Investment Course III – November 2007 Topic Four: Portfolio Risk Analysis

4 - 1 Notion of Tracking Error

4 - 2 Notion of Tracking Error (cont.)

4 - 3 Notion of Tracking Error (cont.) Generally speaking, portfolios can be separated into the following categories by the level of their annualized tracking errors:  Passive (i.e., Indexed): TE < 1.0% (Note: TE < 0.5% is normal)  Structured: 1.0% < TE < 3%  Active: TE > 3% (Note: TE > 5% is normal for active managers)

4 - 4 Index Fund Example: VFINX

4 - 5 ETF Example: SPY

4 - 6 “Large Blend” Active Manager: DGAGX

4 - 7 Tracking Errors for VFINX, SPY, DGAGX

4 - 8 Chile AFP Tracking Errors: Fondo A (Rolling 12-month historical returns relative to Sistema)

4 - 9 Chile AFP Tracking Errors: Fondo E (Rolling 12-month historical returns relative to Sistema)

Risk and Expected Return Within a Portfolio Portfolio Theory begins with the recognition that the total risk and expected return of a portfolio are simple extensions of a few basic statistical concepts. The important insight that emerges is that the risk characteristics of a portfolio become distinct from those of the portfolio’s underlying assets because of diversification. Consequently, investors can only expect compensation for risk that they cannot diversify away by holding a broad-based portfolio of securities (i.e., the systematic risk) Expected Return of a Portfolio: where w i is the percentage investment in the i-th asset Risk of a Portfolio: Total Risk = (Unsystematic Risk) + (Systematic Risk)

Example of Portfolio Diversification: Two-Asset Portfolio Consider the risk and return characteristics of two stock positions: Risk and Return of a 50%-50% Portfolio: E(R p ) = (0.5)(5) + (0.5)(6) = 5.50% and:  p = [(.25)(64) + (.25)(100) + 2(.5)(.5)(8)(10)(.4)] 1/2 = 7.55% Note that the risk of the portfolio is lower than that of either of the individual securities E(R 1 ) = 5%  1 = 8%  1,2 = 0.4 E(R 2 ) = 6%  2 = 10%

Another Two-Asset Class Example:

Example of a Three-Asset Portfolio:

Diversification and Portfolio Size: Graphical Interpretation Total Risk Portfolio Size Systematic Risk

Advanced Portfolio Risk Calculations

Advanced Portfolio Risk Calculations (cont.)

Advanced Portfolio Risk Calculations (cont.)

Advanced Portfolio Risk Calculations (cont.)

Example of Marginal Risk Contribution Calculations

Dollar Allocation vs. Marginal Risk Contribution: UTIMCO - March 2007

Fidelity Investment’s PRISM Risk-Tracking System: Chilean Pension System – March 2004

Chilean Sistema Risk Tracking Example (cont.)

Chilean Sistema Risk Tracking Example (cont.)

Notion of Downside Risk Measures: As we have seen, the variance statistic is a symmetric measure of risk in that it treats a given deviation from the expected outcome the same regardless of whether that deviation is positive of negative. We know, however, that risk-averse investors have asymmetric profiles; they consider only the possibility of achieving outcomes that deliver less than was originally expected as being truly risky. Thus, using variance (or, equivalently, standard deviation) to portray investor risk attitudes may lead to incorrect portfolio analysis whenever the underlying return distribution is not symmetric. Asymmetric return distributions commonly occur when portfolios contain either explicit or implicit derivative positions (e.g., using a put option to provide portfolio insurance). Consequently, a more appropriate way of capturing statistically the subtleties of this dimension must look beyond the variance measure.

Notion of Downside Risk Measures (cont.): We will consider two alternative risk measures: (i) Semi-Variance, and (ii) Lower Partial Moments Semi-Variance: The semi-variance is calculated in the same manner as the variance statistic, but only the potential returns falling below the expected return are used: Lower Partial Moment: The lower partial moment is the sum of the weighted deviations of each potential outcome from a pre-specified threshold level (  ), where each deviation is then raised to some exponential power (n). Like the semi-variance, lower partial moments are asymmetric risk measures in that they consider information for only a portion of the return distribution. The formula for this calculation is given by:

Example of Downside Risk Measures:

Example of Downside Risk Measures (cont.):

Example of Downside Risk Measures (cont.):

Example of Downside Risk Measures (cont.):