Algorithm 5.3 Mean-CVaR ecient portfolio revision
Maximize (1−λ)Return−λ·CV aR (5.11)
subject to Return=X
i
rixi−X
i
xi−T ransCost, (5.12) CV aR=V aR+
P
lplyl+
(1−α), (5.13)
yl++X
i
rlixi+V aR≥0, ∀l, (5.14)
y+l ≥0, ∀l, (5.15)
xi−(1−γ)xBU Yi+ (1 +γ)xSELLi=xHOLDi, ∀i, (5.16) X
i
xBU Yi=X
i
xSELLi, (5.17)
T ransCost=γX
i
(xBU Yi+xSELLi) (5.18) xi, xBU Yi, xSELLi≥0, ∀i, (5.19)
Return, CV aR, V aR∈R. (5.20)
5.3 CVaR optimal portfolios
Feature selection on 2007 data with 7 clusters and STARR criterion are used identify the instruments that are passed to the portfolio optimization framework, hence the results are comparable with the F-STARR portfolios.
The STARR criterion is chosen because it matches the CVaR approach by focusing on down-side volatility..
5.3.1 Scenarios
The scenarios must be inspected before they can be used. This has been done, and a brief overview is given here. The rst 100 scenarios and the quantile scenarios are plotted for IGILT, in gure 5.1, and equivalent plots for the other instruments are found in appendix C. The actual observed result is also plotted, just to show the strength of moment matching, but cannot be used as a predictor when examining the scenarios. In general, the observed scenario is kept within the 1. and 3. quantile scenarios, and often very close to the median. This tells us that the algorithm is good at capturing the true distribution, and that moment matching generates reliable scenarios, at least in the data-period used, which cover several nancial regimes. The scenarios seem to behave reasonable; they don't produce exorbitant values, are slightly volatile and the distribution is a bit skewed with one mode.
It is harder to analyze the correlation structure, as it is a measure across assets for each set of scenarios.
Not much information can be drawn from the plots, but nothing suspicious is observed, hence the correlation structure is assumed to be rather stable.
Based on a manually inspection of the scenarios, they are accepted and used in the optimization model in next section.
5.3.2 Optimal F-STARR portfolios
Three optimal portfolios are found and back-tested, each representing investors with dierent risk tolerance, low (OPT-L), medium, (OPT-M) and high (OPT-H) by specifying the weighting parameterλ= 0.98, 0.5 and
5.3 CVaR optimal portfolios 28
01234Cumulative return on 1 EUR investment
IGLT scenarios
01234
2008 2009 2010 2011 2012 2013 2014 2015 2016 2017
Year Cumulative Scenario Returns
Median 1. & 3. quantile Min & Max Observed
Figure 5.1: IGLT scenarios and observed cumulative returns.
0.02. That theλ−values are equidistant, does not mean that the risk measures (CVaR) are equidistant. The optimal weights for the rst two months are shown in table 5.1.
The risk-seeking investor (OPT-H), does not care about diversication and seeks to hold the asset with the highest expected return unless this prot is vanished by the trading costs. Therefore OPT-H has a higher turnover. The risk-averse investor tends to spread out the investments and overweight low-volatility instruments.
Risk period CVaR Return IGLT H:EGB U:FXS C:CDZ S:DJEU U:KXI U:DXJ OPT-L 1 0.0056 -0.0029 0.0329 0.8268 0.0271 0.0052 0.0487 0.0591
2 0.0065 0.0037 0.9328 0.0609 0.0036 0.0026 OPT-M 1 0.0058 -0.0085 0.0136 0.8201 0.0084 0.0732 0.0845
2 0.0067 0.0034 0.9938 0.0062
OPT-H 1 0.0855 -0.0955 0.8569 0.1431
2 0.1052 -0.0049 0.1343 0.8657
Table 5.1: CVaR, realized return, and the optimal relative weights of the three portfolios rst two periods (months) beginning 01-01-2008.
The consequence is clearly seen on the result of the back-test plotted in gure 5.2, OPT-L and OPT-M both are rather stable with low volatility. During the nancial crisis, they are almost not aected and yield a small positive return. From mid-2015 until ultimo-2016 they seem to pick the same allocation which is very stable with no return. Looking at the scenarios in the same period, all the quantile scenarios are widening, hence the risk are growing and a the CVaR model has no other option than diversifying since none of the assets are expected to have low volatility during this period. In the same period OPT-H does not consider risk and experiences high volatility, but almost no return. As expected OPT-H has the most volatile behavior but throughout the period the it yields a higher return, as expected from the theory of risk and return. The exposure to market risk is also clear from the drawdown plot, where the risk-averse strategies have low drawdown risk compared to the risk-seeking strategy.
In chapter 7 the result of the optimal portfolios is compared to the strategies without optimization on-top.
5.3 CVaR optimal portfolios 29
0.51.01.52.02.5Cumulative return
CVaR optimal portfolios (n=7)
0.51.01.52.02.50.51.01.52.02.5 lambda=0.98 lambda=0.50 lambda=0.02
−0.30−0.20−0.100.00−0.30−0.20−0.100.00−0.30−0.20−0.100.00Drawdown
2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 Year
Figure 5.2: Back-test: cumulated return, Drawdown for CVaR optimal portfolios.
CHAPTER 6
Betting against beta ETF portfolio
6.1 The BAB framework
The Betting against Beta (BAB) strategy is a strategy based on the ndings in Frazzini and Pedersen[2014], which essentially concludes that a beta-neutral portfolio constructed of long leveraged low beta assets and short high beta assets yields a signicant positive risk-adjusted return. The concept of constrained investors holding riskier assets, which increases the prices of these assets and making them less attractive from a risk-adjusted perspective is implemented with an approach very similar the one used inFrazzini and Pedersen[2014].
To estimate the beta of an asset, the market portfolio must be dened. Many denitions have been given in the literature, but a large stock index, e.g. SP500 has often been used as a proxy. An in-depth discussion of the denition is included in chapter 8. In this study, an alternative approach is presented using agglomerative hierarchical clustering. Assume data has been classied into n clusters. Within each cluster a sub-market is dened, such that the beta of a portfolio pi with equally weighted assets from cluster ci has βi = 1 where i ∈ [1;n]. The sub-market portfolios can then be combined to a nal portfolio by assigning each portfolio a weight e.g. 1/n. The nal portfolio can also be seen as a market portfolio, but the for this study the sub-market portfolios are used.
This approach exhibits some advantages. Clustering is not based the size of the volatility, but on assets with the same behavior therefore both high and low volatility assets are present in each cluster. How asset classes should be weighted is also up for discussion. This approach does not consider the asset classes but rather the behavioral classes which are weighted equally in the nal portfolio if1/nare used.
The approach of market portfolio construction also has some advantages which is more specic to this study. As seen in table 2.1 equity-type of ETFs is heavily represented 86 % of the total number of ETFs, but this bias is handled by using the sub-market portfolios to calculate the within cluster betas. A lot of data is available, and is would be very time consuming to analyze each ETF separately this approach by automatically assigning a weight to the fund and includes in the calculation, and assigns a weight based on how many other ETFs it shares properties with.
6.1 The BAB framework 31
If the market portfolio was constructed from equally weighting of all ETFs it might have resulted in portfolio with a damped behavior or even worse, highly correlated to stocks neglecting other asset classes.
The BAB portfolios construction consists of four steps. Assumen clusters and the clusterci containszi ETFs wherei∈[1;n]. Only clusters with more that one asset are considered
Estimate ex-ante beta values
Within each cluster, the market portfolio Mi is dened with βi = 1. With this as reference point, the beta values for ETFi,j are calculated wherej∈[1;zi].
β(ETFi,j) =Cor rMi, rETFi,jσ rETFi,j
σ(rMi) k+ (1−k)β(Mi) (6.1) where k is a constant used to shrink the timer-series estimate of beta towards the sub-market beta. k = 0.6 is used as suggested inFrazzini and Pedersen [2014]. Furthermore it is important to note that the correlation matrix is estimated from 3-days log-returns as in section 3.1, and the standard deviation from 1-day returns. If any assets with negative beta is found, it is removed from the cluster, since it might disturb the idea of BAB, as it is a low beta, but it might as well have a high absolute value, i.e. high exposure to market volatility which is what constrained investors are seeking.
Construct high and low beta portfolios
Within each cluster, a low(Li) and a high(Hi)beta portfolio is constructed. Dene high (low) betas as ETFs with betas larger (smaller) than the median beta value withinci. Each portfolio then containsli=f loor(zi/2) assets. InH (L), the ETFs are order in descending (ascending) order from 1 to li. A weight given by equation 6.2 is assigned to ETFi,j, such that the ETFs with the highest (and lowest) betas is assigned the highest weight.
wi,j =ranki(β(ETFi,j))
j
X
κ=1
κ
(6.2)
Construct zero-beta cluster portfolios
Within each cluster,ci a zero-beta portfolioUi is constructed by buying and leveraging the low beta assets and shorting the high beta assets. The leverage is found by
β(Hi) = ςiβ(Li) (6.3)
ςi = β(Li)
β(Hi) (6.4)
The weights for the zero-beta cluster portfolios is found by
ωi,j =
(−wi,j j ∈Hi
ςiwi,j j ∈Li (6.5)
If ςi >1 then the investor need to lend money for the investment, ifςi <1 then the investor can invest the money in a risk-free rate. 1-year Euro-bond yield plus a 50 BPS spread (annually), is used as proxy for the cost