**Summary**

Many investors, especially retail investors, often remain unaware of the real risks in credit investing. This note highlights the pitfalls of ignoring the low-probability costly events of default while integrating credit risky assets into portfolios. The shape of the risk profile of a credit portfolio is affected substantially by the tails caused by rare credit events. To efficiently tradeoff long-term expected return with risks, we must recognize that the risks are in the tails. It is important that extreme but exceedingly rare credit events be properly simulated with sufficient accuracy. Furthermore, a risk metric must be chosen that accurately captures the impact of the tails. Conditional value-at-risk is one such metric. Its use ensures that long-term investing goals can be met, without suffering catastrophic blows from the tails in the short run. The models for limiting tail effects should also be applicable to portfolios of assets with correlated defaults, such as collateralized loan or debt obligations.

**Introduction**

Credit risky securities and most of their derivatives are characterized by a large chance of positive returns (periodic coupons) and an exceedingly small probability of large investment losses. The distribution of price returns of these instruments is asymmetric and highly skewed, exhibiting very flat tails on the downside. Investors are compensated for assuming the low-probability risk of exceptionally large losses. However, models for integrated risk management in the context of credit risky securities remain scarce and these instruments, in practice are naively included in portfolios.

Properly simulated credit events result in risk profiles with flat tails on downside risk (i.e., losses) and limited upside potential (i.e., gains). Subtle and important observations include:

– Losses are probabilistic events, and without adequate accuracy the low-probability events may be missed.

– Recognizing these low-probability events can lead to different optimal portfolios.

– Standard risk measures do not adequately penalize the low-probability events. The resulting portfolios might be efficient with respect to the standard popular metric of volatility, yet they are improper, because of the probability of achieving substantial losses. Different risk measures such as conditional VaR should therefore be used.

– With appropriate modeling, long-term performance goals can be met without suffering catastrophic blows from the tails in the short run. This is in practice rather hard to do, however.

The causes of loss due to credit assets are many and complex. Credit risk can be described as the changing expectations of an obligator’s ability or willingness to fulfill its obligations on a certain date or at any time beyond. Losses may result from a default or a change in market value due to credit quality migration. In general, credit risk for a single instrument may be decomposed into default risk, migration risk, and security-specific risks that cause idiosyncratic spread changes.

Popular approaches assess credit risk due to default losses only without taking into consideration the term structure of credit spreads; say calculation of the present value of a portfolio of credit risk-sensitive assets depending on existing credit risk only. Market risk is not incorporated explicitly. As a result, no other risks apart from credit risk can be assessed for their impact on the valuation of a portfolio that includes risky credits.

It is important that popular pricing models be extended to the valuation and simulation of portfolios of credit risky securities and their derivatives. These extensions must allow estimating the risk profile of portfolios considering market and spread risk, and the risks of rating migration, defaults, and recovery. Simulations ought to reveal—and quantify—the flat tails due to credit events.

Typically, simulations tend to show a flat lower tail when credit rating migrations are simulated under current economic conditions. This tail is absent when simulating only market and spread changes under constant volatility. The tail however can be quite pronounced when simulated assuming the term structure of risk-free rates. A proper approach should develop the simulation of the same portfolio integrating market and credit spread risk, and then adding the credit events, i.e., credit rating migrations and defaults. Improved simulation may take the form of accurate numerical methods for tail resolution or the use of confidence intervals for the extremes of the loss distribution, or any other technique that allows us to observe the tails that drive low-probability events. This observation has ramifications for the choice of an appropriate risk metric for portfolio optimization.

**Tail Effects on Optimization**

Ignoring the tails has a significant effect on arriving at portfolio efficient frontiers. There is nothing efficient about optimized portfolios obtained by ignoring the tails once the tails are properly accounted for. Tails distort the risk-return frontiers, making seemingly efficient portfolios inefficient ones. Running a mean absolute deviation (MAD) portfolio optimization using the distribution with credit events, may help obtain a frontier that is remarkably close to the out-of-sample frontier and eliminates inefficient portfolios.

Does this imply that it is sufficient to accurately simulate the tails, and then develop portfolios that optimally trade off expected return against risk? The answer is of course affirmative, although the MAD risk measure does not properly account for the tails. The distribution of returns of a minimum-risk portfolio obtained using the MAD model would suggest that there is a small probability of losses, often more than 80% of the portfolio value. These losses are likely to be catastrophic, and when they occur, they will most likely — due to bankruptcy — block the prospects of the long-term expected return. The long-term expected return of the minimum-risk portfolio will be realized only if the portfolio is not ruined in the short term.

**Accounting for Tail Effects**

What then should be done to properly account for the tail effects?

Losses are calculated relative to the exposure that would exist, at a given time horizon, if all obligors maintained their current credit state. Since credit events (i.e., default or a change in credit rating) are relatively rare, the peak of the loss distribution is typically at or near zero. It is important to keep in mind however that the loss distribution is highly skewed; a long-left tail reflects the infrequent, but substantial, losses that can occur when an obligor defaults. Conversely, negative losses (i.e., gains) can result from a net improvement in the credit ratings of obligors.

The answer is to select a risk metric that penalizes appropriately extreme events, and then optimize the portfolio composition with respect to this metric of risk. Relevant measures capture key properties of the loss distribution while tractable measures can be optimized using computationally efficient methods such as linear programming. The expected losses equal the mean of the loss distribution. The maximum losses represent the largest loss that is anticipated to occur with a given probability. The unexpected loss is the difference between the maximum and expected losses. Expected shortfall, also known as tail conditional expectation or conditional VaR, measures the average size of a loss given that it exceeds the Maximum Loss. One attractive feature of expected shortfall is that it considers the entire tail of the loss distribution (beyond the Maximum Loss), rather than just one point on the loss distribution. Thus, expected shortfall is more likely to draw attention to large losses in the extreme tail of the distribution than Maximum Loss which effectively ignores losses beyond the specified quantile level.

Variance is generally not a relevant credit risk measure, while maximum losses and unexpected losses are relevant but not tractable. Value at Risk (VaR) has become an industry standard for measuring extreme events and integrating disparate sources of risk. VaR answers a particular question: What is the maximum loss with a given confidence level (say, α × 100%) over the target horizon? Its calculation also reveals that with probability (1 – α)100% the losses will exceed VaR.

The VaR measure reveals nothing about the extent of the losses beyond the given confidence level. Such losses can be catastrophic and Long-Term Capital Management (LTCM) is a classic historical case in point. LTCM was estimated to have a VaR of only –5% at the 0.95 probability level, but a return of around –80% wiped out a position of $1.85 trillion and threatened a global meltdown of the financial markets. A measure of risk that goes beyond the information revealed by VaR is the expected value of the losses that exceed VaR. This quantity is called expected shortfall, conditional loss, or conditional VaR; CVaR is always greater than or equal to VaR.

It is also important to keep in mind that VaR is difficult to optimize when it is calculated using discrete scenarios. The VaR function is non-convex and non-smooth, and it has multiple local minima. CVaR, however, can be minimized using linear programming formulations. In other words, to avoid distortions of the efficient frontier due to the tail events, we need to optimize a risk metric that appropriately penalizes the tails. CVaR, as mentioned earlier, provides such a risk metric. The risk profile of a portfolio is shaped by the attention the risk manager pays to the tails. Taking a CVaR perspective on risk management substantially reduces the tails. Of course, the choice of a risk metric influences the upside potential of the portfolio and inevitably the upside potential is reduced as the tails are shrunk. There are the usual trade-offs between upside potential and downside risk, but in the context of credit risky securities the downside risk is hidden in the tail and not in the variance or the mean absolute deviation. In this respect CVaR has an important role to play in tracing efficient frontiers for the management of credit risk. CVaR optimization provides the appropriate risk management framework for credit risky portfolios.

**Long Term Performance with Short Term Tails**

Optimization of portfolio performance for the long run ignores the short-term effects. This has been the tradition in myopic single-period optimization models. Ignoring the short-term effects however can be catastrophic in the presence of tails. In particular, the long-term (expected) potential of a portfolio strategy may never be realized if an extreme event in the short run results in bankruptcy. LTCM is a case in point again. When LTCM suffered losses of 80% in 1998, the New York Federal Reserve orchestrated a bailout. Fourteen banks invested $3.6 billion in return for a 90% stake in the firm. The fund eventually recovered its losses and posted positive returns, but the original stakeholders were not there anymore.

**Conclusion**

The risk/return trade-off has been a central tenet of portfolio management since the seminal work of Markowitz. The basic premise, that higher (expected) returns can only be achieved at the expense of greater risk, leads to the concept of an efficient frontier. The efficient frontier defines the maximum return that can be achieved for a given level of risk or, alternatively, the minimum risk that must be incurred to earn a given return. Traditionally, market risk has been measured by the variance (or standard deviation) of portfolio returns, and this measure is now erroneously widely used for credit risk management as well. While this is reasonable when the distribution of gains and losses is normal, variance is an inappropriate measure of risk for the highly skewed, fat-tailed distributions characteristic of portfolios that incur credit risk. A minimum-variance portfolio is far from efficient with respect to unexpected losses that occur in credit investing. In this case, quantile-based measures that focus on the tail of the loss distribution more accurately capture the risk of the portfolio. It is only after taking account of the above, should investors consider the excess return, or “spread,” which represents a risk premium that compensates investors for potential losses due to credit events (and possibly illiquidity).