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1.1 Financial Correlation Two Types of Financial Correlations: Static And 1.1 Financial correlation Two types of financial correlations: static and dynamic (1) Static financial correlations measure how two or more financial assets are associated within a certain time period. Examples are the classic VaR model, the copula approach for CDOs , and the binomial default correlation model of Lucas (1995). (2) Dynamic financial correlation measure how two or more financial assets move together in time. Examples are pairs trading (a type of statistical arbitrage), the deterministic correlation approaches (Heston, 1993), and the stochastic and time-dependent correlation process. 1.2 Financial correlation risk Financial correlation risk is the risk of financial loss due to adverse movements in correlation between two or more variables. (i.e., an increase in the correlation of assets returns) Correlation risk can also involve non-financial variable, such as economic or political events. For example, the correlation between the increasing sovereign debt and currency value can hurt an exporter. L (1) In Figure 7-1, the value of the CDS, i.e., the fixed CDS spread s, is mainly determined by the default probability of the Spain entity and the joint default correlation of BNP Paribas and Spain ( ). (wrong-way risk) (2) In Figure 7-2, if , then the CDS is worthless. 2 1.3 Motivation: correlations and correlation risk are everywhere in finance. Investment and correlation Define the returns of asset and at time are and , average returns are and , and standard deviations are and . The portfolio return is The covariance of returns for assets X and Y is The Pearson correlation coefficient is Example 1: Calculate the covariance and correlation Year Asset X Asset Y Return of X Return of Y 2008 100 200 2009 120 230 20.00% 15.00% 2010 108 460 -10.00% 100.00% 2011 190 410 75.93% -10.87% 2012 160 480 -15.79% 17.07% 2013 280 380 75.00% -20.83% Average 29.03% 20.07% Calculate the covariance and correlation of returns for assets X and Y Ans: -0.1567; -0.7403 3 We can calculate the standard deviation for two-asset portfolio P as In the Example 1, with equal weights, Trading and correlation Traders try to forecast changes in correlation and attempt to financially gain from these changes in correlation, such as pair trading. (1) Multi-asset options (Rainbow options; Mountain range options) B-S Better-of options: Worse-of options: Call on the maximum of two underlying assets: Exchange option: Spread call option: Option on the better of two or cash: Dual-strike call option: Portfolio of basket options: where is the weight of assets 4 (2) Quanto option ( ) Quanto option is an option that allows a domestic investor to exchange his potential option payoff in a foreign currency back into his home currency at a fixed exchange rate. It protects an investor against currency risk. (3) Correlation swap ( ) The correlation between assets can also be traded directly with a correlation swap. Correlation buyer: paying a fixed rate Correlation seller: receiving a fixed rate Fixed percentage (e.g., ) Correlation fixed rate Correlation fixed rate payer receiver Realized The payoff of a correlation swap for the correlation fixed rate payer at maturity is: Example 2: Payoff of a correlation swap What is the payoff of a correlation swap with three assets, a fixed rate of 10%, a notional amount of $1,000,000, and a 1-year maturity? The daily log returns in ln(St/St-1) of the three assets are calculated for 1 year. Assume that the realized pairwise correlations of the log returns at maturity are as follows. Sj=1 Sj=2 Sj=2 Si=1 1 0.5 0.1 Si=2 0.5 1 0.3 Si=3 0.1 0.3 1 Ans: The payoff for the correlation fixed rate payer at swap maturity is 5 (4) Buying call options on an index and selling call options on individual components (A method of buying correlation, i.e., benefiting from an increase in correlation) For example: Index: Dow Jones Industrial Average Index individual components: individual stocks of the Dow Jones Industrial Average Index (5) Paying fixed in a variance swap on an index and receiving fixed on individual components (A further method of buying correlation, i.e., benefiting from an increase in correlation) If correlation increases, so will the variance. Risk management correlation The three main types of financial risk are market, credit, and operational risk. Market risk consists of equity risk, interest rate risk, currency risk, and commodity risk. Measurement tools: VaR, ES, ERM, etc. VaR(X%) = Credit risk consists of migration risk (i.e., downgrade) and default risk. Consider default correlation (the degree to which defaults occur together) Example 3: Deriving VaR of a two-asset portfolio What is the 10-day VaR for a two-asset portfolio with a correlation coefficient of 0.7, daily standard deviation of returns of asset 1 of 2%, of asset 2 of 1%, and $10 million invested in asset 1 and $5 million invested in asset 2, on 99% confidence level? Ans: 6 The global financial crisis of 2007 to 2009 and correlation Reasons: (1) An extremely benign economic and risk environment from 2003 to 2006 with record low credit spreads, low volatility, and low interest rates. (2) Bubble in the housing market, mortgage market, stock market and commodity market in 2007. (3) A new class of structured investment products emerged, such as CDOs, CPPI, CDSs. (4) A moral hazard of rating agencies. (5) Risk managers and regulators lower their standards. (6) Correlation between tranches of CDOs increases during the global crisis. 1.4 Fit correlation risk into the broader picture of risks in finance. Correlation risk and systemic risk (1) Systemic risk: The risk of a financial market or an entire financial system collapsing. (2) Example: the collapse of the entire credit market in 2008. Figure: 7 Correlation risk and concentration risk (1) Concentration risk: The risk of financial loss due to a concentrated exposure to a particular group of counterparties. (2) Concentration ratio: If a creditor has 10 loans of equal size, then the concentration ratio would be 1/10 = 0.1. Case 1: company W has a default probability of 10%, Case 2: Bank C has lent $5m to company X and $5m to company Y, ratio = 1/2. If both X and Y have a default probability of 10%, and the default correlation between X and Y is , than we derive that the worst-case-scenario, , is reduced. Case 3: ratio = 1/3. If the default probabilities of X, Y, and Z are 10% each. The joint probability of default for the two binomial events: 8 .
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