一個簡單的投資模型

假設：
 * f₁(x)：初始投資金額
 * f₂(x)：第一年的預期收益率
 * f₃(x)：第二年的預期收益率
 * ...
 * f₉(x)：第九年的預期收益率
則 F(x) = f₉(f₈(...f₁(x))) 表示經過九年的投資後，初始投資 x 的最終金額。

將函數f₉(f₈(f₇(...f₁(x))))的複合表示為 F：

F(x) ∈ D and F(x) = x₀

集合 D 在函數複合的投資應用中扮演著重要的角色。它代表了所有可能的投資結果，是我們進行風險評估、目標設定和策略制定的基礎。通過對集合 D 的分析，我們可以更好地理解投資的複雜性，做出更明智的投資決策。 下圖加入股市暴跌等因素

Python

import numpy as np

import matplotlib.pyplot as plt

# Simulate random investment returns

def investment_return(num_years, volatility):

  returns = np.random.normal(loc=0.1, scale=volatility, size=num_years)

  return np.cumprod(1 + returns)

# Scenario 1: Normal market conditions

normal_returns = investment_return(10, 0.15)

# Scenario 2: Stock market crash with a sharp decline

crash_year = 5  # Simulate crash in year 5

crash_factor = 0.3  # 70% drop in asset value (more severe crash)

pre_crash_returns = investment_return(crash_year, 0.15)  # Returns before crash

crash_returns = investment_return(10 - crash_year, 0.15) * crash_factor  # Returns after crash (adjusted for crash)

crash_returns = np.concatenate((pre_crash_returns, crash_returns))

# Scenario 3: Possible economic recovery (optimistic scenario)

recovery_year = 7  # Simulate recovery starting in year 7

recovery_factor = 1.15  # 15% annual growth after recovery

pre_recovery_returns = investment_return(recovery_year, 0.15)  # Returns before recovery

recovery_returns = investment_return(10 - recovery_year, 0.2) * recovery_factor  # Returns after recovery (with higher volatility)

recovery_returns = np.concatenate((pre_recovery_returns, recovery_returns))

# Plot all scenarios

plt.plot(normal_returns, label="Normal Market")

plt.plot(crash_returns, label="Stock Market Crash", color='red')  # Highlight crash in red

plt.plot(recovery_returns, label="Possible Recovery", color='green')  # Highlight recovery in green

# Customize plot for better readability

plt.xlabel("Year")

plt.ylabel("Investment Value")

plt.title("Possible Investment Outcomes (Including Stock Market Crash and Recovery)")

plt.legend()

plt.grid(True)

# Adjust x and y axis limits to better fit the data

plt.xlim(0, 10)  # Set x-axis limits to show all 10 years

plt.ylim(min(normal_returns.min(), crash_returns.min(), recovery_returns.min()) * 0.9, max(normal_returns.max(), crash_returns.max(), recovery_returns.max()) * 1.1)  # Adjust y-axis limits to show all data with a 10% buffer

plt.tight_layout()  # Reduce whitespace around the plot

plt.show()

# This code simulates three investment scenarios and plots the possible investment outcomes, 

# including inflation, deflation, stock market crash, global financial crisis, and economic recovery, 

# without the average return line. The set D (represented by the different plots) reflects the range of 

# potential investment results under various market conditions. The plot is optimized for better readability.