Why Currency Hedging Is an Asset-Liability Decision
An insurer that holds foreign-currency assets or owes benefits in a foreign currency has an exposure to changes in the exchange rate. The relevant question is not simply how much foreign currency appears on the asset side. It is how the currencies and payment dates of assets compare with those of liabilities. A suitable forward hedge can lock a domestic-currency value for a specified foreign-currency amount and date; it cannot automatically remove exposure from cash flows that do not match.
Throughout this tutorial, let \(S_t\) denote the spot exchange rate at time \(t\), quoted as domestic currency per unit of foreign currency. For example, \(S_t=1.10\) USD/EUR means one euro exchanges for 1.10 U.S. dollars. Let \(K\) be the forward exchange rate in the same quotation, and let \(T\) be the forward’s maturity. All amounts and rates below are hypothetical.
The relative-value formulas describe the forward’s gain or loss compared with an unhedged spot conversion at maturity. They do not mean that the forward produces the full principal as a separate cash gain. Under physical settlement, the parties exchange currencies at \(K\); under cash settlement, an equivalent net amount may be paid. The examples use the relative-value representation and assume no default, transaction costs, collateral effects, or other funding effects.
Matching a Forward to a Net Exposure
For a single currency and a single payment date, let \(A\) be the foreign-currency amount of assets and \(L\) the foreign-currency amount of liabilities due at that date. The net foreign-currency exposure is \(N=A-L\). A positive \(N\) is a net foreign-currency asset: the insurer benefits, in domestic-currency terms, if the foreign currency strengthens. A negative \(N\) is a net foreign-currency liability: a strengthening foreign currency increases the domestic cost of meeting the liability.
If the insurer sells \(q\) foreign-currency units forward at rate \(K\), the terminal value relative to spot conversion is \(q(K-S_T)\). The combined value is therefore
The coefficient \(N-q\) is the remaining sensitivity to the terminal spot rate. A full hedge sets \(q=N\), eliminating that sensitivity for the matched amount and date. If \(N<0\), this prescription gives \(q<0\), meaning the insurer buys foreign currency forward.
Worked Example: Fully Hedging a Net Euro Asset
Worked Example: Fully Hedging a Net Euro Asset
A hypothetical insurer expects to receive EUR 12.00 million from an asset and pay EUR 7.00 million in claims on the same date. It enters a one-year forward at \(K=1.10\) USD/EUR. The current spot rate is \(S_0=1.08\) USD/EUR. Calculate the full hedge and the combined terminal value if the one-year spot rate is first \(1.00\), then \(1.20\) USD/EUR. Assume both foreign-currency cash flows are fixed and occur on the forward maturity date.
Method. Net the same-date euro cash flows, then sell that net amount forward. The common trap is to hedge the EUR 12 million asset without subtracting the EUR 7 million liability.
Step 1: Calculate the net exposure. \(N=12.00-7.00=5.00\) million EUR. The insurer is net long EUR 5.00 million.
Step 2: Set the full forward position. The full hedge is a sale of \(q=N=5.00\) million EUR at 1.10 USD/EUR. The locked domestic value is \(qK=5.00(1.10)=5.50\) million USD.
Step 3: Check the outcome at \(S_T=1.00\). The unhedged net cash flow is \(5.00(1.00)=5.00\) million USD. The forward’s relative value is \(5.00(1.10-1.00)=0.50\) million USD. Combined value: \(5.00+0.50=5.50\) million USD.
Step 4: Check the outcome at \(S_T=1.20\). The unhedged net cash flow is \(5.00(1.20)=6.00\) million USD. The forward’s relative value is \(5.00(1.10-1.20)=-0.50\) million USD. Combined value: \(6.00-0.50=5.50\) million USD.
Check. Both terminal spot scenarios produce the locked value of USD 5.50 million. The forward offsets the net EUR exposure, not each gross balance separately.
Partial Hedges and Residual Sensitivity
A partial hedge covers only a fraction of the net exposure. If the target hedge fraction is \(h\), where \(0\leq h\leq1\), then the forward sale amount is \(q=hN\). The combined value remains \(qK+(N-q)S_T\), so the residual exposure is \(N-q=(1-h)N\). A partial hedge reduces exchange-rate sensitivity but does not eliminate it. The decision to hedge partially may reflect a risk limit or a deliberate tolerance for currency exposure; it does not change the arithmetic of the residual.
Worked Example: Measuring the Residual After a Partial Hedge
A hypothetical insurer has EUR 9.00 million of assets and EUR 4.00 million of liabilities, both due in one year. It chooses to hedge 70% of its net exposure with a forward rate of \(K=1.12\) USD/EUR. Calculate the forward amount and combined values at terminal spot rates of 1.00 and 1.20 USD/EUR. Also calculate the residual exposure.
Method. Calculate the net amount before applying the hedge fraction. A frequent error is to apply 70% to the asset amount alone and ignore the liability.
Step 1: Net the exposures. \(N=9.00-4.00=5.00\) million EUR.
Step 2: Calculate the partial hedge. \(q=0.70(5.00)=3.50\) million EUR sold forward. The remaining exposure is \(N-q=5.00-3.50=1.50\) million EUR.
Step 3: Evaluate \(S_T=1.00\). The unhedged net value is \(5.00(1.00)=5.00\) million USD. The forward relative value is \(3.50(1.12-1.00)=0.42\) million USD. Combined value is \(5.00+0.42=5.42\) million USD.
Step 4: Evaluate \(S_T=1.20\). The unhedged net value is \(5.00(1.20)=6.00\) million USD. The forward relative value is \(3.50(1.12-1.20)=-0.28\) million USD. Combined value is \(6.00-0.28=5.72\) million USD.
Check. The combined value changes by \(5.72-5.42=0.30\) million USD when spot changes by \(1.20-1.00=0.20\) USD/EUR. The residual amount predicts the same change: \(1.50(0.20)=0.30\) million USD. The full hedge would instead lock \(5.00(1.12)=5.60\) million USD.
Why Dates and Currencies Must Match
Netting foreign-currency amounts is valid only when the relevant cash flows are sufficiently matched. If an asset is received in six months but a liability is payable in one year, they are not a single known net cash flow on one date. The insurer may retain, invest, or convert the earlier proceeds, but each choice introduces an assumption about what happens between the dates. A forward maturing on the liability date does not directly settle an asset cash flow that arrives earlier.
Currency matching matters as well. A U.S.-dollar asset does not directly offset a euro liability merely because their current domestic-currency values are equal. The euro liability remains exposed to the USD/EUR exchange rate. Using a forward in another currency would introduce an additional relationship between exchange rates; the resulting hedge is not an exact euro hedge.
Worked Example: Hedging Euro Cash Flows at Different Dates
A hypothetical insurer expects to receive EUR 6.00 million in six months and pay EUR 5.00 million in one year. It enters a six-month forward to sell EUR 6.00 million at \(K_6=1.10\) USD/EUR and a one-year forward to buy EUR 5.00 million at \(K_{12}=1.08\) USD/EUR. At six months, the spot rate is \(S_6=1.15\); at one year, it is \(S_{12}=1.05\). Calculate the domestic-currency value of each hedged cash flow and the combined value. Assume the specified cash flows are fixed, the forward amounts and dates match them, and there are no intervening investment earnings.
Method. Evaluate each cash flow with its own maturity-matched forward before combining the domestic-currency results. The trap is to subtract EUR 5 million from EUR 6 million and hedge EUR 1 million at just one date.
Step 1: Value the six-month asset receipt. The receipt converted at spot would be \(6.00(1.15)=6.90\) million USD. The forward sale’s relative value is \(6.00(1.10-1.15)=-0.30\) million USD. The combined value is \(6.90-0.30=6.60\) million USD, equal to \(6.00(1.10)\).
Step 2: Value the one-year liability payment. At spot, the liability costs \(5.00(1.05)=5.25\) million USD. The forward purchase’s relative value is \(5.00(1.05-1.08)=-0.15\) million USD. The combined cost is \(5.25+0.15=5.40\) million USD, equal to \(5.00(1.08)\). Equivalently, the insurer has fixed the cost of acquiring the euros at the forward rate.
Step 3: Combine the two domestic-currency amounts. The net value across the two specified cash flows is \(6.60-5.40=1.20\) million USD.
Check. Each flow is separately locked at its own forward rate and maturity; the combined USD 1.20 million is the difference between those locked values. This result does not imply the insurer had EUR 1.00 million available to hedge on one common date.
Deriving the Forward-Hedge Result
The full-hedge formula follows directly from the exchange at maturity. Consider a known net receipt of \(N\) foreign-currency units. Without a forward, converting the receipt at spot produces \(NS_T\) domestic currency. A forward sale of \(q\) units exchanges those units at \(K\), rather than at \(S_T\). Relative to spot conversion, this changes domestic value by \(qK-qS_T=q(K-S_T)\). Adding the two components gives
A full hedge chooses \(q=N\). Substitution gives \(NK+(N-N)S_T=NK\), which contains no \(S_T\). If \(N\) is negative, selling \(N\) units is equivalent to buying \(|N|\) foreign-currency units; the same algebra applies with the signed amount. The conclusion depends on matching the amount, currency, and date and on the cash flow being known. If any of these conditions fails, the remaining exposure must be assessed rather than assumed away.
Common Mistakes and Practical Interpretation
- Hedging gross assets instead of the net same-date exposure. Subtract liabilities in the same currency due on the same date before sizing the forward.
- Reversing the trade direction. A net foreign-currency asset is hedged by selling foreign currency forward; a net foreign-currency liability is hedged by buying it.
- Ignoring the quotation convention. The signs and units here use domestic currency per foreign-currency unit. Reversing the quote changes how rates and sensitivities are expressed.
- Treating a partial hedge as fully protected. The unhedged amount \(N-q\) continues to respond to spot-rate movements.
- Netting cash flows across different dates or currencies without an explicit plan. Similar amounts do not make mismatched flows interchangeable. A direct forward only locks its specified currency amount on its specified maturity date.
Check Your Understanding
For each hypothetical portfolio, identify the forward position or residual exposure. Amounts are in millions unless stated otherwise; use the stated domestic-currency-per-foreign-currency quotation.
- An insurer has CAD 8.0 million of assets and CAD 11.0 million of liabilities due on the same date. It fully hedges at \(K=0.74\) USD/CAD. State the direction and amount of the forward, and the locked net domestic value. (Answer: Buy CAD 3.0 million forward; net value \(=-3.0(0.74)=-2.22\) million USD, a locked liability.)
- A net CHF asset of CHF 4.0 million is hedged 60% at \(K=1.10\) USD/CHF. What amount is sold forward, and what is the residual exposure? (Answer: Sell CHF 2.4 million forward; CHF 1.6 million remains unhedged.)
- A net AUD liability of AUD 2.0 million is hedged with a forward purchase at \(K=0.67\) USD/AUD. Calculate the combined domestic cost if terminal spot is \(0.70\) USD/AUD. (Answer: The liability costs \(2.0(0.70)=1.40\) million USD; the forward relative value is \(2.0(0.70-0.67)=0.06\) million USD in value to the insurer; combined liability cost is \(1.40-0.06=1.34\) million USD.)
- An insurer receives JPY 500 million in six months and owes JPY 300 million in one year. It has a six-month forward sale of JPY 500 million at 0.0068 USD/JPY and a one-year forward purchase of JPY 300 million at 0.0066 USD/JPY. Calculate the net locked domestic value across the two cash flows. (Answer: Asset value \(=500(0.0068)=3.40\) million USD; liability cost \(=300(0.0066)=1.98\) million USD; net locked value \(=3.40-1.98=1.42\) million USD.)
- An insurer has a GBP 5.0 million liability due in one year and a USD 5.0 million asset due on the same date. Explain whether these amounts form a matched net foreign-currency exposure that can be hedged by selling a single net amount of GBP forward. (Answer: No. The USD asset does not cancel the GBP liability in currency units; the insurer remains short GBP 5.0 million and would need a GBP forward purchase for a direct hedge. The USD asset’s value does not remove that GBP exchange-rate exposure.)