Estimating Implied Correlations for Basket FX Options ...



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 stated level 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 [pic] entries is a column vector with [pic] rows and 1 column. Suppose an investor wants to create an optimal portfolio of [pic] asset classes. The covariance between asset returns, given by the [pic] matrix[pic], 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 [pic]. 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 [pic], as follows.

| |Large Value |

|Large Value |5.08% |

|Large Growth |-0.99% |

|SMID Value |18.56% |

|SMID Growth |25.47% |

|EU large |2.83% |

|EU smid |24.40% |

|UK |5.16% |

|Pacific large |6.60% |

|Pacific smid |19.34% |

|EM Equity |35.65% |

The results are very intuitive. Small cap and emerging market equities - asset classes that are unlikely to perform well during a flight to quality – have significantly larger betas than large cap equities in developed markets. The historical correlation between EM Equity and our recession factor is over 40%.

Using historical data and notation from the previous section, we have calculated the following quantities. As expected, asset classes with significant alternative beta exposures have larger equilibrium returns than usual. The table below specifies the vector [pic]. The multiple R squared of the following regression is over 55%.

|  |regression weights |

|Large Value |0.2648 |

|Large Growth |-0.4223 |

|SMID Value |-0.3338 |

|SMID Growth |0.3384 |

|EU large |-0.7260 |

|EU smid |0.5646 |

|UK |0.2067 |

|Pacific large |-0.2707 |

|Pacific smid |0.1550 |

|EM Equity |0.2687 |

This allows us to calibrate [pic] = 4.25 and [pic] = 0.0947. Next, suppose we have the same view as in the one factor example: we believe that SMID Growth equities will outperform European ones by 8% over the next year. Then, [pic] = 0.0630. Since the SMID Growth equilibrium return is already a bit more than 6.30% over the capitalization weighted European equilibrium return, our final allocation to SMID Growth should actually be less than the market’s. Thus, the optimal allocation to SMID Growth moves in the opposite direction from the one factor case.

Using the BL formula, we have the following result.

|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 |6.95% |10.07% |7.71% |14.39% |8.41% |6.92% |7.43% |8.63% |7.60% |10.91% | |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% |7.00% |10.65% |2.98% |10.58% |11.55% |3.56% |4.30% | |

The optimal allocation is vastly different than before, even though the portfolio manager’s view hasn’t changed. In the one factor BL model, the allocation to SMID Growth increases from roughly 7% to over 17%. Conversely, in the two factor model, the optimal SMID Growth weight decreases. While it cannot be said that the two factor model will be a better approximation of reality in every case, it does provide a more complete description of risk. Our model also allocates more conservatively to asset classes that are risky in a way that is not completely captured by volatility.

A Final Note

Our paper can be extended to incorporate uncertain views and more than two factors. While we believe that these generalizations are relatively straightforward, we have not included them in this paper. Our main goal was to explore a simple and concrete two factor example

In principle, our approach should be able to incorporate hedge funds into an overall allocation framework. Hedge fund managers are often able to generate high portfolio Sharpe ratios by “selling insurance” in one form or another. For example, a fixed income arbitrage manager may take a long position in a relatively illiquid basket of securities and short a more expensive but liquid replicating basket. The strategy should consistently make money, while retaining a small probability of a large drawdown. In summary, liquidity providers tend to take recession risk.

Many investors are keen to spend units of risk using hedge funds rather than long only managers. For example, an investor may believe that a leveraged European equity hedge fund will outperform the EU equity index by 2% over the next 12 months, with comparable volatility. The investor may then replace index data with historical returns for the hedge fund in a portfolio optimization routine. While the hedge fund may have a higher expected return than the long only manager, it may also have more exposure to recession risk. Our approach controls the top down allocation to asset classes and active managers that take on systematic risks that are not adequately explained by the traditional CAPM model.

References

[1] Black, F and R Litterman, “Asset Allocation: Combining Investor Views With Market Equilibrium”, Goldman Sachs Fixed Income Research, September 1990.

[2] Fama, EF, “Multifactor Portfolio Efficiency and Multifactor Asset Pricing”, Journal of Financial and Quantitative Analysis 31:4, December 1996, 441-65.

[3] Harvey, C and A Siddique, “Conditional Skewness in Asset Pricing Tests”, Journal of Finance 55, June 2000, 1263-95.

[4] Cochrane, JH, “Portfolio Advice for a Multifactor World”, Economic Perspectives: Federal Reserve Bank of Chicago Q3 1999, 59-78.

[5] Bevan, A and K Winkelmann, “Using the Black-Litterman Global Asset Allocation Model: Three Years of Practical Experience”, Goldman Sachs Fixed Income Portfolio Strategy, June 1998.

[6] Litterman, R and G He, “The Intuition behind Black-Litterman Model Portfolios”, Goldman Sachs Investment Management Research, December 1999.

[7] Koch, W, “Consistent Asset Return Estimates via Black-Litterman – Theory and Application”, ComInvest Asset Management, June 2003.

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[1] The authors would like to thank Hilary F. Till for her helpful comments.

[2] This paper is not for general distribution.

[3] In our analysis, we assume that [pic] and [pic] are annualized quantities.

[4] It should be noted that India, China and Russia are not included in the index, which may lower the EM Equity weight somewhat.

[5] We know that [pic], so we do not have to apply the constraint directly.

[6] We do not believe that a portfolio of global equities is very likely to achieve a Sharpe ratio of 1 over the long term.

[7] While Altman Public Bond Index data goes back to 1987, returns from 1987 to 1997 are only available on a yearly basis and have not been included in the graph.

[8] The Dow Jones Total Market Index is only available from January 1992 onwards, which determines our historical window.

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