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# Correlation Basics: Definitions, Applications, and Terminology

After completing this reading you should be able to:

• 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.

## Financial Correlation Risk

Financial correlation risk is the risk of financial loss due to adverse movements in the correlation between two or more (financial) variables. It is the risk that the correlation between two or more variables changes unfavorably. An increase in asset return correlations increases the risk of financial loss.

The issue of correlation risk came to the fore during the 2007/2009 financial crisis where a dramatic increase in correlation was witnessed across the financial industry. For example, an increase in default correlation between bond issuers and insurers was observed, which represents a wrong-way risk.

A CDS works to transfer credit risk from an investor to a counterparty. It’s more like an insurance policy. Let’s look at a typical scenario.

Assume an investor has invested some $100 million in bonds issued by Italy but is worried that there might be a default event in light of a persistent trade war. To hedge default risk, the investor buys a CDS from a Spanish bank, La Mada. The CDS shields the investor from defaulted-related losses; in case Italy defaults, the counterparty, La Mada, will pay the originally invested$100.0 million to the investor (assuming the recovery rate is zero). The key thing to note here is that the value of the CDS, the fixed CDS spread s, is determined by the default probability of the reference asset – Italy. The “spread” of a CDS is the annual amount the investor must pay the counterparty (protection seller) over the length of the contract, expressed as a percentage of the notional amount

The value of the CDS is also determined by the joint default correlation of La Mada and Italy. If the correlation between Italy and La Mada increases, the present value of the CDS for the investor will decrease in which case they will suffer an unrealized(paper) loss. Thus, the investor is exposed to default correlation risk between the reference asset (Italy) and the counterparty (La Mada). In the worst-case scenario where the reference entity and La Mada jointly default, the investor stands to lose all of his investment.

The fact that both Italy and Spain are in Europe (and the European Union) implies there would most likely be a positive default correlation between them. This implies the investor has a wrong-way risk. The higher the correlation risk, the lower the CDS spread,s. That’s because an increase in correlation means a higher probability of the reference asset and the counterparty defaulting together (the investor would want to pay a lower premium if the probability of losing his investment increases).

Other examples of financial correlation risk include:

• The negative correlation between interest rates and commodity prices. If interest rates rise, losses occur in commodity investments
• During the Greek crisis of 2012, a positive correlation between Greek bonds and Mexican bonds led to heavy losses for investors in Mexican bonds.
• In 2012, US exporters were hurt by a decreasing Euro currency and increasing Euro sovereign debt.

#### Solution

First, we determine the value of the realized correlation:

\begin{align*} \rho_{\text{realized}} & =\cfrac {2}{n^2-n} \sum_{i>j} \rho_{ij} \\& =\cfrac {2}{3^2-3} [0.6+0.3+0.05] \\ & =0.3167 \\ \end{align*}

Next, we now calculate the payoff:

$$\text{Payoff} = N(\rho_{\text{realized}}-\rho_{\text{fixed}} ) = 10(0.3167 – 0.15) = 1,667,000$$

### How else can Investors Buy Correlation?

1. Buying call options on an Index and selling call options on individual components:

There is a positive relationship between correlation and volatility. if correlation increases between stocks making up the index, the implied volatility of call options (on the index) will also increase. The increase in price for the index call options is expected to be greater than the increase in price for individual stocks that have a short call position.

Example: Buying a call option on the S&P 500 index and selling call options on individual stocks of the S&P 500 is a way of buying correlation.

2. Paying fixed in a variance swap on an index and receiving fixed on individual components:

This works much like buying a call on an index and selling a call on the individual components. An increase in correlation for securities within the index causes the variance to increase. As a result, the present value for the variance swap buyer, the fixed variance swap payer, will increase. This increase usually outperforms the potential losses from the short variance swap positions on the individual components.

## Estimating the Impact of Different Correlations between Assets in the Trading Book on the VaR Capital Charge

The overall goal of any risk management process is to mitigate the main types of risk – market risk, credit risk, and operational risk. Other risk types include systemic risk, liquidity risk, volatility risk, and correlation risk. As established across the FRM curriculum, one of the most important risk measures is the value at risk (VaR). It helps us measure the potential loss in value over a given time interval, say, a month, with a specified degree of confidence.

The value at risk of a portfolio is given as:

$$\text{VaR}_\text P=\sigma_\text P \alpha \sqrt {x}$$

where

• $$\alpha$$ is the standard normal deviate corresponding to a specific confidence interval;
• x is the time horizon for the VaR;
• and $$\sigma_P$$ is the volatility of the portfolio

The volatility of the portfolio, $$\sigma_P$$, includes the correlation between assets in the portfolio. We can compute its value as:

$$\sigma_P=\sqrt { \beta_h C\beta_V }$$

where:

• $$\beta_h$$ = horizontal $$\beta$$ vector of investment amount;
• C = covariance matrix of returns;
• $$\beta_V$$ = the vertical $$\beta$$ vector of invested amount

#### Example: Correlation and VaR

An investor holds a two-asset portfolio with $20 million in asset A and$10 million in asset B. The portfolio correlation is 0.7, and the daily standard deviation of returns for assets A and B are 1.8% and 2.2%, respectively. What is the 10-day VaR of this portfolio at a 95% confidence level (i.e., $$\alpha$$ = 1.645)?

#### Solution

Step 1: Construct the variance covariance matrix

Since we have only two assets, we shall have a 2 × 2 matrix that will take the following shape:

$$\begin{bmatrix} Cov(1,1) & Cov(1,2) \\ Cov(2,1) & Cov(2,2) \end{bmatrix}$$

(We use “1” to represent asset A and “2” to represent asset B for convenience)

Recall that:

\begin{align*} \text{Cov}(X,X) & = \text{Var}(X); \\ \text{Cov}(X,Y) & = \text{Cov}(Y,X); \text{ and } \\ \text{Cov}(X,Y) & = \rho_{XY} × \sigma_X×\sigma_Y \\ \end{align*}

Thus,

\begin{align*} \text{Cov}(1,1) & =\sigma_1^2=0.018^2=0.000324; \\ \text{Cov}(2,2) & =\sigma_2^2=0.022^2=0.000484; \\ \text{Cov}(1,2) & = \text{Cov}(2,1) = 0.7×0.018×0.022=0.0002772; \\ \end{align*}

Thus, our covariance matrix is:

$$\begin{bmatrix} 0.000324 & 0.0002772 \\ 0.0002772 & 0.000484 \end{bmatrix}$$

Step 2: Determine the standard deviation of the portfolio

We do this by first solving $$\beta_h C$$ , followed by $$\beta_h C)\beta_V$$, and then finding the square root.

\begin{align*} \beta_h C & =\left[ \begin{matrix} 20 & 10 \end{matrix} \right] \begin{bmatrix} 0.000324 & 0.0002772 \\ 0.0002772 & 0.000484 \end{bmatrix} \\ & = \left[ \begin{matrix} 20×0.000324+10×0.0002772 & 20×0.0002772+10×0.000484 \end{matrix} \right] \\ & = \left[ \begin{matrix} 0.009252 & 0.010384 \end{matrix} \right] \\ (\beta_h C)\beta_V & =\left[ \begin{matrix} 0.009252 & 0.010384 \end{matrix} \right] \begin{bmatrix} 20 \\ 10 \end{bmatrix} \\ & =0.009252×20+0.010384×10 \\ & =0.2889 \\ \end{align*} Thus, $$\sigma_P=\sqrt { \beta_h C \beta_V }=\sqrt {0.2889} =0.5375$$

Step 3: Compute the VaR

$$\text{VaR}_P=\sigma_P \alpha \sqrt {x}=0.5375×1.645×\sqrt {10}=2.7960$$

#### Solution

Since there is only one borrower, the concentration risk is 1.

In the worst-case scenario, a default event by X results in a total loss of loan value. Given that there is a 5% probability that company X defaults, EL for the bank is $$500,000 (= 0.05 x 10,000,000).$$

This time, let’s assume that bank X extends two loans: a $5 million loan to A and a similar amount to B. Let’s assume further that A and B each have a 5% default probability. What is the concentration ratio? What will be the impact on the expected loss (EL) for the bank under the worst-case scenario? Assume the default correlation between companies is bigger than 0 and smaller than 1. #### Solution The bank’s concentration ratio is reduced to½. Since the default correlation between X and Y is bigger than 0 and smaller than 1, the worst-case scenario [i.e., the default of X and Y, P($$X\cap Y$$), with a loss of$500,000 is reduced. This can be seen in figure below:

The exact joint default probability $$P(X \cap Y)$$ depends on the correlation model and correlation parameter values.

If the default correlation between X and Y is 1.0, the expected loss for the bank is $500,000 (0.05 x$10,000,000). In such a case, there is no benefit from the lower concentration ratio. It would be the same as making a $10 million loan to one company. If we further decrease the concentration ratio (by extending loans to more borrowers), the worst-case scenario (i.e., the expected loss of 5%) decreases further. The defaults of companies X and Y are expressed as two binomial variables; they take the value 1 if in default, and 0 otherwise. The joint probability of default for the two binomial events is: $$P(X \cap Y)=\rho_{XY} \sqrt{ P_X (1-P_X)P_Y (1-P_Y) } +P_X P_Y$$ where: • $$\rho_{XY}$$ is the correlation coefficient; • $$P_X (1-P_X)$$ is the standard deviation of the binomially distributed variable X. ## Question 1 Calculate the payoff of a correlation swap if the number assets are 3 and the realized pairwise correlations of the log returns at maturity level are given as 0.95, 0.81 and 0.54. You are also given that the notational amount is$10 million at a 15% fixed rate and 1-year maturity.

1. $6.17 million 2.$3.61 million
3. $13.8 million 4.$7.67 million

The realized correlation is calculated as:

$${ \rho }_{ realized }=\frac { 2 }{ { n }^{ 2 }-n } { \Sigma }_{ i>j }{ \rho }_{ i,j }$$

Therefore, we have:

$${ \rho }_{ realized }=\frac { 2 }{ 3^{ 2 }-3 } \left( 0.54+0.81+0.95 \right) =0.7667$$

Following the equation:

$$N\left( { \rho }_{ realized }-{ \rho }_{ fixed } \right)$$

$$\Rightarrow$$The payoff of the correlation fixed rate player at maturity is

$$\text{\10 million} \times \left( 0.7667–0.15 \right) =6.17 \text{Million}$$

## Question 2

Suppose the correlation coefficient for a two asset portfolios 2.6, the daily standard deviation of return of asset 1 is 6% and 3% for asset 2. Seven million is invested in asset 1 and $9 million for asset 2. Calculate the 9day $$VaR$$ with a 99% confidence level. (Take $$\alpha$$ = 4.369) 1.$11.993
2. $6.3258 3.$23.115
4. \$17.235

Deriving the covariances ($$Cov$$):

\begin{align*}{ Cov }_{ 1,1 }&={ \rho }_{ 1,1 }{ \sigma }_{ 1 }{ \sigma }_{ 1 }=1\times 0.06\times 0.06=0.0036\\ { Cov }_{ 1,2 }&={ \rho }_{ 1,2 }{ \sigma }_{ 1 }{ \sigma }_{ 2 }=2.6\times 0.06\times 0.03=0.00468 \\ { Cov }_{ 2,1 }&={ \rho }_{ 2,1 }{ \sigma }_{ 2 }{ \sigma }_{ 1 }=2.6\times 0.03\times 0.06=0.00468 \\{ Cov }_{ 2,2 }&={ \rho }_{ 2,2 }{ \sigma }_{ 2 }{ \sigma }_{ 2 }=1\times 0.03\times 0.03=0.0009 \end{align*}

Hence our covariance matrix is:

$$A=\begin{bmatrix} 0.0036 & 0.00468 \\ 0.00468 & 0.00090 \end{bmatrix}$$

Finding $${ \sigma }_{ P }$$ we use equation:

$${ \sigma }_{ P }=\sqrt { { \beta }_{ h }c{ \beta }_{ V } }$$

\begin{align*}& \left[ 7\quad 9 \right] \begin{bmatrix} 0.0036 & 0.00468 \\ 0.00468 & 0.00090 \end{bmatrix}\\&=\left[ 7\times 0.0036+9\times 0.00468\quad 7\times 0.00468+9\times 0.0009 \right]\\&=\left[ \begin{matrix} 0.06732 & 0.04086 \end{matrix} \right] \end{align*}

Therefore:

$${ \beta }_{ h }c{ \beta }_{ V }=\left[ \begin{matrix} 0.06732 & 0.04086 \end{matrix} \right] \left[ \begin{matrix} 7 \\ 9 \end{matrix} \right] =7\times 0.06732+9\times 0.04286=83.9\%$$

Thus

\begin{align*}{ \sigma }_{ P }&=\sqrt { { \beta }_{ h }c{ \beta }_{ V } } =\sqrt { 83.9\% } = 91.5\% \\ \Rightarrow VaR _{ P }&={ \sigma }_{ P }\alpha \sqrt { x } =0.915\times 4.369\times \sqrt { 9 } =11.993 \end{align*}

## Question 3

Suppose that institution $$X$$ has a historical default probability of $$P\left( X \right) =7\%$$ and that company $$Y$$ has an historical default probability of $$P\left( Y \right) =5\%$$. What is their joint historical default probability if $$P\left( X \right)$$ and $$P\left( Y \right)$$ are independent?

1. 0.35%
2. 35%
3. 58.33%
4. 5.83%

Recall that events $$A$$ and $$B$$ are independent if their joint probabilities equal the product of their individual probabilities:

\begin{align*} P\left( A\cap B \right) &=P\left( A \right) P\left( B \right)\\ P\left( X\cap Y \right)& =0.35\% \end{align*}

But:

$$P\left( A \right) =\frac { P\left( A\cap B \right) }{ P\left( B \right) }$$

And:

$$P\left( B \right) =\frac { P\left( A\cap B \right) }{ P\left( A \right) } =P\left( B|A \right)$$

Therefore:

$$P\left( X \right) =\frac { 0.35\% }{ 5\% } =7\%$$

And:

\begin{align*}P\left( Y \right)& =\frac { 0.35\% }{ 7\% } =5\% \\ \Rightarrow P\left( X\cap Y \right) &=P\left( X \right) P\left( Y \right) \end{align*}

Since:

$$0.35\%=7\%\times 5\%$$

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