Active Portfolio Construction and Attribution Analysis

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Chapter 10 Examples.xlsx contains historical monthly percentage returns for the Citi BIG bond index (CitiGroup Broad Investment Grade) and the S&P 500 equity index, which serve as benchmarks. There are also returns to the 10 large cap firms we studied in Chapter 7 as well as the four sectors comprising the Citi BIG bond index: Treasuries (TSY), agencies (AGY), collateralized (COLL), and corporate credit (CRED). We are going to construct a fund consisting of two mean-variance efficient portfolios—a bond portfolio and an equity portfolio—and compare their performance to their respective benchmarks. Consider then the following scenario.

It's December 30, 2001. The trustees of a large pension fund are contemplating departing from a passive 70/30 benchmark allocation to the S&P 500 and the Citi BIG indices. They cite their desire to reclaim the fund's losses during the crash of the tech bubble that began in late 2000 and the subsequent decline in the pension's funded status. Hoping to enhance the fund's return performance, the trustees increase the exposure to equities to 80 percent. This move indicates a departure from their strategic benchmark weights. The trustees are now actively managing the fund to the 70/30 benchmark with an active 80/20 allocation. Suppose the trustees hire two active managers to carry out their new mandate. One is a bond manager claiming to deliver 50 basis points annually over the BIG. The second manager runs an active equity strategy and purports that his skill at picking winners will produce an annual 100 basis points in active return over his benchmark, the S&P 500.

Assume that both managers are mean-variance optimizers. Using returns data through December 2001, they estimate covariances and mean returns (monthly mean return plus their monthly alpha estimates) that they then input to their optimizers. There are no short sales and managers must remain fully invested. They solve for their optimal portfolios and invest for the following month (beginning in January 2002). Since they rebalance their portfolios monthly, they repeat this process for the next 60 months, creating a five-year sample of active portfolio returns that we will use to evaluate their performances.

On the sheet labeled “Active Portf,” the first line gives the observed returns to the two benchmarks for the month January 2002. The next two cells (Equity_A, Bonds_A) are returns for January 2002 on two active mean-variance efficient portfolios formed using information through December 2001. Equity_A was built from the returns to the 10 firms, and Bonds_A was built from the four sectors of the Citi BIG. The arguments for the optimizers consisted of the mean returns (plus monthly alpha) and the covariance matrices also using information through December 2001. Subsequent lines report active returns to these portfolios as they are rebalanced. Columns G and H report the returns to two funds: the first, RpA is the active portfolio consisting of an 80/20 mix of Equity_A and Bonds_A while Rb is simply the long-term benchmark 70/30 mix of the two benchmarks. We are interested in the performance of RpA relative to Rb.

The combined active covariance matrix for the evaluation period beginning in January 2002 and ending December 2006 is estimated using the four returns streams given in columns B through E:

These statistics say that the variances on the returns to the S&P 500 and Citi BIG benchmarks were 12.96 percent and 1.23 percent, respectively, with a negative covariance –1.11 percent. The actively managed portfolios are quite different; notice especially that the variance on the equity portfolio, consisting of positions in the 10 firms, is now much higher at 27.51 percent. This is due to two facts: first, 10 stocks don't necessarily offer the diversification opportunities of the broad index consisting of 500 stocks and, second, the manager is forced to take larger positions in the higher returning (and therefore riskier) stocks in order to achieve his aggressive alpha target. With no short positions permitted, this likely means that he is holding just a few stocks with zero weights on the others.

Note that these returns are monthly frequency. Their annualized equivalents would be higher by a factor of 12, for example, the annualized volatility on the active equity portfolio would be . Finally, we note that the off-diagonal covariances are between benchmarks and active portfolios. In any case, there is a negative correlation between stocks and bonds, a characteristic that all optimizers will exploit.