Portfolio Construction and Tail Risk
United Nations; University of Cambridge
In the wake of the financial crisis, investors are increasingly concerned with ways to mitigate extreme losses. We analyze various approaches to enhancing traditional portfolio construction with tail-risk control. Interestingly, we find investors have better managed tail-risk using a minimum-volatility overlay strategy than explicitly penalizing extreme losses via conditional value-at-risk (CVaR). From a practical perspective, this solution can be cheap and easy to implement because it will not result in a rebalancing of the fund, and various minimum-volatility products are readily available on the market.
Portfolio Construction and Tail Risk – Introduction
The Financial Crisis of 2008-2009 resulted in extreme losses for many investors, leading to increased interest in approaches to mitigate so-called “left tail” risk. This paper discusses approaches to enhancing traditional mean-variance portfolio construction with tail-risk control, a factor important for investors concerned with extreme losses.
It is well-known (see, e.g., Markowitz (1952)) that when asset returns are jointly normally distributed, then variance is an appropriate risk measure. Moreover, given a set of expected returns, optimizing a portfolio under the assumption of normality is straightforward, since the first two moments of returns completely characterize the distribution of returns. Simplicity and elegance explains the popularity of mean-variance optimization. Further, minimum-variance portfolios have demonstrated attractive properties, and a wide set of such products are now available to investors.1
More recently, a growing number of financial practitioners and academics have begun to explore alternatives to variance as a measure of risk, and approaches to portfolio construction that go beyond mean-variance optimization.2 On strictly empirical grounds, the empirical distributions of returns of many financial assets do not appear to be consistent with the assumption of normality—exhibiting leftskew and/or fat tails— as exemplified in the Financial Crisis. In this situation, the mean-variance solution is not necessarily optimal. Second, on behavioral grounds, there is a large body of evidence suggesting that investors fear losses more than they value gains—that is, investors are “loss averse.”
The mean-variance approach is symmetric in its treatment of risk—the variance penalty applies to upside risk as much as down-side risk. Post-crisis, regulatory initiatives to guard against systemic risk to the financial system have further motivated interest in tail risk. Banks are required to monitor and manage their capital levels against their potential exposures to large losses.
This paper analyzes the benefits and costs of adding a tail-risk penalty to the standard mean-variance optimization framework using a universe of broadly representative equity and fixed-income exchangetraded funds (ETFs) as proxies for investible indices. We first look at portfolio constructions employing a penalty for conditional value-at-risk (CVaR) which captures the left-tail probability mass of the return distribution. We compare the optimal portfolios obtained in both the minimum-risk context, where expected returns are zero, and in the alpha context where we use forecasts of excess returns, in order to build intuition for how penalizing tail-risk affects the constructions.
By definition, a focus on tail risk means focusing on relatively rare events, putting a premium on the number of time periods over which asset returns and other data points can be measured. We address this issue by building a Monte Carlo simulator that matches the first four moments of the observed returns, and that mimicks the correlation structure and tail risk properties of the observed data. Using the simulator, we investigate the finite-sample distribution of the CVaR estimator, both at the individual instrument level, and at the portfolio level where we study portfolio performance in the out-of-sample back-test setting.
This analysis produces two main results. First, it’s well-known that data limitations are the main issue in forecasting tail-risks. A key result of this paper is that, for simulated data samples of less than about 30 years in length, in the zero expected-return case it is not possible on the basis of ex-post CVaR statistics to differentiate portfolios constructed under minimum-variance from those constructed using minimum-CVaR. A practical implication of this result is that if tail-risk reduction is an investment goal, employing a minimum-volatility portfolio exposure is potentially a better way of achieving this goal.
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