To determine the optimal investment strategy for the Carringtons, we apply the expected value calculation based on the given payoff matrix. Assuming the Carringtons are uncertain about the probabilities of economic expansion and recession, we maximize the payoff under an equal probability assumption or use the given probabilities if specified.

Given Payoff Matrix:

• Stock Market:
  • Expanding economy: +30%
  • Economic recession: -10%
• Commodities:
  • Expanding economy: +10%
  • Economic recession: +25%

Let the fraction of the $100,000 investment in stocks be $x$, and the fraction in commodities be $1 - x$. Their respective payoffs depend on the state of the economy.

Total payoff in each state:

• Expanding economy payoff: $P_{\text{exp}} = x \cdot 30 + (1 - x) \cdot 10$
• Recession economy payoff: $P_{\text{rec}} = x \cdot (-10) + (1 - x) \cdot 25$

Overall expected payoff:

Assuming equal probabilities (50% each for expansion and recession), the expected payoff $E(x)$ is: $E(x) = 0.5 \cdot P_{\text{exp}} + 0.5 \cdot P_{\text{rec}}$ Substituting for $P_{\text{exp}}$ and $P_{\text{rec}}$: $E(x) = 0.5 \cdot \big(30x + 10 - 10x\big) + 0.5 \cdot \big(-10x + 25 - 25x\big)$ Simplify: $E(x) = 0.5 \cdot (20x + 10) + 0.5 \cdot (-35x + 25)$ $E(x) = 10x + 5 - 17.5x + 12.5$ $E(x) = -7.5x + 17.5$

Maximizing the expected payoff:

The function $E(x) = -7.5x + 17.5$ is linear. The payoff decreases as $x$ increases. Therefore, to maximize the payoff, $x$ should be as small as possible ($x = 0$).

Optimal strategy:

• Invest 0% in stocks ($x = 0$), and 100% in commodities ($1 - x = 1$).

Final Investment Allocation:

• Stocks: $100,000 \cdot 0 = 0$
• Commodities: $100,000 \cdot 1 = 100,000$

Expected Profit:

Using the optimal strategy ($x = 0$):

• Expanding economy payoff: $P_{\text{exp}} = 10\% \times 100,000 = 10,000$
• Recession economy payoff: $P_{\text{rec}} = 25\% \times 100,000 = 25,000$

Expected profit: $E(x) = 0.5 \cdot 10,000 + 0.5 \cdot 25,000 = 5,000 + 12,500 = 17,500$

Final Answers:

(a) Investment allocation:

• Stocks: $0
• Commodities: $100,000

(b) Expected profit: $17,500

Would you like additional details or have any follow-up questions? Here are some related questions to consider:

1. How would the strategy change if the probabilities of expansion and recession were different?
2. What happens to the expected profit if more is allocated to stocks?
3. How would diversification (not allocating entirely to commodities) impact risk and profit?
4. What additional information could improve the investment decision?
5. How would this analysis change with transaction fees or market constraints?

Tip: Always consider diversification to mitigate risks when dealing with uncertain economic outcomes.