APPENDIX: CONSTRUCTING OUT-OF-SAMPLE SHORTFALL BETAS

To compute shortfall betas as of a given month, we first simulate a distribution of future, daily portfolio returns using daily historical factor returns from the MSCI Barra Short-Term US Equity Model (USE3S). For each forecast date, we use the factor returns corresponding to the previous 1,000 days.

Since factor volatility is nonstationary, we adjust the historical factor returns to reflect the level of volatility as of the forecast date. There are a few ways to do this. One simple way is to scale the past individual factor return, f_t,i, by the ratio of its volatility as of the forecast month, T, to its volatility at time t; that is

Another approach uses the entire covariance matrix to rescale the factor returns. Let F_t be the factor covariance matrix at t and f_t a vector of factor returns at t. Applying the Cholesky decomposition, we have . Then, the scaled factor return is⁹

After we have 1,000 scaled factor returns describing the distribution of future factor returns, we use the portfolio's current factor exposures X_T together with those factor returns to estimate the distribution of future asset or portfolio returns, that is,

where represents the distribution of future returns. We ignore the specific returns in generating scenario returns.¹⁰

In the main analysis, we forecast the distribution of the MSCI US Prime Market 750 Index every month. Using the distribution, we can calculate the expected shortfall for the index portfolio as well as the marginal contribution of each asset to this shortfall, which along with its weight in the index, forms its shortfall beta.