P2.T5. Market Risk Measurement & Management

Gunter Meissner, Correlation Risk Modeling and Management

Bionic Turtle FRM Study Notes

Meissner, Chapter 1: Some Correlation Basics DESCRIBE FINANCIAL CORRELATION RISK AND THE AREAS IN WHICH IT APPEARS IN . ... 4 EXPLAIN HOW CORRELATION CONTRIBUTED TO THE GLOBAL FINANCIAL CRISIS OF 2007 TO 2009...... 9 DESCRIBE THE STRUCTURE, USES, AND PAYOFFS OF A CORRELATION ...... 13 ESTIMATE THE IMPACT OF DIFFERENT CORRELATIONS BETWEEN ASSETS IN THE TRADING BOOK ON THE VAR CAPITAL CHARGE...... 14 EXPLAIN THE ROLE OF CORRELATION RISK IN MARKET RISK AND CREDIT RISK...... 16 RELATE CORRELATION RISK TO SYSTEMIC AND CONCENTRATION RISK...... 17 CHAPTER 1: MEISSNER’S QUESTIONS AND ANSWERS (Q&A) ...... 21

Meissner, Chapter 2: Empirical Properties of Correlation: How Do Correlations Behave in the Real World? DESCRIBE HOW EQUITY CORRELATIONS AND CORRELATION VOLATILITIES BEHAVE THROUGHOUT VARIOUS ECONOMIC STATES...... 26 CALCULATE A MEAN REVERSION RATE USING STANDARD REGRESSION AND CALCULATE THE CORRESPONDING AUTOCORRELATION...... 28 IDENTIFY THE BEST-FIT DISTRIBUTION FOR EQUITY, BOND, AND DEFAULT CORRELATIONS...... 32 CHAPTER 2: MEISSNER’S QUESTIONS AND ANSWERS (Q&A) ...... 34

Meissner, Chapter 5 :Financial Correlation Modeling – Bottom-Up Approaches (pages 126-134 only) EXPLAIN THE PURPOSE OF COPULA FUNCTIONS AND THE TRANSLATION OF THE COPULA EQUATION...... 36 DESCRIBE THE GAUSSIAN COPULA AND EXPLAIN HOW TO USE IT TO DERIVE THE JOINT PROBABILITY OF DEFAULT OF TWO ASSETS...... 37 SUMMARIZE THE PROCESS OF FINDING THE DEFAULT TIME OF AN ASSET CORRELATED TO ALL OTHER ASSETS IN A PORTFOLIO USING THE GAUSSIAN COPULA...... 40 CHAPTER 5: MEISSNER’S QUESTIONS AND ANSWERS ...... 41

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Meissner, Chapter 1: Some Correlation Basics  Describe financial correlation risk and the areas in which it appears in finance.  Explain how correlation contributed to the global financial crisis of 2007 to 2009.  Describe the structure, uses, and payoffs of a correlation swap.  Estimate the impact of different correlations between assets in the trading book on the VaR capital charge.  Explain the role of correlation risk in market risk and credit risk.  Relate correlation risk to systemic and concentration risk.

Key ideas and/or definitions in this chapter

 To be developed on next revision  TBD-2

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Describe financial correlation risk and the areas in which it appears in finance.

Financial correlation risk is the risk of loss due to an adverse change in correlation between two or more variables.

For example:  Wrong-way risk is adverse correlation between a bond issuer (issued bond is the reference asset below) and the bond insurer (bond insurer is the CDS Seller below)1

Correlation risk includes co-movement between non-financial variables (e.g., economic, political, technical). For most risk measures (e.g., , ), an increase in correlation implies an increase in the risk of financial loss. During the global financial crisis (GFC), famously the correlations spiked. This is a typical symptom of a crisis: correlation increase, or spike, even approaching 1.0 in severe situations.

Examples from Meissner2:  Sovereign debt and currency value: In 2012, US exporters were hurt by a decreasing Euro currency and increasing Euro sovereign debt  Geopolitical tensions: Middle East tensions can hurt demand and increase oil prices  Correlated markets and economies: A slowing gross domestic product (GDP) in the United States can hurt Asian and European exporters and investors since economies and financial markets are correlated worldwide.

1 Inspired by Meissner’s Figure 1.1 (Gunter Meissner, Correlation Risk Modeling and Management, New York: Wiley, 2014) 2 Gunter Meissner, Correlation Risk Modeling and Management, New York: Wiley, 2014

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Correlation risk via Meissner’s example3 of a (CDS)

A CDS transfers credit risk from an investor to a counterparty. For example,  An investor invested $1.0 million in a bond issued by Spain (aka, the reference asset) but worries about its default. To default risk, he bought a CDS from a French bank, BNP Paribas (aka, the counterparty who sells the CDS and is synthetically long the reference). o The investor is protected against default from Spain: in case of default, the CDS counterparty (BNP Paribas) will pay the originally invested $1.0 million to the investor (if we assume the recovery rate and accrued interest are zero) o The value of the CDS, the fixed CDS spread s, is determined by the default probability of the reference entity Spain.  The worst-case scenario is the joint default of Spain and BNP Paribas, in which case the investor will lose his entire investment in the Spanish bond of $1.0 million.  The investor is exposed to default correlation risk between the reference asset r (Spain) and the counterparty c (BNP Paribas). o The spread s is also determined by the joint default correlation of BNP Paribas and Spain. If that default correlation increases, the present value of the CDS for the investor will decrease and the investor will suffer a paper loss. o Because both Spain and BNP Paribas are in Europe, we assume that there is a positive default correlation between the two. This investor has wrong-way correlation risk (aka, short wrong-way risk). Meissner’s4 example continued:  Below (our render of Meissner’s Figure 1.25), we see that above ρ = −0.30, higher correlation implies a lower CDS spread. Why? An increasing ρ indicates a higher probability that both the reference and counterparty simultaneously default. In extremis (ie, ρ = 1.0) the CDS is worthless and deserves no spread.  For a correlation from −1.0 to about −0.30, the CDS spread increases slightly, which is perhaps non-intuitive. But a less negative correlation indicates a higher probability of either Spain or BNP Paribas defaulting. If only Spain defaults, the CDS buyer collects the payoff; If only BNP defaults, the CDS buyer must repurchase insurance, possibly at a higher price.  The relationship is non- monotonous; the CDS spread alternates between increasing and decreasing.

3 Gunter Meissner, Correlation Risk Modeling and Management, New York: Wiley, 2014 4 Gunter Meissner, Correlation Risk Modeling and Management, New York: Wiley, 2014 5 Ibid but actual spreadsheet constructed by David Harper

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Financial correlations are common in finance. Meissner considers five areas: (1) investments and correlation, (2) trading and correlation, (3) risk management and correlation, (4) the global financial crisis and correlation, and (5) regulation and correlation.

Investments and correlation

In Markowitz's portfolio theory (MPT, including CAPM), diversification is a benefit and diversification is imperfect (less than one) correlation. For given asset returns and volatilities, a lower correlation will decrease for the same level of expected return.

For example:

Consider a two-asset portfolio that performed as below:

The simple return is given by, (St – St-1)/St-1. Over the illustrated time frame (from 2014 to 2019) the average returns are: µX= 13.70% and µY = 5.22%.

The portfolio’s return depends on the weights and is given by the following (in this example, if the weights are equal, then the portfolio return is +9.46%):

= + The sample standard deviation of returns (aka, volatility) is given by:

= ( − ) −

In this example, the volatilities for each asset are: = 17.20% and = 10.32%.

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For example (continued):

Covariance. The covariance measures how two variables “covary” or move together. The covariance measures the strength of the linear relationship between two variables. The covariance of returns for assets X and Y is derived with equation: = ( − )( − ) −

In our example, CovXY = −0.0117. But the covariance is not easy to interpret.

It is much more convenient to use the Pearson correlation coefficient, ρ(x,y), which returns scale-free (ie, unitless) values between −1.0 and +1.0. This correlation coefficient is:

=

In this example rXY = −0.6594; ie, these asset returns are negatively correlated.

We can calculate the standard deviation for our two-asset portfolio P as:

= + +

Under equal (50/50) weights, the above portfolio volatility is , = . %.

The standard deviation is probably the most common measure of risk in finance. Put simply, a higher (lower) standard deviation implies higher (lower) risk. Classical finance assumes risk-averse investors who prefer, all other things being equal (aka, ceteris paribus), lower risk and therefore lower standard deviations. But a risk-seeking investor would prefer a higher standard deviation because it provides the upside possibility of higher payoffs! This is also a criticism of standard deviation as a risk measure: it is indifferent to whether the volatility is upside or downside. Yet, realistically, most of us do not mind if the high volatility consists of upside movements in price!

Risk-adjusted return: We can divide the (expected) return by standard deviation for one version of a risk-adjusted return. For a portfolio it is /.

For the same (example) portfolio, to the right is a plot of this return/risk ratio as a function of increasing correlation. Higher correlation corresponds to a lower return/risk ratio. We can see the dramatic impact of correlation on the portfolio’s return/risk ratio! A high negative correlation results in a return/risk ratio of above 2.5, whereas a high positive correlation results in a ratio markedly less than 1.0.

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Trading and correlation

Correlation trades take a view on prices influenced by asset co-movements.  Multi-Asset Options: They are a popular group of correlation options. Many different types are traded.

6 The most popular ones according to Meissner are listed below. Here, S1 is the price of asset 1 and S2 is the price of asset 2 at maturity. K is the .

o Option on the better of two. Payoff = max (S1, S2).

o Option on the worse of two. Payoff = min (S1, S2).

o Call on the maximum of two. Payoff = max [0, max (S1, S2) – K].

o Exchange option (as a convertible bond). Payoff = max (0, S2−S1).

o Spread . Payoff = max [0, (S2−S1) −K].

o Option on the better of two or cash. Payoff = max (S1, S2, cash).

o Dual-strike call option. Payoff = max (0, S1−K1, S2−K2). o Portfolio of basket options. Payoff = [Σ − , 0], where ni is the weight of assets i.

These option values are sensitive to the correlation between the prices S1 and S2. For the above-listed options, a lower (higher) correlation implies a higher (lower) option price; except for the “option on the worse of two.”  Quanto option gives the holder the right (but not the obligation) to exchange the foreign currency option payoff into home currency at a fixed . Consequently, the quanto option hedges currency risk. The more positive the correlation coefficient, the lower the price for the quanto option. The lower the correlation coefficient, the more expensive the quanto option. Correlation Swap: A fixed correlation is exchanged for the actual realized correlation, which is measured during the swap’s tenor. The payoff of a correlation swap for the correlation fixed rate payer at maturity, if N is the notional amount is 7 (Meissner’s equation 1.7): − . Indirectly, correlation swaps can hedge stock price drops: because correlation tends to increase when stocks decrease, a fixed correlation payer indirectly hedges against a market decline.  Buying call options on an index and selling call options on individual components: This is to buy call option(s) on an index (e.g., S&P 500), plus sell call options on individual stocks in the index. If correlation between the stocks in the index increases, so will the of the call on the index (i.e., positive relationship between correlation and volatility). The gain on the call should outpace the loss on individual short calls.  Paying fixed in a swap on an index and receiving fixed on individual components: This is to pay fixed in a on an index; plus receive fixed in variance swaps on individual components in the index. Same idea as above.

6 Gunter Meissner, Correlation Risk Modeling and Management, New York: Wiley, 2014 7 Gunter Meissner, Correlation Risk Modeling and Management, New York: Wiley, 2014

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Risk Management and Correlation

The three classic financial risk buckets are market risk, credit risk, and operational risk. Additional types include systemic risk, liquidity risk, volatility risk, and correlation risk. Market risk includes: (1) equity risk, (2) risk, (3) currency risk, and (4) commodity risk.

VaR is the most common risk management measure. VaR is the worst expected loss over some time horizon with some confidence level.

Portfolio VaR is given by = √ where is the standard normal quantile; most typically 1.645 corresponding to 95.0% confidence and 2.33 corresponding to 99.0% confidence, is the time horizon, and is the standard deviation (aka, volatility). Of course, portfolio volatility incorporations correlation between asset returns. In this basic delta-normal portfolio VaR, a higher correlation increases the portfolio VaR. If the goal is to reduce portfolio VaR, then (assuming long positions only), a negative correlation is ideal.

Regulation and Correlation

Regulatory frameworks increasingly incorporate correlation as an explicit factor or variable to be addressed. This includes the Basel accords. Basel III introduced, or at least explored, several concepts related to correlation, including: double defaults in insured risk transactions; correlated defaults in a multi-asset portfolio via Gaussian copula; correlations in derivatives transactions via credit value adjustment (CVA); and correlations in wrong-way risk (WWR).

Explain how correlation contributed to the global financial crisis of 2007 to 2009.

While there were multiple causes of the crisis, the main ones according to Meissner8:

 Increased fueled by loose credit: An extremely benign economic and risk environment from 2003 to 2006 with record low credit spreads, low volatility, and low-interest rates. Increasing risk-taking and speculation of traders and investors who tried to benefit in these presumably calm times. In 2007, U.S. investors had borrowed 470% of the U.S. national income to invest and speculate in the real estate, financial, and commodity markets.  Complex credit products: A new class of structured investment products, such as collateralized debt obligations (CDOs), CDO-squareds, constant-proportion debt obligations (CPDOs), constant-proportion portfolio insurance (CPPI), and new products like options on credit default swaps (CDSs), credit indexes.  Model risk: The new copula correlation model, which was trusted by many investors and which could presumably correlate the n(n−1)/2 assets in a structured product. Most CDOs contained 125 assets. There are 125(125−1)/2=7,750 asset correlation pairs to be quantified and managed.  Moral hazards: Rating agencies were paid by the same companies whose assets they rated. As a consequence, many structured products received AAA ratings and gave the illusion of little price and default risk. Risk managers and regulators who lowered their standards in light of the greed and profit frenzy. Meissner recommends an excellent paper in the Economist: “A Personal View of the Crisis, Confessions of a Risk Manager.”

8 Gunter Meissner, Correlation Risk Modeling and Management, New York: Wiley, 2014

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