Interest rate parity, IRP. It is defined as the difference in interest
rates between two countries is equal to the percentage difference between the forward
exchange rate and the spot exchange rate. Let me give you an example of how this definition
should be understood. For example, if Japan’s interest rate is 4%
higher than the U.S. interest rate, then the forward rate should be 4% higher than the
current spot rate as well. Otherwise, there’s an arbitrage opportunity,
arbitrage means making guaranteed profit without taking any risk. So, it’s probably still pretty confusing what
this whole interest rate parities about Let me explain it using the following example:
Let’s say you have $1, there are two things you could do with your $1. One; you could invest it domestically. Two; you could invest it, let’s say, in Japan. Let’s look at the option number one, you want
to invest it domestically, within the United States. You want to make a risk-free investment with
a guaranteed return after one year. What is the name of, like, a risk-free asset
in this country? It’s basically, you know, the short-term government
bond called “Treasury Bill” and the return you will get on is it known as the risk-free
rate. So, let’s the risk-free rate on such short-term
government bond in the United States is 4% so the notion we use for the risk-free rate
in the United States is capital R, subscript U.S.A. and that’s 4% that’s one thing you
can do to get your guaranteed 4% return on your $1. Instead, you could use your $1 and invest
it overseas and, again, you want to make a very safe investment that will bring you a
guaranteed return, should be a risk-free investment. So, whatever the name of the Japanese government
short term bond is, that’s the risk-free asset you would buy in Japan for one year. And let’s say the risk-free rate there is
6% so capital R, subscript, Japan equal 6% 4% here, 6% there. One more thing we know in this example is
that the current spot exchange rate is S0, equals 120 Yen per one U.S. dollar, the question
in this problem is, under which condition will it be indifferent between the two investments? Okay. So, the question may sound very vague and
kind of unclear what needs to be done. So, let’s say, you know, let’s kind of dig
a bit deeper and do some math, how much are going to have on today’s $1 after one year? Well, if we invest $1 into the U.S. risk free
asset, with the return of 4% then in one year this $1 investment will go to $1 times 1 plus
the interest rate, R U.S.A. which equals $1 times 1 plus .04, in decimals, which gives
$1.04 right? It’s basically, you know, the future value
calculations, you could also do it in the financial calculator where $1 would be your
PV, change to a negative number, N would be 1 because we want to find the value of the
1 here, IY would be the return, you would put 4 and you would be computing FV and the
calculator would give you 1.04 as the answer. So that’s how much you would have after one
year from today’s $1 if this $1 is invested domestically, into the risk-free asset that
brings a guaranteed 4% return. The second option was, you instead invest
your $1 into the Japan’s risk-free asset equivalent. The return on the Japan’s risk-free asses
is 6% okay, well, first of all, we’re kind of not really investing dollars in Japan,
we should be investing Yen, if you have $1 to invest Yen, we need to first convert it
into Yen, what do we use for the conversion? Today’s exchange rate, the spot exchange rate,
which was given, right? 120 Yen per 1 U.S. dollar. So, $1 is multiplied by 120 Yen, per $1, which
gives 120 Yen, so really, you’re investing 120 Yen into the Japanese financial market
and now we want to do a similar math. In one year, this investment will grow to
120 Yen, times 1 plus .06, the return that’s guaranteed after one year, which is 127.2
Japanese Yen. And then we want to compare apples with apples,
so what does it mean in terms of dollars, right? So, we can then compare, you know, the value
of this investment to what you would get if you instead invested your money in the United
States, right? So, we want to go back from Yen to dollars,
right? In one year, 127.2 Yen will be exchanged back
into dollars, how? So, what is the name of the exchange rate
that we can already know today that we know today already that will be used in one year
to convert 1 currency into another? What was the name of such trade where we know
what exchange rate will be used at some point in time in the future? That’s the forward exchange rate, the one
year forward exchange rate agreed upon today. So whatever that rate is that is already known
today that would be used to exchange our Yen into dollars in one year, that’s what we would
need to use to convert next year’s 127.2 Yen back into dollars. So, in terms of dollars, this investment made
in Japan will be worth this much money in dollars, right? So, it will be worth 127.2 Yen divided by
the one year forward and one year forward will be expressed as how many Yen it is per
dollar, right? So, you would divide next year’s Yen by Yen
per dollar and you will be left with dollar, right? The dollar amount, right? And now, let’s summarize our results. So if you invest your $1 in the U.S. risk
free asset at 4% you will have $1.04 if you instead, invest your $1 into Japan’s risk
free asset at 6% then in one year, in terms of dollars, you will have 127.2 Yen divided
by the one year forward rate, so whatever that rate shelf gives you, that’s how much
your investment will be worth in dollars, when it’s made in Japan. And now we can answer the main question in
this problem, which was, under which circumstances would be indifferent between the two investments? What does be indifferent between the two investments
really mean? It means, you don’t care whether your $1 is
invested in the United States or in Japan, what does it mean you don’t care? In this case, we do not care. Well, if you make the same money in one year,
right? So how much money are we talking about in
one year? We already have it for option number one,
$1.04 we, kind of, don’t have it yet for option number two, it’s not very clear because we
don’t know what F1 is, the one year forward rate, but what we’re basically saying is that
regardless of whether your $1 is invested here or there, either way it should give you
$1.04 in one year. So, you must make the same money, $1.04 in
both the U.S.A. and in Japan and we can now calculate in which case this will be true. What is our unknown? The F1, for the value of the investment in
Japan. So, $1.04 that’s in one year in the United
States must equal 1267.2 Yen, divided by the one year forward rate and then we rearrange
to solve for the one year forward rate. The one year forward rate should be 127.2
divided by 1.01, which gives 122.3 Yen per dollar, this is the condition under which
we will be indifferent between whether our $1 is invested in the United States or in
Japan, the one year forward rate Must be exactly 122.3 for us to be indifferent between the
two countries investments. If the one year forward rate is actually not
that number, it’s not 122.3, but something else, then we, again, have what we, you know,
kind of keep coming back to when we learn every new formula in this chapter. We will have an opportunity to make profit
a guaranteed profit, in other words, an arbitrage opportunity. In this particular context with the interest
rates it’s called a covered interest rate arbitrage. Let me go back by one slide and give you an
example. Let’s say the one year forward rate is not
122.3, but only 121; so, it’s less than it should be, if it’s only 121, less than it
should be, which country are you going to prefer? Are you going to want to invest in the United
States or in Japan? Well, mathematically, if you’ll look at where
the one year forward rate makes a difference, it makes a difference for the investment made
in Japan, so mathematically, if we divide 127.2 Yen by a smaller number, we’re going
to get a bigger number as a result, which should mean that everybody would want to invest
their money in Japan where they would make more money after one year, rather than in
the United States, that’s if the forward rate is lower than what it should be for that indifference
between the two countries, so lower than 122.3, such as, for example, 121 Yen per dollar and
the other way around, what if the one year forward rate is higher than how high it should
be? So, let’s say its 125 Yen per dollar, then
mathematically, for Japan, when we divide by the larger number, we get the smaller answer,
smaller means we will make less than $1.04 if we invest our money in Japan. So where would all investors be investigating
their money to? Not into Japan, but into the United States
where they get more money after one year. So in order for investors to not, kind of,
all invest into just one country, in order for there to be, sort of, a balance in the
investments made in this world, we must have these, you know, condition that the one year
forward rate between two specific countries should be calculated, you know, the way we
did it in this example, and the actual forward rate should be, should exactly match this
calculated number to prevent the covered interest arbitrage opportunity.