CQF Comprehensive Cheatsheet - Commodities
Energy, Metals, Agriculture: Futures, Options, Storage, and Convenience Yield
COMMODITY FUNDAMENTALS
Commodity Classifications
Energy:
- Crude oil (WTI, Brent)
- Natural gas (Henry Hub)
- Heating oil, gasoline, diesel
- Coal, electricity
Precious Metals:
- Gold, silver, platinum, palladium
- Investment assets (low storage cost)
- Backwardation typically
Base Metals:
- Copper, aluminum, zinc, nickel, lead
- Industrial demand drivers
- High storage costs
Agriculture:
- Grains: Corn, wheat, soybeans
- Softs: Coffee, sugar, cotton, cocoa
- Livestock: Cattle, hogs
- Seasonality important
Key Differences from Financial Assets
| Feature | Financial Assets | Commodities |
|---|---|---|
| Dividends | Cash flows | No dividends |
| Storage | Free/negligible | Costly (physical) |
| Convenience | None | Yield from holding |
| Seasonality | Minimal | Strong (agriculture) |
| Delivery | Cash settled | Physical possible |
| Location | Irrelevant | Critical (transport) |
Spot vs Futures Prices
Spot Price (\(S_t\)): Current physical delivery price
Futures Price (\(F_t\)): Agreed price for future delivery
Basis:
$$\text{Basis} = S_t - F_t$$
- Contango: \(F > S\) (negative basis, upward-sloping curve)
- Backwardation: \(F < S\) (positive basis, downward-sloping curve)
Example: Crude oil spot = $80/bbl, 1-year futures = $85/bbl
Basis = $80 - $85 = -$5
Market in contango by $5/bbl
Basis = $80 - $85 = -$5
Market in contango by $5/bbl
COST OF CARRY MODEL
Basic Cost of Carry
No-arbitrage relationship:
$$F(t,T) = S_t e^{(r+u-c)(T-t)}$$
where:
- \(r\) = risk-free rate
- \(u\) = storage cost (as % of spot price per annum)
- \(c\) = convenience yield
Interpretation:
- \(r\): Financing cost (opportunity cost of capital)
- \(u\): Cost of physically storing commodity
- \(c\): Benefit of holding physical commodity
Storage Costs
Components:
- Warehouse rental
- Insurance
- Transportation
- Quality deterioration/spoilage
Modeling approaches:
- Proportional: \(u\) as % of spot (used above)
- Fixed: \(U\) dollars per unit per time
With fixed storage cost:
$$F(t,T) = (S_t + U) e^{r(T-t)}$$
Example: Gold spot = $2000/oz, \(r = 5\%\), storage = 1% p.a.
No convenience yield (\(c = 0\)) for gold
1-year forward: \(F = 2000 \times e^{(0.05+0.01) \times 1} = 2000 \times 1.0618 = \boxed{\$2123.60}\)
No convenience yield (\(c = 0\)) for gold
1-year forward: \(F = 2000 \times e^{(0.05+0.01) \times 1} = 2000 \times 1.0618 = \boxed{\$2123.60}\)
Convenience Yield
Definition: Implicit benefit from holding physical commodity
Sources:
- Ability to meet unexpected demand
- Maintain production processes
- Avoid stockouts
- Profit from temporary shortages
Properties:
- Non-tradeable benefit (can't be monetized directly)
- Higher when inventories low (scarcity value)
- Lower when inventories high (abundant supply)
- Varies by commodity and market conditions
Implied convenience yield (from market prices):
$$c = r + u - \frac{1}{T-t}\ln\frac{F(t,T)}{S_t}$$
Example: Crude oil spot = $80, 1Y futures = $75
\(r = 4\%\), \(u = 2\%\)
\(c = 0.04 + 0.02 - \ln(75/80) = 0.06 - (-0.0645) = 0.1245\) or 12.45%
High convenience yield → backwardation
\(r = 4\%\), \(u = 2\%\)
\(c = 0.04 + 0.02 - \ln(75/80) = 0.06 - (-0.0645) = 0.1245\) or 12.45%
High convenience yield → backwardation
Theory of Storage (Kaldor, Working)
Relationship between inventory and convenience yield:
$$c = c(\text{Inventory level})$$- Low inventory → High \(c\) (scarcity premium)
- High inventory → Low \(c\) (abundant supply)
Typical functional form:
$$c(I) = \alpha e^{-\beta I}$$where \(I\) = inventory level, \(\alpha, \beta\) = parameters
Further Reading:
Wikipedia: Cost of Carry |
Investopedia: Cost of Carry |
CME: Introduction to Commodities
COMMODITY FUTURES CURVE SHAPES
Contango
Condition: \(F > S\) (futures above spot)
Cause: \(r + u > c\) (carry costs exceed convenience)
Typical for:
- Precious metals (gold, silver)
- Industrial metals with high storage costs
- Markets with ample inventory
Roll yield: Negative (futures converge down to spot)
Contango Example:
Spot: $100, 3M: $102, 6M: $104, 12M: $107
Upward-sloping curve, typical of well-supplied market
Spot: $100, 3M: $102, 6M: $104, 12M: $107
Upward-sloping curve, typical of well-supplied market
Backwardation
Condition: \(F < S\) (futures below spot)
Cause: \(c > r + u\) (convenience exceeds carry costs)
Typical for:
- Energy (crude oil) during supply disruptions
- Agricultural commodities pre-harvest
- Markets with tight supply
Roll yield: Positive (futures converge up to spot)
Backwardation Example:
Spot: $100, 3M: $97, 6M: $95, 12M: $92
Downward-sloping curve, typical of tight supply
Spot: $100, 3M: $97, 6M: $95, 12M: $92
Downward-sloping curve, typical of tight supply
Roll Yield
Definition: Return from rolling futures positions
Contango (negative roll yield):
- Sell expiring contract at spot
- Buy new contract at higher price
- Loss = \(F_{new} - S\)
Backwardation (positive roll yield):
- Sell expiring at spot
- Buy new at lower price
- Gain = \(S - F_{new}\)
Total futures return:
$$R_{total} = R_{spot} + R_{roll} + R_{collateral}$$
Example: Hold 1M futures, roll monthly for 12 months
If consistently in backwardation (5% per roll):
Roll yield ≈ 5% × 12 = 60% annual return from roll alone
(Plus spot price changes and collateral interest)
If consistently in backwardation (5% per roll):
Roll yield ≈ 5% × 12 = 60% annual return from roll alone
(Plus spot price changes and collateral interest)
COMMODITY FUTURES PRICING MODELS
Gibson-Schwartz Two-Factor Model
Spot dynamics:
$$dS = \mu S dt + \sigma_S S dW_S$$Convenience yield dynamics:
$$d\delta = \kappa(\alpha - \delta)dt + \sigma_\delta dW_\delta$$where \(\delta = c - u\) is net convenience yield
Correlation: \(dW_S dW_\delta = \rho dt\) (typically negative)
Futures price:
$$F(t,T) = S_t \exp\left[A(T-t) - B(T-t)\delta_t\right]$$
where \(A(t)\) and \(B(t)\) are known functions
Key insight: Two sources of uncertainty (spot and convenience)
Schwartz Three-Factor Model
Three state variables:
- Spot price \(S_t\)
- Convenience yield \(\delta_t\)
- Interest rate \(r_t\)
Spot dynamics:
$$\frac{dS}{S} = (\mu - \delta)dt + \sigma_S dW_S$$Convenience yield:
$$d\delta = \kappa(\alpha - \delta)dt + \sigma_\delta dW_\delta$$Interest rate (Vasicek):
$$dr = a(b-r)dt + \sigma_r dW_r$$Correlations: \(\rho_{S\delta}, \rho_{Sr}, \rho_{\delta r}\)
Spot Price Models with Mean Reversion
Ornstein-Uhlenbeck (OU):
$$dX_t = \kappa(\mu - X_t)dt + \sigma dW_t$$where \(S_t = e^{X_t}\) (log spot)
Properties:
- Mean reversion to long-term level \(\mu\)
- Speed \(\kappa\)
- Common for energy commodities
Alternative (geometric mean reversion):
$$dS_t = \kappa(\mu - S_t)S_t dt + \sigma S_t dW_t$$
Example: Natural gas spot = $4/MMBtu, long-term mean = $5
\(\kappa = 2\) (fast mean reversion due to weather/seasonal factors)
Expected to revert toward $5 relatively quickly
\(\kappa = 2\) (fast mean reversion due to weather/seasonal factors)
Expected to revert toward $5 relatively quickly
Further Reading:
Schwartz: Stochastic Behavior of Commodity Prices |
Wikipedia: Convenience Yield
COMMODITY OPTIONS
Black-76 Formula
Standard for commodity options (on futures):
Call option:
$$C = e^{-rT}[F \cdot N(d_1) - K \cdot N(d_2)]$$
$$d_1 = \frac{\ln(F/K) + \sigma^2T/2}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$$
Put option:
$$P = e^{-rT}[K \cdot N(-d_2) - F \cdot N(-d_1)]$$where \(F\) = futures price, \(K\) = strike, \(\sigma\) = volatility
Example: Crude oil call option
\(F = \$80\), \(K = \$85\), \(\sigma = 30\%\), \(T = 0.25\), \(r = 5\%\)
\(d_1 = \frac{\ln(80/85) + 0.045 \times 0.25}{0.15} = -0.3367\)
\(d_2 = -0.4867\)
\(C = e^{-0.0125}[80 \times 0.3682 - 85 \times 0.3134] = \boxed{\$2.51}\)
\(F = \$80\), \(K = \$85\), \(\sigma = 30\%\), \(T = 0.25\), \(r = 5\%\)
\(d_1 = \frac{\ln(80/85) + 0.045 \times 0.25}{0.15} = -0.3367\)
\(d_2 = -0.4867\)
\(C = e^{-0.0125}[80 \times 0.3682 - 85 \times 0.3134] = \boxed{\$2.51}\)
Greeks for Commodity Options
Delta (option on futures):
$$\Delta_C = e^{-rT} N(d_1), \quad \Delta_P = -e^{-rT} N(-d_1)$$Gamma:
$$\Gamma = \frac{e^{-rT} N'(d_1)}{F \sigma \sqrt{T}}$$Vega:
$$\mathcal{V} = F e^{-rT} \sqrt{T} N'(d_1)$$Theta:
$$\Theta = -\frac{F e^{-rT} N'(d_1) \sigma}{2\sqrt{T}} + rC$$Spread Options
Definition: Options on price difference between two commodities
Crack spread: Refining margin (crude oil → gasoline + heating oil)
$$\text{Payoff} = \max(aP_{gas} + bP_{heat} - P_{crude} - K, 0)$$Spark spread: Power generation margin (electricity - gas)
$$\text{Payoff} = \max(P_{electricity} - HR \times P_{gas} - K, 0)$$where HR = heat rate (efficiency factor)
Calendar spread:
$$\text{Payoff} = \max(F_2 - F_1 - K, 0)$$Options on contango/backwardation changes
Margrabe Formula (exchange option, \(K=0\)):
$$C = S_1 N(d_1) - S_2 N(d_2)$$ $$d_1 = \frac{\ln(S_1/S_2) + \sigma^2T/2}{\sigma\sqrt{T}}$$where \(\sigma^2 = \sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2\)
Asian Options
Common in commodities: Average price over period
Use case: Hedge average purchase/sale price
Payoff (arithmetic average):
$$\max\left(\frac{1}{n}\sum_{i=1}^n S_{t_i} - K, 0\right)$$Properties:
- Lower premium than vanilla (reduced volatility)
- No closed form (use Monte Carlo)
- Popular for energy hedging
ENERGY COMMODITIES
Crude Oil Market Structure
Major benchmarks:
- WTI (West Texas Intermediate): Light, sweet crude, Cushing OK delivery
- Brent: North Sea crude, global benchmark
- Dubai/Oman: Middle East/Asia pricing
Price relationships:
$$P_{WTI} \approx P_{Brent} + \text{Quality differential} - \text{Transport cost}$$Typical spread: Brent premium to WTI ($2-5/bbl normally)
Oil Market Participants:
- Producers: Short futures (hedge production)
- Refiners: Long crude, short products (crack spread hedge)
- Airlines: Long jet fuel futures (hedge costs)
- Speculators: Directional or spread trades
Natural Gas Pricing
Henry Hub: US benchmark (Louisiana)
Seasonality: Strong winter heating demand
Storage dynamics:
- Injection season: April-October
- Withdrawal season: November-March
- Storage capacity constraints
Mean reversion: Faster than oil (less storable, regional)
Example: Natural gas futures curve (summer)
Summer (injection): $3/MMBtu
Winter (withdrawal): $5/MMBtu
Strong seasonal backwardation into winter
Summer (injection): $3/MMBtu
Winter (withdrawal): $5/MMBtu
Strong seasonal backwardation into winter
Electricity
Unique properties:
- Cannot be stored (economically)
- Must balance supply-demand instantaneously
- Extreme volatility (spikes to $1000+/MWh)
- Location-specific (transmission constraints)
Pricing models:
Jump-diffusion:
$$dS = \kappa(\mu - S)dt + \sigma S dW + J dN$$where \(dN\) = Poisson jump (captures spikes)
Regime-switching:
- Normal regime: Low volatility
- Spike regime: High volatility, mean reversion
METALS
Precious Metals
Gold:
- Store of value, safe haven asset
- Low/zero convenience yield
- Typically in contango (storage > convenience)
- Negative correlation with USD
Gold lease rate:
$$\text{Lease rate} = r - \frac{1}{T}\ln\frac{F}{S}$$Rate to borrow gold (miners, jewelers)
Silver:
- Hybrid: Monetary + industrial use
- Higher volatility than gold
- Gold/silver ratio: Typically 50-80
Example: Gold = $2000/oz, Silver = $25/oz
Gold/Silver ratio = 2000/25 = 80
Historical mean ≈ 60-70, currently silver "cheap" relative to gold
Gold/Silver ratio = 2000/25 = 80
Historical mean ≈ 60-70, currently silver "cheap" relative to gold
Base Metals
Copper ("Dr. Copper" - economic indicator):
- Industrial demand (construction, electrical)
- Often in backwardation (high convenience)
- LME (London Metal Exchange) benchmark
Aluminum:
- High production costs (energy-intensive)
- Storage costs significant
- Typically contango
LME Contract Specifications:
- 3-month forward most liquid
- Physical delivery possible
- Warehouse system (Rotterdam, Singapore, etc.)
Metal Spreads
Time spreads (same metal, different dates):
$$\text{3M-15M spread} = F_{3M} - F_{15M}$$Inter-commodity spreads:
- Copper/Gold: Risk-on/risk-off indicator
- Platinum/Palladium: Auto catalyst demand
AGRICULTURAL COMMODITIES
Grain Markets
Corn, Wheat, Soybeans:
- Planting: Spring (April-May)
- Growing season: Summer
- Harvest: Fall (September-November)
- Storage: Winter/Spring
Seasonality patterns:
- Pre-harvest: High uncertainty, weather risk, backwardation
- Harvest: Prices fall (supply surge), contango emerges
- Post-harvest: Contango reflects storage costs
Example: Corn futures curve (June)
July (new crop): $4.50/bu
Dec (storage): $4.80/bu
March (pre-planting): $5.00/bu
Contango reflects storage costs ≈ 5 cents/month
July (new crop): $4.50/bu
Dec (storage): $4.80/bu
March (pre-planting): $5.00/bu
Contango reflects storage costs ≈ 5 cents/month
Weather Risk
Yield risk:
$$\text{Yield} = f(\text{Weather, Technology, Inputs})$$Critical periods:
- Planting: Soil moisture
- Pollination: Temperature, rain
- Harvest: Avoid frost, excess rain
Weather derivatives:
- Temperature (HDD/CDD - Heating/Cooling Degree Days)
- Precipitation
- Index-based (no delivery)
Crop Yield Models
Trend-adjusted yield:
$$Y_t = a + bt + \epsilon_t$$where \(a + bt\) = technology trend, \(\epsilon_t\) = weather shock
USDA Reports (major market movers):
- WASDE: World Agricultural Supply/Demand Estimates (monthly)
- Prospective Plantings: March (planting intentions)
- Crop Progress: Weekly during season
Soft Commodities
Coffee:
- Arabica (high quality) vs Robusta
- Brazil weather crucial (frost risk)
- 4-5 year production cycles
Sugar:
- Energy linkage (ethanol production from sugarcane)
- Brazil, India, EU major producers
- Government policies significant
Cotton:
- Textile demand driver
- Competes with grains for acreage
- Weather-sensitive (US, China, India)
COMMODITY TRADING STRATEGIES
Momentum Strategies
Trend following:
- Commodities exhibit persistence (momentum)
- Time-series momentum: past 12M return predicts next 1M
- Cross-sectional momentum: relative performance
Signal:
$$\text{Position}_t = \text{sign}(R_{t-12,t-1})$$Long if positive return, short if negative
Carry Strategies
Long backwardation, short contango:
Carry signal:
$$\text{Carry} = \frac{F_1 - F_2}{F_2}$$where \(F_1\) = near contract, \(F_2\) = far contract
- Positive carry (backwardation) → Long
- Negative carry (contango) → Short
Calendar Spreads
Definition: Long one maturity, short another
Example: Long Dec corn, short July corn
Profit drivers:
- Change in storage costs
- Inventory dynamics
- Supply shocks
Example: Crude oil calendar spread
Initially: Front month $80, 6M $85 (contango $5)
Supply shock → tightness
New: Front $90, 6M $88 (backwardation $2)
Front gained $10, far gained $3
Long front, short far → profit $7/bbl
Initially: Front month $80, 6M $85 (contango $5)
Supply shock → tightness
New: Front $90, 6M $88 (backwardation $2)
Front gained $10, far gained $3
Long front, short far → profit $7/bbl
Crack Spreads
3:2:1 Crack spread:
$$\text{Spread} = \frac{2 \times P_{gasoline} + 1 \times P_{diesel}}{3} - P_{crude}$$Represents refinery margin (3 barrels crude → 2 gasoline + 1 diesel)
Hedging:
- Refiners: Long spread (long products, short crude)
- Speculators: Trade margin expectations
Pairs Trading
Gold/Silver ratio:
$$\text{Ratio} = \frac{P_{gold}}{P_{silver}}$$Mean-reverting around 65 historically
- Ratio high (80+): Long silver, short gold
- Ratio low (50-): Long gold, short silver
Copper/Gold:
- Risk-on: Copper outperforms (industrial demand)
- Risk-off: Gold outperforms (safe haven)
Further Reading:
Wikipedia: Commodity Markets |
Investopedia: Spread Trading |
CFTC: Commitments of Traders
COMMODITY RISK MANAGEMENT
Producer Hedging
Natural position: Long physical commodity
Hedge: Short futures
Example: Oil producer expects 100k barrels in 6 months
Current 6M futures: $80/bbl
Hedge: Sell 100k barrels of 6M futures at $80
Scenario 1: Spot falls to $70
Physical sale: 100k × $70 = $7.0M
Futures gain: 100k × ($80-$70) = $1.0M
Total: $8.0M (locked in $80)
Scenario 2: Spot rises to $90
Physical sale: 100k × $90 = $9.0M
Futures loss: 100k × ($80-$90) = -$1.0M
Total: $8.0M (locked in $80)
Current 6M futures: $80/bbl
Hedge: Sell 100k barrels of 6M futures at $80
Scenario 1: Spot falls to $70
Physical sale: 100k × $70 = $7.0M
Futures gain: 100k × ($80-$70) = $1.0M
Total: $8.0M (locked in $80)
Scenario 2: Spot rises to $90
Physical sale: 100k × $90 = $9.0M
Futures loss: 100k × ($80-$90) = -$1.0M
Total: $8.0M (locked in $80)
Consumer Hedging
Natural position: Short physical (need to buy)
Hedge: Long futures
Examples:
- Airlines: Hedge jet fuel costs
- Food processors: Hedge grain costs
- Utilities: Hedge natural gas for power generation
Minimum Variance Hedge Ratio
Optimal hedge ratio:
$$h^* = \rho \frac{\sigma_S}{\sigma_F}$$
where \(\rho\) = correlation, \(\sigma_S\) = spot volatility, \(\sigma_F\) = futures volatility
Alternatively (regression):
$$h^* = \frac{\text{Cov}(\Delta S, \Delta F)}{\text{Var}(\Delta F)} = \beta_{S,F}$$
Example: Hedge 10,000 barrels crude
\(\sigma_S = 35\%\), \(\sigma_F = 30\%\), \(\rho = 0.9\)
\(h^* = 0.9 \times \frac{0.35}{0.30} = 1.05\)
Optimal hedge: Sell 10,500 barrels of futures
\(\sigma_S = 35\%\), \(\sigma_F = 30\%\), \(\rho = 0.9\)
\(h^* = 0.9 \times \frac{0.35}{0.30} = 1.05\)
Optimal hedge: Sell 10,500 barrels of futures
Basis Risk
Definition: Risk that spot-futures relationship changes
Sources:
- Quality differences (hedge WTI with Brent)
- Location differences (Chicago wheat vs Kansas)
- Timing mismatch (hedge 3M ahead with 6M futures)
Hedged position value:
$$V = S + h(F - F_0) = S - hF_0 + hF$$where \(F_0\) = initial futures price
Basis risk: \(\text{Var}(S - hF)\)
COMMODITY INDICES
Major Indices
| Index | Weighting | Rebalancing | Roll |
|---|---|---|---|
| S&P GSCI | Production-weighted | Annual | 5-9th business day |
| Bloomberg Commodity | Liquidity, diversification | Annual | 5th-9th business day |
| RICI (Rogers) | Fundamental importance | Annual | Month-end |
| CRB | Equal/tiered | Annual | Variable |
S&P GSCI composition (approximate):
- Energy: ~65% (crude, gas, products)
- Agriculture: ~15%
- Industrial metals: ~10%
- Precious metals: ~5%
- Livestock: ~5%
Index Construction Issues
Roll yield drag:
- Indices typically hold front-month contracts
- Must roll monthly to avoid delivery
- In contango: Sell low (expiring), buy high (new) → loss
- In backwardation: Positive roll yield
Roll timing predictability:
- Indices roll on fixed schedule
- Front-running by hedge funds
- Increases roll costs
Enhanced Index Strategies
Optimal roll timing:
- Spread roll over multiple days
- Roll when spread narrows
Dynamic contract selection:
- Hold contracts with best roll yield
- May hold 2nd or 3rd month instead of front
Momentum overlay:
- Overweight commodities in uptrend
- Underweight/short those in downtrend
STORAGE AND INFRASTRUCTURE
Storage Economics
Optimal storage decision:
Store if: \(F - S > u + \frac{rS}{1}\) (simplified)
More precisely, storage value:
$$V_{storage} = \mathbb{E}[F_T - S_t] - (u \times t + r \times S_t \times t)$$Storage capacity constraints:
- Scarcity when near capacity → high convenience yield
- Contango collapses (no arbitrage possible)
- Seen in crude oil 2020 (negative prices)
Infrastructure Constraints
Pipeline capacity (natural gas):
- Regional price differentials
- Basis risk between hubs
Refining bottlenecks (crude oil):
- Crude quality mismatches
- Seasonal maintenance (turnarounds)
Port/shipping constraints:
- Grain exports (limited port capacity)
- LNG shipping (limited vessels)
KEY FORMULAS SUMMARY
| Concept | Formula |
|---|---|
| Cost of Carry | \(F = Se^{(r+u-c)(T-t)}\) |
| Convenience Yield | \(c = r+u - \frac{1}{T-t}\ln\frac{F}{S}\) |
| Basis | \(B = S - F\) |
| Black-76 Call | \(C = e^{-rT}[FN(d_1) - KN(d_2)]\) |
| Hedge Ratio | \(h^* = \rho\frac{\sigma_S}{\sigma_F}\) |
| Spot (OU) | \(dX = \kappa(\mu-X)dt + \sigma dW\) |
| Gibson-Schwartz | \(dS = \mu S dt + \sigma_S SdW_S\) \(d\delta = \kappa(\alpha-\delta)dt + \sigma_\delta dW_\delta\) |
| Roll Yield | \(R_{roll} = \frac{F_{near} - F_{far}}{F_{far}}\) |
| Crack Spread (3:2:1) | \(\frac{2P_{gas} + P_{diesel}}{3} - P_{crude}\) |
| Gold/Silver Ratio | \(\frac{P_{gold}}{P_{silver}}\) |
COMMON MISTAKES & TIPS
Common Mistakes:
- Ignoring convenience yield: Not all commodities follow simple cost-of-carry
- Confusing contango direction: Contango = futures above spot (upward slope)
- Roll yield sign: Backwardation gives positive roll yield, not contango
- Storage cost units: Sometimes % of spot, sometimes $/unit/time
- Seasonality neglect: Agriculture has strong seasonal patterns
- Basis risk underestimation: Location, quality, timing all matter
- Negative convenience yield: Possible but rare (abundant supply)
- Forward vs futures: Commodities mostly futures (marked-to-market daily)
- Physical delivery: Most contracts cash-settled, but can deliver
- Inventory data importance: Critical for convenience yield estimation
Quick Interview Tips:
- Contango intuition: Storage costs > scarcity value (abundant supply)
- Backwardation intuition: Scarcity value > storage (tight supply)
- Convenience yield high when: Low inventories, production disruptions
- Gold special case: Low/zero convenience (doesn't spoil, abundant above-ground)
- Energy mean reversion: Faster for gas than oil (less storable)
- Agriculture seasonality: Harvest → contango, pre-harvest → backwardation
- WTI vs Brent: Quality similar, location different (Cushing vs North Sea)
- Electricity unique: Cannot store → extreme volatility, jump processes
- Roll yield matters: Can dominate returns in commodity indices
- Crack spread = refining margin: Products minus crude
- Spark spread = power margin: Electricity minus gas (adjusted for heat rate)
- LME 3M most liquid: Base metals standard maturity
- USDA reports: Major market movers for agriculture
- Minimum variance hedge: Not always 1:1 due to basis risk
PRACTICAL EXAMPLE: COMPLETE COMMODITY HEDGE
Real-World Commodity Hedging Example:
Situation:
Historical data shows:
\(\sigma_{jet} = 40\%\), \(\sigma_{crude} = 35\%\), \(\rho = 0.85\)
Minimum variance hedge ratio:
\(h^* = 0.85 \times \frac{0.40}{0.35} = 0.971\)
Alternatively, from regression: \(\Delta P_{jet} = \alpha + 1.05 \times \Delta P_{crude} + \epsilon\)
Use \(\beta = 1.05\) as hedge ratio
Step 2: Convert gallons to barrels
1 million gallons ÷ 42 gal/bbl = 23,810 barrels
Step 3: Determine futures contracts
Crude futures contract size: 1,000 barrels
Adjusted for hedge ratio: 23,810 × 1.05 = 25,000 barrels
Number of contracts: 25,000 ÷ 1,000 = 25 contracts
Action: Buy 25 crude oil futures at $80/bbl
Step 4: Scenario analysis (6 months later)
Scenario A: Prices rise
Jet fuel spot: $3.60/gal ($151/bbl) - up 20%
Crude futures settlement: $95/bbl - up 18.75%
Physical purchase: 1M gal × $3.60 = -$3,600,000
Futures gain: 25,000 bbl × ($95 - $80) = +$375,000
Net cost: $3,600,000 - $375,000 = $3,225,000
Effective price: $3.225/gal
Scenario B: Prices fall
Jet fuel spot: $2.40/gal ($101/bbl) - down 20%
Crude futures settlement: $64/bbl - down 20%
Physical purchase: 1M gal × $2.40 = -$2,400,000
Futures loss: 25,000 bbl × ($64 - $80) = -$400,000
Net cost: $2,400,000 + $400,000 = $2,800,000
Effective price: $2.80/gal
Step 5: Hedge effectiveness
Without hedge:
Scenario A: $3.60/gal (up from $3.00)
Scenario B: $2.40/gal (down from $3.00)
Range: $1.20/gal
With hedge:
Scenario A: $3.225/gal
Scenario B: $2.80/gal
Range: $0.425/gal
Variance reduction: (1.20 - 0.425)/1.20 = 64.6%
Remaining risk is basis risk (jet fuel vs crude correlation imperfect)
Situation:
- Airline needs to hedge jet fuel costs
- Expected consumption: 1 million gallons over next 6 months
- Current jet fuel spot: $3.00/gallon
- Available: Crude oil futures (no jet fuel futures)
- 6-month crude futures: $80/barrel
- Jet fuel ≈ $3.00/gal = $126/barrel (42 gal/bbl)
Historical data shows:
\(\sigma_{jet} = 40\%\), \(\sigma_{crude} = 35\%\), \(\rho = 0.85\)
Minimum variance hedge ratio:
\(h^* = 0.85 \times \frac{0.40}{0.35} = 0.971\)
Alternatively, from regression: \(\Delta P_{jet} = \alpha + 1.05 \times \Delta P_{crude} + \epsilon\)
Use \(\beta = 1.05\) as hedge ratio
Step 2: Convert gallons to barrels
1 million gallons ÷ 42 gal/bbl = 23,810 barrels
Step 3: Determine futures contracts
Crude futures contract size: 1,000 barrels
Adjusted for hedge ratio: 23,810 × 1.05 = 25,000 barrels
Number of contracts: 25,000 ÷ 1,000 = 25 contracts
Action: Buy 25 crude oil futures at $80/bbl
Step 4: Scenario analysis (6 months later)
Scenario A: Prices rise
Jet fuel spot: $3.60/gal ($151/bbl) - up 20%
Crude futures settlement: $95/bbl - up 18.75%
Physical purchase: 1M gal × $3.60 = -$3,600,000
Futures gain: 25,000 bbl × ($95 - $80) = +$375,000
Net cost: $3,600,000 - $375,000 = $3,225,000
Effective price: $3.225/gal
Scenario B: Prices fall
Jet fuel spot: $2.40/gal ($101/bbl) - down 20%
Crude futures settlement: $64/bbl - down 20%
Physical purchase: 1M gal × $2.40 = -$2,400,000
Futures loss: 25,000 bbl × ($64 - $80) = -$400,000
Net cost: $2,400,000 + $400,000 = $2,800,000
Effective price: $2.80/gal
Step 5: Hedge effectiveness
Without hedge:
Scenario A: $3.60/gal (up from $3.00)
Scenario B: $2.40/gal (down from $3.00)
Range: $1.20/gal
With hedge:
Scenario A: $3.225/gal
Scenario B: $2.80/gal
Range: $0.425/gal
Variance reduction: (1.20 - 0.425)/1.20 = 64.6%
Remaining risk is basis risk (jet fuel vs crude correlation imperfect)