MR 8 – Correlation Basics – Definitions, Applications and Terminology

Two asset prices rarely move independently of each other, and the whole field of financial correlation is about measuring and managing that shared movement. Before getting to the mechanics, it helps to see where the idea came from, because the statistical tools used on trading floors today were built more than a century ago for very different purposes.

Where the idea of correlation came from

The French mathematician Auguste Bravais studied the behaviour of error terms in 1846 and, in doing so, arrived at what we now call the regression line. Helen Walker (1929) later described him as a sort of Columbus who found correlation without realising what he had found. In 1886 Sir Walter Galton built a simple linear regression and noticed the tendency he called “Regression to Mediocrity”, which modern finance knows as mean reversion. His student Karl Pearson, starting in 1900, turned the idea into a formal tool: he framed the correlation coefficient as a product moment measure, and his wider output included the method of moments, principal component analysis and the early machinery of hypothesis testing, with P-values and Chi-squared distances. The Pearson coefficient remains the default meaning of the word “correlation” in most textbooks.

Static financial correlations

Speaking informally, without heavy mathematics, financial correlations fall into two families. A static correlation describes how strongly two or more assets are associated at one point in time or over one fixed window. Four familiar cases show the range:

• Bond prices and their own yields, observed at a single moment, are negatively associated: when the yield is higher, the price is lower.
• The classic value at risk (VaR) model asks how much a portfolio of correlated assets could lose, at a stated probability, over a stated horizon. The correlations between the assets are fixed inputs for that horizon.
• The copula approach for collateralised debt obligations (CDOs) assigns default correlations to every asset in the pool, which typically holds 125 names, for a given period.
• Lucas (1995) proposed a binomial model of default correlation, itself a special case of the Pearson framework, which estimates how likely two assets are to default together within a short window.

Dynamic financial correlations

A dynamic correlation, by contrast, describes how assets move together through time. Pairs trading is the simplest example. Suppose the returns x and y of two assets have tracked each other closely for a long period. If asset X then lags asset Y, the trader buys X and sells Y, betting that the gap will close again. The profit depends entirely on the co-movement continuing. If the historical relationship breaks down, the long position in X and the short position in Y can both lose money at the same time, so a pairs trader is exposed to a change in correlation rather than to the direction of the market as a whole.

A second example comes from option pricing. The Heston model (1993), one of the deterministic correlation approaches, links the Brownian motions of two assets so that their random shocks are correlated through time with parameter ρ. The core relation builds the first shock from the second plus an independent third shock:

The weight on the independent shock is chosen so that the variance of dz₁ stays the same whatever value ρ takes. When ρ is 1 the two motions are identical, and when ρ is 0 they are unrelated.

A third view treats correlation itself as random. Observed correlations jump around in ways that are hard to forecast, so a natural response is to let correlation follow its own random process. Models built this way are time dependent by construction, which lets them reproduce the way real correlations drift and spike.

Financial correlation risk is the possibility of losing money because the correlation between two or more variables moves against you. The variables can be almost anything. A holder of Mexican bonds was hurt in 2012 when Greek bond prices fell, because the two markets were positively correlated during the Greek crisis. A commodity investor can suffer when interest rates rise, because commodity prices and rates tend to be negatively correlated. A bond investor can be hurt by the link between the issuer of a bond and the company insuring it.

The variables need not be financial at all. Rising sovereign debt and a weakening currency hurt exporters in Europe in 2012, where a falling euro damaged US exporters. Tension in the Middle East can push up the oil price and squeeze airline profits, and slower US growth can damage exporters and investors in Asia and Europe, because economic activity and markets across the world are linked.

Correlation risk matters most in risk management. When asset returns become more correlated, losses tend to arrive together, and the portfolio’s VaR rises. That is exactly what happens in a severe systemic crisis: during the great recession of 2007 to 2009, assets that risk managers had treated as lowly or even negatively correlated fell at the same time.

The hedged Spanish bond

A credit default swap (CDS) moves credit risk from an investor, the CDS buyer, to a counterparty, the CDS seller. Imagine an investor who holds USD 1 million of a Spanish government bond and, fearing a Spanish default, buys a CDS from the French bank BNP Paribas. If Spain defaults, BNP Paribas pays the investor the full USD 1 million. For simplicity, set both the recovery rate and any accrued interest to zero.

The investor pays a fixed CDS spread s for this protection. The spread is the insurance premium, and it is called a spread because it roughly equals the yield on the risky Spanish bond minus the yield on a riskless bond. Its level depends mainly on how likely Spain is to default, but it also depends on the default correlation between Spain (the reference entity r) and BNP Paribas (the counterparty c). If that correlation rises, the protection is worth less, so the present value of the CDS falls for the investor and they record a paper loss. In the worst case Spain and BNP Paribas default together, and the investor loses the whole USD 1 million.

Because both names are European, a positive default correlation between them is a reasonable assumption. This is wrong-way correlation risk, usually shortened to wrong-way risk (WWR). If both default probabilities rise, the exposure to Spain increases because the CDS is now more valuable to the investor, and at the same moment the chance that BNP Paribas can actually pay grows smaller. The exposure and the credit quality of the protection seller move in the wrong direction together.

It helps to lay out every combination of events at maturity and ask what the investor ends up with. The bond is worth USD 1 million if Spain pays, and the CDS is worth something only if Spain fails while BNP Paribas survives.

Only the last row produces a total loss, and its probability is governed by the default correlation between the two names. A positive correlation fattens that row, which is why the investor’s real exposure is not to Spain alone but to Spain and BNP Paribas failing together.

Why the curve is non-monotonic

For correlations from about -0.3 upward, a higher correlation means a higher chance that Spain and BNP Paribas fail together, so the fair spread falls, and at a perfect correlation of 1 the protection is worthless because BNP Paribas would fail whenever Spain does. More surprising is the left part of the curve, where the spread falls slightly as correlation moves from about -0.3 toward -1. A strongly negative correlation raises the chance that exactly one of the two names defaults. If Spain defaults alone, BNP Paribas pays out as promised. If BNP Paribas defaults alone, the investor loses the protection and has to buy it again, possibly at a higher price if Spain’s credit has weakened since the first trade, for example from a spread of 3% to 6%. These two outcomes together produce the slight decline. Pricing CDSs with counterparty risk and this correlation is treated in Kettunen and Meissner (2006).

The broader lesson is that a price need not move in one direction as correlation rises. The CDS spread sometimes increases and sometimes decreases, and that non-monotonic behaviour is a warning against assuming that “more correlation means more risk” in every product.

Correlation runs through five broad areas of finance: investments, trading, risk management, the global financial crisis and regulation. Any entity exposed to correlation carries correlation risk, the risk that the correlation changes. Investments come first because portfolio theory makes the link plain. The Capital Asset Pricing Model work of Markowitz (1952) and Sharpe (1964) shows that greater diversification improves the ratio of return to risk, and high diversification goes hand in hand with low correlation.

A two-asset portfolio

Take two assets, X and Y, whose year-end prices are shown below. A return is a percentage change, (S_t – S_t-1)/S_t-1, where S is a price or a rate. The table adds a final column, the return of a portfolio holding equal weights in both assets, to show how much the two series offset each other.

Portfolio column derived here as the simple average of the two asset returns.

Over 2014 to 2018 the mean return of X is μ_X = 29.03% and the mean return of Y is μ_Y = 20.07%. With weights w_X and w_Y that sum to 1, the expected portfolio return is the weighted average:

Volatility, the standard deviation of returns, uses the n observed returns of each asset. The volatility of Y follows the same form.

For the data above, σ_X = 44.51% and σ_Y = 47.58%.

Covariance and the Pearson coefficient

Covariance captures how two variables co-vary, or more precisely the strength of their linear relationship. It averages the product of each pair of deviations from the means:

Here COV_XY = -0.1567. The trouble with covariance is scale: it can take any value between minus and plus infinity, so its size alone says little. Dividing by both volatilities gives the Pearson correlation coefficient, a standardised covariance bounded by -1 and +1:

For the two assets, ρ_XY = -0.7403, a strong negative correlation. The same three quantities feed the volatility of the portfolio:

Finance treats standard deviation, and its square, variance, as the measure of risk: a portfolio with a larger standard deviation is treated as riskier. This measure is imperfect. A wide spread of returns includes large gains as well as large losses, so standard deviation penalises upside surprises just as much as downside ones. Even so, it remains one of the most useful measures available, and risk-averse investors genuinely dislike large fluctuations in their returns.

A more informative yardstick is the risk-adjusted return, the ratio μ_P/σ_P built from equations 8.1 and 8.5. Correlation does not appear in the numerator, since the expected return is just a weighted average of the asset means. It appears only in the denominator, so lowering the correlation shrinks σ_P and lifts the ratio. That is why diversification is often called a rare “free lunch” for investors: the investor keeps the same expected return and carries less risk. Markowitz (1952) and Sharpe (1964) give the rigorous proof.

Holding the two volatilities and the 24.55% expected return fixed, and replacing the observed correlation with other values, shows how strong the effect is. The rows below are recomputed from the same inputs with equal weights.

A high positive correlation leaves the ratio near 50%, while the observed strong negative correlation roughly triples it. As ρ approaches -1 the ratio climbs very steeply, because the two volatilities are almost equal and nearly cancel: at exactly -1 the portfolio volatility would fall to about 1.5%. All of these results sit inside the Pearson correlation framework that underlies equations 8.1 to 8.5.

The practical message for portfolio construction follows directly. Adding an asset whose returns tend to move against the existing holdings can raise the risk-adjusted return even if that asset has a lower expected return on its own, because the reduction in volatility outweighs the small loss in average return. This is why an asset’s correlation with the rest of the portfolio, not just its stand-alone return and risk, decides whether it earns a place.

Every risk is also a trading opportunity, so most large investment banks and hedge funds run correlation desks. Their traders forecast changes in correlation and position for them. Correlation trading means dealing in assets whose price depends, at least partly, on how one or more assets move together through time. Pairs trading is one strategy; multi-asset options are another large group.

Payoffs of rainbow options

Traders refer to multi-asset options as rainbow options, and sometimes as mountain-range options. In the payoffs below, S₁ and S₂ are the two asset prices at maturity and K is the strike, the price fixed at the start at which the underlying can be bought under a call or sold under a put.

In the basket option, n_i is the weight of asset i, and the payoff can also be written as a single formula:

The pattern in the last column has a simple reason. For most of these options the holder gains from whichever asset ends highest, or from the gap between two assets. A low, ideally negative, correlation means that when one asset falls the other tends to rise, so at least one of them is likely to finish high, and the payoff is likely to be large. The two exceptions run the other way. The option on the worse of two pays the lower price, and diverging assets drag that minimum down. A basket call is effectively an option on a diversified portfolio, and lower correlation lowers the volatility of the basket, which lowers the value of any option on it. Analytical prices for all of them come from extended versions of the Black-Scholes-Merton model (1973).

Pricing an exchange option

An exchange option with payoff max(0, S₂ – S₁) gives the holder the right to hand over asset 1 and receive asset 2 at maturity, which they will do only when S₂ > S₁. Rewriting the payoff as S₁ max(0, S₂/S₁ – 1) turns it into a call on the ratio S₂/S₁ with a strike of 1, measured in units of asset 1. The volatility of that ratio comes from a variation of equation 8.5, writing A and B for the two assets:

The covariance enters with a minus sign. A higher covariance makes the ratio less volatile and the option cheaper, while a negative covariance adds to the volatility and raises the price. The source example shows the exchange option price falling close to 0 at high correlation and rising to USD 15.08 at a correlation of -1, the same negative relationship seen in the CDS and portfolio examples, again within the Pearson framework.

A quanto option lets a domestic investor convert a payoff earned in a foreign currency back into the home currency at an exchange rate fixed in advance, so it removes currency risk from a foreign bet. Consider an American who expects the Nikkei to rise but fears a weaker yen, which would erode or wipe out the dollar value of any gain on an ordinary Nikkei call. A quanto call on the Nikkei solves the problem: the yen payoff is turned into dollars at a fixed rate, usually the spot rate when the trade is struck.

The name comes from “quantity”. The seller of the quanto call does not know how many yen will need converting, because that depends on the option’s future payoff. In fact the seller faces two unknowns at once:

• how far in the money the call will be at maturity, which fixes the yen amount to be converted; and
• what the exchange rate will be at maturity, at which that uncertain yen amount must be turned into dollars.

The correlation between these two, the Nikkei price S’ and the exchange rate X quoted as dollars per yen, drives the quanto price. Suppose the correlation is positive. A rising Nikkei then tends to come with a stronger yen, which suits the seller: they owe a larger payoff, but each yen buys more dollars, so a smaller yen amount covers the dollar payment. So, the more positive the correlation, the cheaper the quanto call.

With a negative correlation the opposite happens. When the Nikkei rises the yen tends to weaken, and the seller needs more yen to meet the same dollar obligation. The lower the correlation, the more expensive the quanto call. The price therefore falls as correlation rises, the same negative relationship found for the exchange option. Closed-form quanto prices are available from an adapted Black-Scholes-Merton model (1973).

A correlation swap trades correlation directly. One side pays a fixed correlation rate agreed today, for instance 10%, and the other side pays the realised correlation, the correlation that actually occurs between a set of assets over the life of the swap. Because realised correlation is unknown at the start, it is also called stochastic correlation.

The fixed-rate payer is said to be “buying correlation”, since the swap gains value for them when realised correlation rises. The fixed-rate receiver is “selling correlation”. Realised correlation for a portfolio of n assets is the average of all distinct pairwise Pearson correlations:

The factor 2/(n² – n) is just one divided by the number of distinct pairs, n(n – 1)/2. With three assets there are three pairs; with 30 stocks, as in a Dow-based swap, there are 435, so realised correlation is an average over a large set of pairwise estimates. At maturity the fixed-rate payer receives:

Uses of a correlation swap

Correlation between stocks tends to rise when stock prices fall, a pattern visible in the Dow data discussed in the systemic risk section. So a long correlation position, paying fixed, gives an indirect hedge against a falling stock market. Valuing these swaps is not settled. There is no industry-standard model, traders often lean on historical data to anticipate ρ_realised, and ordinary swap valuation would need a term structure of correlation that does not yet exist. Stochastic correlation models are an emerging alternative.

Correlation can also be bought through options. A trader can go long puts on a stock index, say the Dow Jones Industrial Average, and short puts on the index’s member stocks. Correlation and volatility tend to rise together, so if correlation among Dow stocks increases, perhaps in a downturn, the implied volatility of the index put rises too. The hope is that the gain on the long index put outweighs any loss on the short single-stock puts.

The London whale

Bruno Iksil, a trader in the London office of JP Morgan who earned the nickname “London whale” for the sheer size of his CDS positions, ran the mirror image of that idea in 2012. He sold CDSs on the bond index CDX.NA.IG.9 and “hedged” by purchasing single-name CDS protection. When an economy is recovering, volatility and correlation usually fall, so the short index protection should gain more than the single-name protection loses. The trade may have made sense over a longer horizon, but the position was so large that hedge funds squeezed it by buying the index aggressively. Index CDS values rose, the whale showed a huge paper loss, and JP Morgan had to buy back the index positions at a loss of over USD 2 billion.

Managing financial risk means finding it, putting a number on it and, if the firm chooses, cutting it down. Since the crisis of 2007 to 2009, banks and many nonfinancial companies have invested heavily in it. The main risk types are market risk, credit risk and operational risk, and others such as systemic, liquidity, volatility and correlation risk are often added. Market risk itself splits into equity, interest-rate, currency and commodity risk, and the toolkit for measuring it includes VaR, expected shortfall (ES) and enterprise risk management (ERM), among others. VaR was described as the most widely used measure as of 2018.

VaR is the largest loss a portfolio should suffer from market moves over a stated horizon at a stated probability. In the variance-covariance approach used here, it is:

Here α is the point on the standard normal distribution for the chosen confidence level (NORMSINV in Excel, norminv in MATLAB), x is the horizon in days and σ_P is the portfolio volatility, which contains the correlations. With β_h the horizontal vector of amounts invested (price times quantity), β_v its vertical transpose and C the covariance matrix of returns:

The covariances come from correlations and volatilities as Cov_ij = ρ_ijσ_iσ_j, which differs from equation 8.3 only in that ρ and σ are assumed known rather than built from raw returns. Because β_h is simply the transpose of β_v, the same product is often written β_hCβ_h^T; the matrix form scales to any number of assets, with every pairwise correlation sitting inside C. A non-parametric alternative instead derives VaR from simulated historical data.

Changing the correlation

Repeating the calculation with only the correlation changed traces how strongly VaR depends on it. The line below is recomputed from the example’s inputs.

Lower correlation means lower risk, and a negative correlation is better still, because when one asset falls the other tends to rise and the losses partly offset. At a perfect negative correlation VaR is about USD 1.1 million, and at a perfect positive correlation it is about USD 1.84 million, so the correlation assumption alone moves the risk figure by roughly two thirds. Since a bank’s market risk capital is built on VaR, any change in the assumed correlation flows straight through to the capital it must hold.

By 2018 the crisis felt distant. The Dow Jones Industrial Average had climbed from its March 2009 low of 6,547 points to over 25,000 by October 2018, world growth was a moderate 2.5% and US unemployment was a historically low 3.7%. The cost of recovery, however, was large stimulus spending and heavy sovereign deficits, with US gross debt at about 107% of GDP.

Causes of the crisis

A collapse of that size had many causes, not one:

• Very benign conditions over 2003 to 2006: credit spreads at record lows, and both volatility and interest rates subdued.
• Growing speculation by traders and investors trying to profit from those calm conditions, which inflated bubbles in housing, mortgages (especially subprime), stocks and commodities. By 2007 US investors had borrowed 470% of US national income to invest in real estate, financial and commodity markets.
• A wave of new structured products such as CDOs, CDO squared, constant-proportion debt obligations (CPDOs) and constant proportion portfolio insurance (CPPI), along with options on CDSs and credit indices.
• Naive trust in the copula correlation model, which was supposed to handle the n(n – 1)/2 pairwise correlations in a structured product. With 125 assets in a typical CDO, that is 125 × 124/2 = 7,750 correlation pairs.
• A moral hazard at rating agencies, which were paid by the issuers whose products they rated, so many structured products carried AAA ratings and looked safer than they were.
• Risk managers and regulators who relaxed their standards amid the profit boom.

CDO tranches and the May 2005 correlation shock

From around 2003 the appetite for risk returned, and residential mortgages in particular became an investment product. They were pooled into CDOs and sold in tranches, so that investors could pick their level of default risk: the equity tranche absorbs the first 3% of losses in the pool, the mezzanine tranche the losses from 3% to 7%, and so on. The Gaussian copula, devised by Abe Sklar in 1959 and brought into finance by David Li in 2000, was used to model the default correlations in these pools.

Tranche spreads respond to correlation in different directions. When default correlation in the pool is low, defaults are scattered and a few losses are almost certain each year, so the thin equity tranche is very likely to be hit and its spread is high. As correlation rises, defaults bunch together into rare bad scenarios and the pool more often suffers no losses at all, so the equity spread falls sharply. The mezzanine spread is far less sensitive and tends to fall when correlation drops. Investors who treated these two tranches as a hedged pair were really holding two positions with different, and in one direction aligned, exposures to correlation.

A first correlation crisis, a forerunner of the larger one, hit in May 2005. General Motors was cut to BB and Ford to BB+, pushing both into junk status, and junk downgrades usually cause steep price falls because many mutual and pension funds may not hold junk bonds. Bonds of different credit quality tend to be less correlated, so correlation inside CDOs that had held only investment-grade bonds fell. Many hedge funds were long the equity tranche, collecting its high spread, and short the mezzanine tranche as a supposed hedge. The lower correlation hurt both legs. The equity tranche spread jumped well above the spread the funds were receiving, a paper loss, and the mezzanine spread fell below the spread they were paying, a second paper loss. Several funds failed as a result, Marin Capital, Aman Capital and Baily Coates Cromwell among them. The losses came from misunderstanding how the tranches respond to correlation, not from anything inherently toxic in the CDOs themselves.

From the housing bubble to the great recession

The CDO market grew from USD 64 billion in 2003 to USD 455 billion in 2006, and feeding it required ever weaker subprime lending, including so-called NINJA loans to borrowers with “no income, no job or assets”. House prices stopped rising in 2006, defaults began, and in 2007 they accelerated into a property collapse that took the CDO market, stocks and commodities with it and froze credit markets worldwide. The downturn that followed is known as the great recession.

Many correlations climb at once during a systemic crash. Default correlation between mortgages in CDOs went up, which on its own would have lowered the equity tranche spread and helped equity investors. That benefit was swamped by a sharp rise in default probabilities, so spreads on all tranches widened and investors throughout the structure lost heavily. Correlation between the tranches themselves also rose. Super-senior tranches, previously seen as extremely safe because they were rated AAA and cushioned by the tranches beneath them, were suddenly seen as risky, and their value fell by as much as 20%.

Leverage made it worse. Many investors held leveraged super-senior tranches (LSS), typically levered 10 or 20 times, so a USD 10,000,000 payment carried exposure of USD 100,000,000 or USD 200,000,000. These structures let investors unwind if spreads widened, and the resulting forced selling pushed LSS spreads wider still. The CDS market had meanwhile ballooned from about USD 8 trillion in 2004 to almost USD 60 trillion in 2007. A CDS used as insurance is useful, since spreading risk across many holders is the whole point of insurance. Used speculatively it is different: a seller hopes the insured event never happens and keeps the premium, as AIG tried to do, and a buyer who does not own the bond is betting on its deterioration, much as a holder of a naked put bets on a falling stock.

Who was to blame

Responsibility was shared among the quants who designed the products and models, the managers and traders who authorised and ran the excessive positions, the rating agencies, the regulators and the risk managers who allowed it. The common thread was greed. AIG’s London unit had written nearly USD 500 billion of CDS protection while buying little reinsurance against it. Iceland’s banks had borrowed 10 times the country’s GDP, and Iceland was effectively bankrupt in 2008. Lehman Brothers, before its September 2008 filing, reported leverage of 30.7, with USD 691 billion of assets against only USD 22 billion of equity, and it used misleading repurchase (repo) transactions, a form of short-term collateralised loan, to hide even higher true leverage. It also carried 1.5 million derivatives transactions with 8,000 counterparties. Calling the products “toxic assets” shifted blame from the people who misused them; most traders knew the risks they were running, and where they did not, the fault lay with their competence rather than with the instrument.

Correlations are central inputs to regulatory frameworks, above all in the Basel accords’ treatment of market risk and credit risk. Basel I was implemented in 1988 and Basel II in 2006, and Basel III was being developed and phased in through 2019. All three are guidelines meant to keep the banking system stable. They take their name from the Swiss city where the regulators meet, and none of them has legal force by itself. Most countries nonetheless pass laws to apply them to their banks, around 100 in the case of Basel II, because a common standard rewards sound risk measurement and supports the safety of the whole system.

Basel III was written to fix the weaknesses the crisis of 2007 to 2009 exposed. Alongside new liquidity and leverage ratios, it brings in correlation models for double default in insured transactions like the hedged bond in the CDS example, for correlated defaults in multi-asset portfolios using the Gaussian copula, for correlation in derivatives exposures through the credit value adjustment (CVA), and for wrong-way risk (WWR).

Each of these responds to a lesson from the crisis. Double default treatment recognises that a guarantee is only as good as the guarantor’s ability to survive the borrower’s failure, which is the same problem as the Spanish bond protected by a European bank. CVA captures the fact that the value of a derivative to a bank depends on whether the counterparty survives to pay, and wrong-way risk flags the cases where exposure grows exactly as the counterparty weakens.

Beyond the three main risk types, finance recognises systemic, concentration, liquidity, volatility, legal and reputational risk among others. Correlation risk feeds directly into both market and credit risk, and it is bound up with systemic and concentration risk too.

Market risk

Market risk is usually measured with VaR, and VaR takes the covariance matrix of the portfolio’s assets as an input. It therefore carries correlation risk inside it: if the correlations in that matrix change, the measured risk changes, as the two-asset example showed. Expected shortfall, also called conditional VaR or tail risk, measures losses in extreme scenarios, usually the worst 0.1%, 1% or 5% of outcomes, and any careful estimate of it also needs the correlations between asset returns.

Credit risk

Credit risk has two parts. Migration risk is the risk that a debtor’s credit quality falls to a lower state, which usually lowers the price of its debt and creates a paper loss for the creditor. Default risk is the risk that the debtor fails to pay. The hedged Spanish bond is a correlation problem of the second kind: the higher the default correlation between reference entity and protection seller, the larger the investor’s paper loss on the CDS and the higher the chance of losing the whole investment.

How often defaults happen together matters greatly to commercial banks, credit unions, mortgage lenders, trusts and insurers, all of which hold claims on many debtors. These lenders want low default correlation among their borrowers so that the credit risk is diversified. The table below gives average one-year default correlations by industry over 1981 to 2001 for non-investment-grade US bonds, drawn from 6,907 companies of which 674 defaulted.

Source: one-year US default correlations, non-investment-grade bonds 1981 to 2001, as printed. The Tele and Trans rows against Util show the opposite sign to the Util row (0.8% against -0.8%, 0.2% against -0.2%).

Reading the table: the 3.8% in the Auto row and Auto column means that if one auto-industry bond defaults, there is a 3.8% probability that another auto bond defaults as well. The -2.5% where Energy meets Finance means that an energy default actually lowers the chance of a financial-sector default, and the reverse. Three patterns stand out. Cross-industry correlations are mostly positive, the energy sector being the main exception, since it is seen as stable and resistant to recession. Within-industry correlations, on the diagonal, are higher than cross-industry ones, which suggests that systematic forces such as a recession or the decline of a whole sector drive defaults more than firm-specific events: if General Motors defaults, Ford is more likely to follow than to benefit. So a lender should spread loans across sectors to reduce default correlation risk.

Term structure of default

Because a borrower either defaults or does not, defaults are binomial events, and simple binomial models are widely used. A richer view looks at how default probability changes year by year. Suppose a creditor has lent to one A-rated debtor and one CC-rated debtor; the table gives their historical default probabilities by year, with a derived row comparing the two.

Source: Moody’s. The difference row is derived here.

The A-rated bond has, for example, a 0.15% chance of defaulting in year 5. For investment-grade names the default probability generally rises with time, because a longer horizon leaves more room for internal problems such as mismanagement or external ones such as new competitors or a recession. Distressed names show the reverse, an inverted term structure: the next year or two decide everything, and a company that survives it becomes progressively less likely to fail. A creditor gains from low default correlation between its debtors, and the two term structures can be linked with the copula model to answer joint questions such as the probability that the first debtor defaults in year 3 and the second in year 5.

Systemic risk is the risk that a whole financial market, or the entire financial system, breaks down. The credit market in 2008 is the clearest case. When Lehman Brothers failed in September 2008, lending virtually stopped, and even after the Federal Reserve guaranteed interbank loans, lending came back only slowly.

The stock market fall that began in October 2007 was another systemic collapse. The Dow dropped 53.54%, from 14,093 points to 6,547 points by March 2009. Only one of the 30 Dow stocks, Walmart, gained over that period, and 489 of the S&P 500 stocks fell. The few that rose clustered in a handful of areas:

• Consumer staples, which sell food and basic household goods: Family Dollar, Ross Stores and Walmart.
• Education, which tends to do well when unemployed people return to study: Apollo Group and DeVry.
• Healthcare and pharmaceuticals: Edwards Lifesciences and Gilead.
• Others with steady demand: Autozone (replacement auto parts), CF Industries (fertiliser), Netflix (internet subscriptions) and Southwestern Energy.

Systemic failures of this kind spread into the real economy through falling GDP and rising unemployment, which lowers living standards. The short list of winners also shows how little room for diversification remained: in a market where 489 of 500 large stocks fell, the only shelter came from a few sectors whose demand does not depend on the business cycle.

Correlation in the Dow, 2004 to 2009

Systemic risk and correlation risk feed each other. When almost the whole market falls at once, the stocks in it move together, and their correlation jumps. One way to see this is to take daily closes for all 30 Dow stocks, group them by month, compute a 30 × 30 Pearson correlation matrix for each month, average the entries, and smooth the result with a one-year moving average. Over 2004 to 2006, while the Dow rose gently, correlation was fairly stable. From January 2007 to February 2008, correlation rose as the Dow rose more strongly. Then, during the steep fall from August 2008 to March 2009, average correlation jumped from a non-crisis level of about 27% to over 50%, and at the height of the crisis in February 2009 it reached 96.97%.

The consequence for investors is direct. A portfolio that looked well diversified in calm markets suddenly behaved like a single position, and its holders took unexpected losses from many stocks falling together. The diversification benefit measured with calm-period correlations was largely an illusion at the moment it was needed, so a risk manager who calibrates VaR only on benign data understates the loss in exactly the scenario that matters most. This is the same pattern that makes a long position in a correlation swap a hedge against a crash.

Concentration risk is a fairly new category without a single agreed definition. A sensible one is the risk of financial loss from a concentrated exposure to a particular group of counterparties. It can be measured by the concentration ratio: a creditor with 10 equal loans has a ratio of 1/10 = 0.1, while a creditor with one loan to one borrower has a ratio of 1. The lower the ratio, the better diversified the default risk, provided the default correlation between borrowers is below 1. Grouping borrowers by sector gives sector concentration risk, and lending across many sectors helps because defaults within a sector are more correlated than defaults across sectors, as the industry table showed.

Splitting one loan into several

Consider a bank C with USD 10,000,000 to lend, and borrowers that each have a 10% default probability.

• Case (a): all USD 10,000,000 goes to company W. The concentration ratio is 1, and expected loss is USD 10,000,000 × 0.1 = USD 1,000,000.
• Case (b): USD 5,000,000 each goes to companies X and Y. The ratio falls to 1/2.
• Case (c): USD 3.33 million each goes to X, Y and Z. The ratio falls to a third.

Expected loss is USD 1,000,000 in every case, because each dollar lent still has a 10% chance of being lost. What changes is the worst case, losing the whole USD 10,000,000, which requires every borrower to default together. In case (a) that happens with probability 10%. In case (b) it requires the joint default of X and Y, and as long as their default correlation is between 0 and 1, the joint probability is below 10%. In case (c) it requires all three to fail together, which is less likely still. To get a feel for the size of the effect, take the special case of independent defaults: the chance that both X and Y fail is 0.1 × 0.1 = 1%, and the chance that X, Y and Z all fail is 0.1 × 0.1 × 0.1 = 0.1%, against 10% for the single loan. Positive correlation pushes these numbers back up toward 10%. If the default correlation were 1, however, splitting the loan would give no benefit at all, since all borrowers would default or survive together.

The concentration ratio says nothing about who the borrowers are, so two portfolios with the same ratio can carry very different risk. Ten equal loans to ten telecom companies and ten equal loans spread over ten industries both have a ratio of 0.1, yet the telecom book has a within-industry default correlation of 10.7% while the cross-industry entries in the table range from -2.6% to 6.5%, with more than four in five of them below 3%. The sector view therefore adds information that the simple ratio misses.

Joint default probability of two binomial events

Treat each default as a binomial variable equal to 1 on default and 0 otherwise. The joint default probability is:

The square-root term contains the standard deviation of each binomial variable:

Lowering the concentration ratio and lowering the correlation coefficient therefore work in the same direction: both shrink the joint probability of default, which is the creditor’s worst case. The same idea extends to portfolios through credit value-at-risk (CVaR), the largest loss on a portfolio of correlated debt at a given probability over a given period. CVaR is the credit counterpart of VaR for correlated market risk, and a higher copula correlation between the assets produces a higher CVaR.

The word “correlation” means different things to different people. On trading desks, terms like “correlation desk” and “correlation trading” refer to any strategy whose value depends at least partly on two or more assets moving together, including pairs trading, exchange options and quanto options. In that setting correlation covers any co-movement of prices through time.

Much of the recent theory, for example Cherubini et al (2004), Nelsen (2006) and Gregory (2010), uses the word narrowly for the linear Pearson coefficient alone, and calls other coefficients dependence measures or measures of association. Yet theory also uses “correlation” loosely in phrases such as default correlation, credit correlation and volatility-asset return correlation, quantities that are estimated with non-Pearson models such as Heston (1993), Lucas (1995) and Li (2000). A workable convention is to call the Pearson number the correlation coefficient, call coefficients from other models dependency coefficients, and call every method that measures some kind of dependence a correlation model or dependency model.

Independence

Two events are dependent if one happening changes the probability of the other, and independent if it does not. Formally, A and B are independent exactly when the probability of both occurring is the product of their separate probabilities:

Dividing by P(B) and using the Kolmogorov definition of conditional probability shows what independence means in practice: knowing that B happened leaves the probability of A unchanged.

Covariance, correlation and uncorrelatedness

For random variables, the covariance and the Pearson coefficient are written with expectations:

Independence of two random variables implies E(XY) = E(X)E(Y), and by A4 that means zero covariance. So independent variables are always uncorrelated. The reverse fails. Take Y = X², where Y is plainly determined by X, and let X be uniform on [-1, +1]:

Both E(X) and E(X³) are zero because the distribution is symmetric about zero, so the covariance and the correlation are zero even though Y depends completely on X. The Pearson measure captures only linear dependence and misses nonlinear links, which is a real limitation in finance, where many relationships are nonlinear.

The payoff of an option against the price of its underlying asset is a simple financial case of the same problem: the relationship is curved, so a linear coefficient summarises it poorly. A zero or small Pearson correlation should therefore never be read as proof that two risks are unrelated. It only shows that no straight-line relationship is present, and checking for other forms of dependence, for example in the tails, is a separate task.

Growth in finance is usually quoted as a relative change, (S_t – S_t-1)/S_t-1. A move from S_t-1 = 100 to S_t = 110 is a relative change of (110 – 100)/100 = 0.1, or 10%. For small moves, the natural logarithm of the price ratio gives nearly the same number:

Log-returns have one big practical advantage, which is why correlation swaps and many risk models use them: they add up over time, while percentage returns do not.

The same property explains why daily log-returns are used to estimate the realised correlation in a correlation swap: returns over a long period can be built up by adding daily figures, and the daily series behaves consistently whatever horizon the analysis later needs. Working backwards is just as easy. If a log growth rate of 9.531% is applied to 110, then ln(x) = ln(110) + 0.09531, and raising e to that power gives x = 121.