Chapter 8 of 16

Silver cross-hedging, ratio risk and hedge-ratio instability

Choose physical, value or statistical hedge ratios deliberately, and test their out-of-sample behaviour.

3 min read · Original educational chapter

In this chapter

Name the risk measure you want to reduce
Use consistent units in the estimator
Distinguish fitted risk reduction from realised protection
Round contracts against the stated objective
Stress a relationship failure
Worked example
Common mistakes
Check your understanding
Desk assignment

Name the risk measure you want to reduce #

A fabricator may want to reduce variance of a silver purchase price; a relative-value investor may want to isolate silver performance versus gold. Those are different objectives. Matched physical mass, equal rupee value and regression beta define different hedge quantities.

A raw gold–silver ratio is not a hedge ratio and does not establish stationarity. Correlation of returns does not prove a stable long-run equilibrium in price levels. Separate descriptive relative prices from a tested mean-reversion strategy.

Use consistent units in the estimator #

In a return-based linear hedge, a silver exposure with value V and beta h against gold returns targets hV of opposite gold value exposure. In a price-change hedge, the covariance and variance must use compatible currency/mass units before scaling physical quantity.

The minimum-variance coefficient under a chosen linear model is Cov(ΔS,ΔF)/Var(ΔF), or the analogous return covariance ratio for a value hedge. Using one version’s coefficient in the other version’s quantity formula can silently change exposure by orders of magnitude.

Distinguish fitted risk reduction from realised protection #

Under the fitted model, residual variance is Var(S) − Cov(S,F)²/Var(F). It is an in-sample quantity conditional on the data and model. It does not promise the same reduction after a structural break or under a liquidity shock.

Report estimation window, sampling frequency, timestamp alignment, outliers and the uncertainty of the coefficient. Compare coefficients across regimes, then evaluate a frozen hedge in later data. Re-estimating on the future path before calling it hedged introduces look-ahead.

Round contracts against the stated objective #

After converting target exposure to actual contracts, compare adjacent whole-contract choices. Include FX and contract maturity. Minimise the declared risk/error measure subject to cash, notional, concentration and operational constraints rather than rounding the displayed beta to a comfortable number.

If the model produces a negative coefficient, inspect the economics and sample before mechanically taking the opposite of the intended hedge. An unstable result can be a reason to reject the proxy. A hedge is not obligatory merely because regression software returns a number.

Stress a relationship failure #

Construct paths in which silver and gold both fall, diverge sharply or become more volatile. A dollar-neutral pair can lose when the return ratio differs from the hedge assumption. Add a currency or local-premium dislocation if the commercial exposure is rupee-denominated.

Maintain both the residual economic-risk ledger and cash obligations. A gold gain that offsets a silver commercial loss later does not necessarily meet today’s collateral call on the other leg.

Hypothetical worked example

A fictional return hedge

Silver return variance is 0.0009, gold return variance 0.0004 and covariance 0.00048. The fitted coefficient is 1.2. Residual variance is 0.0009 − 0.00048²/0.0004 = 0.000324, giving fitted residual standard deviation 1.8%, versus 3% unhedged. This is an in-sample model result.

For a 3-million-dollar long silver value exposure, a 1.2 coefficient targets 3.6 million dollars of short gold value exposure. At a fictional 100,000 dollars per gold contract, that is 36 contracts. Contract access, margin, FX, basis and stress can invalidate its practical feasibility.

Common mistakes

Applying a return beta directly to physical kilograms.
Calling the fitted variance reduction a future guarantee.
Equating an extreme ratio with a tradable equilibrium.

Check your understanding

A return beta of 1.2 hedges a 3-million value exposure against gold returns. What is its model target?

Make this chapter useful

Can you defend the reasoning?

Before moving on, answer the self-check in your own words and write why each alternative fails. Use the worked example to test that explanation. Then apply it in the module research task; keep the assumptions and the evidence that could change your answer.

Desk assignment

What to submit

Submit a units-and-objective sheet, coefficient stability test, adjacent-contract rounding analysis and correlation-break scenario.

Use the research ledger for dated evidence, or the module workbooks for calculations. Separate observations, assumptions and interpretation.

Sources and further reading

Further-reading references from the curriculum source list. Contract sizes, expiry calendars, settlement references, margin parameters and tax treatment can change; verify the current operative documents for the actual product. Worked figures are illustrative, not current quotes.

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