How Do Firms Manage Financial Risk?: formula sheet

Key formulas

MM benchmark for hedging

Perfect markets ⇒ hedging does not change firm value

Needs no taxes, no transaction or bankruptcy costs, symmetric information. Shareholders can replicate any hedge.

Expected distress cost

Expected distress cost = P(distress) × cost if distress occurs

Hedging lowers P(distress), and so lowers the expected cost. Use this to explain the value gain.

Convex tax effect

E[tax(income)] ≥ tax(E[income]) when the tax function is convex

This is Jensen's inequality. Smoother pre-tax income lowers expected tax.

Value of hedging in practice

Value added ≈ PV(distress cost saved + tax saved + investment protected) − PV(hedging costs)

A framing for the case for and against, not a formula you compute from a standard source.

Forward hedge outcome (long exposure to buy)

Effective price = S_T − (S_T − F) = F

S_T is the spot price at maturity. The payoff to a long forward bought at F is S_T − F. Subtracting it from the spot price paid gives an effective purchase price of F whatever the spot price, ignoring credit risk and basis.

Futures hedge number of contracts

N = h × (Exposure size ÷ Contract size)

h is the hedge ratio. Exposure size and contract size must be in the same units, either both in units (such as barrels) or both in value (such as dollars). Do not mix them. With h = 1 the futures position matches the exposure in size. This removes price-level exposure only when the futures asset matches the exposure and the futures maturity matches the hedge date. Otherwise basis risk remains. Short futures hedge a long exposure; long futures hedge a future purchase.

Minimum variance hedge ratio

h* = ρ × (σS ÷ σF)

ρ is the correlation between spot and futures price changes. σS and σF are the standard deviations of those changes.

Basis

Basis = Spot price − Futures price

Basis risk is uncertainty in the basis when the hedge is closed. Let F1 be the futures price when the hedge is set up and b2 the basis (S2 − F2) when it is closed. With this definition, a short hedge on an asset you hold and sell at the close gives effective price = F1 + b2.

Fixed-for-floating swap, payer's net cost

Net cost = Floating paid to lender + Fixed paid − Floating received = Fixed rate + loan spread

If the swap floating leg matches the loan's floating rate, the floating receipt cancels the floating loan payment. The borrower pays the swap fixed rate plus the loan spread.

Protective put floor

Minimum net proceeds per unit = K − premium (ignoring financing); with financing, K − premium × (1 + r)^T (annual compounding) or K − premium × e^(rT) (continuous compounding)

K is the strike, r the financing rate and T the time in years. Use the compounding convention the question states: (1 + r)^T for annual compounding, e^(rT) for continuous compounding. This is per unit of asset held. The upside stays open but is reduced by the premium.

Duration-based hedge (DV01 matching)

N = DV01 of exposure ÷ DV01 of one hedging contract

Use a short position in the contract to hedge a long bond position.

Basis

Basis = Spot price − Futures price

Sign convention follows Hull: spot minus futures. Check the question's definition.

Minimum variance hedge ratio

h* = ρ × σS ÷ σF

σS and σF are standard deviations of changes in spot and futures prices over the hedge life. ρ is their correlation. Equals the regression slope of ΔS on ΔF.

Number of futures contracts

N* = h* × QA ÷ QF

QA = size of exposure in units. QF = size of one futures contract in units. Round to the nearest whole contract.

Hedge effectiveness

Effectiveness = ρ² = h*² × σF² ÷ σS²

Proportion of the variance of the unhedged position eliminated.

Variance of hedged position (per unit of exposure)

Var = σS² × (1 − ρ²)

Valid at the optimal h*. Residual risk is its square root: σS × √(1 − ρ²).

Variance for any hedge ratio h

Var(h) = σS² + h² × σF² − 2 × h × ρ × σS × σF

Minimised at h = ρ × σS ÷ σF.

Exposure coefficient (regression)

ΔV = a + b × ΔS + ε, where exposure b = Cov(ΔV, ΔS) ÷ Var(ΔS)

V is firm value, cash flow or stock return. S is the exchange rate in domestic currency per unit of foreign currency. b is the exposure.

Value change from exposure

Expected ΔV ≈ b × ΔS

Use consistent units. If b is in currency units, ΔS is in absolute terms. If b is an elasticity, use percentage changes.

Forward hedge of a receivable

Domestic proceeds = Foreign amount × forward rate

Locks in the domestic value of a contracted flow. The hedge removes spot risk but fixes the outcome at the forward rate.

Net exposure after hedging

Net exposure = Original exposure − Hedge position

A full hedge of a known flow brings net exposure to zero. A hedge of a flow that may not occur can create a new exposure.

Covered interest parity forward

F = S × (1 + r_d × T) ÷ (1 + r_f × T)

S and F are domestic per foreign unit. Simple interest shown for short maturities. Use the stated convention in the question.

Parametric VaR (normal)

VaR = z × σ × √t × Value, where z = 1.645 (95%) or 2.33 (99%)

Use the loss tail. Scale daily σ by √t for t days, with zero mean assumed unless told otherwise.

Parametric CFaR (normal)

CFaR = z × σ(cash flow) , measured relative to expected cash flow

The σ is the standard deviation of the cash flow over the planning period, not a daily figure.

Hedged position volatility

σ(hedged) = σ(S) × √(1 − ρ²) for the minimum variance hedge

Residual risk remains whenever correlation ρ is below 1.

Minimum variance hedge ratio

h* = ρ × σ(S) ÷ σ(F)

Number of futures units = h* × exposure ÷ contract size.

Hedge ratio (proportion hedged)

Hedge % = hedged amount ÷ total exposure

Use below 100% when exposure is uncertain or upside is wanted.

Quick revision

• Minimum variance hedge ratio: h* = ρ × σS ÷ σF.
• Number of contracts: N = h* × QA ÷ QF.
• If ρ = 1 and σS = σF, the hedge ratio is 1.
• Basis = spot price − futures price. Basis risk is uncertainty in how the basis changes.
• Forwards are customised over-the-counter contracts with counterparty credit risk. Futures are standardised, exchange-traded and margined daily.
• A swap is an agreement to exchange cash flows, for example fixed for floating interest payments.
• A bought option gives a right, not an obligation. The loss is limited to the premium paid.
• Transaction exposure comes from specific contracted foreign-currency cash flows.
• Economic exposure is the effect of rate or price changes on a firm's future competitive position and cash flows.
• Translation exposure arises when foreign subsidiary results are converted into the reporting currency.
• A hedge reduces risk but rarely removes it. Residual risk remains.
• Read the question carefully for sign, units and contract size before calculating.

Common mistakes

• Saying MM proves firms should never hedge. Fix: MM is a benchmark under perfect markets. Real frictions are what make hedging valuable.
• Claiming hedging always increases firm value. Fix: Net value depends on benefits minus costs. Hedging can be costly and can create liquidity and basis risk.
• Taking the same direction as the exposure instead of the opposite. Fix: Ask what hurts you first, then choose a derivative that gains in that scenario.
• Saying forwards have no cost so they dominate options. Fix: Compare outcomes in both price scenarios. The forward locks the rate and removes gains. The option keeps gains for a premium.
• Inverting the volatility ratio and using σF ÷ σS. Fix: Remember the regression logic: spot is the dependent variable, so the slope has σS on top. h* = ρ × σS ÷ σF.
• Reporting hedge effectiveness as ρ instead of ρ². Fix: Effectiveness is the variance reduction, which is R² = ρ². A ρ of 0.8 means 64% effectiveness.
• Treating translation exposure as a cash-flow risk. Fix: Translation is an accounting effect with no direct cash impact. Only cash-based effects belong to transaction or economic exposure.
• Saying economic exposure is the same as transaction exposure over a longer period. Fix: Transaction exposure covers contracted flows. Economic exposure covers uncontracted flows and competitive effects, including for firms with no foreign sales.
• Using VaR for a cash flow question without adjusting the horizon or variable Fix: If the question is about cash flows or earnings over a planning period, use CFaR or EaR.
• Assuming a full hedge always minimises risk Fix: For uncertain exposures, a full hedge can create an over-hedged position. Consider a partial hedge.

Exam tips

• Always state first whether the question is in a perfect-market (MM) setting or not.
• Match each friction to its mechanism: distress, tax convexity, underinvestment, agency.
• Watch for answers saying hedging 'always' or 'never' adds value; they are usually wrong.
• For numerical questions, compute expected cost as probability times cost, then net off hedging costs.
• Remember the arguments against hedging: cost, basis risk, margin liquidity, speculation, and investors' ability to hedge themselves.
• Questions often ask which instrument best fits a stated need such as flexibility, low credit risk or keeping upside. Match the keyword to the instrument.
• Always state the residual risk: basis for futures, counterparty for forwards and swaps, premium for options.
• For hedge ratio questions, check the formula direction and round contracts to the nearest whole number.