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Lareina · 2025年06月11日

为什么SFR能体现probability 呢？

  1. CFA III
  2. Asset Allocation

NO.PZ2022122801000024

问题如下：

Remington and Montgomery’s first meeting of the day is with a new client, Spencer Shipman, who recently won $900,000 in the lottery. Shipman wants to fund a comfortable retirement. Earning a return on his investment portfolio that outpaces inflation over the long term is critical to him. He plans to withdraw $54,000 from the lottery winnings investment portfolio in one year to help fund the purchase of a vacation home and states that it is important that he be able to withdraw the $54,000 without reducing the initial $900,000 principal. Montgomery suggests they use a risk-adjusted expected return approach in selecting one of the portfolios provided in Exhibit 1.

Exhibit 1 Investment Portfolio One-Year Projections

Which of the portfolios provided in Exhibit 1 has the highest probability of enabling Shipman to meet his goal for the vacation home?

选项：

A.

Portfolio 1

B.

Portfolio 2

C.

Portfolio 3

解释：

Portfolio 2 has the highest probability of enabling Shipman to meet his goal for the vacation home. All three of the portfolios’ expected returns over the next year exceed the 6.0% (see calculations below) required return threshold to avoid reducing the portfolio. However, on a risk-adjusted basis, Portfolio 2 (probability ratio of 0.231) has a higher probability of meeting and surpassing the threshold than either Portfolio 1 (probability ratio of 0.175) or Portfolio 3 (probability ratio of 0.225).

Step 1: Calculate the required return threshold: 54,000 ÷ 900,000 = 0.06 = 6.0%.

Step 2: To decide which allocation is best for Shipman, calculate the probability ratio:

[E(Rp ) – RL ] ÷ σp

Portfolio 1: (10.50% – 6.0%) ÷ 20.0% = 4.50% ÷ 20.0% = 0.225.

Portfolio 2: (9.00% – 6.0%) ÷ 13.0% = 3.00% ÷ 13.0% = 0.231. (Highest)

Portfolio 3: (7.75% – 6.0%) ÷ 10.0% = 1.75% ÷ 10.0% = 0.175.

为什么SFR能体现probability 呢？

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1 个答案

Lucky_品职助教 · 2025年06月12日

嗨，从没放弃的小努力你好：


在投资决策中，Safe First Ratio之所以能代表 Probability Ratio，核心在于其通过风险调整后的收益，来反映投资组合收益率超过最低要求的概率。

该比率的计算公式为 [E (Rp)−RL]/σp，其中 E (Rp) 是投资组合预期收益率，RL 是最低必要收益率，σp 为收益率标准差。

这一公式本质上等同于统计学中的标准正态变量（Z 分数），当收益率近似服从正态分布时，Z 分数可直接反应为 “收益率超过 RL” 的累积概率 —— 比率越高，对应的 Z 分数越大，右侧尾部概率（即满足收益目标的概率）也越高。

如题目中，Portfolio 2 的 Safe First Ratio（0.231）最高，意味着其收益率超过 6% 临界值的概率在三个组合中最大，因此被判定为最可能帮助投资者实现目标的选择。

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