Question13 Suppose an Australian firm will receive USD 10 million in one year. Given the following conditions, compare the expected value of our net payoff in AUD (in one year) using the following methods: 1) hedging with a one-year forward; 2) hedging by buying an option; and 3) staying unhedged. Assume option premiums are paid today and no discounting. Spot rate: AUD1.5/USD.Spot rate in one year: AUD1.4/USD with 50% probability and AUD1.6/USD with 50% probability.1-year forward rate: AUD1.48/USD.1-year call option: AUD 0.03 mil as premium, to buy USD 1 mil at 1.55.1-year put option: AUD 0.04 mil as premium, to sell USD 1 mil at 1.45. Which of the following statements is TRUE? Select one alternative: The expected value of our net payoff is the highest hedging with a put option contract, yielding AUD 1.485 mil. The expected value of our net payoff is the highest hedging with a forward contract, yielding AUD 1.48 mil. The expected value of our net payoff is the highest hedging with a put option contract, yielding AUD 1.525 mil. The expected value of our net payoff is the highest staying unhedged. ResetMaximum marks: 2 Flag question undefined

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BlackTom AI · Accepted Answer

We start by restating the setup in our own words and listing the options clearly so we can compare them step by step.
Question: An Australian firm will receive USD 10 million in one year. We compare three hedging approaches for converting that USD amount into AUD in one year, with no discounting today:
- (1) hedge with a one-year forward at 1.48 AUD per USD
- (2) hedge by buying a call option on USD with strike 1.55 (premium 0.03 AUD mil per 1m USD, i.e., 0.30 AUD mil for 10m USD to be able to buy USD at 1.55)
- (3) hedge by buying a put option on USD with strike 1.45 (premium 0.04 AUD mil per 1m USD, i.e., 0.40 AUD mil for 10m USD to be able to sell USD at 1.45)
- (4) stay unhedged and convert in the market at the spot rate in one year
Spot rate today: 1.50 AUD/USD. Spot rate in one year: 1.40 AUD/USD with 50% probability, or 1.60 AUD/USD with 50% probability. One-year forward rate: 1.48 AUD/USD.
Question asks which statement about the expected net payoff in AUD is true, with options giving specific numeric EVs.

Now, we analyze each option's claim and compute or reason about the expected payoffs step by step.

Option 1: The expected value of our net payoff is the highest hedging with a put option contract, yielding AUD 1.485 mil.
- First, interpret the claim: it says the best EV comes from buying put options on USD (covering 10m USD) and that the EV is 1.485 million AUD. This figure is far smaller than the scale of the underlying exposure (10 million USD). Even if we only consider incremental numbers after premium expenditure, the put premium is 0.40 AUD mil for 1m USD, i.e., 4.0e5 AUD per 1m USD, and for 10m USD it would be 0.40 AUD mil × 10 = 4.0 AUD mil in premium. That already dwarfs a purported EV of 1.485 AUD mil. In addition, the put protects downside by guaranteeing a floor price (1.45 AUD/USD) but limits upside to 1.60 AUD/USD in the other state, and, crucially, you have paid upfront premium. Thus the raw EV must be higher than the forward’s 14.8m AUD minus the option premium, and certainly not as small as 1.485m AUD. Therefore this option’s numeric EV figure is inconsistent with the scale of the exposure and with the effect of paying a nontrivial premium, making it implausible as the highest EV. Consequently, this option is not correct.

Option 2: The expected value of our net payoff is the highest hedging with a forward contract, yielding AUD 1.48 mil.
- This says the forward hedge gives the highest EV and specifies 1.48 million AUD. If you hedge with a forward, you lock in 1.48 AUD per USD for the 10m USD, so the payoff in AUD is 10m × 1.48 = 14.8m AUD, regardless of future spot outcomes. In other words, the EV here is 14.8m AUD, not 1.48m AUD. The figure stated (1.48 mil) is numerically inconsistent with the actual scale of the exposure (we’re dealing with tens of millions of AUD, not single-digit millions). Even if the problem’s answer options shorthand numbers, the forward hedge yields a clearly lower EV than the unhedged expectation, which is 15m AUD (since E[spot in one year] = (1.40 + 1.60)/2 = 1.50 AUD/USD, so E[AUD received] = 10m × 1.50 = 15m). Therefore this option is incorrect both in magnitude and in ranking EVs relative to the unhedged case.

Option 3: The expected value of our net payoff is the highest hedging with a put option contract, yielding AUD 1.525 mil.
- This is similar to Option 1 but with a slightly different EV figure (1.525 million AUD). As reasoned for Option 1, putting on a USD put costs a substantial premium (0.40 AUD mil per 1m USD, i.e., 4.0 AUD mil for 10m USD). The protection provided by the put ensures a floor of 1.45 AUD/USD, but at a heavy upfront cost. The resulting EV, after accounting for the premium, would still be far below the unhedged EV of 15m AUD, and certainly far from a figure around 1.525 million AUD. Thus this option is also incorrect.

Option 4: The expected value of our net payoff is the highest staying unhedged.
- This option claims the unhedged path yields the highest EV. Let’s compare numerically. If we stay unhedged, we simply convert USD 10m at the spot rate in one year, which is a random rate with two equiprobable states: 1.40 or 1.60 AUD/USD. The expected spot rate in one year equals (1.40 + 1.60)/2 = 1.50 AUD/USD. Therefore the expected AUD payoff from 10m USD is 10m × 1.50 = 15.0m AUD. If we hedge with a forward, we lock in 1.48 AUD/USD, giving exactly 14.8m AUD in all states, which is lower than 15.0m AUD. If we hedge with options, we pay upfront premiums (0.30 AUD mil for calls, or 0.40 AUD mil for puts, scaled to 10m USD) and only potentially achieve higher payoffs in favorable states, but after paying these premiums, the EV is still generally below the unhedged 15.0m AUD in expectation (and the problem’s provided numerical options suggest much smaller EV readings for hedged cases). Consequently, the unhedged EV of 15.0m AUD is the largest, making this option the correct statement.

Synthesis of the analysis across all options shows a common pattern: hedging (forward or options) reduces the expected payoff in comparison to the unhedged case when the expected future spot rate equals the forward rate (or is around the forward’s level) and when option premiums are nonzero. Since the unhedged path yields an expected payoff of 15.0m AUD, which dominates the other hedging paths in expectation, the correct statement is that the highest expected value comes from staying unhedged.

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