# On January 31, 2017, you purchased 2,500 Australian Exchange-traded Treasury Bonds denominated in the Australian dollar (AUD) at 102.50 AUD per bond. In order to make this purchase, you had to exchange your USD into AUD at the rate of 1.3450 AUD per 1 USD. Each of the bonds has a face value of 100 AUD, carries an annual coupon rate of 5.5%, pays coupons semiannually, and matures on January 31, 2018. At the time of pu

On January 31, 2017, you purchased 2,500 Australian Exchange-traded Treasury Bonds denominated in the Australian dollar (AUD) at 102.50 AUD per bond. In order to make this purchase, you had to exchange your USD into AUD at the rate of 1.3450 AUD per 1 USD. Each of the bonds has a face value of 100 AUD, carries an annual coupon rate of 5.5%, pays coupons semiannually, and matures on January 31, 2018. At the time of purchase, the bonds had just paid their semiannual coupon immediately before you purchased them.

You worry about the exchange rate risk and consider two exit scenarios. In both cases, assume that the Australian Treasury will not default on its bonds. Also, when you collect multiple coupons during your holding period (as in scenario (a) below), assume that the coupons are not reinvested and that all coupons collected are kept in AUD until you sell the bonds. At the time of exit (bond sale), all payments obtained from the bonds during the holding period are exchanged into USD in one transaction at the exchange rate prevailing at the time of exit.

You consider two exit scenarios:

a) You will hold the bonds until maturity. What is the break-even exchange rate of AUD to USD (i.e., the maximum number of AUD per USD) on January 31, 2018 such that your total holding period return in USD on this investment is at least not negative? State the exchange rate as AUD per USD.

b) You will collect the July coupon and sell the bonds immediately thereafter on July 31, 2017. Assume that there will be no major interest rate shocks in the Australian market and that the YTM on the bonds will be the same on July 31, 2017 as it was on January 31, 2017. What is the exchange rate (i.e., the maximum number of AUD per USD) at the end of July 2017 such that your effective annualized return in USD on this investment is at least 2%? State the exchange rate as AUD per USD.

Solution

Notice that the number of bonds purchased does not matter for returns, so we will solve the problem for one bond to avoid redundant calculations.

Scenario (a)

First, find the holding period return in AUD (for this, we don’t need the exchange rates):

rHPR in AUD = (face value collected + two coupons collected)/(purchase price) – 1

= (100 + 2*(100*5.5%/2))/102.5 – 1

= 2.9268%

Next, write down the holding period return in USD:

rHPR in USD = (1+rHPR in AUD)*(1+∆ AUD exchange rate) – 1 = 1.029268*(1+∆ AUD exchange rate) – 1

Solve for the hurdle ∆ AUD exchange rate by equating the above expression to 0:

1.029268*(1+∆ AUD exchange rate) – 1 = 0

1+∆ AUD exchange rate = 1/1.029268

∆ AUD exchange rate = 1/1.029268 – 1

∆ AUD exchange rate = -2.8436%

Finally, invert the quotes and express ∆ AUD exchange rate through the rates at the time of purchase and sale and solve for the rate of AUD per USD at sale:

∆ AUD exchange rate = (USD per AUD at sale – USD per AUD at purchase)/USD per AUD at purchase

-2.8436% = [1/(AUD per USD at sale) – 1/1.3450] / [1/1.3450]

From this expression, we need to find AUD per USD at sale. Solving the equation gives:

1/(AUD per USD at sale) – 1/1.3450 = -2.8436%/1.3450

1/(AUD per USD at sale) = -2.8436%/1.3450 + 1/1.3450

AUD per USD at sale = 1/(-2.8436%/1.3450 + 1/1.3450)

AUD per USD at sale = 1.3844

Scenario (b)

As before, we begin by calculating the holding period return in AUD. For this, we need to know the price at which the bond will be sold on July 31, 2017. We can calculate this price if we know the YTM in July. But the YTM at the end of July is equal to the YTM at the time of purchase, which can be computed as follows:

YTM at the time of purchase

= 2*RATE(number of coupon payments left, (face value*coupon rate)/2, -purchase price, face value)

= 2*RATE(2, (100*5.5%)/2,-102.5, 100)

= 2.9447%

Sale price on July 31, 2017

= -PV(YTM/2, number of coupon payments left, (face value*coupon rate)/2, face value)

= -PV(2.9447%/2, 1, (100*5.5%)/2, 100)

= 101.2591

Now, the holding period return in AUD:

rHPR in AUD

= (price of sale + one coupon collected)/(purchase price) – 1

= (101.2591 + 1*(100*5.5%/2))/102.5 – 1

= 1.4723%

We require that the EAR in USD be at least 2%. This means that the holding period (six-month) return in USD should be at least (solve for rHPR in USD):

(1+ rHPR in USD)2 – 1 = 2%

rHPR in USD = (1+0.02)1/2-1

rHPR in USD = 0.9950%

Solve for the hurdle ∆ AUD exchange rate by equating rHPR in USD to 0.9950%:

(1 + 0.014723)*(1+∆ AUD exchange rate) – 1 = 0.009950

∆ AUD exchange rate = (1+0.009950)/1.014723-1

∆ AUD exchange rate = -0.4704%

Express ∆ AUD exchange rate through the rates at the time of purchase and sale and solve for the rate of AUD per USD at sale:

∆ AUD exchange rate = (USD per AUD at sale – USD per AUD at purchase)/USD per AUD at purchase

-0.004704 = [1/(AUD per USD at sale) – 1/1.3450] / [1/1.3450]

1/(AUD per USD at sale) – 1/1.3450 = -0.004704/1.3450

1/(AUD per USD at sale) = -0.004704/1.3450 + 1/1.3450

AUD per USD at sale = 1/(-0.004704/1.3450 + 1/1.3450)

AUD per USD at sale = 1.3514

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