RISK AND ASSET MANAGEMENT MODELS : THEORETICAL AND PRACTICAL ASPECTS

Abstract

In the last 20 years the financial markets have been characterized by several crisis, that have produced crashes, corrections and bear markets. Most recently we can remember one of the most intense crises, in 2007, the first signs of something going wrong with the American residential mortgage market could be observed. Bond markets had been flooded with securitizations of all kinds. From December 31, 1999 to March 31, 2007 the American real estate market, measured by the index FHFA US House Price, had grown almost like NASDAQ had done in the late 1990s (+68.46%). Suddenly house prices collapsed, from March 31, 2007 to September 30, 2009 that index lost 12.5% and it come back to the March 2005 price. In that period borrowers did not repay loans that banks had entrusted them and the delinquencies reached 10% of the total loans. This unexpected event led banks into a liquidity crisis. It was the beginning of a domino effect that led to: • the introduction of liquidity by the FED; • the expansive monetary policy with interest rates close to zero; • the acquisition of Fannie Mae and Freddie Mac by the government; • the rescue of Bear Stearns and AIG by the government; • the acquisition of Merrill Lynch by Bank of America; • the bankruptcy of Lehman Brothers. In 2008, as a result of this big bubble, the market crumbled. The Standard and Poor’s 500 index during the crisis fell roughly by 52%. Before the subprime crises, traditional strategic asset allocation theory was deeply rooted in the mean-variance portfolio optimization framework developed by Markowitz and, therefore, the risk-gain analysis has become a key issue for portfolio selection. How ever, the mean-variance optimization methodology can be very sensitive to the input parameters required by the model. Therefore, the problem of estimation errors becomes very relevant, particularly for the expected returns of the assets. In addition, subjective estimates of expected future returns can frequently be influenced and modified by indi vidual biases or by a priori investor view, such as the overestimation of expected returns due to the really strong momentum of a particular asset class or the economic cycle. In vestors can also underestimate risk when they consider one particular type of distribution of returns which may result in ignoring fat tails when markets collapse. As such, param eter estimation based on past realized observations can contain a certain level of noise, especially if the risk premia and correlations are time-varying. The concept of diversification was better explored and the attention was shifted from capital diversification to risk diversification. Some investors do not seem concerned by risk contribution and are thinking that a traditional 60/40 portfolio offers a real diversified portfolio. This thesis will deal with the problem facing investors in seeking to determine a ’well’ diversified asset allocation strategy. The thesis reviews the widely used models to build up a portfolio avoiding the explicit use of expected returns, so as minimum Risk, Risk Diversifications, and Capital Diversification strategies. Our contribution to the literature is to provide the Equal Risk Contribution (ERC) strategy based on Conditional Value at Risk (CVaR) and on Conditional Value at Risk-deviation (CVaR-deviation) as a risk measure. More precisely, we first tackle these problems by means of a least-squares approach and, then, under appropriate conditions, we provide an alternative approach to finding a CVaRERC portfolio as a solution to a convex optimization problem. From a theoretical and practical viewpoint, the solution of the convex problem can be easy to find (at least for a discrete probability space), but another issue still remains: how do we can generate the discrete probability space? Is it sufficient to use past realized asset returns or can we use a more clever approach? In this thesis we consider both historical and simulated scenarios as the input to portfolio selection problem specification. This means that we use two different Risk Management approaches: Historical Simulation and Historical Filtered Bootstrap. Summing up, we provide an anwer to the following questions: 1. Could we formulate and solve an ERC problem based on a coherent risk measure as CVaR? 2. How intense is the effect of estimation errors on the ERCCVaR model? 3. Could we mitigate this effect? 4. Is there any difference between a historical scenario and a simulated scenario? In order to address such questions, we discuss the motivations of our research. So we review the risk-based portfolio models, after that we give our first contribution to literature: the closed form solution of the Conditional Value at Risk Equal Risk Con tribution. We compare the new model with respect to the other smart beta portfolios like equal weight or minimum risk portfolios on seven datasets composed of equity, bonds and commodities, as base for the backtests. We found that there is not clear evidence of 3 dominance among those method but each one should be preferred according the investor’s risk aversion behavior. We extend the ERC to CVaR-deviation and we investigate the stability of the ERC CVaR and ERC CVaR-deviation solutions with respect to the input parameters, in partic ular we find that the unconstrained portfolios shown the highest effect of the estimation error in particular on minimum CVaR. When we introduced positiveness of portfolio weights and no leverage condition in order to test how the perturbation of the input date afflict the ERC results, we observed that the optimal solutions are much more stable for minimun CVaR and minimum CVaR-deviation. The second contribution to the literature is that we conduct a deep analysis of stability of ERC CVaR portfolio solutions, we high light that those methods shown less effect of the estimation error, so they provide more stable solution with respect to the minimum CVaR. The key problem is due to the ratio N/T, where N is the number of the investable asset and T is the number of available returns. Clearly when N increases not always there are enough data in order to maintain small the ratio N/T. In order to find a solution with the aim to minimize N/T (increas ing T obviously) we review the Historical Filtered Bootstrap procedure that allows us to generate future scenarios. We show how to use those scenarios for portfolio selection and we also validate the Historical Filtered Bootstrap model using both statistical accuracy and efficiency evaluation tests. Our third contribution to the literature is the by-passing the problem of lack in historical data using a robust e consistent procedure that allows to generate a huge number of data increasing the size of T. The HFB is a well know risk management model but it requires intensive estimation of parameters of the ARMA-GARCH model that describe the asset’s hidden generating process. We provide a new risk management model that is a nice tool to estimate risk as good as the sophisticated Historical Filtered Bootstrap strategy that need only two input parameters: 1. implied volatility (that is quoted in the market); 2. realized volatility (calculated on twenty realized returns). The new model, called Shrunk Volatility VaR, is faster than HFB, it does not require 4 estimation model nor simulation procedure. We perform a backtest of many models among the most common into the financial industry (Historical Simulation, Risk Metrics, Historical Filtered Boostrap, Extreme Value Theory Historical Filtered Bootstrap) versus the Shrunk Volatility VaR. We evaluate the statistical accuracy of one-day-ahead VaR estimates by means of the following: • The unconditional coverage (UC) test, which analyzes the statistical significance of the observed frequency of violations with respect to the expected one; • the independence (IND) test which gauges the independence of violations, namely the absence of violation clustering; • the conditional coverage (CC) test, which combines these two desirable properties. Further, to examine the performance of the VaR models, we also perform backtesting based on loss functions that take into account both the regulators’ and the investors’ viewpoints. We found that Shrunk Volatility VaR is able to achieve the same performances estimate as the sophisticated Historical Filtered Bootstrap strategies do. Talking in practical terms, and considering the case of a portfolio manager who ad ministers a flexible UCITS fund, aiming to obtain the maximum return with a constraint on risk, measured by VaR. Since the portfolio managers must face transaction costs when they buy or sell assets, they could use our quick forecasting tool as a what-if scenario analysis before they start trading. If the portfolio VaR is within specific risk bounds, the portfolio managers can purchase and sell; otherwise, they should revise their investment. Therefore, the pre-analysis obtained by our model may allow for the control of risk both upstream and downstream of the investment process. Indeed, it is typical for the asset manager to construct their portfolio, and only after wards for the risk manager to ensure compliance with the risk limit. So, if the portfolio VaR goes beyond the regulators’ limitations, then the portfolio manager has to change their investment strategy, thus leading to carry the trading costs twice. Finally the answers to the initial questions are as follows: 1. Q: Could we formulate and solve an ERC problem based on a coherent risk measure as CVaR? A: yes we could. CVaR is a homogeneous function of degree 1 and it admits Euler decomposition. In addition we formulate the ERC problem as a convex problem in a convex set. 2. Q: How intense is the effect of estimation errors on the ERCCVaR model? A: it is higher with respect to the one we record with the minimum variance ap proach when ε < 5%. It is quite comparable when ε = 5%, and we notice a smaller effect when ε > 5%. We also recall that ERC procedure always has a smaller effect than Minimum CVaR and Minimum CVaR-deviation do. 3. Q: Could we mitigate this effect? A: yes we could. We can consider introducing constraints to portfolio weights and reducing the ratio N/T by increasing the amount of data we use to estimate parameters. We can also consider evaluating CVaR, not in the extreme tail, so ε > 10%. 4. Q: Is there any difference between a historical scenario and a simulated scenario? A: yes there is. Using a historical scenario we need too many observation data points in order to reduce the estimation error effect, whereas using simulated scenarios (that are simulated using a quite good estimation of the hidden data generating process) we can force the effect of the estimation error to be negligible. 1 A stock market crash is when a stock index drops severely in a day or two of trading. A correction is when the market falls 10% from its 52-week high over days, weeks, or even months. A bear market is defined when the market falls another 10%, for a total decline of 20% or more. 2 On September 7, 2008 were being placed into conservator-ship of the Federal Housing Finance Agency (FHFA). This action was seen as one of the most relevant government intervention in the capital and bond markets.