##### Systemic Risk

# Systemic Risk Measures and Portfolio Choice

### Authors:

**Agostino Capponi, ***Assistant Professor, Columbia University & An independent contributor to the Global Risk Institute*

Alex LaPlante,* Managing Director of Research, Global Risk Institute*

Alexey Rubtsov,* Research Associate, Global Risk Institute*

## EXECUTIVE SUMMARY

As opposed to a firm's individual risk of failure, which can be contained without harming the entire financial system, *systemic risk* is the risk of collapse of the entire financial system or market. Since the 2007–2009 financial crisis numerous attempts have been made to identify and measure the systemic risk of financial institutions. In this respect, the following question arises: how can a given systemic risk measure be used to construct portfolios that perform relatively well when systemic risk materializes? In this paper, we develop a framework for the optimal portfolio choice based on an exogenous systemic risk measure.

In his pioneering work on portfolio choice, Markowitz developed a theory of portfolio selection based on the “risk-return” characteristics of stocks in the portfolio. Markowitz’s investor was assumed to be minimizing the variance (risk) of a portfolio’s returns subject to meeting a given level of expected returns. In other words, Markowitz answered the question:

### What portfolio of stocks will deliver a specified expected return and at the same time have the lowest variance of future returns?

In this current research, we focus on adverse return scenarios and attempt to answer the question:

### What portfolio of stocks will deliver the highest expected returns in the case of a financial crisis?

A “financial crisis”, or a systemic event, is defined as a prolonged market decline. For the purpose of this paper, a systemic event is defined as the drop in a broad market index below its 5% VaR[1] level over a 6-month period (Figure 1).

#### Figure 1: 5% VaR allows us to make statements of the form: We are 95% certain that we will not lose more than VaR over, say, the next 6 months.

In addition to stressed market conditions, we consider only those portfolio’s returns that are below the so-called, Conditional VaR (CoVaR), which is defined as the VaR of the portfolio’s returns given that the market is in a crisis situation (market returns are below VaR). In this sense, 5% CoVaR allows us to make statements of the form: We are 95% certain that our portfolio will not lose more than CoVaR if market returns fall below VaR over, say, the next 6 months (Figure 2).

*Figure 2: The segment of market and portfolio’s returns distribution that correspond to stressed market and portfolio scenarios. The segment can be viewed as a two-dimensional analogy to the more commonly used “tail of return distribution”.*

The goal of our investor is to construct a portfolio that delivers maximal expected returns in the stressed market and portfolio’s return scenarios (the red segment in Figure 2).

In the numerical part of our analysis, we use five banks and three insurance companies: TD Bank, CIBC, RBC, Scotiabank, BMO, Manulife, Great-West Lifeco, and Sunlife. As a benchmark portfolio we use the tangency portfolio, that is, the portfolio on the Markowitz efficient frontier that has the highest expected return per unit risk (standard deviation). To avoid large negative portfolio positions during stressed market conditions, we preclude short sales. We compare the benchmark and developed portfolio construction methodologies by backtesting the portfolios on daily data that covers the period 2007-2017. The portfolios’ performance is shown in Figure 3.

*Figure 3: The* performance* of the Markowitz tangency portfolio and the developed portfolio choice methodology (CoVaR) based on the initial investment of $1 on January 31, 2007. The y-axis shows the USD value of the portfolios.*

There are several notable features in Figure 3. First, both portfolios perform poorly during the 2007-2009 financial crisis and lose almost 50% of their value. The values of both portfolios significantly decline during this period because *all* stocks that we consider substantially lose in value. Second, and most importantly, our methodology has a superior performance during 2009-2012 European Sovereign Debt crisis when compared with the benchmark portfolio that loses almost 50%. Similarly, the benchmark portfolio value declines in the beginning of 2016 due to the declining price of oil, concerns regarding China’s economic slowdown, and a weaker Canadian dollar. On the other hand, the CoVaR portfolio value is fairly stable during this period. Third, the Markowitz tangency portfolio has higher volatility than the CoVaR portfolio.

#### Figure* 4: Composition of (a) CoVaR and (b) Markowitz tangency portfolios.*

Importantly, from Figure 4 it follows that the CoVaR portfolio is more diversified than the Markowitz portfolio for which Manulife is the stock with the highest value of investment concentration starting from 2009. On the other hand, CoVaR portfolios imply relatively high investment in TD Bank and Great-West Lifeco which have been well known to be among the most consistent performers.

One of the most significant outcomes of the Markowitz portfolio analysis is the so-called “mutual fund separation theorem” which states that any portfolio on the Markowitz efficient frontier can be replicated by any two portfolios on the efficient frontier. This result implies that an investor can achieve a desired “risk-return” trade-off on the efficient frontier by trading in only two mutual funds, thereby reducing the transaction costs. In our research, we show an extension of the mutual fund theorem, which requires three mutual funds. This result is due to the fact that in addition to risk and return characteristics of the portfolio, our approach also includes portfolio’s correlation with the market, and therefore, more portfolios are required to obtain a desired “risk-return-correlation” trade-off.

#### Footnotes

[1] Recall, Value-at-Risk estimates how much a set of investments might lose over a certain time period.

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Systemic-Risk-Measures-and-Portfolio-Choice-Executive-Summary-and-Report.pdf