Download Week 1 Handout - UCLA Anderson

UCLA Anderson – MGMT234A: International Financial Markets (Winter 2013)
Week #1
Professor Andrea Eisfeldt
January 14, 2013
Handout written by Shenje Hshieh†
Reminder: Read chapters 2 and 6.1 - 6.3 in Bekaert and Hodrick (2012). Complete problem set
1 for your own self-study.
1
Key Concepts from Textbook
1.1
Terminology
Bid/Ask price/rate: From the market maker’s perspective, the bid (ask) price is the price at
which he would like to buy (sell). For rates, from the bank’s perspective, the bid (ask) rate is the
borrowing/deposit (lending/loan) rate.
Base/quote currency: Denominator (numerator) of the exchange rate. Base is domestic; quote
is foreign.
Periodic rate: The interest that is charged (and subsequently compounded) for each period.
(Annual) Nominal interest rate: Periodic rate multiplied by number of compounding periods
(frequency) per year.
1.2
Compound Interest Formulas
Changing compounding from discrete to another discrete basis:


nn1
2
r1
r2 =  1 +
− 1 n2
n1
(1)
where r1 is the stated annual nominal interest rate with compounding frequency n1 , and r2 is the
stated annual nominal rate with compounding frequency n2 . Under this notation, frequency refers
to the number of times the interest is compounded per year. In the textbook, n1 = 1 (annually),
n2 = 12 (monthly), and r1 is given. Then, r2 is the “annualized” compound monthly rate.
Changing compounding from discrete to continuous basis:
r
R = nln 1 +
n
(2)
where R is the annual nominal interest rate on continuous compounding basis and r is the stated
annual nominal rate with a compounding frequency n. In the textbook, n = 1 and r is the
percentage rate of change. R is the annualized continuous compounded rate. See chapter 2.5 for
further explanation.
†
Please email me at [email protected] if there are any errors.
1
1.3
Covered Interest Rate Parity Formulas
1+r =
F
(1 + r∗ )
S
r − r∗
F −S
=
1 + r∗
S
er =
F r∗
e
S
r − r∗ = ln(F ) − ln(S)
(3)
(4)
(5)
(6)
where r is the domestic currency interest rate for 1 period, r∗ is the foreign currency interest rate
for 1 period, S is the spot exchange rate (domestic per foreign currency), and F is the one-period
forward exchange rate (domestic per foreign currency). Note that Equations 3 and 4 are equivalent
– so are Equations 5 and 6. Equations 5 and 6 assume continuous interest compounding for r and
r∗ .
Notice that r > r∗ if and only if F > S from Equations 4 (or 6). Thus, the intuition for Equations
3 and 5 should be clear. Assuming the covered interest rate partity holds, an investor should
be indifferent between domestic and foreign currency deposits. Any excess return on domestic
deposits (i.e. r > r∗ ) must be offset by the ability to lock in a domestic currency capital gain (i.e.
buy foreign currency in spot market and contract to sell it forward for more than initial domestic
currency holding).
1.4
Constructing Simple Proofs of Arbitrage/No-Arbitrage
In chapter 2, we are introduced to “Triangular Abritage” of currency. The no-abitrage condition
implies that
CUR1
CUR3
CUR1
=Y
×Z
(7)
X
CUR2
CUR3
CUR2
If Equation 7 does not hold, then arbitrage is possible. If X > Y Z, one could short 1 unit of CUR2
for X units of CUR1 and then convert it back to CUR2 via Y and Z (i.e. YXZ > 1). Similarly, if
X < Y Z, one could short one unit of CUR3 for Y units of CUR1, convert it to CUR2 by dividing
by X, and finally converting it back to CUR3 by multiplying by Z (i.e. 1 < YXZ ). In both cases, we
end up with more than the amount we borrowed.
In chapter 6, we learn about the no-arbitrage condition in the interest rate market. The idea
is analogous: we borrow cheaply and lend/deposit expensively. Assuming no transaction costs,
suppose the interest rate parity fails in Equation 3:
1+r >
F
(1 + r∗ )
S
(8)
We borrow 1 unit of the foreign currency at an interest rate of r∗ , convert it to the domestic currency
by multiplying by S, deposit it at a domestic bank at r, and cover the transaction foreign exchange
risk by dividing by F . At the end of the period, the returns (in terms of the foreign currency) will
2
exceed the borrowing cost. A similar argument can be constructed when the inequality goes the
other direction.
When we have both bid and ask rates/prices, the argument is practically identical. Two inequalities
must hold for no-arbitrage:
S bid
1 + r∗,ask > ask (1 + rbid )
(9)
F
F bid
(1 + r∗,bid )
(10)
S ask
Can we borrow one currency, convert to another and deposit it to make excess returns? The ask
rate is the bank’s lending rate. Thus, we first borrow at its lending rate. For Equation 9, we first
borrow the foreign currency; second, sell it at the spot bid price (i.e. we want a market maker to
buy our foreign currency in exchange for domestic currency); third, deposit/lend it domestically at
the bid rate; lastly, cover our transaction foreign exchange risk at the forward ask price (i.e. we
want a market maker to sell us foreign currency in exchange for domestic currency). If at the end of
the period our foreign currency returns exceeds our borrowing costs, then we would have exploited
an arbitrage opportunity. A similar argument can be posed for Equation 10.
1 + rask >
2
2.1
Key Points from Lecture
Currency Quoting Convention
Internet and trading platforms tend to use EUR/USD instead of $/e(i.e. price of the euro in terms
of dollars or how many dollars you can buy with 1 euro). Prof. Eisfeldt wants to emphasize that
these two quoting conventions refer to the same exchange rate. However, in this course, we use the
latter convention (i.e. $/e) as in the textbook.
For example, e0.77/$ (USD/EUR = 0.77) is the price dollars in terms of euros (i.e. how many
euros does it cost to buy 1 dollar). $ 1.30/e(EUR/USD = 1.30) is the price euros in terms of dollars
(i.e. how many dollars does it cost to buy 1 euro).
2.2
Unit Conversion of Goods and Currency Example
Suppose the spot currency exchange rate of dollars per euro is $ 1.30/e= S$/e (i.e. this is the
price of euros in terms of dollars; that means we buy/sell euros for dollars). The price of a liter of
gasoline in France on February, 2012 was e2. What was the price of a gallon of gas in dollars?
1gal ×
e
1 liter
×2
× S$/e ≈ $9.84
0.264172 gal
liter
(11)
After canceling out all the units from the numerator and the denominator, we are only left with
dollars.
3
3
Problem Set 1 Solutions (TA version)
3.1
Read through example 1 of chapter 2 of the textbook on your own.
3.2
Note that the r’s in the interest rate parity given in Equation 3 are periodic rates. However, we
are given (annualized) nominal interest rates. We want to convert these rates to periodic rates over
. Thus, from
six-months. Since the r’s in Equation 1 are annual nominal rates, rperiodic = rnominal
n
Equation 1, we can derive the following:
r2
1+
=(1 + r)
n2
(12)
n1
r1 n2
= 1+
n1
where r is the periodic rate. Using this property, we find the following periodic rates for r$, periodic
and re, periodic :
1
(13)
(1 + r$, periodic ) = (1 + 0.03215) 2
1
(1 + re, periodic ) = (1 + re, nominal ) 2
(14)
Substituting Equations 13 and 14 into Equation 3 yields:
1
1 + re, periodic (1 + re, nominal ) 2
=
1
1 + r$, periodic
(1 + 0.03215) 2
S
=
F
1.2834
=
1.2779
(15)
Solving for re, nominal , the annualized nominal six-month Euroeuro interest rate (compounded annually), gives us re, nominal = 4.015%.
3.3
Again, we need to first convert the given 3-month nominal interest rates into periodic rates. Similar
to Question 2, we do not need to solve explicitly for the periodic rates; we just need to replace the
(1 + rperiodic ) terms.
4
Recall that when we have both bid and ask prices/rates, there are two inequalities that need to
hold to guarantee no arbitrage. From Equations 9 and 10, we replace the following:
1
1 + rU, ask, periodic = (1 + 0.014375) 4
1
1 + r$, bid, periodic = (1 + 0.051875) 4
1
1 + r$, ask, periodic = (1 + 0.053125) 4
1
1 + rU, bid, periodic = (1 + 0.013125) 4
(16)
(17)
(18)
(19)
Substituting Equations 16 and 17 into Equation 9, the inequality does not hold, which means we
can borrow any amount of Yen, convert it to dollars, deposit it at a domestic bank and hedge it
against transaction foreign exchange risk, and at the end obtain excess Yen returns after paying the
Yen loan principal plus interest.
The inequality in Equation 10 holds after substituting Equations 18 and 19 into Equation 10.
3.4
We lock in a dollar price per barrel with the following steps:
1
1. Borrow e122 × 1+0.0475
× 1.2834 e$ ≈ $149.48 at an interest rate of 4.25%. After 1 year, you
will owe 149.48 × (1 + 1.0425) ≈ $155.83.
2. Sell $ 149.48 for 149.48 ×
e
1
1.2834 $
≈ e116.47 via spot exchange rate.
3. Lend/deposit e116.47 at an interest rate of 4.75%. After 1 year, you should receive e122.
122
From the steps above, we have effectively locked in a forward exchange rate of 155.83
≈ 0.78292 e$ .
That is, after 1 year, we pay a borrowing cost of approximately $155.83 and receive exactly e122
from our deposit.
5