Financial Engineering – IC302
Autumn Term 2020/1
Seminar 1: Introduction to Financial Engineering Answers
1. Suppose that the spot EURUSD exchange rate is 1.50, the one-year USD interest rate is 3%, and the one-year EUR interest rate is 5%. For the purpose of this question, you may ignore any day-count conventions specific to interest rates in these currencies.
(a) Calculate the fair-value one-year EURUSD forward exchange rate consistent with covered interest arbitrage.
The fair-value forward exchange rate consistent with covered interest arbitrage is:
1+𝑟 1+0.03
𝐹=𝑆 !”# =1.50* ,=1.4714
1+𝑟 1+0.05 $!%
(b) Suppose that the market forward exchange rate is 1.50. Use replication arguments to construct an arbitrage portfolio and calculate the arbitrage profit.
The market forward exchange rate is higher than the cost of constructing a synthetic equivalent to the FX forward, so we should go short the FX forward (i.e. sell EUR at the market forward price) and long the synthetic. The arbitrage portfolio and the resulting cash flows are summarized in the following table:
𝑡&
𝑡’
Borrow USD
𝑈𝑆𝐷 𝑆
−𝑈𝑆𝐷𝑆(1+𝑟 ) !”#
Buy EUR spot
−𝑈𝑆𝐷 𝑆
𝐸𝑈𝑅 1
Lend EUR
−𝐸𝑈𝑅 1
𝐸𝑈𝑅(1+𝑟 ) $!%
Sell EUR forward
−𝐸𝑈𝑅 (1 + 𝑟 ) $!%
𝑈𝑆𝐷𝐹(1+𝑟 ) $!%
Net cash flows
0
𝑈𝑆𝐷𝐹(1+𝑟 )−𝑈𝑆𝐷𝑆(1+𝑟 ) $!% !”#
The arbitrage portfolio costs nothing at date 𝑡& and at date 𝑡’ it will generate a cash flow of:
𝑈𝑆𝐷𝐹(1+𝑟 )−𝑈𝑆𝐷𝑆(1+𝑟 ) $!% !”#
= 𝑈𝑆𝐷 1.50(1 + 0.05) − 𝑈𝑆𝐷 1.50(1 + 0.03) = 𝑈𝑆𝐷 0.03
This is the arbitrage profit. Note that this is the arbitrage profit from selling EUR 1.05 forward, since will be the proceeds of the EUR deposit. The arbitrage profit per Euro is USD 0.0286, which is equal to the difference between the market and fair-value forward exchange rates.
(c) Suppose that the market forward exchange rate is 1.45. Use replication arguments to construct an arbitrage portfolio and calculate the arbitrage profit.
In this case, the market forward exchange rate is lower than the cost of constructing a synthetic equivalent to the FX forward, so we should go long the FX forward (i.e. buy EUR at the market forward price) and short the synthetic. The arbitrage portfolio and the resulting cash flows are summarized in the following table:
𝑡&
𝑡’
Borrow EUR
𝐸𝑈𝑅 1
−𝐸𝑈𝑅 (1 + 𝑟 ) $!%
Sell EUR spot
𝑈𝑆𝐷 𝑆
−𝐸𝑈𝑅 1
Lend USD
−𝑈𝑆𝐷 𝑆
𝑈𝑆𝐷𝑆(1+𝑟 ) !”#
Buy EUR forward
𝐸𝑈𝑅
𝑆 (1 + 𝑟 ) !”#
𝐹
−𝑈𝑆𝐷𝑆(1+𝑟 ) !”#
Net cash flows
0
𝑆(1+𝑟 )
𝐸𝑈𝑅
𝐹
!”# −𝐸𝑈𝑅(1+𝑟 ) $!%
The arbitrage portfolio costs nothing at date 𝑡& and at date 𝑡’ it will generate a cash flow of:
𝐹
= 𝐸𝑈𝑅 1.50(1 + 0.03) − 𝐸𝑈𝑅 (1 + 0.05) = 𝐸𝑈𝑅 0.0155 1.45
This is the arbitrage profit. Note that this is the arbitrage profit from buying EUR 1.0655 forward, since this is what can be purchased with the proceeds of the USD loan. The arbitrage profit per Euro is EUR 0.0146, which is equivalent to USD 0.0211 at the market
𝐸𝑈𝑅
!”# −𝐸𝑈𝑅(1+𝑟 ) $!%
𝑆(1+𝑟 )
forward rate. The small difference between this value and the difference between the market and synthetic forward exchange rates (USD 0.0214) is a rounding error that results from the exchange rate being quoted to only four decimals.
2. Use the contractual equation for an EURUSD FX forward to explain how we might create a synthetic loan in which we borrow USD.
The contractual equation for the EURUSD FX forward is:
=++
This can be rearranged to give a replicating portfolio for a USD loan:
=–
How are we to interpret the negative signs on the right-hand-side of this equation? What does it mean to be short a deposit in which we lend Euros or short an FX spot transaction in which we buy Euros? A more natural way to express these positions would be as a loan in which we borrow Euros and a spot transaction in which sell Euros, respectively. Making these substitutions gives the following replicating portfolio for a USD loan:
=++
This is the replicating portfolio that we seek.
3. How does the synthetic USD loan created in the previous question compare to synthetic borrowing of USD using an FX swap, as described in the lecture?
The replicating portfolio for the USD loan in the previous question was:
=++
FX Forward Buy EUR for USD at !!
Loan
Borrow USD from !! to !”
Deposit
Lend EUR from !” to !!
FX Forward Buy EUR for USD at !!
Loan
Borrow USD from !! to !”
FX Spot Buy EUR for USD at !”
Loan
Borrow USD from !! to !”
Deposit
Lend EUR from !” to !!
FX Spot Buy EUR for USD at !”
FX Forward Buy EUR for USD at !!
Loan
Borrow EUR from !” to !!
FX Spot Sell EUR for USD at !”
Loan
Borrow USD from !! to !”
FX Forward Buy EUR for USD at !!
Loan
Borrow EUR from !” to !!
FX Spot Sell EUR for USD at !”
An FX swap is a spot purchase and forward sale of a currency pair. We can therefore rewrite the replicating portfolio for the synthetic USD loan as:
=+
This is exactly the strategy we described in the lecture, in which we borrow EUR and at the same time enter into an FX swap in which we sell EUR spot and buy EUR forward, in both cases against USD. We will have the use of the USD for the period of the FX, so we have in effect borrowed USD. The EUR that we give up to our counterparty in the FX swap acts as collateral for the synthetic USD loan.
4. The Refinitiv Eikon screen image below reports spot and forward exchange rates for USDJPY on 28 September 2020. If covered interest parity holds for this currency pair, what do the forward points and outright forward exchange rates shown here imply about the relative level of money market interest rates in the United States and Japan?
Loan
Borrow USD from !” to !!
Loan
Borrow EUR from !! to !”
FX Swap
Sell EUR for USD at !! Buy EUR for USD at !”
Source: Refinitiv Eikon
The forward points are negative and increasing in magnitude with the forward horizon. Equivalently, the outright forward exchange rates are lower than the spot exchange rate and decreasing with the forward horizon. Under covered interest parity, this would be imply that the on the base currency (USD) was positive, which in turn would imply that USD money market interest rates were higher than JPY money market interest rates. This was in fact the case on 28 September 2020: money market interest rates in the United States were near zero but still positive, whereas money market interest rates in Japan were negative.