4.2. Portfolio optimization
I start by constructing a portfolio without cryptocurrencies,
which will be referred to as the basis portfolio. Furthermore, I investigate
the options of including cryptocurrencies to the traditional assets' portfolio.
I construct two sets of portfolios. The first portfolio includes traditional
assets and only Bitcoin and the second one includes traditional assets and the
four cryptocurrencies. The benefits of adding cryptocurrencies are assessed in
terms of risk-return profiles, cumulative wealth and downside risk.
In order to calculate these performance metrics, I use the
out-of-sample backtesting method, which evaluates trading strategies using
historical data. The models' parameters are assessed via a rolling window
approach under the following steps: I use the 200 last days observations before
the rebalancing date for the parameters' estimation. Then, the resulting
weights are rebalanced on a monthly basis for the whole out of sample
period.
Thus, the optimized weights are subject to different
parameters depending on the optimization frameworks presented below.
11
4.2.1. Minimum risk approaches
Minimum variance portfolio is the Markowitz least variance
framework. It is set out as the portfolio that maximizes the use of
diversification to achieve the lowest risk. The portfolio weights are optimized
for each point in time using the subsequent formula:
min 6P = En1 Ey 1 wiw1611 s. t.
wl >_ 0, En1 wl = 1 Where weights are estimated by using
the historical variance and covariance matrix.
Nevertheless, a strong shortcoming of the mean variance
analysis is the assumption of normal distribution of returns. In this context,
cryptocurrencies' excess volatility infers a heavy tail distribution as already
stated by Eisl et al. (2015) and Chuen et al. (2017).
To cope with this issue, I follow Rockafeller and Uryasev
(2002) to construct the conditional value at risk strategy (CVaR). The strategy
uses the expected shortfall, which is a more coherent risk measure contrasting
to the variance since it aims to quantify only the downside risk. Log returns
are simulated via a T-student distribution.
Therefore, conditional value at risk portfolio weights are
given by solving the following optimization problem:
min CVARa(wt) s. t. up,t(wt) = rtarget
; wt1p = 1 , wl,t » 0 wtERP
1
CVARa(w) = (1 - a) if(w,r)<VARa(w)
f(w,r)p(r)dr
Where f (w, r) is the probability density function of
portfolio returns with weights w, a is the confidence level,
VARa is the loss to be expected in a.100% of the
times.
Short selling is constrained under the two strategies since
Bitcoin futures were only introduced recently on Chicago Board Options Exchange
(CBOE) and Chicago Mercantile Exchange
12
(CME) in December 2017. In addition, I impose a maximum weight
constraint of 50% for each asset in order to omit extreme weight
allocations.
0% < ???? < 50%
4.2.2. Risk budgeting approaches
Requiring only the estimation of volatility, risk budgeting
approaches are becoming a popular solution for risk adverse investors. Booth
and Fama (1992) argue that these models put diversification at the heart of the
investment strategy and are a good alternative to Markowitz least variance
framework when the assumption of normal returns is not solid. Therefore, I
adopt the subsequent risk budgeting approaches:
The inverse of the volatility is used to determine the weight
of each asset. Highly volatile assets will be given a lower weight in
comparison to low volatility assets. Hence, each asset contributes different
amount of risk to the overall portfolio. The optimization problem takes the
following form:
1
|
???? =
|
????????
|
|
? ( 1
?? ??=1 ????????)
|
Introduced by Choueifaty and Cognard (2008) maximum
diversification seeks to maximize the diversification ratio of the weighted
average assets volatilities to the total portfolio volatility. The
diversification ratio is given as:
Maximize DR = ? ????????
??
??=1 s. t. ? ????
?? ??=1 = 1 and ???? = 0
? ????
??
??=1
13
| 
