Читать книгу Asset Allocation - William Kinlaw, Mark P. Kritzman - Страница 62

1 1. Markowitz (1952).

2 2. Levy and Markowitz (1979).

3 3. Kurtosis refers to the peakedness of a distribution. Kurtosis greater than 3 indicates that extreme returns are more likely than what one would expect from a normal distribution. Returns that are independent and identically distributed are likely to be normal.

4 4. This concept applies as well to any number of asset classes, but it is easier to visualize with only two asset classes.

5 5. See Chapter 1 for a detailed discussion of these characteristics.

6 6. Our empirical analysis is meant for illustration, and we do not intend to offer conclusions about any specific portfolio, investment universe, or data set. We calibrate models and assumptions using reasonable market proxies such as the S&P 500 for US equities; MSCI World ex USA and MSCI emerging markets for foreign and emerging market equities; Bloomberg Barclays aggregate US Treasury and corporate bond indexes; Bloomberg commodities index; and the risk-free rate from Kenneth French's data website.

7 7. For those who care deeply about maximizing the portfolio's geometric mean, the arithmetic approach may still offer a reasonable approximation that can be tested for efficacy and compared to optimization with other more complex numerical procedures.

8 8. See Sharpe (1987). Sharpe's algorithm can easily be adapted to accommodate transaction costs and allocation constraints.

9 9. A continuous return equals the natural logarithm of 1 plus the discrete return. It is the return that if compounded continuously would give the discrete return. Continuous returns that are independent and identically distributed are normally distributed. It is therefore common practice to convert discrete returns, which are lognormally distributed owing to the effect of compounding, to continuous returns in order to estimate probabilities. We then convert the continuous return back to a discrete return by raising the base of the natural logarithm to the power of 1 plus the continuous return and subtracting 1.