Module 14: Commodities, Gold & Oil

1. Commodities: Physical Assets vs. Financial Assets

Commodities are physical goods that are traded on markets: metals (gold, silver, copper), energy (crude oil, natural gas), agriculture (wheat, corn, soybeans), and livestock. They differ fundamentally from stocks and bonds because they are real, tangible assets that must be stored, transported, and physically delivered.

1.1 Key Differences from Financial Assets

1.2 Major Commodity Categories

2. Spot Price vs. Futures Price

Every commodity has two types of prices that coexist in the market:

• Spot price (S): The price for immediate delivery of the physical commodity. Pay now, get the commodity now.
• Futures price (F): The price agreed upon today for delivery at a specified future date. No payment or delivery occurs until the contract expires.

2.1 Contango and Backwardation

The relationship between spot and futures prices defines two market regimes:

3. The Cost of Carry Model

The fundamental pricing relationship for commodity futures is the cost of carry model. It links the futures price to the spot price through the economics of physically holding the commodity:

where:

• F = futures price for delivery at time T
• S = current spot price
• r = risk-free interest rate (financing cost)
• u = storage cost rate (annualized, as fraction of spot)
• y = convenience yield (annualized)
• T = time to delivery (years)

3.1 Components of the Carry Cost

3.2 Implied Convenience Yield

We can invert the cost of carry model to extract the market's implied convenience yield, just as we extracted implied volatility from option prices:

4. Gold as an Inflation Hedge: A Hypothesis Test

One of the most persistent claims in finance is that gold is an inflation hedge — that gold returns are positively correlated with inflation, preserving purchasing power when the value of money declines. As statisticians, we should test this claim rather than accept it as folklore.

4.1 The Hypothesis

where R_gold is the gold return and π is the inflation rate (CPI change).

4.2 Empirical Evidence by Time Horizon

4.3 Gold and Real Interest Rates

A more nuanced model recognizes that gold responds not to inflation itself but to real interest rates (nominal rate minus inflation). When real rates are low or negative, the opportunity cost of holding gold (which pays no interest) is low, making gold more attractive.

This multivariate regression typically finds β₂ < 0 (negative relationship with real rates) to be the strongest predictor, often more significant than β₁ (direct inflation effect).

5. Oil's Unique Dynamics

Crude oil is the most economically significant commodity. Its price affects virtually every sector of the global economy. Oil prices exhibit dynamics that are distinct from other commodities due to three factors: cartel behavior, geopolitical risk, and storage constraints.

5.1 OPEC as a Cartel

The Organization of the Petroleum Exporting Countries (OPEC) is a cartel of oil-producing nations that collectively control approximately 35–40% of global oil production. OPEC's production decisions directly affect global supply and therefore prices.

5.2 Supply Shocks and Fat Tails

Oil prices are subject to sudden supply disruptions caused by:

• Geopolitical events: Wars in the Middle East, sanctions on major producers (Iran, Russia, Venezuela)
• Natural disasters: Hurricanes in the Gulf of Mexico disrupting offshore production and refining
• OPEC decisions: Production cuts or increases decided in OPEC meetings
• Demand shocks: Recessions reduce demand; recoveries increase it

These shocks create a return distribution with extreme tails. Oil prices have experienced moves of ±30% in a single month multiple times in the past 50 years. The 2020 COVID demand shock briefly pushed oil futures to negative prices — a previously unthinkable event.

5.3 Oil's Statistical Properties

6. The Gold-to-Oil Ratio: A Mean-Reverting Spread

The gold-to-oil ratio — the price of one ounce of gold divided by the price of one barrel of crude oil — is one of the most watched inter-commodity ratios in finance:

Historically, this ratio has averaged about 15–20, meaning one ounce of gold buys 15–20 barrels of oil. The ratio is mean-reverting: when it deviates far from its historical average, it tends to revert back.

6.1 Interpreting Extreme Ratios

7. Connecting to Our Gold Prediction Model

Earlier in this course, we built a statistical model to predict gold prices. Now we can place that model in the broader context of commodity pricing theory:

7.1 Model Features Revisited

The features we used in the gold prediction model correspond to the fundamental drivers identified in commodity pricing theory:

8. Commodity Indices and Their Construction

Commodity indices aggregate the prices of multiple commodities into a single index, providing a benchmark for the asset class. However, the construction methodology matters enormously because it determines what the index actually measures.

8.1 Major Commodity Indices

8.2 Return Components of Commodity Indices

The total return of a commodity futures index has three components:

9. Python: Commodity Analysis

9.1 Testing the Gold-Inflation Hedge Hypothesis

Pythonimport numpy as np
from scipy import stats

# Simulated monthly data: gold returns and inflation (CPI change)
# In practice, you would download this from FRED or similar
np.random.seed(42)
n_months = 360  # 30 years of monthly data

# Generate correlated gold returns and inflation
# True correlation is weak at monthly frequency (~0.05)
rho_monthly = 0.05
cov_matrix = np.array([[0.05**2, rho_monthly * 0.05 * 0.003],
                        [rho_monthly * 0.05 * 0.003, 0.003**2]])
data = np.random.multivariate_normal([0.007, 0.002], cov_matrix, n_months)
gold_returns = data[:, 0]
inflation = data[:, 1]

# Test 1: Monthly correlation
r_monthly, p_monthly = stats.pearsonr(gold_returns, inflation)
print("=== Gold-Inflation Correlation by Time Horizon ===")
print(f"Monthly:   r = {r_monthly:.4f}, p-value = {p_monthly:.4f}")

# Test 2: Quarterly (aggregate to 3-month periods)
n_quarters = n_months // 3
gold_q = gold_returns[:n_quarters*3].reshape(n_quarters, 3).sum(axis=1)
infl_q = inflation[:n_quarters*3].reshape(n_quarters, 3).sum(axis=1)
r_q, p_q = stats.pearsonr(gold_q, infl_q)
print(f"Quarterly: r = {r_q:.4f}, p-value = {p_q:.4f}")

# Test 3: Annual (aggregate to 12-month periods)
n_years = n_months // 12
gold_y = gold_returns[:n_years*12].reshape(n_years, 12).sum(axis=1)
infl_y = inflation[:n_years*12].reshape(n_years, 12).sum(axis=1)
r_y, p_y = stats.pearsonr(gold_y, infl_y)
print(f"Annual:    r = {r_y:.4f}, p-value = {p_y:.4f}")

# Test 4: 5-year rolling (aggregate to 60-month periods)
n_5y = n_months // 60
gold_5y = gold_returns[:n_5y*60].reshape(n_5y, 60).sum(axis=1)
infl_5y = inflation[:n_5y*60].reshape(n_5y, 60).sum(axis=1)
if len(gold_5y) > 2:
    r_5y, p_5y = stats.pearsonr(gold_5y, infl_5y)
    print(f"5-Year:    r = {r_5y:.4f}, p-value = {p_5y:.4f}")

print("\nConclusion: Correlation strengthens with time horizon.")
print("Gold is a long-run inflation hedge, not a short-run one.")

9.2 Cost of Carry and Contango/Backwardation

Pythonimport numpy as np

def futures_price(spot, r, storage, convenience_yield, T):
    """Cost of carry model for commodity futures."""
    return spot * np.exp((r + storage - convenience_yield) * T)

def implied_convenience_yield(spot, futures, r, storage, T):
    """Extract implied convenience yield from observed prices."""
    return r + storage - np.log(futures / spot) / T

# === Gold: Near-zero convenience yield, always in contango ===
print("=== Gold Futures Term Structure ===")
gold_spot = 2000  # $/oz
r = 0.05           # 5% risk-free rate
gold_storage = 0.002  # 0.2% storage cost
gold_cy = 0.0       # zero convenience yield for gold

print(f"{'Delivery':>10} {'Futures Price':>15} {'Basis':>10} {'State':>15}")
for months in [1, 3, 6, 12, 24]:
    T = months / 12
    F = futures_price(gold_spot, r, gold_storage, gold_cy, T)
    basis = F - gold_spot
    state = "Contango" if F > gold_spot else "Backwardation"
    print(f"  {months:>3}mo     ${F:>12.2f}   ${basis:>+8.2f}   {state}")

# === Oil: Varies between contango and backwardation ===
print("\n=== Oil Futures: Contango vs Backwardation ===")
oil_spot = 75  # $/bbl
oil_storage = 0.04  # 4% storage cost

# Scenario 1: Low convenience yield (ample supply) = Contango
print("\nScenario 1: Ample supply (convenience yield = 2%)")
print(f"{'Delivery':>10} {'Futures':>10} {'Basis':>10} {'State':>15}")
for months in [1, 3, 6, 12]:
    T = months / 12
    F = futures_price(oil_spot, r, oil_storage, 0.02, T)
    basis = F - oil_spot
    state = "Contango" if F > oil_spot else "Backwardation"
    print(f"  {months:>3}mo     ${F:>7.2f}   ${basis:>+7.2f}   {state}")

# Scenario 2: High convenience yield (supply crisis) = Backwardation
print("\nScenario 2: Supply crisis (convenience yield = 20%)")
print(f"{'Delivery':>10} {'Futures':>10} {'Basis':>10} {'State':>15}")
for months in [1, 3, 6, 12]:
    T = months / 12
    F = futures_price(oil_spot, r, oil_storage, 0.20, T)
    basis = F - oil_spot
    state = "Contango" if F > oil_spot else "Backwardation"
    print(f"  {months:>3}mo     ${F:>7.2f}   ${basis:>+7.2f}   {state}")

9.3 Gold-to-Oil Ratio and Mean Reversion

Pythonimport numpy as np
from scipy import stats

# Simulate 30 years of monthly gold and oil prices
# Both are non-stationary, but their ratio is mean-reverting
np.random.seed(99)
n = 360

# Gold: random walk with drift (non-stationary)
gold_log_returns = np.random.normal(0.005, 0.04, n)
gold_prices = 800 * np.exp(np.cumsum(gold_log_returns))

# Oil: random walk with drift, correlated with gold
oil_log_returns = 0.3 * gold_log_returns + np.random.normal(0.003, 0.06, n)
oil_prices = 50 * np.exp(np.cumsum(oil_log_returns))

# Compute ratio
ratio = gold_prices / oil_prices

# Summary statistics
print("=== Gold-to-Oil Ratio Analysis ===")
print(f"Mean ratio:   {ratio.mean():.1f}")
print(f"Median ratio: {np.median(ratio):.1f}")
print(f"Std dev:      {ratio.std():.1f}")
print(f"Min:          {ratio.min():.1f}")
print(f"Max:          {ratio.max():.1f}")

# Test for stationarity (ADF test approximation)
# H0: ratio has a unit root (non-stationary)
# H1: ratio is stationary (mean-reverting)
# Simple check: regression of change on lagged level
delta_ratio = np.diff(ratio)
lagged_ratio = ratio[:-1]
slope, intercept, r_val, p_val, se = stats.linregress(lagged_ratio, delta_ratio)

print(f"\n=== Mean Reversion Test ===")
print(f"AR(1) coefficient on lagged level: {slope:.4f}")
print(f"  (Negative = mean-reverting, should be significantly < 0)")
print(f"t-statistic: {slope / se:.2f}")

if slope < 0:
    half_life = -np.log(2) / np.log(1 + slope)
    print(f"Estimated half-life of mean reversion: {half_life:.1f} months")
    print(f"  ({half_life / 12:.1f} years)")

# Trading signal: z-score of the ratio
z_score = (ratio - ratio.mean()) / ratio.std()
print(f"\nCurrent ratio:  {ratio[-1]:.1f}")
print(f"Current z-score: {z_score[-1]:+.2f}")
if z_score[-1] > 2:
    print("Signal: Gold expensive vs oil. Consider long oil / short gold.")
elif z_score[-1] < -2:
    print("Signal: Oil expensive vs gold. Consider long gold / short oil.")
else:
    print("Signal: Ratio within normal range. No trade.")

9.4 Commodity Return Distribution Analysis

Pythonimport numpy as np
from scipy import stats

# Compare return distributions across asset classes
np.random.seed(55)
n = 2520  # ~10 years of daily data

# Simulate daily returns with realistic properties
stock_returns = np.random.standard_t(df=5, size=n) * 0.01 + 0.0003
bond_returns  = np.random.normal(0.0001, 0.003, n)
gold_returns  = np.random.standard_t(df=7, size=n) * 0.008 + 0.0002
oil_returns   = np.random.standard_t(df=4, size=n) * 0.015 + 0.0001

assets = {
    'Stocks (S&P 500)': stock_returns,
    'Bonds (10Y Treasury)': bond_returns,
    'Gold': gold_returns,
    'Crude Oil': oil_returns,
}

print("=== Daily Return Distribution Comparison ===")
print(f"{'Asset':>22} {'Mean':>8} {'Std':>8} {'Skew':>8} "
      f"{'Kurt':>8} {'Min':>8} {'Max':>8}")
print("-" * 74)

for name, rets in assets.items():
    print(f"{name:>22} {rets.mean():>+7.4f} {rets.std():>7.4f} "
          f"{stats.skew(rets):>+7.2f}  {stats.kurtosis(rets):>6.1f} "
          f"{rets.min():>+7.3f} {rets.max():>+7.3f}")

print("\n(Kurtosis shown is excess kurtosis; normal distribution = 0)")
print("Note: Oil has the fattest tails and highest volatility.")

# Normality test (Jarque-Bera)
print("\n=== Jarque-Bera Normality Test ===")
for name, rets in assets.items():
    jb_stat, jb_p = stats.jarque_bera(rets)
    normal = "Normal" if jb_p > 0.05 else "Non-normal"
    print(f"  {name:>22}: JB = {jb_stat:>10.1f}, p = {jb_p:.4e}  [{normal}]")

# Correlation matrix
print("\n=== Cross-Asset Correlation Matrix ===")
returns_matrix = np.column_stack([stock_returns, bond_returns, gold_returns, oil_returns])
corr = np.corrcoef(returns_matrix.T)
labels = ['Stocks', 'Bonds', 'Gold', 'Oil']

print(f"{'':>10}" + "".join(f"{l:>10}" for l in labels))
for i, label in enumerate(labels):
    print(f"{label:>10}" + "".join(f"{corr[i,j]:>10.3f}" for j in range(4)))

print("\nKey: Gold's low correlation with stocks makes it a diversifier.")

10. Summary

This module has covered the statistical foundations of commodity pricing and analysis:

With stocks, bonds, options, and commodities now covered, you have a comprehensive understanding of the four major asset classes. Each has distinct statistical properties, return distributions, and pricing mechanisms. In the next module, we will explore how to combine these assets into portfolios using the tools of modern portfolio theory.