two times standard deviation计算

问题如下：

Two years later, the Voorts ask Lenard to construct a new long-term strategic asset allocation with a more aggressive goal of achieving at least 3.5% annualized growth in the after-tax purchasing power of the portfolio. They indicate that the portfolio should have only a small probability of declining more than 10% in nominal pre-tax terms in any one year. Lenard explains to the Voorts that a normal distribution can be used to model the portfolio returns. The Voorts agree to use a two-standard-deviation approach to monitor the shortfall risk of the portfolio.

Expected inflation remains 2.5% per year and the tax rate remains 30%. Based on his current market outlook, Lenard considers three potential portfolio allocations for the Voorts as shown in Exhibit 1.

Determine the most appropriate portfolio from Exhibit 1 for the Voorts, given their objectives and constraints. Justify your response with two reasons.

选项：

解释：

The most appropriate portfolio for the Voorts must meet the following requirements:

Real after-tax return of 3.5% or more ((pre-tax return × (1 – tax rate)) – inflation rate)

Shortfall risk of no lower than –10% in any one year (equal to nominal pre-tax expected return minus two times standard deviation)

The following analysis shows whether each portfolio meets (pass/fail) the specified return and risk requirements:

Portfolio X

Return Objective:

Real after-tax return: 9.3% × (1 – 30%) – 2.5% = 4.0% > 3.5%; pass

Shortfall risk constraint:

Shortfall risk: 9.3% – (2 × 11.0%) = –12.7% < –10.0%; fail

Portfolio Y

Return Objective:

Real after-tax return: 8.4% × (1 – 30%) – 2.5% = 3.4% < 3.5%; fail

Shortfall risk constraint:

Shortfall risk: 8.4% – (2 × 8.7%) = –9.0% < –10.0%; pass

Portfolio Z

Return Objective:

Real after-tax return: 8.8% × (1 – 30%) – 2.5 = 3.7 > 3.5; pass

Shortfall risk constraint:

Shortfall risk: 8.8% – (2 × 9.3%) = –9.8% > –10.0%; pass

Porfolio X does not meet the shortfall risk constraint and Portfolio Y does not meet the return objective. Portfolio Z is the only one of the three proposed portfolios that meets both the return objective and the shortfall risk constraint.