 ### Some Correlation Basics: Properties Motivation Terminologies

Financial correlation and financial correlation risk are the main subjects of focus for this chapter. The relationship between correlation risk and other financial risks are also explored.

# Financial Correlations

There are two types of financial correlations namely: Static and Dynamic Financial Correlations. In Static financial correlations, two or more financial assets are measured to determine their association within a given time. In Dynamic, the measure is on how the two, or more, assets move jointly with time. The behavior of correlations is quite random and unpredictable and hence should be modeled as stochastic processes.

# Financial Correlation Risk

When the correlations between two or more variables move adversely, the risk of financial loss caused is called Financial Correlation Risk. In particular, the $$VaR$$ concept will measure the risk of financial loss as a result of increased correlation of assets returns. Non-financial variables such as political events can also be involved in the correlation risk.

# Motivation: In Finance Correlation and Correlation Risks are Everywhere

Any entity that is exposed to correlation has a correlation risk. We focus on several areas of finance that financial correlation appears.

## Investments and correlation

Studies have revealed that increased diversification will lead to a $${ return }/{ risk }$$ ratio increase. There exists an inverse relation between correlation and diversification as high diversification is associated with low correlation.

To further explain this, consider a return on asset $$X$$ at time $$t$$ as $${ x }_{ t }$$, and $${ y }_{ t }$$ be the return of asset $$Y$$ at time $$t$$. The percentage change $${ \left( { S }_{ t }-{ S }_{ t-1 } \right) }/{ { S }_{ t-1 } }$$ is the return where $$S$$ is a price or a rate. Given that the average return of asset $$X$$ from 2009 to 2013 is $${ \mu }_{ x }=29.03\%$$ and $${ \mu }_{ y }=20.07\%$$ return for asset $$Y$$. Assigning the weight $${ W }_{ x }$$ to asset $$X$$ and $${ W }_{ Y }$$ be weight assigned to asset $$Y$$. The portfolio return is:

$${ \mu }_{ P }={ W }_{ X }{ \mu }_{ X }+{ W }_{ Y }{ \mu }_{ Y }\quad and:\quad { W }_{ X }+{ W }_{ Y }=1$$

The following equation derived for asset $$X$$ is the standard deviation of returns which can similarly be applied for asset $$Y$$:

$${ \sigma }_{ X }=\sqrt { \frac { 1 }{ n-1 } \sum _{ t=1 }^{ n }{ { \left( { x }_{ t }-{ \mu }_{ X } \right) }^{ 2 } } }$$

Where $$n$$ is the number of observed points in time.

In a covariance, the strength of the relationship between two variables is measured. For assets $$X$$ and $$Y$$ the covariance of returns is derived as:

$${ Cov }_{ XY }=\frac { 1 }{ n-1 } \sum _{ t=1 }^{ n }{ \left( { x }_{ t }-{ \mu }_{ X } \right) \left( { y }_{ t }-{ \mu }_{ Y } \right) }$$

Since values taken in covariance computation are between $$-\infty$$ and $$+\infty$$ it is not easy to interpret covariance. We, therefore, apply the Pearson correlation coefficient $${ \rho }_{ XY }$$:

$${ \rho }_{ XY }=\frac { { Cov }_{ XY } }{ { \sigma }_{ X }{ \sigma }_{ Y } }$$

The standard deviation for the two asset portfolio $$P$$ is derived as:

$${ \sigma }_{ P }=\sqrt { { W }_{ X }^{ 2 }{ \sigma }_{ Y }^{ 2 }+{ W }_{ Y }^{ 2 }{ \sigma }_{ Y }^{ 2 }+2{ W }_{ X }^{ 2 }{ W }_{ Y }^{ 2 }Co{ v }_{ XY } }$$

A higher standard deviation implies a higher risk of an asset portfolio. Risk-averse investors do not like high standard deviation as it implies high returns fluctuation.

## Trading and Correlation

Traders often forecast changes in correlation and try to gain financially from these changes in correlation. In correlation trading, assets whose prices are determined at least in part by the co-movement of one or more asset in time are traded.

### Multi-Asset Options:

In this group of correlation options many types of assets are traded. Let $${ S }_{ 1 }$$ be price of asset 1 and $${ S }_{ 2 }$$ be the price of asset 2 at option maturity and $$K$$ be the strike price. Therefore:

• $$Payoff=max\left( { S }_{ 1 },{ S }_{ 2 } \right)$$ for options on the better of two.
• $$Payoff=min\left( { S }_{ 1 },{ S }_{ 2 } \right)$$ for options on the worse of two.
• $$Payoff=max\left[ 0,max\left( { S }_{ 1 },{ S }_{ 2 } \right) -K \right]$$ for calloptions on the maximum of two
• $$Payoff=max\left( 0,{ S }_{ 2 }-{ S }_{ 1 } \right)$$ for exchange options.
• $$Payoff=max\left[ 0,\left( { S }_{ 2 }-{ S }_{ 1 } \right) -K \right]$$ for spread call options
• $$Payoff=max\left( { S }_{ 1 },{ S }_{ 2 },cash \right)$$ for options on the better of two or cash.
• $$Payoff=max\left( { 0,S }_{ 1 }-{ K }_{ 1 },{ S }_{ 2 }-{ K }_{ 2 } \right)$$ for Dual-strike call options.
• $$Payoff=max\left[ \sum _{ i=1 }^{ n }{ { n }_{ i }{ S }_{ i }-K,0 } \right]$$ for a portfolio of basket options, where $$n_{ i }$$ is the weight of assets $$i$$.

Consider an exchange option with $$Payoff=max\left( 0,{ S }_{ 2 }-{ S }_{ 1 } \right)$$, which implies that a buyer can exchange asset 1 and get asset 2 at option maturity if $${ S }_{ 2 }>{ S }_{ 1 }$$. To derive the exchange option price, we rewrite the payoff equation: $$max\left( 0,{ S }_{ 2 }-{ S }_{ 1 } \right) ={ S }_{ 1 }\quad max\left[ 0,\left( { { S }_{ 2 } }/{ { S }_{ 1 } } \right) -1 \right]$$ and input the covariance between asset $${ S }_{ 1 }$$ and $${ S }_{ 2 }$$n the implied volatility function of the exchange option using a variation equation:

$${ \sigma }_{ E }=\sqrt { { \sigma }_{ A }^{ 2 }+{ \sigma }_{ B }^{ 2 }-2Co{ v }_{ AB } }$$

Where $${ \sigma }_{ E }$$ is the implied volatility between asset prices $${ S }_{ 1 }$$ and $${ S }_{ 2 }$$ .

### Quanto Option:

The term quanto is derived from the word quantity. Domestic investors are allowed to exchange potential option payoff in a foreign currency bank into his home currency at a fixed exchange rate, by this option and therefore protecting them against currency risk. The institutions selling quanto calls do not know what the depth of the money the quanto call is going to be, and the option maturity exchange rate at which the conversion of the stochastic foreign currency pay off to domestic currency will happen.

## Correlation Swap:

The exchange between a fixed correlation and the $${ realized }/{ stochastic }$$ correlation occurs in a correlation swap. In a correlation swap, paying a fixed rate is called buying correlation. Realized correlation $$\rho$$ is calculated as:

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

Where $${ \rho }_{ i,j }$$ is the Pearson correlation between assets $$I$$ and $$j$$ and $$n$$ is the number of assets in the portfolio.

The correlation payoff swap for the correlation fixed rate payer at maturity is:

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

### Buying call options on an Index and selling call options on individual components:

Since there is a positive relation between correlation and volatility, therefore, if an index and stocks increase, the implied volatility of the call of the index will also increase. The potential loss from the increase in short call positions is likely to be outperformed by this increase. So buying a call option on an index like the S&P 500 and selling the call options on individual stocks of the S&P 500 is a way of buying correlation.

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

An increase in correlation will lead to an increase in the variance hence leading to an increase in the present value for the variance swap buyer and the fixed variance swap payer. The potential losses from the short variance swap positions are expected to be outperformed by the said increase.

## Risk Management and Correlation

The process of identifying, quantifying and possibly reducing financial risk is called Financial Risk Management. These risks are:market, credit,liquidity, systemic, volatility, correlation and operational risks.$$VaR$$ measures the maximum loss of a portfolio with respect to a certain probability for a certain time frame by the following equation:

$${ VaR }_{ P }={ \sigma }_{ P }\alpha \sqrt { x }$$

Where $${ VaR }_{ P }$$ is the $$VaR$$ for portfolio $$P$$, $$\alpha$$ is the abscise value for a standard normal distribution corresponding to a given confidence level, and $$x$$ is the time horizon for the $$VaR$$.

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

Where $${ \beta }_{ h }$$ is the horizontal $$\beta$$ vector,$${ \beta }_{ V }$$ is the vertical $$\beta$$ vector of invested amounts and $$C$$ is the covariance matrix of the returns of assets.

## Regulation and Correlation

In regulatory frameworks like the Basel accords, correlations are critical inputs.

### Basel I, II and III:

These are regulatory guidelines to ensure the stability of banking systems. To enforce the Basel accords for their banks, most countries have created regulations. In an effort to ensure banks get incentives for risk measurement and management systems enhancement, the Basel accords have been put in place.

# How does Correlation Risk Fit into the Broader Picture of Risks in Finance?

Correlation risk is closely related to systemic risk and plays an important role in market and credit risks which constitutes the main types of financial risk.

## Correlation Risk and Market Risk

Comprising of equity risk, interest rate risk, currency risk and commodity risk, market risk is greatly influenced by correlation risk. The concept of $$VaR$$ is typically applied for market risk measurement since it has as an input, the covariance matrix of assets in the portfolio thereby implicitly incorporating correlation risk. For extreme events, Expected Shortfall will also measure the market risk since just like $$VaR$$,a thorough $$ES$$ valuation will naturally include the correlation between asset returns in the portfolio.

## Correlation Risk and Credit Risk

Comprising of migration and default risk, credit risk is affected by correlation risk. The risk that a debtor’s credit quality will decrease is termed as the migration risk. Financial lending is greatly influenced by the degree of occurrence of defaults. Default correlations are also crucial for insurers. To diversify the credit risk, a low debtor’s default correlation is desired.To reduce default correlation risk, it is advisable for lenders to have a loan portfolio that is sector diversified since intra-sector default correlations are than inter-sector default correlations.

## Correlation and Systemic Risk

Systemic risk can be defined as the risk of collapse of an entire system of finance or financial market like the 2008 total collapse of the credit market. These financial failures will typically spread to the economy leading to a GDP decrease, increased unemployment and consequently reduced living standards. There is interdependence between systemic risk and correlation as a systemic decline in stocks will sharply increase correlations almost throughout the entire stock market.

## Correlation Risk and Concentration Risk

As a result of concentrated exposure to a particular group of counterparties, the risk of financial loss encountered is referred to as concentration risk which can be quantified with the concentration ratio. A lower concentration ratio means a more diversified default risk of the creditor. Counterparts can also be categorized into groups and then analyze the sector concentration risk. Higher sector diversification is as a result of higher number of different sectors a creditor has lent to.

## Dependence and Correlation

If the occurrence of an event affects the probability of another, then the two events are statistically dependent. If the occurrence of one event does not affect the probability of another, then the two events are independent. Statistically, events $$A$$ and $$B$$ are independent if their joint probabilities equal the product of their individual probabilities:

$$P\left( A\cap B \right) =P\left( A \right) P\left( B \right)$$

Therefore;

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

From the Kolmogorov definition:

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

And:

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

Where: $$P\left( A|B \right)$$ Is the conditional probability of $$A$$ with respect to $$B$$ and is read as probability of $$A$$ given $$B$$ and $$P\left( B|A \right)$$ is the probability of $$B$$ with respect to $$A$$ and is read as the probability of $$B$$ given $$A$$.

### Correlation:

In trading, correlation is defined as any co-movement of asset prices in time. In statistics, it is defined as the linear dependency derived from the Pearson correlation model. We relate this dependence to the Pearson covariance. The strength of the linear relationship between two variables is measured by covariance. The Pearson covariance is derived with expectation values:

$$Cov\left( X,Y \right) =\left[ E\left( X \right) -E\left( X \right) \left( Y \right) -E\left( Y \right) \right] =E\left( XY \right) -E\left( X \right) E\left( Y \right) \quad \quad \quad \quad \quad \left( a \right)$$

Where $$E\left( X \right)$$ and $$E\left( Y \right)$$ are the expected values of $$X$$ and $$Y$$ respectively and are also known as the mean.$$E\left( XY \right)$$ is the mean of the product of the random variables $$X$$ and $$Y$$.

Since covariance is not easily interpreted, a correlation coefficient is usually used, and the Pearson correlation coefficient is defined as:

$$\rho \left( X,Y \right) =\frac { Cov\left( X,Y \right) }{ \sigma \left( X \right) \sigma \left( Y \right) } \quad \quad \quad \quad \quad \left( b \right)$$

Where $$\sigma \left( X \right)$$ and $$\sigma \left( Y \right)$$ are the standard deviations of $$X$$ and $$Y$$ respectively.

### Independence and Un-correlatedness:

The covariance of two independent variables is zero,but zero covariance does not mean independence since variables can have zero covariance even with independence.

From equation ($$a$$) we can have that:

$$Cov\left( X,Y \right) =E\left( XY \right) -E\left( X \right) E\left( Y \right)$$

Inputting $$Y={ X }^{ 2 }$$ ,We derive:

$$COV\left( X,Y \right) =E\left( X{ X }^{ 2 } \right) -E\left( X \right) E\left( { X }^{ 2 } \right)$$

$$=E\left( { X }^{ 3 } \right) -E\left( X \right) E\left( { X }^{ 2 } \right)$$

For a uniform variable $$X$$ bounded in $$\left[ -1,+1 \right]$$, the mean E(X) is zero and:

$$Cov\left( X,Y \right) =0-0E\left( { X }^{ 2 } \right)$$

$$=0$$

Since the Pearson correlation concept measures linear dependence only, the zero given by the Pearson covariance (Correlation coefficient) then the variables are uncorrelated despite being dependent.

### Percentage and Logarithmic Changes:

Relative changes, in finance, will always express growth rates, $${ \left( { S }_{ t }-{ { S }_{ t-1 } } \right) }/{ { S }_{ t-1 } }$$ ,Where $${ S }_{ t }$$ and $${ S }_{ t-1 }$$ are asset prices at time $$t$$ and $$t – 1$$ respectively.

With the help of natural logarithms, relative changes can be estimated as:

$${ \left( { S }_{ t }-{ { S }_{ t-1 } } \right) }/{ { S }_{ t-1 } }\approx ln\left( { { S }_{ t } }/{ { S }_{ t-1 } } \right)$$

For small differences, between $${ S }_{ t }$$ and $${ S }_{ t-1 }$$ , $$ln\left( { { S }_{ t } }/{ { S }_{ t-1 } } \right)$$ is the log return whose merit is that it can be added over time whereas relative changes are not additive over time.

# Practice Questions

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 is 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 correct answer is A. The realized correlation is calculated as: $${ \rho }_{ realized }=\frac { 2 }{ { n }^{ 2 }-n } { \Sigma }_{ i>j }{ \rho }_{ i,j }$$ Therefor,$$e$$ 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 $$10\quad million\times \left( 0.7667–0.15 \right) =6.17\quad Million$$ 2) Suppose the correlation coefficient for a two asset portfoliois 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

The correct answer is A.

Deriving the covariances ($$Cov$$):

$${ Cov }_{ 11 }={ \rho }_{ 11 }{ \sigma }_{ 1 }{ \sigma }_{ 1 }=1\times 0.06\times 0.06=0.0036$$

$${ Cov }_{ 12 }={ \rho }_{ 12 }{ \sigma }_{ 1 }{ \sigma }_{ 2 }=2.6\times 0.06\times 0.03=0.00468$$

$${ Cov }_{ 21 }={ \rho }_{ 21 }{ \sigma }_{ 2 }{ \sigma }_{ 1 }=2.6\times 0.03\times 0.06=0.00468$$

$${ Cov }_{ 22 }={ \rho }_{ 22 }{ \sigma }_{ 2 }{ \sigma }_{ 2 }=1\times 0.03\times 0.03=0.0009$$

Hence our covariance matrix is

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

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

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

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

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

$${ \sigma }_{ P }=\sqrt { { \beta }_{ h }c{ \beta }_{ V } } =\sqrt { 83.9\% } = 91.5\%$$

$$\Rightarrow Va{ R }_{ P }={ \sigma }_{ P }\alpha \sqrt { x } =0.915\times 4.369\times \sqrt { 9 } =11.993$$

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. 35%
2. 40%
3. 58.33%
4. 41.67%

The correct answer is A.

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

$$P\left( A\cap B \right) =P\left( A \right) P\left( B \right)$$

$$P\left( X\cap Y \right) =35\%$$

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:

$$P\left( X \right) =\frac { 35\% }{ 5\% } =7\%$$

And:

$$P\left( Y \right) =\frac { 35\% }{ 7\% } =5\%$$

$$\Rightarrow P\left( X\cap Y \right) =P\left( X \right) P\left( Y \right)$$

Since:

$$35\%=7\%\times 5\%$$