Derivatives, Hedging and Portfolio Risk

Introduction

Financial institutions face market risks arising from interest rates, exchange rates, equity prices, commodity prices and credit spreads. These exposures can be retained, diversified, insured or transferred through derivatives. Hedging does not normally eliminate risk; it changes the form, timing or ownership of risk. A futures hedge may replace price risk with basis and liquidity risk, while a swap may replace interest-rate risk with counterparty, collateral and model risk.

Effective risk management therefore requires more than selecting a derivative. Managers must identify the exposure, determine the desired risk reduction, calculate the position correctly, assess residual risks and maintain governance over leverage, collateral and hedge effectiveness.

Learning Objectives

After studying this chapter, students should be able to:

1. distinguish forwards, futures, options and swaps;
2. calculate correctly signed futures hedge positions;
3. assess basis risk and hedge effectiveness;
4. construct option-based and swap-based hedges;
5. apply diversification and mean–variance portfolio optimization;
6. immunize interest-rate-sensitive assets and liabilities;
7. evaluate leverage, liquidity and counterparty consequences of hedging; and
8. design an integrated hedging program with appropriate governance.

1. Risk Exposure and the Purpose of Hedging

An exposure is a position whose value or cash flow changes when a specified risk factor changes. Examples include:

• a bank holding fixed-rate bonds that lose value when yields rise;
• an exporter expecting foreign-currency revenue;
• an insurer whose assets and liabilities have different durations;
• an investment fund exposed to a broad equity-market decline; and
• a borrower paying a floating interest rate.

A hedge is a position intended to offset some or all of the adverse change in an exposure. The main economic objective is usually to reduce variability in value, profit, cash flow or regulatory capital—not necessarily to maximize profit.

A risk manager should distinguish three decisions:

1. Hedge direction: long or short the hedging instrument.
2. Hedge size: the quantity or notional amount.
3. Hedge horizon: how long the hedge should remain in place.

Figure: A hedge replaces the original exposure with a residual risk profile that must still be managed.

2. Forwards and Futures

2.1 Forward contracts

A forward is an over-the-counter agreement to buy or sell an asset at a specified future date for a price fixed today. It can be customized, but customization introduces counterparty credit and liquidity concerns.

For a non-income-producing investment asset, the no-arbitrage forward price under continuous compounding is:

Here, F₀ is the delivery price, S₀ is the current spot price, r is the continuously compounded risk-free financing rate per year, and T is time to delivery in years. The formula assumes frictionless borrowing and lending, no transaction costs, no default and an asset that can be stored and short-sold.

If the asset provides a known continuous income yield q, the special-case formula becomes:

Commodity forwards may additionally require storage costs and convenience yields. Consequently, the simple financial-asset formula should not be applied mechanically to commodities.

2.2 Futures contracts

Futures are standardized exchange-traded contracts. Gains and losses are settled daily through variation margin. Central clearing reduces bilateral counterparty exposure but does not eliminate default, liquidity or operational risk.

Forward and futures prices can differ when interest rates are stochastic and contract gains or losses are correlated with rates. Treating them as identical is most defensible when rates are deterministic or the marking-to-market effect is immaterial.

3. Correct Futures Hedge Sizing

Consider an existing long exposure of Q_A units whose price change is ΔS. One futures contract covers Q_F units, and n denotes the signed number of futures contracts:

• n > 0 means a long futures position;
• n < 0 means a short futures position.

The change in the combined position is:

Minimizing Var(ΔH) with respect to n gives:

Define the minimum-variance hedge ratio as:

Therefore:

The minus sign is essential. For a long asset exposure with positive spot–futures covariance, n^* is negative, indicating a short futures hedge. The number of short contracts as a positive magnitude is h^*Q_A/Q_F. Direction and magnitude must not be confused.

The derivation assumes that price changes and quantities use compatible units, the covariance estimate is relevant for the hedge horizon, and a linear variance objective is appropriate. If the exposure itself is signed, the same formula can be used with a positive quantity for a long exposure and a negative quantity for a short exposure.

Worked Example: Short hedge

A fund holds an asset exposure equivalent to 2,000,000 units of currency. Each futures contract has a notional value of 100,000. Over the intended hedge horizon, ρ_SF=0.90, σ_S=2.5\% and σ_F=3.0\%.

The fund should take a short position in 15 contracts. Reporting “15 contracts” without stating the direction would be incomplete.

For an anticipated future purchase, rising prices are harmful. A long futures position is therefore generally appropriate. For an anticipated sale or an existing long inventory, falling prices are harmful, so a short hedge is generally appropriate.

4. Basis Risk and Hedge Effectiveness

The basis is commonly defined as:

For a short hedge initiated at time 0 and closed at time T, the effective sale price per unit is:

If the basis at hedge termination is uncertain, the effective price remains uncertain. This is basis risk. It arises from mismatched assets, locations, grades, maturities, reset dates or contract specifications. Basis normally converges toward zero at contract expiration when the futures contract and deliverable spot asset match, but convergence may be imperfect before expiration.

A common ex post measure of variance reduction is:

Here, HE is hedge effectiveness, ΔH is the hedged change and ΔU is the unhedged change. A value of 0.80 indicates an 80% reduction in variance. Under an unconstrained minimum-variance hedge with stable covariance estimates and a single linear hedging instrument, the theoretical variance reduction equals ρ_SF². This is a model-based special case, not a universal result.

Effectiveness should also be evaluated through cash-flow-at-risk, value-at-risk, stress losses, tracking error and achievement of the economic objective. A low-variance hedge may still create severe margin calls.

5. Options as Asymmetric Hedges

An option gives its buyer a right without an obligation. A call gives the right to buy; a put gives the right to sell. For strike price K and terminal underlying price S_T, European option payoffs at expiration are:

These are payoffs, not profits. Buyer profit must deduct the premium and its financing cost.

A protective put combines a long asset with a long put. Its terminal value before considering the premium is:

Suppose an investor buys an asset for 100 and a three-month put with strike 95 for a premium of 2. If the asset falls to 80, the asset-plus-option terminal value is 95, while the premium reduces the overall economic return. If the asset rises to 120, the put expires unexercised and the investor retains the upside, less the premium.

A collar combines a protective put with a written call. The call premium can reduce the hedge cost, but gains above the call strike are surrendered. Writing uncovered options is not equivalent to insurance because losses may be very large.

Option risk changes nonlinearly. Important sensitivities include:

• Delta: change in option value for a small change in the underlying;
• Gamma: change in delta as the underlying changes;
• Vega: sensitivity to implied volatility;
• Theta: sensitivity to time decay; and
• Rho: sensitivity to interest rates.

Delta hedging is only locally effective and must be rebalanced. Transaction costs, price jumps and discrete trading prevent perfect replication.

6. Swaps

A swap exchanges specified cash-flow streams. The principal amount is usually not exchanged in a standard interest-rate swap; it serves as the notional amount for calculating interest.

6.1 Interest-rate swaps

In a plain-vanilla swap, one party pays a fixed rate and receives a floating rate. A floating-rate borrower can synthetically fix its rate by paying fixed and receiving floating.

Assume a company has a 50 million loan priced at a benchmark floating rate plus 1.5%. It enters a swap to pay 3.2% fixed and receive the same benchmark rate. Ignoring timing differences, fees, credit adjustments and basis mismatch, the net annualized borrowing rate is:

If the loan and swap use different benchmarks or reset dates, basis risk remains.

6.2 Currency and commodity swaps

Currency swaps exchange cash flows denominated in different currencies and may exchange principal at inception and maturity. Commodity swaps exchange a fixed commodity price for a floating market price. These instruments can stabilize cash flows but create counterparty exposure, collateral needs and valuation complexity.

7. Diversification and Portfolio Risk

Diversification reduces exposure to asset-specific risk when returns are not perfectly positively correlated. For a two-asset portfolio with weights w_A and w_B, where w_A+w_B=1, expected return is:

Portfolio variance is:

The terms use decimal returns, so variance is measured in squared return units. Diversification benefits increase as correlation declines. However, correlations often rise during market stress, weakening protection when it is most needed.

For N assets, with weight vector w and covariance matrix Σ, portfolio variance is:

The weights normally satisfy 1'w=1. Additional constraints may prohibit short selling, impose concentration limits or require minimum liquidity.

8. Mean–Variance Portfolio Optimization

Mean–variance optimization selects portfolio weights by balancing expected return and variance. A common formulation minimizes variance for a target expected return μ^*:

subject to:

Here, μ is the vector of expected returns. Without a target-return constraint and without short-sale restrictions, the global minimum-variance portfolio is:

This expression requires an invertible covariance matrix. Regulatory, liquidity and short-selling constraints require numerical optimization rather than this unrestricted closed form.

Worked Application: Two-asset optimization

Asset A has an expected annual return of 7% and volatility of 12%. Asset B has an expected annual return of 11% and volatility of 20%. Their correlation is 0.25.

First calculate covariance:

For two assets, the unrestricted global minimum-variance weight in Asset A is:

Substitution gives:

Thus w_B^*=1-0.8019=0.1981. The expected portfolio return is:

The expected return is approximately 7.79%. Portfolio variance is approximately 0.012736, giving:

The optimized volatility is approximately 11.29%, below Asset A’s 12% volatility.

Suppose instead that the investor requires a 9% expected return. The return constraint gives:

Therefore, w_A=0.50 and w_B=0.50. Its variance is:

Its volatility is \sqrt{0.0166}=12.88\%. Achieving the higher expected return therefore requires accepting more estimated risk.

Optimization outputs are highly sensitive to expected-return estimates. Robust practice includes weight constraints, estimation-error analysis, scenario tests, shrinkage covariance estimates and turnover penalties.

9. Interest-Rate Immunization

Immunization seeks to make an asset portfolio’s value respond to interest-rate changes in a way that offsets the response of liabilities. Modified duration approximates the percentage price change for a small yield change:

Here, D_mod is modified duration in years and Δy is the yield change in decimal form. The approximation assumes a small, approximately parallel yield-curve shift. Convexity improves the approximation for larger changes:

A basic immunization strategy matches:

1. the present value of assets and liabilities; and
2. asset and liability duration.

Worked Example

An insurer has liabilities with a present value of 10 million and modified duration of 6.2 years. It can invest in two bond funds with durations of 4 and 8 years. Let x be the proportion invested in the four-year fund.

Solving gives x=0.45. The insurer should invest 4.5 million in the four-year fund and 5.5 million in the eight-year fund. This matches present value and duration, but not necessarily convexity or sensitivity to non-parallel yield-curve changes. Durations also change over time, requiring rebalancing.

10. Leverage and the Unintended Consequences of Risk Transfer

Derivatives create economic exposure with relatively little initial cash. This leverage is useful but can amplify losses and liquidity demands. A position described as a hedge may become speculative if its notional exceeds the exposure, its maturity is mismatched or the original exposure disappears.

Major residual risks include:

Risk transfer can also create systemic effects. Similar hedging strategies may force many institutions to sell simultaneously. Dynamic option hedging can intensify price movements, while margin calls can trigger fire sales. Consequently, micro-level risk reduction does not always produce system-wide stability.

11. Integrated Illustrative Case: Meridian Insurance

Background

Meridian Insurance is a fictional European insurer. It faces three exposures:

• a USD 6 million receivable due in three months;
• a EUR 20 million equity portfolio with beta 1.15; and
• EUR 10 million of liabilities with modified duration of 6.2 years.

Management wants to protect budgeted cash flows, reduce equity beta temporarily to 0.35 and immunize the liability portfolio.

Analysis and approach

Foreign exchange exposure: The three-month forward rate is EUR 0.916 per USD. Because Meridian will receive dollars and is harmed by dollar depreciation, it sells USD 6 million forward.

If the terminal spot rate is EUR 0.880 per USD, the unhedged receipt would be EUR 5,280,000. The forward therefore protects EUR 216,000 relative to that spot outcome, before transaction and credit costs.

Equity exposure: Each index futures contract has a notional value of EUR 200,000. The signed number of contracts needed to change beta from β_P to β_T is:

Thus:

Meridian shorts 80 futures contracts. The hedge is not perfect because the portfolio may not track the index and beta may change.

Liability immunization: Using the earlier duration calculation, Meridian invests EUR 4.5 million in the four-year-duration fund and EUR 5.5 million in the eight-year-duration fund.

Outcome and residual risks

The forward stabilizes the euro value of the receivable, the futures reduce systematic equity sensitivity, and the bond allocation approximately matches liability duration. Nevertheless, Meridian retains counterparty exposure on the forward, basis and margin-liquidity risk on the futures, and yield-curve and convexity risk in the immunization strategy.

Learning points

1. Hedge direction must follow the economic exposure.
2. Hedge size should be derived from the stated objective rather than from notional amounts alone.
3. Several instruments may be required because one derivative rarely addresses every risk.
4. Residual risks and liquidity demands must be reported alongside the expected risk reduction.

12. Governance, Regulation and Professional Practice

A sound derivatives policy specifies authorized products, permissible purposes, position limits, approved counterparties, collateral rules, valuation methods and escalation procedures. Front-office trading should be separated from risk measurement, confirmation, settlement and accounting.

Key governance practices include:

• documenting the exposure and hedge objective before execution;
• using independent price verification;
• monitoring gross and net notionals;
• conducting margin and collateral stress tests;
• validating valuation and risk models;
• testing hedge effectiveness over time; and
• reporting exceptions promptly to senior management and the board.

Accounting treatment does not determine economic effectiveness. Under applicable hedge-accounting standards, institutions generally need formal documentation and evidence of an economic relationship between the hedging instrument and the hedged item. Regulatory requirements may also mandate central clearing, trade reporting, margin and capital for derivatives, depending on jurisdiction and counterparty type.

Ethical conduct requires transparent reporting of losses and limitations. Calling a speculative position a hedge, hiding leverage through net presentation or manipulating model assumptions misleads stakeholders.

13. Technology and Contemporary Developments

Modern risk platforms aggregate positions, value derivatives, calculate sensitivities and simulate collateral requirements. Machine learning may improve volatility, correlation or liquidity forecasts, but unstable relationships and opaque models can create false confidence. Human oversight, explainability, data governance and out-of-sample validation remain essential.

Contemporary developments include increased central clearing, transition to alternative interest-rate benchmarks, climate-linked derivatives, automated collateral optimization and greater use of scenario-based risk measures. These innovations do not replace fundamental controls over direction, notional, basis and liquidity.

Chapter Summary

Derivatives and portfolio techniques allow institutions to reshape financial exposures. Forwards lock customized future prices, futures provide standardized and margined exposure, options create asymmetric protection, and swaps exchange cash-flow structures. Correct futures hedging requires an explicit sign convention: with positive covariance, an existing long asset exposure normally requires a negative signed futures position—a short hedge.

Hedge effectiveness depends on covariance, basis behavior, horizon and the chosen risk measure. Diversification reduces portfolio variance when returns are imperfectly correlated, while mean–variance optimization identifies portfolios that satisfy return and risk objectives. Duration matching provides approximate protection against small parallel interest-rate changes but requires rebalancing and may leave convexity and yield-curve risk.

Risk transfer does not eliminate risk. It can generate leverage, margin calls, counterparty exposure, model error and systemic feedback. Effective practice therefore integrates quantitative analysis with governance, liquidity planning, independent validation and ethical reporting.

Key Terms

Basis: The difference between spot and futures prices, commonly S_t-F_t.

Basis risk: Uncertainty caused by changes in the relationship between an exposure and its hedge.

Beta: Sensitivity of an asset or portfolio return to a market index return.

Collar: A hedge combining a long put with a written call.

Convexity: The curvature of a bond’s price–yield relationship.

Counterparty risk: The risk that the other party to a contract fails to perform.

Delta: The sensitivity of an option’s value to a small change in the underlying price.

Diversification: Combining imperfectly correlated assets to reduce portfolio risk.

Duration: A measure of the sensitivity of a fixed-income value to yield changes.

Forward: A customized agreement to transact at a specified future price and date.

Futures contract: A standardized, exchange-traded contract settled through daily marking to market.

Hedge effectiveness: The extent to which a hedge reduces the targeted risk.

Hedge ratio: The amount of hedging-instrument exposure per unit of underlying exposure.

Immunization: Structuring assets to offset the interest-rate sensitivity of liabilities.

Leverage: Economic exposure that is large relative to the cash initially committed.

Mean–variance optimization: Selection of portfolio weights using expected returns and covariances.

Protective put: A long asset combined with a long put option.

Swap: A contract that exchanges defined streams of cash flows.

Review Questions

Short Answer Questions

1. Distinguish a forward contract from a futures contract.
2. What is basis risk?
3. Why is an option payoff different from option profit?
4. What conditions are matched in basic duration immunization?
5. Why can a hedge create liquidity risk?

Long Answer Questions

1. Explain the derivation and interpretation of the minimum-variance futures hedge ratio.
2. Compare forwards, futures, options and swaps as risk-management instruments.
3. Explain how diversification and covariance affect portfolio variance.
4. Critically evaluate the assumptions and limitations of mean–variance optimization.
5. Discuss the governance, ethical and regulatory requirements of a derivatives program.

Analytical / Application Questions

1. A fund holds an equity portfolio worth 30 million with beta 1.2. Index futures have a notional value of 250,000. Calculate the signed futures position required to reduce beta to 0.6, and interpret its direction.
2. Construct and compare the terminal outcomes of an unhedged asset position, a protective put and a collar under rising, unchanged and falling prices.
3. An insurer has liabilities with a present value of 40 million and duration of 7 years. Determine how two bond portfolios with durations of 5 and 9 years can be combined to match the liabilities, and discuss the remaining risks.

Multiple Choice Questions with Answers

Question 1

A fund has a long exposure of 2,000,000, each futures contract represents 100,000, and the minimum-variance hedge ratio is 0.75. Under the chapter’s signed convention, what is the optimal futures position?

A. Long 15 contracts

B. Short 15 contracts

C. Long 20 contracts

D. Short 20 contracts

Correct Answer: B. Short 15 contracts

Question 2

An exporter will receive USD 6 million and sells the dollars forward at EUR 0.916 per USD. What euro receipt is locked in?

A. EUR 5.280 million

B. EUR 5.496 million

C. EUR 6.550 million

D. EUR 6.916 million

Correct Answer: B. EUR 5.496 million

Question 3

For a short futures hedge closed at time T, which expression gives the effective sale price per unit?

A. F₀+b_T

B. F_T-b_T

C. S_T+F_T-F₀

D. F₀-b_T

Correct Answer: A. F₀+b_T

Question 4

Which position creates a floor under an owned asset while preserving upside, before considering the option premium?

A. Short asset plus long call

B. Long asset plus short put

C. Long asset plus long put

D. Long asset plus short call

Correct Answer: C. Long asset plus long put

Question 5

A borrower pays a benchmark rate plus 1.5% on a loan and enters a swap paying 3.2% fixed while receiving the same benchmark. Ignoring fees and basis risk, what is the resulting annualized rate?

A. 1.7%

B. 3.2%

C. 4.7%

D. Benchmark plus 4.7%

Correct Answer: C. 4.7%

Question 6

Two assets have volatilities of 12% and 20% and correlation 0.25. What is their covariance when returns are expressed as decimals?

A. 0.006

B. 0.024

C. 0.060

D. 0.600

Correct Answer: A. 0.006

Question 7

In the chapter’s two-asset optimization example, approximately what proportion is invested in Asset A in the global minimum-variance portfolio?

A. 19.8%

B. 50.0%

C. 68.0%

D. 80.2%

Correct Answer: D. 80.2%

Question 8

Liabilities have duration 6.2 years. What proportion should be invested in a four-year-duration fund when the remainder is invested in an eight-year-duration fund?

A. 35%

B. 45%

C. 55%

D. 65%

Correct Answer: B. 45%

Question 9

Under the special assumptions of an unconstrained optimal linear hedge, what is the theoretical variance reduction when spot–futures correlation is 0.90?

A. 10%

B. 19%

C. 81%

D. 90%

Correct Answer: C. 81%

Question 10

Which consequence arises most directly from daily marking to market on a futures hedge?

A. The hedge can never experience basis risk

B. The hedger may face cash margin calls before the exposure realizes offsetting gains

C. The futures position becomes an option

D. The underlying exposure is legally transferred to the clearinghouse

Correct Answer: B. The hedger may face cash margin calls before the exposure realizes offsetting gains