However, some investors can't use leverage. Others not only may be precluded from using leverage, but they also must have some of their assets in cash. For example, a mutual fund may need cash to be able to meet daily redemptions, an insurance company needs to pay claims, and individual investors may need cash for unforeseen expenses. And investors face margin requirements, limiting the amount of leverage that can be applied. For mutual funds, a need to hold cash creates an incentive to overweight high-beta securities in order to avoid lagging their benchmark in a bull market because of the cash holdings.

Because of these constraints, risk-seeking investors will overweight risky securities instead of using leverage. That can lead to overpricing of risky securities. And there are impediments to short-selling that play a significant role in limiting the ability of rational traders to exploit overpricing:

- Many institutional investors -- such as pension plans, endowments, and mutual funds -- are prohibited by their charters from taking short positions.
- Shorting can be expensive -- you have to borrow a stock to go short, and many stocks are costly to borrow because there are low supplies of available stock from institutional investors.
- Investors are unwilling to accept the risks of shorting because of the potential for unlimited losses. Even traders who believe that a stock's price is too high know that they can be correct (the price may eventually fall), but they know they face the risk that the price will go up before it goes down. Such a price move, requiring additional capital, can force the traders to liquidate at a loss.

### CAPM: It's all about beta

In the CAPM, beta is the measure of the volatility, or systematic risk, of a security or a portfolio in comparison to the market as a whole. High-beta securities have more risk than the market and low-beta securities less. Thus, under CAPM high-beta stocks should have higher returns to compensate investors for their higher risk. However, the behavior of tilting toward high-beta assets could result in investors accepting lower returns from high-beta assets because low-beta stocks require the use of leverage.

The authors of 2011 study, "Betting Against Beta," sought the answers to the following questions:

- How can an unconstrained arbitrageur exploit this effect? How do you bet against beta?
- What is the magnitude of this anomaly relative to the size, value and momentum effects?
- Is betting against beta rewarded in other countries and asset classes?
- Which investors are constraint and thus bet on beta?
- Who bets against beta?

To determine the answers, they analyzed the returns of a market-neutral "betting against beta" (BAB) strategy. The BAB factor is a portfolio that holds low-beta assets, leveraged to a beta of 1, and that shorts high-beta assets, de-leveraged to a beta of 1. For instance, the BAB factor for U.S. stocks achieves a zero beta by holding $1.5 of low-beta stocks and short-selling $0.7 of high-beta stocks.

The following is a summary of their findings, which covered the U.S., 19 developed markets, and more than 50,000 stocks:

- The higher the beta, the lower the alpha (excess returns) and the lower the Sharpe ratio (risk-adjusted return).
- For the period 1926-2009, and for each of the four 20-year sub periods, the BAB factor has highly significant risk-adjusted returns after accounting for its realized exposure to market, value, size, momentum, and liquidity factors, and realizes a significant positive return in each of the four 20-year sub-periods between 1926 and 2009.
- A long-low-beta/short-high-beta portfolio measured against the Fama-French three-factor model (beta, size and value) produced abnormal returns of 0.69 percent per month with a t-statistic of 6.55 (above 2 indicates statistical significance). Adjusting returns for Carhart's momentum-factor (the fourth factor), the BAB portfolio earns abnormal returns of 0.55 percent per month with a t-statistic of 5.12.
- Adjusting returns for a fifth factor, liquidity risk, produced an abnormal BAB return of 0.46 percent per month with a t-statistic of 2.93.
- The U.S. BAB factor produced a Sharpe ratio of 0.75, higher than that of the size, value, and momentum factors.
- For the period 1984-2009, international stocks produced similar results with positive Sharpe ratios in 18 of the 19 MSCI developed countries. The BAB factor earned risk-adjusted returns between 0.41 percent and 0.74 percent per month depending on the choice of risk adjustment, with t-statistics ranging from 2.48 to 4.15.
- Relative to high-beta stocks, low-beta stocks are likely to be larger, have higher book-to-market ratios, and have higher returns over the prior 12 months (the momentum factor), although none of the loadings can explain the large and significant abnormal returns.

The authors found similar results in the bond market.

- For U.S. Treasuries, the BAB factor is a portfolio that holds leveraged low-beta (short-maturity) bonds and short-sells deleveraged high-beta long-term bonds. This portfolio produces highly significant risk-adjusted returns with a Sharpe ratio of 0.85.
- The Sharpe ratios decline monotonically from 0.73 for low-beta (short maturity) bonds to 0.27 for high-beta (long maturity) bonds. The bond BAB portfolio delivers abnormal returns of 0.16 percent per month, with a t-statistic of 6.37 and a large annual Sharpe ratio of 0.85.

The authors noted that the profitability of short-selling long-term bonds seems to contradict the term premium in the bond market. But there's no paradox. The term premium means that investors are compensated on average for holding long-term bonds rather than Treasury bills. However, the term premium exists at all horizons.

They explain:

Just like investors are compensated for holding 10-year bonds over T-bills, they are also compensated for holding one-year bonds. Our finding is that the compensation per unit of risk is in fact larger for the one-year bond than for the 10-year bond. Hence, a portfolio that has a leveraged long position in one-year (and other short-term) bonds and a short position in long-term bonds produces positive returns. This result is consistent with our model in which some investors are leverage constrained in their bond exposure and, therefore, require lower risk-adjusted returns for long-term bonds that give more "bang for the buck."

The authors also provided an example to illustrate the constraint effect at work:

Suppose a pension fund has $1 to allocate to Treasuries with a target excess return of 2.5 percent per year. One way to achieve this return target is to invest $1 in a portfolio of Treasuries with maturity above 10 years. If the agent invests in one-year Treasuries instead, then he would need to invest $11 if all maturities had the same Sharpe ratio. This higher leverage is needed because the long-term Treasures are 11 times more volatile than the short-term Treasuries. Hence, the fund would need to borrow an additional $10 to lever his investment in one-year bonds. If the fund has leverage limits (or prefers lower leverage), then it would strictly prefer the 10-year Treasuries.

According to their hypothesis, the one-year Treasuries must offer higher returns and higher Sharpe ratios, flattening the security market line for bonds. That is exactly what they found. "Short-term Treasuries did provide higher risk-adjusted returns -- so the return target can be achieved by investing about $4 in one-year bonds. While a constrained investor may still prefer an un-leveraged investment in 10-year bonds, unconstrained investors now prefer the leveraged low-beta bonds, and the market can clear."

They also found similar evidence in the credit markets. Corporate credits were sorted by ranking by rating, from AAA to D (distressed). For the period 1973-2009, a leveraged portfolio of high-rated corporate bonds outperforms a deleveraged portfolio of low-rated bonds. Similarly, using a BAB factor based on corporate bond indexes by maturity produced large abnormal returns (0.56 percent per month with a t-statistics of 4.02) and declining alphas and Sharpe ratios across beta-sorted portfolios.

Turning to the question of who bets on and against beta, the authors found that constrained investors such as mutual funds and individual investors hold portfolios with betas of more than one. On the other hand, leveraged buyout (LBO) funds, which can apply leverage, acquire firms with betas below 1. Similarly, they found that Berkshire Hathaway (Warren Buffett) bets against beta by buying low-beta stocks and applying leverage (helping to explain Buffett's legendary alpha).

The authors concluded that real-world investors face funding constraints such as leverage constraints and margin requirements. These constraints influence investors' required returns across securities and over time. Because investors prefer un-leveraged risky assets to leveraged safe assets, they hold portfolios of high-beta assets that have lower alphas and Sharpe ratios than portfolios of low-beta assets.