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Page 3 of 5
Data
The following illustrations use daily market data for the seven-year period January 2, 2002, through December 31, 2008. The source for long-term bond data is daily yields for Moody’s Seasoned Baa Corporate Bond Yields reported at stlouisfed.org by the Federal Reserve Bank of St. Louis. The SPDR S&P 500 ETF (SPY) serves as the U.S. large-capitalization proxy; the iShares Russell 2000 Index Fund (IWM) serves as the small-capitalization proxy; and the iShares MSCI EAFE Index Fund (EFA) serves as the non-U.S. large-capitalization proxy. The commodities data source is the Dow Jones-AIG Commodity Total Return index reported at djindexes.com by Dow Jones.
When credit, equity or commodities markets recognized different holidays or closed for other reasons, we use linear interpolation to fill in missing data. Daily return calculations apply dividend and coupon reinvestment, and the process assumes assets are held for the entire seven-year period without rebalancing. The process of calculating returns for the Moody’s Seasoned Baa Corporate Bond Yields assumes bonds were purchased at par on January 2, 2002, and coupons were paid semiannually on the first trading day on or after July 1 and January 1, each year. For all assets, daily return and standard deviation calculations are based on daily value changes that include cumulative change in the value of reinvested coupons and dividends. We omit the effects of taxes and transaction fees, although exchange-traded fund values are net of expense ratios and other costs incurred by the fund. Betas are calculated using the standard deviation of the asset or portfolio, standard deviation of the S&P 500 Index and the corresponding correlation coefficient. Sharpe ratios are calculated using the daily five-year geometric mean yield of three-month Treasury bills, which serves as the riskless rate of return.
The three illustrations that follow use historical data and make changes in portfolio components the way a portfolio manager might evaluate prospective portfolio changes in a real investment portfolio. However, investors cannot directly invest in the indexes used in these illustrations. The reader should bear in mind that all data used in portfolio analysis and projections are historical. Extrapolations from historical data, although practically necessary, must be done with caution, particularly when the extrapolations are for relatively short periods beyond the historical database and when the historical database is a relatively short period. One of the implications of these facts is that investors should regularly reevaluate their data and assumptions. And, of course, past performance cannot guarantee future results.
Diversification Effect Illustration
The first illustration applies the diversification effect metric to a theoretical, equally allocated three-asset portfolio consisting of proxies for U.S. large-capitalization stocks, long-term bonds and non-U.S large-capitalization stocks. Later we add the proxy for the Russell 2000 Small-Capitalization Index and add the Dow Jones-AIG Commodity Total Return Index. Our interest rests primarily with diversification effect metrics, though we calculate other useful portfolio statistics.
Figure 1 reports performance data for assets used in the present and subsequent illustrations and Figure 2 displays correlation coefficients for the present and subsequent illustrations. Figure 3 illustrates the effects of equally allocating the SPY, 30-year bond and EFA into a three-asset portfolio. This allocation yields a diversification effect (DE) of 19.19%, which means that correlation coefficients between all three portfolio assets decrease portfolio risk by 19.19% compared to the allocation-weighted mean standard deviation that assumes +1.00 correlations. The portfolio beta falls below the arithmetic mean beta (see Figure 1) of the three assets, also because of the influence of these correlations. The portfolio Sharpe ratio exceeds the arithmetic mean Sharpe ratio because the portfolio standard deviation has been reduced by imperfect correlations between the assets.
This illustration explains the effect of correlation coefficients on at least three commonly used portfolio metrics, and the DE statistic explains the other two. Clearly, the statistics are related by the effect correlations have on them. Next we change portfolio composition and monitor the results.
Incremental Diversification Effect Illustration: Adding An Asset With Higher Correlations
In this illustration, we add a fourth asset class with historically high returns and higher correlation coefficients with other assets in the portfolio. The rationale for adding this asset might be to increase expected returns, reduce beta, increase the Sharpe ratio or increase DE. We add incremental diversification effect (IDE) in the analysis to evaluate the diversification benefit of adding a fourth asset. We select IWM as a proxy for the Russell 2000 small-capitalization index. Again, we impose the equal allocation constraint. Figure 4 displays results.
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