Optimal Portfolio Planning for Gold and Bitcoin Investments

Introduction

Background

The combination of traditional investments and alternative assets has become a significant trend. Gold, with its stable value and scarcity, has long been recognized as a safe-haven asset. In contrast, Bitcoin—introduced in 2008—reached an all-time high of $69,000 per coin in 2021. Unlike gold, Bitcoin offers high returns, volatility, and decentralization. Studies using GARCH models ([1], [2]) highlight Bitcoin’s utility in portfolio management and risk analysis. The inverse risk trends between gold and Bitcoin markets enable effective hedging strategies [3].

Research Objective

This study develops a model to maximize total returns over five years (2016–2021) using a $1,000 initial investment. Daily decisions on buying/selling gold and Bitcoin are based on historical price data, with the portfolio represented as [Cash (C), Gold (G), Bitcoin (B)]. Key goals:

• Optimize the asset allocation strategy.
• Assess sensitivity to transaction costs.

Methodology

1. Data Mining: Analyzed historical prices for gold and Bitcoin.
2. Modeling: Utilized gray prediction, dynamic programming, and Monte Carlo simulations.
3. Validation: Verified optimality and sensitivity via MATLAB and Excel.

Key Assumptions and Variables

Data Preprocessing

• Bitcoin trades daily; gold trades 5 days/week with occasional missing data.
• Price predictions were derived using gray forecasting (30-day cycles for gold, 20-day for Bitcoin).

Assumptions

1. Daily prices are known; future prices are unpredictable.
2. No trading occurs during non-price days.
3. Transactions involve simultaneous gold and Bitcoin trades.
4. Cash earns no interest.

Symbols

| Variable | Description |
|----------|-------------|
| C | Cash (USD) |
| G | Gold (troy oz) |
| B | Bitcoin (BTC) |

Gray Forecasting Model

Methodology

1. Data Generation: Transformed raw price data into cumulative sequences.
2. GM(1,1) Model: Solved differential equations to predict prices.
3. Validation: Compared predicted vs. actual prices (Tables 3–4).

Results

• High accuracy in price predictions (Figures 2–3).
• Predicted prices informed trading strategies.

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Risk Assessment Model

Venture Capital Framework

• Returns: Calculated daily for gold (r₁) and Bitcoin (r₂).
• Risk: Measured via variance (σ²) and covariance (σ₁₂).

Optimization

Minimize portfolio risk (σₚ²) while ensuring positive returns:

min σₚ² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂σ₁₂  
s.t. w₁ + w₂ = 1, E(rₚ) > 0  

Monte Carlo Simulation

• Generated 10,000 weight combinations (w₁, w₂).
• Identified optimal allocations to minimize risk.

Dynamic Programming Model

Trading Strategy

1. Buy/Sell Signals:

   • Sell gold/Bitcoin if predicted涨幅 > median rise.
   • Buy if predicted跌幅 > median fall.
2. Monthly Transactions: Executed on valid trading days.

Example:

• Month 1: Invest $500 (38% gold, 62% Bitcoin).
• Month 5: Sell gold, netting $948.77.
• Final Portfolio: [$113.31, 0.542 oz, 0.055 BTC\] → **Total Value: $3,636.26**.

Performance

• 43 profitable transactions (Figure 5).
• Gold/Bitcoin trade volumes shown in Figures 6–7.

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Proof of Optimality

Fixed gold/Bitcoin weights (95.6%/4.4%) yielded $1,431.16—61% lower than our model’s $3,636.26. Conclusion: Dynamic allocation outperforms static strategies.

Sensitivity Analysis

| Commission (Gold, Bitcoin) | Total Assets | Change |
|----------------------------|--------------|--------|
| (1%, 2%) | $3,636.26 | — |
| (0.5%, 1%) | $3,659.59 | +0.64% |

• Low sensitivity confirms model stability (Figure 9).
• Reduced commissions slightly boost returns (Figure 10).

FAQs

1. Why combine gold and Bitcoin?

Gold provides stability; Bitcoin offers high returns. Their inverse risk trends enable hedging.

2. How often should trades occur?

Monthly transactions balance risk and profit, avoiding excessive fees.

3. What’s the optimal gold/Bitcoin ratio?

Dynamic allocation (via Monte Carlo) outperforms fixed ratios (e.g., 95.6%/4.4%).

4. How do commissions impact results?

A 50% fee reduction increased profits by only 0.64%, indicating robustness.

5. Can this model adapt to new data?

Yes—gray forecasting updates predictions with recent price data.

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Conclusion

1. Gray forecasting accurately predicts prices.
2. Dynamic programming maximizes returns ($3,636.26) vs. static strategies ($1,431.16).
3. Low sensitivity to fees ensures reliability.

Final Advice: Diversify with gold and Bitcoin, but adjust allocations dynamically to mitigate risks.