BBA 4226 University of Southern California Risk On and VIX Discussion read RISK-ON, RISK-OFF (Neuberger and Berman). QUESTION: When risk is “on,” what do y

BBA 4226 University of Southern California Risk On and VIX Discussion read RISK-ON, RISK-OFF (Neuberger and Berman). QUESTION: When risk is “on,” what do you think would happen to the VIX index? Go beyond saying something obvious such as “Fear is up.”I attched the artical Risk On, Risk Off
Wai Lee, Ph.D.
The phrase “Risk On, Risk Off” may have become ingrained in the vernacular of
global investors. We attempt to crystallize what “Risk On, Risk Off” really means by
analyzing an extreme state of the market when correlation of all assets are perfect.
We derive a set of normative results in relation to the investment opportunity set on
how assets should behave. While our results do not provide guidance on interpolation
between a particular state and the extreme state of perfect correlation, we believe
that our analysis can serve as a compass for our investment decision-making process
in the event we believe that we are moving towards or away from a “Risk On, Risk
Off” environment. Investment implications in relation to a risk-parity portfolio,
global asset allocation, and active portfolio management are discussed.
Managing Director
Chief Investment Officer and
Director of Research
Quantitative Investment Group
Neuberger Berman
November 2011
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Forthcoming in the Journal of
Portfolio Management
Ever since the global financial crisis came to a head in 2008, “Risk On, Risk Off ” and
“correlations go to one” have become the most widely used phrases in describing
investment and asset price behavior. Generally speaking, 2008 was a “risk off ” year
in which investors were said to “de-risk” either by deleveraging or by selling existing
risky positions across the board and going to cash. For a large part of the year, 2009
was a “risk on” year during which some investors’ appetite for risk was back, leading to
a synchronized strong rebound in returns of risky assets ranging from global equities,
credit, emerging market debt to commodities and others. Most recently, the downgrade
of the long-term sovereign credit rating of the United States by Standard & Poor’s on
August
5, 2011—in
middle
global
85 100
30 67the100
48of a100
100growth
55 slowdown—triggered
55 100 100 83 a massive
31 72 global
90 100 3
sell-off of risky assets for days, a “risk off ” scenario.
But what does “Risk On, Risk Off ” really mean? Based on our anecdotal observations,
“Risk On, Risk Off ” generally refers to an investment environment in which asset
price behavior is largely driven by how the appetite for risk advances or retreats over
time, usually in a synchronized way across global regions and assets at a faster-thannormal pace. Depending on the environment, investors will tend to buy or sell risky
assets across the board, paying less attention to the unique characteristics of these assets.
Volatilities and, most noticeably, correlations of assets that are perceived as risky jump,
particularly during the “risk off ” periods, as we often hear the comment “correlations
go to one,” during a crisis. On the other hand, assets such as U.S. Treasury bonds and
some currencies such as the Japanese yen tend to move in the opposite direction of
risky assets, as they are generally perceived as the safer assets to hold in the event of a
flight to safety.
In this paper, we attempt to put more formality on “Risk On, Risk Off,” analyzing this
widely accepted phrase in describing investment behavior in conjunction with jumps
in correlations. In doing so, we formulate a situation where we push the markets to an
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1
Risk On, Risk Off
extreme in which assets are perfectly correlated, that is, where “correlations go to one.”
We acknowledge that such a world does not exist. However, by analyzing this extreme
state of the market, we derive a set of normative results in relation to the investment
opportunity set on how assets should behave. While our results do not provide guidance
on interpolation between a particular state and the extreme state of perfect correlation,
we believe that our analysis can serve as a compass for our investment decision-making
process in the event we believe that we are moving towards or away from a “Risk On,
Risk Off ” environment. We demonstrate in a normative sense that expectations of
perfect correlation imply that expected Sharpe ratios of all assets must be identical.
In such a world, all assets are statistically redundant. Implications for investments are
also discussed.
Given a goal of outperforming the market capitalization-weighted index in the active
investment industry, professional investors often comment that there are not many
opportunities to add alpha when stocks are highly correlated—“it is a macro-driven
market” is a common response to such an environment. For example, in his August 11,
2011 report, Mezrich (2011, p.2) wrote “Stock correlation is now at the highest level
since January 1987, posing yet another challenge to equity investing. U.S. stocks are
moving together in lockstep fashion, rendering stock selection extremely difficult.”
If the conventional wisdom is true, how do some managers outperform in such a highly
correlated world? In addressing this question, we turn the established framework into a
linear factor model structure. Through this structure of correlation and volatilities, we
derive that, indeed, when all assets are highly correlated, the only way to outperform is
to successfully estimate the factor structure, time the factor returns, and then adjust the
portfolio’s factor exposures accordingly.
W h e n C orr e l at i o n s G o To O n e
In this section, we study what the investment opportunity set should be like when
assets are expected to be perfectly correlated. All performance statistics in this paper are
conditional expectations based on the investor’s information set, unless otherwise stated.
Without loss of generality, we analyze a world with only two assets. Our results and the
implications can easily be generalized to a world with multiple assets.
Let x and y be two risky assets with returns ?x and ?y, volatilities ?x and ?y where ?y
> ?x, with correlation coefficient ?. The risk-free rate of return is denoted by r. In the
appendix, we demonstrate that Sharpe ratios of two perfectly correlated assets must be
identical. That is,
(1)
µx-r µy-r
if ? = 1
?x = ?y
Therefore, when the two assets are perfectly correlated,
(2)
[
]
?
?
µx = x µy + 1- x r
?y
?y
In other words, when correlation is perfect, one asset, the less risky asset x in the case of
equation (2), can be synthetically created by a portfolio of another asset, the more risky
asset y in this case, and the risk-free asset. The weight of y in the synthetic portfolio is
equal to the ratio of the volatility of x to be replicated to the volatility of y, while the rest
of the portfolio goes to the risk-free asset, r. Similarly, the more risky asset y can also be
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Risk On, Risk Off
synthetically created by a leveraged investment into x, with the weight of x higher than
one and a negative weight on the risk-free asset representing borrowing. The degree of
leverage is determined by the ratio of their volatilities. Put another way, assets x and y are
statistically redundant since one can be synthetically created by a portfolio of the other
and the risk-free asset. For example, suppose the volatility of y is twice the volatility of
x. According to equation (2), x can be created by a portfolio with 50% invested in y and
50% invested in the risk-free asset. Similarly, y can be created by borrowing at the riskfree rate and investing 200% in x. Linking this result to the real world when correlations
increase—albeit not to a perfectly correlated state—means that assets are perceived to
be very similar in terms of the tradeoff between their returns and risks. Therefore,
investors tend to buy or sell across the board; in other words, they exhibit the “Risk On,
Risk Off ” behavior.
Exhibit 1 graphically illustrates, hypothetically, the different investment opportunity set
when the market is in a normal state with correlation less than one versus the extreme
“Risk On, Risk Off ” state with perfect correlation among all assets. In the normal state,
assets will offer different return-risk tradeoffs, or Sharpe ratios, and different correlations.
As correlations go to one, either returns or risks, or both returns and risks, of all assets will
adjust to an extent such that their return-risk profiles fall on a straight line on this chart,
offering the same Sharpe ratio.
E x h ib it 1 : Hy poth etica l In v estm en t Oppor tu n ity Set
Normal Scenario: Correlation < +1 vs. “Risk On, Risk Off” State: Correlation = +1 35% Corr < +1 Corr = +1 30% Expected Return 25% 20% 15% 10% 5% 0% 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% Volatility Source: Neuberger Berman Quantitative Investment Group. A l p h a I n a “ R i s k O n , R i s k O f f ” W or l d When one hears the term “alpha” in the investment industry, clarification should follow. Unlike in the academic literature in which “alpha” is understood to be risk-adjusted, such as Jensen’s alpha being adjusted for the market premium or, the intercept term in the regression of portfolio return on the Fama-French factors (Fama and French, 1992), “alpha” in the investment management industry often refers to the difference of return of a portfolio versus its benchmark. Presumably and loosely speaking, one can deliver Not for retail client use in Europe, Asia and the Middle East. 3 Risk On, Risk Off “alpha” just by taking more risks when risks are rewarded, and vice versa. Along the same line of argument, the “alpha” of asset y over asset x, defined as the difference of their returns, in a “Risk On, Risk Off ” environment so that their Sharpe ratios are identical as in equation (2), is given by, (3) ?xy = µy - µx = (?y - ?x) µy r ?y [ ] Since ?y > ?x, thus, ?y – ?x > 0 when ?y > r and the “alpha” of y over x is therefore positive
when y delivers positive premium over the risk-free rate. However, as made clear by
equation (3), the apparent “alpha” of y over x is merely a result of the fact that y is more
risky than x, and indeed there is no outperformance from a risk-adjusted perspective as
measured by their identical Sharpe ratios, since the two assets are statistically redundant,
as discussed above. The case of ?y < r can be interpreted similarly. Consider the real world, when conventional wisdom often says “there is no opportunity when correlations go to one.” While we have established the result that in a perfectly correlated world Sharpe ratios of all assets are identical and, therefore, there is no riskadjusted outperformance of any assets over the others, it is still entirely possible to see different returns achieved by different assets as long as the risks of these assets are not the same; see equation (3). In other words, an investor can achieve a higher return, or outperformance, simply by holding more (less) risky assets in her portfolio when risky assets have positive (negative) returns. Higher return or outperformance in such a world can, however, also be achieved simply by leveraging up (down) any asset in this universe. Does this necessarily mean that the higher return has nothing to do with investment skill and, therefore, the investor should not be compensated? T h e R e a l W or l d : A “ M a cro - Dr i v e n ” M a r k e t W i t h H i g h C orr e l at i o n s In an attempt to answer the question above, we move back to the real world in which correlations can at times be very high, but never really be perfect at a value of +1. To provide a framework for tractable analysis, we make use of a factor model that drives asset returns and risks. In the appendix, we provide a more formal description of the framework in a realistic, multi-asset, multi-factor world, and delineate how correlations are driven by the systematic, factor-related components versus the nonsystematic, idiosyncratic components. For ease of illustration and without loss of generality, below we consider a world with only one factor. An example is the market factor in the Capital Asset Pricing Model (CAPM). The only factor exposures in this case are the market betas, ?i. That is, for any asset i, its excess return can be represented as (4) Ri-r = ?i (Rf-r) + ?i Where Ri is the return of asset i, ?i is the idiosyncratic return of asset i with idiosyncratic volatility ??i which is independent of the factor f, and Rf is the factor return with factor volatility ?f. The covariance matrix and correlation for two assets, asset 1 and asset 2, can be given as, respectively, (5) [ [[ ?21 ?12 ?1 ?2 ?21 ?2f + ??2 = ?= ? ?1 ?2?2f ?22 12 ?1 ?2 Not for retail client use in Europe, Asia and the Middle East. 1 ?1 ?2?2f ?22 ?2f + ??2 2 [ 4 Risk On, Risk Off (6) ?12 = ?1 ?2?2f ( ?1?2+ ??2) ( ? ?2 + ??2) 2 2 f 1 2 f 2 Consider the extreme case when ??2 = ??2 = 0 where volatilities of assets are all attributable to their factor-related components, as there is no idiosyncratic risk. Equation (6) demonstrates that the correlation between the two assets must be perfect at +1. This is intuitive as when assets are completely driven by the same common factor, and there is absolutely nothing unique about these assets to have any impact on their return dynamics, then their correlation must be perfect. As discussed in the previous section, the Sharpe ratios of the two assets must be identical in this case. 1 2 Equation (6) also demonstrates that when magnitudes of idiosyncratic risks are dominated by the factor-related risks, correlations among assets are higher. For example, consider the universe of individual stocks and suppose the factors are sectors. Investors often refer to an environment when risks of stocks are largely driven by sector exposures—particularly during distinct economic regimes—as a “macro-driven market.” For instance, during an economic upturn, cyclical sectors such as Industrial and Consumer Discretionary tend to outperform. During an economic downturn, investors prefer to hold more defensive sectors such as Utilities. In this market environment, performance of a portfolio is largely determined by exposures to sectors that are out- or underperforming, and individual stock selection within sectors can become less of a concern. In other words, in a world with highly correlated assets when risks and correlations are largely dictated by their factor-related components, investment performance becomes increasingly driven by factors. In order to outperform in such an environment, the two key investment skills are (1) precise estimation of the factor structure, including identification of factors, estimation of factor exposures, and estimation of the factor covariance matrix;1 and (2) factor timing. As an example, consider the one-factor CAPM world again in which assets are highly correlated such that the idiosyncratic components are insignificant in determining returns of assets. In order to outperform in this world, the investor, having correctly identified the market factor as the only factor, must precisely estimate asset betas as well as the sign of the market factor return. If the investor is bullish on the market factor, the portfolio should be steered to hold high beta and, therefore, high risk assets, and vice versa. Similarly, when assets are highly correlated in a multi-factor world, with factors including sectors and other characteristics, the investor must shift efforts into factorrelated research, including estimation and timing, in order to outperform. Carrying the argument to the extreme case, the only skill that should be rewarded when correlations go to one is factor timing. I n v e s t m e n t I m p l i c at i o n s Derman (2010) made the following observations: • “ The world is multi-dimensional. Models allow us to project the object into a smaller space and then extrapolate or interpolate within it. At some point the extrapolation will break down.” • “ Theories tell you what something is. Models tell you only what something is more or less like.” Not for retail client use in Europe, Asia and the Middle East. 5 Risk On, Risk Off • “ The one law you can rely on in finance is the law of one price, … The law of one price is not a law of nature. It’s a general reflection on the practices of human beings, who, when they have enough time and enough information, will grab a bargain when they see one. The law usually holds in the long run, in well-oiled markets with enough savvy participants, but there are always short- or even longer-term exceptions that persist.” The world with perfect correlations and identical Sharpe ratios does not exist. Even if it did, and our set of normative results in such a state were indeed derived in the spirit of the law of one price, it could not be done without the same set of assumptions on human beings’ behavior underlying the Modern Portfolio Theory. As Derman pointed out, exceptions can persist for various reasons. Furthermore, our analysis only offers insights into the end point—the scenario of perfectly correlated assets—and tells us nothing about the scenarios approaching such an end point. Therefore, this paper does not provide any guidance for interpolation. Nevertheless, we take the insights developed so far as our compass and apply them to three different cases that we believe are of interest. We leave the interpolation exercise to the creativity of the readers. Case 1: Risk Parity In Kaya and Lee (2011), the risk-parity portfolio in which all assets contribute equal risk is determined to be the mean-variance optimal portfolio when (1) correlations of all assets are the same, and (2) Sharpe ratios are identical. Note that the optimality of risk parity does not require perfect correlations. But in the very special case with perfect correlations among all assets, which means correlations are the same by definition, we have shown above that Sharpe ratios of all assets must be identical. Therefore, optimality conditions of the risk-parity portfolio are satisfied in this case. In other words, in this nonexistent world, one can argue that the risk-parity portfolio is the mean-variance optimal portfolio. However, as all assets are statistically equivalent in this special case, portfolios of these assets, regardless of their weights, are also statistically equivalent to the single, nonredundant asset. Therefore, any portfolio in such a world must also satisfy the risk parity efficiency conditions, as risks in all portfolios are derived from one nonredundant asset. In this paper, we make no predictions on the efficiency of risk parity as a function of the level and direction of correlations. Kaya and Lee (2011) shows that the relative efficiency of risk parity can depend on a number of parameters in the underlying multivariate return distribution, including the Sharpe ratios, correlations, volatilities, deviation from the normal distribution, uncertainty of parameters estimates, among others. We leave the more in-depth analysis of empirical performance of the risk-parity portfolios under different conditions to future research. Instead, we construct a risk-parity (RP) portfolio in the universe of the largest 500 stocks at the end of each calendar year in the U.S. stock market from 1979 to July 2011 by using the Barra USE3L model as our covariance matrix.2,3 The RP portfolio is rebalanced at the beginning of every month. Following Lee (2011), we do not impose any constraints on the portfolio construction process in order to preserve the original characteristics of the portfolio. The average turnover per month of the unconstrained RP portfolio is about 6%, which makes it a very reasonable and implementable portfolio. The widely applied W statistic of Gibbons, Ross, and Shaken (1989) provides a formal t... Purchase answer to see full attachment