In what follows, I summarize how my different research papers tie in together. My primary research interest is investment management. Investment management is inherently a question of asset pricing, and I therefore also conduct research in asset pricing.

The central tenet of investment management is to find the appropriate trade-off between the expected payoff from an investment and its risk. Most of my research so far has focused on the latter part: how we can properly measure and manage risk, especially when an investor has many assets to choose from.

We first examined the behaviour of the market, size, value, and momentum portfolios. These portfolios are referred to as *risk factors *by academics and are used to explain the differences in average returns across stocks. For example, stocks with *value* characteristics (e.g. high accounting value compared to its market value) have had higher returns on average over the past century. A *value *factor*, *or more precisely a *value-minus-growth* factor, captures this source of performance.

These portfolios, which involve long and short positions in stocks, are also the basic elements of strategies used by some hedge funds. For example, hedge funds such as AQR Capital Management follow, among other strategies, a combination of value and momentum. Risk factor investing is also common in mutual funds. Firms such as Dimensional Fund Advisors were early small cap investors and Warren Buffet is probably the best-known value investor. More recently, the practice of combining risk factors, or more simply tilting a diversified portfolio towards stocks with certain attributes, has become known as smart beta investing.

An investor combining these sources of performance should be concerned with how they co-move together. If they all move in the same direction at the same time, then your portfolio is risky because you can simultaneously lose money on all fronts. Part of what makes them attractive is the fact that these long-short portfolios have low correlations between themselves, meaning that their returns do not *usually* go in the same direction. These low cross-correlations are not surprising: these factors are generally constructed to behave this way. Given these low cross-correlations, an investor can gain large diversification benefits by combining these long-short strategies in a portfolio.

But here the word *usually* is key; we show in Christoffersen and Langlois (2013) that these cross-correlations vary wildly over time and they hide striking dependence between extreme events. For example, we find extreme dependence despite close-to-zero cross-correlations, which means that when one risk factor is having its worst day in history there is a chance that another risk factor is also having its worst day in history. This may remind some of the Quant meltdown in August 2007 during which the returns of many long-short strategies favoured by hedge funds tanked despite the overall market barely moving.

What is problematic is that the standard statistical measures of risk—volatility and correlation—do not allow for this possibility. Even worse, there were few flexible econometric methodologies that could model both close-to-zero cross-correlations *and* extreme dependence.

In Christoffersen and Langlois (2013), we propose a flexible econometric methodology that can account for multiple facets of risk in the data.[1] Our methodology better captures how risk factor returns actually behave. We show that **funds combining long-short strategies based on these risk factors can increase their risk-adjusted returns by more than 1% per year** when using our methodology. This added value is impressive given that it comes from simply using a better *risk model*, not better forecasting the profitability of these strategies.

We then went beyond risk factors in the U.S. equity market and turned our attention to international equity markets. The structure of international financial markets has changed over the past few decades; barriers to international investment have fallen and financial markets have become more integrated. We show in Christoffersen, Errunza, Jacobs, and Langlois (2012) that measures of dependence between national equity markets have at the same time greatly increased. Higher correlations and higher occurrence of joint extreme events (e.g. crises) imply that the benefit for an investor of diversifying his portfolio internationally has decreased over time. For example, a U.S. investor allocating a part of his wealth in Germany and Japan would have improved his portfolio risk-return profile a lot in the 1980s. Nowadays, these gains are lower.

To model these patterns we generalize the econometric methodology of Christoffersen and Langlois (2013) in two different ways. First, we introduce a trend in long-run correlation. Correlations of short-term returns (i.e. weekly or monthly) depend on current market conditions, but in addition, they hover around a long-run value that slowly increases over time, a feature robustly supported by the data. Second, we adopt an estimation methodology that allows us to estimate our model on a wide set of countries. It is this large set of countries that allows us to infer the time trend in average dependence measures.

We show that correlations, but also dependence among extreme events, have increased between developed country equity markets (e.g. between the U.S. and U.K.), between emerging country equity markets (e.g. between Brazil and India), and between developed and emerging country equity markets (e.g. between Japan and Russia). While a large proportion of the benefits from diversifying across developed countries has disappeared, larger gains remain in diversifying across emerging markets. The intuition is that while crises may be more frequent in emerging markets, they are more country-specific.

To better illustrate how these diversification benefits in international equity markets have gone down, I developed a new measure of **conditional diversification benefit**. The objective was to find one measure that could summarize several risk measures (volatility, correlations, downside risk, tail risk, etc.) into one meaningful measure of portfolio diversification.

We start with a well-known and widely used risk measure: the expected shortfall. In a nutshell, the expected shortfall is the average portfolio loss one can expect given that the loss on the portfolio is equal to or larger than a pre-chosen level. If you determine that you are in trouble if you lose 5% or more on your portfolio, then the expected shortfall tells you how much you should expect to lose in such case (is it closer to 5% or to 20%?). We use this risk measure because it respects a set of nice mathematical properties a risk measure should have.[2]

The expected shortfall of a portfolio has two natural bounds. It cannot be lower than the pre-chosen threshold (i.e. 5%), a situation in which all portfolio losses equal to or larger than this level actually are equal to this threshold. Think of buying an insurance contract on the value of your portfolio which would pay you the extra loss in such case. At the other end of the spectrum, the expected portfolio shortfall cannot be higher than the average of the expected shortfalls of each asset weighted by their weight in your portfolio. This situation corresponds to the case in which combining these assets in a portfolio does not bring *any* diversification benefits.

The **Conditional Diversification Benefit **measure, **CDB**, is the distance of our portfolio’s expected shortfall, ES, to its upper bound, max(ES), standardized by the distance between the two bounds: ( max(ES) – ES ) / ( max(ES) – min(ES) ).

The conditional diversification benefit measure is extremely useful and has intuitive properties: it does not depend for example on the asset expected returns which are notoriously hard to measure[3] and it varies between zero (no diversification benefits) and one (perfect diversification). By measuring its value at different points in time, we can describe the evolution of diversification benefits obtained from a portfolio.

Using this measure, we show that diversification benefits have been halved since the 1970s for developed countries. In contrast, diversification benefits among emerging markets have decreased by around 25% during the two decades up to the end of the 2000s.

What about **diversification in other asset classes**? In Christoffersen, Jacobs, Jin, and Langlois (2017) we show how to measure time variations in risk of a large portfolio of corporate bonds. Whereas we considered up to 33 equity markets in the previous paper, here we show how we can handle credit securities of more than 200 U.S. firms. Our methodology is useful when allocating a portfolio, assessing the risk of a portfolio, or even pricing a structured product on a large set of bonds.

We find that the dependence between the prices of default insurance (i.e. credit spreads) on large U.S. companies’ debt has spiked during the financial crisis of 2007-2009 and has remained elevated since. In comparison, the correlation between the stocks of these same companies has spiked later in the crisis but has since come back down. When measuring the Conditional Diversification Benefit measure for an equal-weighted portfolio of investment grade corporate bonds, we find that they have decreased by more than one third during the 2000s.

We find further evidence that our dependence measure impacts the level of credit spreads for these U.S. corporations; higher dependence between a firm’s credit spread and the average credit spread in the market is associated with a higher credit spread level.

[1] We model time-varying volatilities and cross-correlations, and univariate and multivariate fat tails and asymmetries.

[2] See Artzner, Delbaen, Eber, and Heath (1999) for a thorough discussion of coherent risk measures, and Basak and Shapiro (2001).

[3] They cancel out when computing the ratio.