LO 6.1: Describe financial correlation risk and the areas in which it appears in

LO 6.1: Describe financial correlation risk and the areas in which it appears in finance.
Correlation risk measures the risk of financial loss resulting from adverse changes in correlations between financial or nonfinancial assets. An example of financial correlation risk is the negative correlation between interest rates and commodity prices. If interest rates rise, losses occur in commodity investments. Another example of this risk occurred during the 2012 Greek crisis. The positive correlation between Mexican bonds and Greek bonds caused losses for investors of Mexican bonds.
The financial crisis beginning in 2007 illustrated how financial correlation risk can impact global markets. During this time period, correlations across global markets became highly correlated. Assets that previously had very low or negative correlations suddenly become very highly positively correlated and fell in value together.
Nonfinancial assets can also be impacted by correlation risk. For example, the correlation of sovereign debt levels and currency values can result in financial losses for exporters. In 2012, U.S. exporters experienced losses due to the devaluation of the euro. Similarly, a low gross domestic product (GDP) for the United States has major adverse impacts on Asian and European exporters who rely heavily on the U.S. market. Another nonfinancial example is related to political events, such as uprisings in the Middle East that cause airline travel to decrease due to rising oil prices.
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Financial correlations can be categorized as static or dynamic. Static financial correlations do not change and measure the relationship between assets for a specific time period. Examples of static correlation measures are value at risk (VaR), correlation copulas for collateralized debt obligations (CDOs), and the binomial default correlation model. Dynamic financial correlations measure the comovement of assets over time. Examples of dynamic financial correlations are pairs trading, deterministic correlation approaches, and stochastic correlation processes.
Structured products are becoming an increasing area of concern regarding correlation risk. The following example demonstrates the role correlation risk plays in credit default swaps (CDS). A CDS transfers credit risk from the investor (CDS buyer) to a counterparty (CDS seller).
Suppose an investor purchases $1 million of French bonds and is concerned about France defaulting. The investor (CDS buyer) can transfer the default risk to a counterparty (CDS seller). Figure 1 illustrates the process for an investor transferring credit default risk by purchasing a CDS from Deutsche Bank (a large European bank).
Figure 1: CDS Buyer Hedging Risk in Foreign Bonds
Fixed CDS spread, s ———————–
……………………….. $ 1 million payout
* 1 1 * if France defaults
Deutsche Bank (Counterparty)
(CDS buyer)
Coupon payment k
$1 million French Bond
(Reference asset)
Assume the recovery rate is zero with no accrued interest in the event of default. The investor (CDS buyer) is protected if France defaults because the investor receives a $1 million payment from Deutsche Bank. The fixed CDS spread is valued based on the default probability of the reference asset (French Bond) and the joint default correlation of Deutsche Bank and France. A paper loss occurs if the correlation risk between Deutsche Bank and France increases because the value of the CDS will decrease. If Deutsche Bank and France default (worst case scenario), the investor loses the entire $ 1 million investment.
If there is positive correlation risk between Deutsche Bank and France, the investor has wrong-way risk (WWR). The higher the correlation risk, the lower the CDS spread, s. The increasing correlation risk increases the probability that both the French bond (reference asset) and Deutsche Bank (counterparty) default.
The dependencies between the CDS spread, s, and correlation risk may be nonmonotonous. This means that the CDS spread may sometimes increase and sometimes decrease if correlation risk increases. For example, for a correlation o f1 to 0.4, the CDS spread may
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increase slightly. This is due to the fact that a high negative correlation implies either France or Deutsche Bank will default, but not both. If France defaults, the $1 million is recovered from Deutsche Bank. If Deutsche Bank defaults, the investor loses the value of the CDS spread and the investor will need to repurchase a CDS spread to hedge the position. The new CDS spread cost will most likely increase in the event that Deutsche Bank defaults or if the credit quality of France decreases.
There are many areas in finance that have financial correlations. Five common finance areas where correlations play an im portant role are (1) investments, (2) trading, (3) risk management, (4) global markets, and (3) regulation.
Correlations in Financial Investments
In 1932, Flarry Markowitz provided the foundation of modern investment theory by demonstrating the role that correlation plays in reducing risk. The portfolio return is simply the weighted average of the individual returns where the weights are the percentage of investment in each asset. The following equation defines the average return (i.e., mean) for a portfolio, pp, comprised of assets X and Y Asset 26 has a weight of wx and an average return of px, and asset Thas a weight of wY and an average return of pY.
Fp = wxFx + w y Fy
The standard deviation of a portfolio is determined by the variances of each asset, the weights of each asset, and the covariance between assets. The risk or standard deviation (i.e., volatility) for a two-asset portfolio is calculated as follows:
XX “F WyCTy T 2wy W y C O V ^ y
V 9
2 2
Let us review how variances, covariance, and correlation are calculated using the following example. Suppose an analyst gathers historical prices for two assets, X and Y, and calculates their average returns as illustrated in Figure 2.
Figure 2: Prices and Returns for Assets X and Y
Average Return
Return X
Return Y
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The calculations for determining the standard deviations, variances, covariance, and correlation for assets X and Y are illustrated in Figure 3.
Figure 3: Variances and Covariance for Assets X and Y
> 1 :
0.0000 0.4701 0.0687 0.0061 0.1170 0.6620 0.1655 0.4068
(0.0001) (0.2927) (0.0831) 0.0328 (0.1704) (0.5135) (0.1284)
Return X
Return Y
2010 2011 2012 2013 2014 Mean
0.3333 (0.1250) 0.6190 (0.1176) 0.8000 0.3019
0.2000 0.8889 (0.0588) 0.1250 (0.1389) 0.2032
Xt~ JAx 0.0314 (0.4269) 0.3171 (0.4196) 0.4981
Variance Standard Deviation Correlation
t~^Y (x, – Y (0.0032) 0.6857 (0.2621) (0.0782) (0.3421)
0.0010 0.1823 0.1006 0.1761 0.2481 0.7079 0.1770 0.4207 (0.7501)
Notice that the sixth and seventh columns of Figure 3 are used to calculate the variance of X and Y, respectively. The deviation from each respective mean is squared to calculate the variance for each asset: (Xt px)2 for AT and (Y pY)2 fr T. The sum of the deviations is then divided by four (i.e., the number of observations minus one for degrees of freedom). For example, the asset AT variance is calculated by taking 0.7079 and dividing by 4 (i.e., n 1) to get 0.1770.
Covariance is a measure of how two assets move together over time. The last column of Figure 3 illustrates that the calculation for covariance is similar to the calculation for variance. Flowever, instead of squaring each deviation from the mean, the last column multiplies the deviations from the mean for each respective asset together. This not only captures the magnitude of movement but also the direction of movement. Thus, when asset returns are moving in opposite directions for the same time period, the product of their deviations is negative. The following equation defines the calculation for covariance. The sum of the products of the deviations from the means is 0.3133 in the last column of Figure 3. Covariance is calculated as -0.1284 by dividing -0.5135 by 4 (i.e., n – 1).
( X t nx )(Yt – n Y)
c o v x y = 1=1—————–;—————-
In finance, the correlation coefficient is often used to standardize the comovement or covariance between assets. The following equation defines the correlation for two assets, X and Y, by dividing covariance, cov^y* by the product of the asset standard deviations, <Txoy
PXY covXY C T Xa Y
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The correlation in this example is 0.7501, which is calculated as:
-0.1284 / (0.4207 x 0.4068) = -0.7501
In his research, Markowitz emphasized the importance of focusing on risk-adjusted returns. The return/risk ratio measures the average return for a portfolio, pp, by the risk of the portfolio, Op. Figure 2 provided the average return for Xand Y as 0.3019 and 0.2032, respectively. If we assume the portfolio is equally weighted, the average return for the portfolio is 0.2526, the correlation between assets X and Kis -0.7501, and the standard deviations for Xand Yare 0.4207 and 0.4068, respectively. The standard deviation for an equally-weighted portfolio is determined using the following expression:
x 0.42072) + (o.52 x 0.40682) + (2 x 0.5 x 0.5 x -0.1284)
= V0.02142 = 0.1464
The return/risk ratio of this equally-weighted two-asset portfolio is 1.725 (calculated as 0.2526 divided by 0.1464). Figure 4 illustrates the relationship of the return/risk ratio and correlation. The lower the correlation between the two assets, the higher the return/risk ratio. Avery high negative correlation (e.g., 0.9) results in a return/risk ratio greater than 250%. Avery high positive correlation (e.g., +0.9) results in a return/risk ratio near 50%.
Figure 4: Relationship of Return/Risk Ratio and Correlation
Correlation in Trading with Multi-Asset Options
Correlation trading strategies involve trading assets that have prices determined by the comovement of one or more assets over time. Correlation options have prices that are very sensitive to the correlation between two assets and are often referred to as multi-asset options.
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A quick review of the common notation for options is helpful. Assume the price of asset one and two are noted as S l and S2, respectively, and that the strike price, K, for a call option is the predetermined price an asset can be purchased. Likewise, the strike price, K, for a put option is the predetermined price an asset can be sold for.
The correlation between the two assets price of correlation options. Figure 5 lists a number of multi-asset correlation strategies along with their payoffs. For all of these strategies, a lower correlation results in a higher option price. A low correlation is expected to result in one asset price going higher while the other is lower. Thus, there is a better chance of a higher payout.
and S2 is an important factor in determining the
Figure 5: Payoffs for Multi-Asset Correlation Strategies
Correlation strategies
Option on higher of two stocks Call option on maximum of two stocks max[0, max(S15 S2) K] Exchange option Spread call option Dual-strike call option
max(0, S2 Sj) max(0, S2 S l K) max(0, S l K p S2 K2)
max(Sp S2)
Portfolio of basket options
n V 'n j xSi K,0 i=l
, where ^ = weight of asset i
.Another correlation strategy that is not listed in Figure 5 is a correlation option on the worse of two stocks where the payoff is the minimum of the two stock prices. This is the only correlation option where a lower correlation is not desirable because it reduces the correlation option price.
We can better understand the role correlation plays by taking a closer look at the valuation of the exchange option. The exchange option has a payoff of max(0, S2 Sj). The buyer of the option has the right to receive asset 2 and give away asset 1 when the option matures. The standard deviation of the exchange option, a , is the implied volatility of S2 / S p which is defined as:
+ CTy 2 c OVx y
Implied volatility is an important determinant of the options price. Thus, the exchange option price is highly sensitive to the covariance or correlation between the two assets. The price of the exchange option is close to zero when the correlation is close to 1 because the two asset prices move together, and the spread between them does not change. The price of the exchange option increases as the correlation between the two assets decreases because the spread between the two assets is more likely to be greater.
Quanto Option
The quanto option is another investment strategy using correlation options. It protects a domestic investor from foreign currency risk. However, the financial institution selling the
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quanto call does not know how deep in the money the call will be or what the exchange rate will be when the option is exercised to convert foreign currency to domestic currency. Lower correlations between currencies result in higher prices for quanto options.
Example: Quanto option
Suppose a U.S. investor buys a quanto call to invest in the Nikkei index and protect potential gains by setting a fixed currency exchange rate (USD/JPY). How does the correlation between the call on the Nikkei index and the exchange rate impact the price of the quanto option?
The U.S. investor buys a quanto call on the Nikkei index that has a fixed exchange rate for converting yen to dollars. If the correlation coefficient is positive (negative) between the Nikkei index and the yen relative to the dollar, an increasing Nikkei index results in an increasing (decreasing) value of the yen. Thus, the lower the correlation, the higher the price for the quanto option. If the Nikkei index increases and the yen decreases, the financial institution will need more yen to convert the profits in yen from the Nikkei investment into dollars.
Correlation Swap