Black-Litterman + 1: Generalizing to the Two Factor Case[1][2]

Hari P. Krishnan and Norman E. Mains

Introduction

Since the 1950s, mean-variance optimization (MVO) has been widely used to make asset allocation decisions. In practice, however, MVO is rarely used without modification, for a variety of reasons. We list two here.

  • The results of MVO can be very sensitive to small changes in the expected return of a given asset class, particularly when the asset has a high correlation to other assets. Usually, we do not know the expected return of any asset class with certainty. Thus, portfolio managers sometimes wind up making return estimates simply to generate portfolios that look reasonable. This does not seem to be a very scientific approach.
  • Classical MVO does not allow a portfolio manager to express tactical views in a consistent way. Suppose that an investor has a reasonable estimate of the correlation between asset classes in a portfolio and believes that the correlation will not change too much over time. Then, if the investor has a tactical view on a given class, he or she implicitly has a view on all classes that are highly correlated (either positively or negatively) to it. If one asset is predicted to go up, then every correlated asset will typically have a higher than usual chance of going up as well. This is not accounted for in the original Markowitz model.

The Black-Litterman (BL) model [1], developed around 1990, addresses both of these issues in an innovative and intellectually appealing way. First, it uses a piece of market information that MVO does not: the expected return for each asset class is partly dependent upon the amount of capital that has flowed into it over time. Assuming that the global market is close to equilibrium, the BL model provides a mechanism for calculating an implied return for every asset class as a function of its size and covariance with other assets. This set of implied returns constitutes an equilibrium return vector. As in the CAPM model, assets with high market betas are assumed to have relatively high expected returns. Since these assets generally contribute to portfolio risk, an investor needs to be compensated with a larger than usual return. Asset classes with large capitalization weights also have relatively large equilibrium returns. If investors are allocating a large amount of money to a given asset class, the market outlook for that asset class is likely to be good.

The BL model also offers a consistent framework for implementing tactical views. A portfolio manager can express an opinion about the future return of a collection of assets (a mini-portfolio) with a statedlevel of confidence. In the absence of any views, market implied returns are used. When a view is taken, BL returns are simultaneously chosen so that tactical views are expressed and optimal portfolio weights are close to the market. If a portfolio manager does not have a view about a particular asset class, its equilibrium and optimal BL weights are identical. The BL return vector, moreover, has a nice analytical solution which can be used as an optimizer input.

While the BL model significantly improves upon MVO, it makes several restrictive assumptions as well. In this paper, we focus on one of these: the BL model assumes that risk can be completely characterized by covariance. Near equilibrium, the expected return of an asset with fixed correlation to others should increase purely as a function of its volatility. As we have noted, a similar assumption is made in the single factor CAPM model. Recently, several researchers have argued that the standard CAPM model does not adequately describe risk. It is possible to collect a premium for taking on systematic exposures that are uncorrelated to the market and that investors wish to avoid. Fama [2] has estimated the premium for taking a long position in the (value – growth) and (small cap – large cap) spread for US equities. Moreover, Harvey [3] has suggested that over the long term, investors can get paid for including negatively skewed assets in a portfolio. If this is the case, the expected return of an asset class should increase not only as a function of its volatility but also its beta to these alternative risk factors. In his expository article, Cochrane [4] collectively refers to these exposures as recession risk. Most investors are willing to pay a premium for insurance during an economic downturn; this premium can be collected by others who have longer investment horizons.

In this paper, we incorporate recession risk as a second factor into the Black-Litterman model. This is why we call our paper “Black-Litterman + 1”. We start with a review of the one factor BL framework and then show how to extend it. An intuitive factor is then introduced that tracks the performance of riskier asset classes, such as emerging markets. Our factor is uncorrelated with the market, yet has generated a premium in the past. We generalize the quadratic utility function in the BL model and calibrate the market’s risk aversion to volatility and alternative beta risk (exposure to the second factor). We then calculate a two factor equilibrium return vector for a set of assets. Next, we introduce a simple transformation that allows us to use the standard BL formula in the two factor case. Finally, we describe by way of example how optimal portfolio weights vary from the standard BL model to ours.

Our technique should be useful whenever risky assets with low historical volatilities are added to a portfolio. In particular, it offers a quantitative way to incorporate hedge funds and other active managers who take systematic risks that are not explained by CAPM into an overall portfolio.

The BL Model: Equilibrium Returns

We start with a review of the BL model. We use the convention that any vector containing entries is a column vector with rows and 1 column. Suppose an investor wants to create an optimal portfolio of asset classes. The covariance between asset returns, given by the matrix, is assumed to be known.[3] In addition, the investor has information about the market portfolio, equivalently, the size of each asset class relative to the total market is given by the weight vector. For example, suppose that we want to allocate to the asset classes given in the table below. These are a rough approximation for the global equity market.

Equity Category / Equity Market Weight / Comments
Large Value / 20.14% / US only
Large Growth / 21.41% / US only
SMID Value / 7.83% / "SMID" = small and mid cap US
SMID Growth / 7.15% / "SMID" = small and mid cap US
Large EU / 10.53% / "EU" = European Union
SMID EU / 2.94%
UK / 10.58%
Large Pacific / 11.55%
SMID Pacific / 3.56%
EM Equity / 4.30% / "EM" = Emerging Markets
100.00%

The weights in the table above have been calculated from the Dow Jones Total Market Index as of June 2004.[4]

The covariance between returns is given by , as follows.

Large Value / Large Growth / SMID Value / SMID Growth / EU large / EU smid / UK / Pacific large / Pacific smid / EM Equity
Large Value / 1.90% / 1.80% / 1.70% / 2.11% / 1.51% / 1.20% / 1.39% / 1.11% / 0.93% / 1.86%
Large Growth / 3.40% / 1.82% / 4.39% / 2.07% / 1.47% / 1.82% / 1.68% / 1.69% / 2.72%
SMID Value / 2.43% / 3.29% / 1.72% / 1.74% / 1.38% / 1.22% / 1.00% / 2.44%
SMID Growth / 7.40% / 3.03% / 2.71% / 2.36% / 2.69% / 2.34% / 4.56%
EU large / 2.68% / 2.09% / 1.82% / 1.88% / 1.08% / 2.04%
EU smid / 2.63% / 1.42% / 1.33% / 1.17% / 2.29%
UK / 2.80% / 1.32% / 1.05% / 1.89%
Pacific large / 4.73% / 3.74% / 2.44%
Pacific smid / 5.22% / 2.74%
EM Equity / 5.46%

We have estimated using historical returns over the longest available period, using Russell indices for US equities, MSCI indices for European, Pacific and emerging market equities and the FTSE 350 for the UK stock market.

The BL model assumes that, on average, investors make allocation decisions relative to the utility function. Here, is any vector of portfolio weights, not necessarily equal to the market weight vector. This is the same quadratic functionof used in standard MVO, where is the vector of expected returns for each asset class. is a risk aversion parameter that varies from investor to investor. The level of risk aversion increases with, since the portfolio variance plays a larger role.

In the usual MVO approach, is thought of as a function of given a set of returns. Unfortunately, we do not know exactly what or even will be in the future, so whenever we try to optimize we wind up with a noisy estimate. The BL model uses market implied information to come up with a consensus estimate of . Thus, becomes the dependent variable in . If the global market is close to equilibrium, then the vector of market weights should be nearly optimal at any point in time. This means that the current allocation of capital in the global market is reflective of all available information. We then ask: what set of returns would make the vector optimal? These are the BL equilibrium returns.

We can solve for explicitly. Since is concave in, we know it contains a unique maximum. For any vector, is maximized over when , subject to. However, if we know that is optimal, we can solve the above equation for as .[5] This formula is similar to CAPM in the sense that volatile assets with a high correlation to other assets have a high expected return. These assets typically have high market betas. In the following discussion, we will refer to as the equilibrium return vector .

It remains to specify the risk aversion parameter. Unfortunately, this parameter is not directly observable in the market and there are many ways to calibrate it. We have decided to adapt Bevan and Winkelmann’s [5] convention. We can define the expected Sharpe ratio of any portfolio as , where and are the portfolio’s expected return and risk-free rate, respectively. Once we specify (the Sharpe ratio we think the portfolio will generate), we can solve for . Bevan and Winkelmann set equal to 1 in their paper, but in general any choice can be made. If we believe that the market portfolio will have Sharpe ratio over some horizon, we can solve for . However, we know that in an equilibrium setting, which implies that .

We now calculate directly. Using historical data, we have . Assuming that = 0.015 and that a capitalization weighted portfolio of equities will generate a Sharpe ratio of 0.5 in the future, we have = 0.09.[6] Thus, = 4.25. then takes the following form.

Large Value / Large Growth / SMID Value / SMID Growth / EU large / EU smid / UK / Pacific large / Pacific smid / EM Equity
equilibrium return / 6.95% / 10.09% / 7.72% / 14.42% / 8.41% / 6.93% / 7.43% / 8.64% / 7.61% / 10.93%

The Black-Litterman Model: Taking a View

In the previous section, we calculated BL equilibrium returns for each asset class as a function of the covariance matrix and the market weight vector. If an individual did not have an opinion about any class, the vector would be used as a default. However, most active managers do have opinions about future returns and are in fact paid to have them. The BL model offers a way to perturb relative to a set of market views. The views may be either absolute or relative. An absolute view specifies the future return of a single asset class and a relative view specifies the return of a sub-portfolio. For example, an active manager may believe that India's expected return is 10% larger than China's over the next year. This is a relative view that is equivalent to saying that an appropriately weighted (long India, short China) portfolio has an expected return of 10%. As Litterman and He [6] argue, portfolio managers often think in terms of relative, rather than absolute, valuation.

The BL model also allows a portfolio manager to associate a confidence level with a given view. 100% confidence indicates that the portfolio manager is certain that the view is correct. Conversely, a confidence level close to 0 indicates that the manager is very unsure of the view. Intuitively, if a manager is sure that asset class has expected return , then . If the manager has no confidence in a view, then (ignoring correlation effects) . For simplicity, we will assume that every view is specified with 100% confidence in our analysis. The BL model provides a method for perturbing in a systematic way when calculating the BL return vector . The problem is set up as a constrained optimization satisfying the following properties.

  • If asset class is sensitive to small changes in expected return, then is small. This stabilizes the optimization results.
  • The covariance matrix is taken into account. Thus, if is thought to be larger (smaller) than and assets and are highly correlated, will generally be larger (smaller) than . The reverse will hold true if and have a large negative correlation.
  • Views are implemented as constraints. If asset is known to have expected return, then . Otherwise, (ignoring correlation effects) will lie somewhere between and .
  • If no view is taken about asset class , then needs to be chosen so that .

Formally, the BL model chooses so that is minimized, subject to a set of views. Here, is a constant. Since and are necessary conditions for optimality, and are also necessary. Thus, has an informal interpretation as, where , are optimal portfolio weights relative to any return vector and are equilibrium weights. If is very unstable as a function of, the optimization will penalize for small changes in . We note that while is a function of , is not.

Now suppose we have a view about sub-portfolios, where . Using the matrix and the vector , we can express these views as . In the BL formulation, is a vector of uncorrelated, normally distributed random variables. If every view is certain, . In summary, we want to solve the following problem:

  • choose so that is maximized;
  • if each view is certain, satisfy the constraint .

The solution to the above problem takes the following analytical form:

. (The Black-Litterman Formula)

This is known as the Black-Litterman formula with certain views. A concise proof of the BL formula can be found in Koch’s [7] presentation, using the method of Lagrange multipliers.

We now consider a concrete example. Suppose a portfolio manager takes the view that the SMID growthindex (i.e., the Russell 2000 Growth Index) will outperform Europe by 8% over the next year. Using the market capitalization weights for Europegiven above, this is equivalent to saying that the portfolio (1 * SMID Growth – 0.78 * EU Large – 0.22 * EU SMID) has an expected return of =0.08. Thus, the vector has the following form. Recall from above that the EU Large Cap market is about 0.78 / 0.22 times as large as the EU SMID market.

Large Value / Large Growth / SMID Value / SMID Growth / EU large / EU smid / UK / Pacific large / Pacific smid / EM Equity
P / 0.00 / 0.00 / 0.00 / 1.00 / -0.78 / -0.22 / 0.00 / 0.00 / 0.00 / 0.00

Using the BL formula, we can find BL returns and optimal weights (relative to the BL returns) for each asset class.

Large Value / Large Growth / SMID Value / SMID Growth / EU large / EU smid / UK / Pacific large / Pacific smid / EM Equity
equilibrium return / 6.95% / 10.09% / 7.72% / 14.42% / 8.41% / 6.93% / 7.43% / 8.64% / 7.61% / 10.93%
Black-Litterman return / 7.23% / 11.12% / 8.38% / 16.29% / 8.61% / 7.14% / 7.70% / 9.03% / 8.14% / 11.97%
equilbrium weight / 20.14% / 21.41% / 7.83% / 7.15% / 10.53% / 2.94% / 10.58% / 11.55% / 3.56% / 4.30%
optimal portfolio weight / 20.14% / 21.41% / 7.83% / 17.11% / 2.75% / 0.77% / 10.58% / 11.55% / 3.56% / 4.30%

The portfolio weights that differ from equilibrium are highlighted in bold. Importantly, the BL model does not change the optimal weights for asset classes where the portfolio manager has not taken a view. This property will also hold in the two factor case.

Adding a Second Factor

As mentioned above, Black and Litterman use the quadratic utility function in their analysis. Although this is a natural choice, it does not fully characterize investor behavior. Investors generally do not believe that risk can be completely characterized by a single parameter.Rather, theyare worried about extreme event or recession risk. In the meantime, investors who are willing to suffer through difficult market conditions can earn a premium.

In the next section, we will explicitly define a factor that has the following properties:

  • it is uncorrelated with the market;
  • it adequately captures recession and event risk;
  • it generates a premium for taking these risks.

For now, suppose that is a vector containing the beta of each asset class to the factorand assume that the above properties hold. In particular, set for . The market utility function is now given by . The term measures the beta of a portfolio with weight vector to our second factor. We can calculate the equilibrium return vector using the equation . In particular, if we set in , we have . (Two Factor Equilibrium Return Formula)

This equation is quite intuitive and suggests that assets that are exposed to risk should have a relatively large expected return. More precisely, one factor equilibrium returns are simply translated by . We can calibrate and using the following argument. Since the market portfolio is uncorrelated to , we know that . Thus, , as before. From above, we have .

For , we need to construct a portfolio that replicates exposure as closely as possible.If we parameterize by time, we can replicate using a linear combination of for each asset class. This amounts to performing a regression, i.e. solving for so that is minimized. Now suppose that the expected return of is given by . This is the premium an investor receives for taking risk in the dimension , with volatility .

Now suppose that the expected return of is given by . Note that is not necessarily equal to and that the regression is only used to estimate . is the premium an investor receives for taking risk in the dimension , with volatility . We then have , which can be solved for as .

Before we can apply the Black-Litterman formula directly, we need to make the following transformation. Recall that investors are making decisions relative to the utility rather than . However, if we set and (so that is the same set of equilibrium returns as in the one factor case), we have . The Black-Litterman formula then applies to , namely . If an investor takes the view that , this needs to be transformed to .