177 | 0.930 |

Gold | 2.037 | 2.111 | 0.819 |

TC = $20 | |||

BIT | 4.415 | 4.353 | 1.860 |

Gold | 4.075 | 4.221 | 1.638 |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Henriques, I.; Sadorsky, P.
Can Bitcoin Replace Gold in an Investment Portfolio? *J. Risk Financial Manag.* **2018**, *11*, 48.
https://doi.org/10.3390/jrfm11030048

**AMA Style**

Henriques I, Sadorsky P.
Can Bitcoin Replace Gold in an Investment Portfolio? *Journal of Risk and Financial Management*. 2018; 11(3):48.
https://doi.org/10.3390/jrfm11030048

**Chicago/Turabian Style**

Henriques, Irene, and Perry Sadorsky.
2018. "Can Bitcoin Replace Gold in an Investment Portfolio?" *Journal of Risk and Financial Management* 11, no. 3: 48.
https://doi.org/10.3390/jrfm11030048