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Suppose the investor is Australian, and there is a single, 3-month, USD-denominated zero-coupon bond with a face value of \$1 million USD. The AUD/USD exchange rate is \$1.2AUD/USD, and the 3-month US interest rate is 2% per annum compounded quarterly.

Question is: what are the Australian dollar values of the two basic positions that the bond can be mapped onto? That is, what are $X_1$ and $X_2$?

My attempt (assuming risk factors are the spot exchange rate and 3-month US zero rate): one of the positions would be the present value of the bond in AUD:

PV = $\frac{10^{6}}{1+\frac{0.02}{4}}\cdot1.2 = \\\$1194030$

however I'm not sure what the other position would be? Since we receive \$1 million USD at maturity, would that just be \$1.2 million AUD?

Hence $X_1 = -\\\$1194030$, $X_2 = \\\$1200000$?

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    $\begingroup$ please reformulate your question: there is a lack of clarity $\endgroup$
    – lehalle
    Nov 21 '21 at 4:29
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While I don't fully understand the question (please edit and clarify), I'll try anyway

Scenario: your accounting is in AUD. You're long a USD-denominated treasury bond, so no credit risk. Let us ignore bid-ask spread.

Please note that you don't know exctly how much the AUDUSD currency exchange rate will be at the time of your bond's cash flow(s). You have some approximate idea based on the spot rate, and on USD and AUD interest rates, but it's very likely that the market will move and the actual spot rate on that day will differ somewhat. You know exactly how much USD you'll receive when, but you don't know exactly how much AUD that USD will be worth. If the USD or AUD interest rates move, or the cross-currency swap spreads move, this will cause you P&L.

The fair price in AUD that you'd get in the secondary market if you sold the bond before its maturity depends on the AUDUSD rate when you sell it, and on the bond's USD price; and the latter in turn depends on USD interest rates.

Scenario: in addition to the above, you hedge some of our market risks by entering into an FX forward for each of the bond's cash flows. Since you know how much USD you will receive, you arrange with a counterparty (e.g. a bank) that on some future dates you'll pay them the same USD amounts that the bond will pay you, and you will receive from the counterparty some predetermined fixed amount of AUD. Effectively, you lock in the exchange rates. (This is known as an asset swap.)

Now you no longer have any exposure to USD interest rates. However the present value of the AUD cash flows that you will receive in the future can still change if the AUD interest rates change.

Scenario: instead of fixed AUD, you swap into floating AUD, so you get paid more AUD if AUD interest rates go up. This way, you have less sensitivity to AUD interest rates.

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