Harry Markowitz, one of the original founders of modern portfolio theory, recently explored in a simple yet elegant paper how the capital asset pricing model (CAPM) crumbles in the real world (Markowitz, 2006). A striking consequence of this paper, which he does not mention but is relevant for investors today, is that the increasing availability of leverage for some investors may actually drive all risky security prices higher, even those not held by levered investors, potentially leading to a market far from equilibrium and with an ultimately destabilizing outcome.

To recap Markowitz’s new conclusions briefly: If you leave out the ability to lever as an assumption, or limit borrowing, the following consequences follow:

1. The market portfolio is no longer the mean-variance efficient portfolio.
2. The returns of securities are no longer proportional to their beta with the market portfolio.

The purpose of this note is to summarize his insight and also take the analysis one step further regarding what it means for prospective market risks and returns, and in particular for one possible outcome as a consequence of increased leverage. 

Markowitz’s New Analysis: An Example

Markowitz’s analysis is strikingly simple. Assume that there are three investable securities. For simplicity, the securities are taken to be uncorrelated and their expected returns, standard deviations and Sharpe ratios are given in Table 1 (we assume risk-free rates are zero):

The figure below (taken from his paper) illustrates the difference in portfolio allocation between two investors, one who can use leverage (i.e., does not have the constraint that all weights be positive), versus another investor, who cannot use leverage.  

Copyright, 2005 CFA Institute.

Reproduced and republished from Financial Analysts Journal with permission from CFA Institute.

The horizontal axis is the allocation to asset 1 and the vertical axis is the allocation to asset 2. The allocation to the third asset is determined by the budget constraint X₁+X₂+X₃=1 where X₁, X₂ and X₃ are the fractions in the three assets.  For an un-levered investor, the point (1,0) denotes full allocation to asset 1 only; point (0,1) full allocation to asset 2 only, and point (0,0) full allocation to asset 3 only.

The following consequences follow (and the qualitative results do not change if investors have only the ability to lever limited amounts):

1. c is the minimum variance (risk) portfolio and is unique. Being on c pre-determines the portfolio return (12.4%, obtained by plugging in the weights (0.28, 0.62, 0.10)). To increase return, one has to take more risk. The optimal portfolio with more return (and risk) lies along the line moving down towards b.
2. As return needs increase, a levered investor can move down beyond point b (say to point P), but an unlevered investor has no choice but to move along from point b to point e.  The unlevered investor cannot go any further than point (0,0), at which his return is maximized. If an unlevered investor tries to compete with a levered investor for returns, he can only access higher yielding, risky securities. As a consequence, the presence of leverage for some investors drives down the risk-premium for ALL securities, including the riskiest securities with worse risk-reward profiles.
3. The market portfolio is obtained by taking the wealth-weighted average of the portfolio allocation of various investors. If everyone remains on the line , then the market portfolio also lies along . The market portfolio is efficient (i.e., it has the least risk for the desired return). However, if some unlevered investors lie along the line connecting (0,0) and (1,0), and some investors lie along between points c and b, then the market portfolio is at M, which is not mean-variance optimal. Further, assuming the “sharks” (using a term coined by Stephen Ross and applicable in the current context to those who can lever), are based at P, the overall market portfolio in the presence of all three types of investors (unlevered lying on mean-variance frontier, levered lying on mean-variance frontier, and unlevered off the mean variance frontier), is somewhere like the point M^a, M^b or M^c. This portfolio is off , so it is clearly not optimal for the sharks.
4. Security returns: Markowitz shows in his paper that the returns of securities are no longer proportional to their betas relative to the portfolio M, since M is not the mean-variance efficient market portfolio. Rather, returns are now proportional to the levered optimal portfolio P.

Taking Markowitz One Step Beyond

We can now take the Markowitz paradigm one step beyond to explore the dynamic impact of leveraging on the market portfolio and security prices.  As more and more investors are able to lever, they buy securities that can be levered and move further out and down the  line. This puts pressure on investors who cannot lever, and they move further out to the left on the (0,0), (1,0) line, until they are all the way at (0,0), (i.e., they hold only the riskiest asset).  Clearly, easy Fed policy of years past and the implicit put provided by a faith in the government, along with affordability products that have enhanced main-street access to this leverage has created a perfect environment in which leveraging was easy.  We want to ask: What are the consequences?

First, unlevered investors are forced to hold the riskiest security 3, which has the highest return but also the lowest information ratio. The market portfolio of the unlevered investor lies along the line connecting d and e, which in the limit lies right on top of the point (0,0).  In this limit, there is zero net demand for both security 2 and security 3, which are ex ante more attractive on a risk-reward basis. The market is distorted in that high information ratio securities are not held at all by unlevered investors.  Note that the point labelled M^c cannot be an equilibrium because there is net negative demand for security 2. The consequence is that the demand for the riskiest security drives up its price, and drives down the price of the securities with better risk-reward profiles. 

Second, unlevered investors eventually begin to realize that leverage constraints are forcing them to hold the wrong securities, so they begin to relax their leverage constraints either explicitly or implicitly (e.g., with “packaged” solutions that allow leverage to be had via a structured note). They would like to do what the levered investors do, but the only way to do this is to (1) sell off the large unlevered holdings of risky securities and exchange them for a more optimal, levered mix; or (2) wait until the levered investors de-lever and come back inside the triangle. Of course, if they choose path (2), they run the risk of holding an underperforming portfolio that is suboptimal. To avoid this, they might throw in the towel, sell the risky securities and start to lever. At this point, the levered investors and the unlevered investors are all completely committed to holding market optimal but highly levered portfolios. However, the very definition of risk using variance entails unhedged, embedded vulnerability to fat-tail shocks, such as a crisis of confidence or liquidity/financing potholes. If unrealized fat-tail events occur, the weaker hands, or those with implicit leverage will de-lever first, and rapidly move the holdings back into the triangle towards point c. Market equilibrium returns with a vengeance, and depending on the overall magnitude of shorts in security 2, may set the stage for the next set of security price distortions. 

One way of dealing prospectively with the threat of fat-tail events in the presence of leverage is to build in a return penalty, ex ante, as the leveraging dial of the investor’s optimal allocation is dialled up. Another approach is to not chase the mean-variance optimal portfolio but set hard limits on the maximum amount of leverage allowed at security level. A final approach is to purchase options, either explicit or implicit, as embedded in security prices.  In selecting between these choices an investor has to carefully analyze the tradeoffs between the prospective loss of returns from not being fully invested in risky securities, versus the immediate cost of purchasing options.  Once in a while, an investor with patience is given the opportunity to get paid to purchase insurance and invest cash in prospectively higher returning securities with  lower risk: today an investment in short term treasury notes, which carry the highest yield in the yield curve, combined with an underweight in credit, the main beneficiary of the availability of leverage through creative structured products, may provide just such an opportunity. As we have seen from Markowitz’s new analysis, the selective ability to lever can create significant market distortions, and astute investors are well served by positioning themselves for the inevitable state when the unravelling of these distortions bites back those who have not paid attention to the underlying structures that have enabled them to arise in the first place.

References:

Markowitz, Harry M. “Market Efficiency: A Theoretical Distinction and So What?” Financial Analysts Journal, Vol. 61, No. 5, pp 17-30, 2005. Also presented at the 25th anniversary of the Q-Group, Fall 2006, Santa Barbara.