问题如下：
Liu
presents the following hypothetical scenario during a lecture on behavioral
portfolio theory (BPT).
Ann
Lundstrom, a fictitious technology entrepreneur, is a BPT investor who is
developing her portfolio. This portfolio will contain two layers: a layer of
riskless investments and a layer of speculative investments. The riskless layer
will earn 0.50%, and the probability distribution of the expected return on the
speculative layer is shown in Exhibit 2.
Lundstrom plans to invest $1,000,000 and has an aspirational level of $1,050,000 with a probability of 75%. She can tolerate some potential loss in wealth but not more than $100,000 (minimum portfolio value of $900,000). Exhibit 3 presents two potential portfolio allocations for this scenario.
Determine which portfolio allocation in Exhibit 3 is closest to the BPT optimal portfolio for Lundstrom. Justify your response.
选项：
解释：
Allocation 1.
Justify your response:
● Both portfolio allocations meet the safety objective of $900,000.
● Allocation 1 has a 90% chance of exceeding the aspirational level of $1,050,000, whereas Allocation 2 only has a 30% chance of exceeding it.
A BPT investor constructs a portfolio in layers to satisfy investor goals rather than be mean–variance efficient. The investor’s expectations of returns and attitudes toward risk vary between the layers. In this case, Lundstrom has a safety objective of $900,000 and aspirational level of return of 5% ($50,000) with a 75% probability.
Given the expected returns for the riskless and speculative layers, Allocation 1 will result in the following amounts:
10% chance: (59% × $1,000,000) × 1.005 + (41% × $1,000,000) × (1 – 0.25) = $900,450
60% chance: (59% × $1,000,000) × 1.005 + (41% × $1,000,000) × (1.12) = $1,052,150
30% chance: (59% × $1,000,000) × 1.005 + (41% × $1,000,000) × (1.50) = $1,207,950.
Given the expected returns for the riskless and speculative layers, Allocation 2 will result in the following amounts:
10% chance: (90% × $1,000,000) × 1.005 + (10% × $1,000,000) × (1 – 0.25) = $979,500
60% chance: (90% × $1,000,000) × 1.005 + (10% × $1,000,000) × (1.12) = $1,016,500
30% chance: (90% × $1,000,000) × 1.005 + (10% × $1,000,000) × (1.50) = $1,054,500
Both portfolio allocations meet the safety objective of $900,000 (minimum value of $900,450 for Allocation 1 and $979,500 for Allocation 2).
Allocation 1 has a 90% chance of exceeding the aspirational level of $1,050,000, however, whereas Allocation 2 has only a 30% chance of exceeding it. As a result, only Allocation 1 meets both the safety objective and the 75% probability of reaching the aspirational level. Thus, Allocation 1 is closest to the BPT optimal portfolio for Lundstrom.
请问这题为什么不能算speculative layer return的期望 然后再加上riskless layer呢？谢谢！
client 1: [(1-25%) * 10% + 1.12 * 60% + 1.5 * 30% ] * 1,000,000 * 41% + 59% * 1,000,000 *0.5% = 493720
client 2: [(1-25%) * 10% + 1.12 * 60% + 1.5 * 30% ] * 1,000,000 * 10% + 90% * 1,000,000 *0.5% = 124200
