I-3. FX AND INTEREST RATES
FORWARD FX RATES, EXPECTED FX RATES, AND INTEREST RATES
It is often asserted that an observed forward FX rate is a “rational forecast” of
the future spot FX rate for the corresponding horizon, where a rational
forecast is the “true” expected spot rate, given all available information. This
economic theory is difficult to prove or disprove in reality, and is therefore very
controversial. The theory rests on the twin assertions that the FX market as a
whole holds a consensus forecast of the “true” expected future spot FX rate,
and that the FX market has already achieved equilibrium.
The argument that is used to support the theory is that if the
market knows the “true” expected spot rate at time N, E(XNSf/$), and if FNSf/$ ≠
E(XNSf/$), then there is a speculative profit opportunity for an informed trader to
“buy low/sell high”. For example, if the forward FX value of the US dollar is
lower than the expected future spot FX value, a trader would be inclined to go
long forward US dollars. The trader’s motivation is to lock in a low price for
buying US dollars in the future, with the expectation of a profit from a higher
realized value of the US dollar. The economic logic behind the assertion that
FNSf/$ = E(XNSf/$) is thus the “efficient market” argument that alert traders will
have already exploited any such profit opportunities, and that the associated
trading has forced the forward FX rate into alignment with the expectation.
Since the CIRP is a reliable no-arbitrage condition, then the assertion that
E(XNSf/$) = FNSf/$ is the same as the assertion that E(XNSf/$) = X0Sf/$[(1 + rSf)/(1
+ r$)]N.
Using * notation to explicitly denote that technically equilibrium
must have already been achieved for the condition to hold, equation (3-1a)
gives the Uncovered Interest Rate Parity (UIRP) condition, also called the
International Fisher Equation or the Fisher Open Equation, after the
economist Irving Fisher. In equation (3-1), rSf and r$ represent the annualized
interest rate on a zero-coupon instrument in Swiss francs and US dollars,
respectively, between now an time N.
Uncovered Interest Rate Parity (UIRP) Condition
“FX Rate Form”
E(XN*Sf/$) = X0*Sf/$[(1 + rSf)/(1 + r$)]N
(3-1a)
There is a corresponding “percentage form” UIRP in terms of the
equilibrium expected annualized percentage change in the FX rate. The
percentage change in the FX value of a currency must be computed with FX
rates expressed with the currency as the “denominator.” For example, if the
spot FX value of the pound at time 0 is X0$/£ = 1.60 $/£, and if the spot FX
value of the pound appreciates to 2.00 $/£, then the percentage change in the
spot FX value of the pound is (2.00 $/£ – 1.60 $/£)/(1.60 $/£) = 0.25, or 25%.
In general, the percentage change in the spot FX value of the pound over the
period from time 0 to time N is [XN$/£ - X0$/£]/X0$/£, or more easily XN$/£/X0$/£ - 1.
Let the notation xSf/$ (lower case x) represent the percentage
change in value of the US dollar (relative to the Swiss franc). The notation
E(xN*Sf/$) represents the expected (annualized) equilibrium percentage change
in the value of the US dollar (relative to the Swiss franc), between now and
2
time N. The “percentage form” UIRP is shown in equation (3-1b):
1 + E(xN*Sf/$) = (1 + rSf)/(1 + r$)
(3-1b)
And there is a corresponding linear approximation version in terms
of the annualized percentage forward premium/discount, which is shown in
equation (3-1c).
E(xN*Sf/$) = fNSf/$ = rSf – r$
(3-1c)
In the UIRP theory, the currency with the higher interest rate is
expected to depreciate (just as the currency with the higher interest rate is at a
forward discount in the CIRP condition.) This result can seem counter-intuitive
to some, but remember the assumption of UIRP is that the current spot FX
rate is already in equilibrium. The logic of UIRP is easier to see by
remembering the implicit assumption that all potential profit opportunities have
already been exploited, if the theory holds. If the currency with the higher
interest rate were expected to appreciate, and thus if the currency with the
lower interest rate were expected to depreciate, global “hot money” would shift
from the low interest rate currency to the high interest rate currency. This
shifting would cause the spot value of the high interest rate currency to rise
until equilibrium is reached. In equilibrium, the spot FX value of the high
interest rate currency has already been bid up high enough that the expected
future depreciation of the currency leaves “hot money” investors indifferent
between interest-bearing deposits in the two currencies.
While this economic logic is clear, the ability of the UIRP to “fit”
real-world data has been highly questioned. Many empirical studies designed
to test the UIRP theory check whether the forward discount/premium is a good
3
predictor of actual future FX rate movements. These studies have generally
found results that are exactly the opposite of those predicted by the
equilibrium UIRP relationship. That is, on average, currencies at a forward
premium (lower interest rate) have tended to depreciate in FX value, exactly
the opposite of the UIRP prediction; similarly, currencies at a forward discount
(higher interest rate) have on average appreciated, contrary to the UIRP
prediction. As an example, the euro was at a forward premium (lower euro
interest rates than US dollar interest rates), while the euro depreciated in
1999-2000. Thus, the empirical evidence that has been reported generally
supports the notion that FX rates are often not in equilibrium. One problem is
likely to be the market’s ability to form, as a consensus of all traders
influencing FX rates, a rational forecast the future spot FX rate. Another
problem is that dynamic markets never really allow a good test of an
equilibrium relationship with actual historical data.
Note that the lack of “empirical validity” of the UIRP relationship is
in stark contrast with the CIRP. Recall that CIRP is a no-arbitrage financial
relationship, the violation of which represents an immediate profit that can be
captured via an easy and inexpensive arbitrage with traded instruments and
no risk. Thus the CIRP condition reliably fits actual data for developed country
currencies. The UIRP condition is an economic theory based upon much
more vague and less realistic notions. And the UIRP is not supported by
empirical research. Because they look similar, CIRP and UIRP are often
equated, but they are fundamentally different. Again, the CIRP condition is a
‘no-arbitrage’ condition whose empirical validity is as easy to enforce as to
observe, while the UIRP condition is a questionable theory.
The UIRP condition is also known to suffer from the annoyance of
Siegel’s Paradox, which is explained later in the chapter. The UIRP condition
is also in need of a risk-adjustment. It turns out that when the risk-adjustment
is made (later in the text), the Siegel’s Paradox problem will simultaneously be
solved. The adjustment for risk (and Siegel’s Paradox) is relatively minor,
4
however, and not sufficient to create a model that “fits” the real-world data and
would be supported by empirical research any more than the unadjusted
UIRP of equation (3-3) has been supported.
Nevertheless, the UIRP condition is very useful, even though it
does not fit actual FX data! There are three reasons for this usefulness. The
first is that the interest rate differential has been found to have some partial
FX forecasting power, when used in forecasting models in combination with
several other factors, even though the interest rate differential by itself is not a
reliable forecaster. Second, the UIRP condition’s equilibrium expected FX
changes are useful in cost of capital estimation, which we do later in the text,
even if not good forecasts of actual FX changes. This point is similar to the
CAPM being useful for cost of capital estimation even if is not useful as a
forecaster of actual stock prices. Third, the UIRP condition is useful as a
logical model for analyzing the general idea of how interest rate changes
affect FX rates, as demonstrated in the next section.
INTEREST RATE CHANGES AND FX RATES
Although the traditional UIRP of equation (3-1a) has not been validated
empirically, the model has sufficient economic content to be useful in
analyzing the general way that interest rate changes affect FX rates. In this
regard, we perform so-called ‘comparative statics’ and ask the question: If an
interest rate suddenly changes, what will be the impact on the spot FX rate?
We start with the assumptions that the 1-year rSf = 8%, the 1year r$ = 10%, and the current spot FX rate X0Sf/$ = 1.50 Sf/$. Suppose that
suddenly the 1-year Swiss franc interest rate rises from 8% to 8.50%. What
would one expect to happen to the spot FX rate? This question can be
explored with the UIRP condition as a model, but NOT with either of the
UIRP conditions expressed in “% form” for E(xN*Sf/$) (equations (3-1b) and
(3-1c)). Instead, the “FX rate form” of equation (3-1a) should be used, where
E(X1*Sf/$) = X0*Sf/$[(1 + rSf)/(1 + r$)]. Thus we assume as a starting point that
5
the FX market, in aggregate, currently expects the rationally-forecasted
equilibrium spot FX rate a year from now to be E(X1*Sf/$) = 1.50
Sf/$[1.08/1.10] = 1.473 Sf/$.
The impact of the change in the Swiss franc interest rate will
depend on a) the cause of the interest rate change, and b) the anticipated
collateral impact of the interest rate change on other economic variables that
bear on FX rates. The ‘polar extremes’ are referred to as the asset market
theory and the Fisher theory.
In the asset market theory, the current spot FX rate changes in
response to the interest rate change.
In the Fisher theory, the forecasted FX rate changes, not the current
spot FX rate, with an interest rate change.
Suppose, for example, that the sudden increase in the Swiss franc
interest rate from 8% to 8.50% is caused by a demand for capital by
productive Swiss firms. In this case, the asset market theory applies, and the
spot FX value of the Swiss franc will immediately appreciate. The reason is
that capital of the asset market will be flow immediately into the Swiss franc to
capture the high potential returns offered by Swiss companies. If E(X1*Sf/$)
stays at 1.473 Sf/$, the new spot FX rate that will re-equilibrate the UIRP
condition, given the new Swiss franc interest rate of 8.50%, can be found
using equation (3-1a), 1.473 Sf/$ = X0*Sf/$[1.085/1.10]  new X0*Sf/$ = 1.493
Sf/$. In this scenario, the spot FX value of the Swiss franc appreciates
immediately, since the spot FX rate rises from 1.50 Sf/$ to 1.493 Sf/$, when
the Swiss franc interest rate suddenly rises 50 basis points from 8% to 8.50%.
A similar asset market scenario may occur if the increase in the
Swiss franc interest rate is due to an increase in the Swiss discount rate by
the central Bank of Switzerland. Often, the increase in the short-term interest
6
rate by a central bank is designed to help increase the spot FX value of the
currency. The higher interest rate will attract foreign short-term investors; their
movement of funds into the currency will indeed cause the spot FX value of
the currency to increase. If the interest rate rise has no other collateral effects,
the new spot FX rate would be the same as in the previous paragraph.
Now assume that, on the other hand, the cause of the sudden
interest rate increase, from rSf = 8% to rSf = 8.50%, is new information of a
sudden upward revision in the anticipated future Swiss inflation rate. A change
in inflation expectations is an instance of the Fisher theory. In this case, there
is (theoretically) no immediate reaction in the spot FX market. Instead, the
increase in inflation will dictate a lower rational expectation of the future FX
value of the Swiss franc. Before the increase in the anticipated inflation rate
and the Swiss franc interest rate, the Swiss franc was expected to gradually
increase in FX value, from a spot FX rate of 1.50 Sf/$ today to 1.473 Sf/$ a
year from now. If the anticipated Swiss inflation rate suddenly increases, the
expectation of the Swiss franc’s future FX value will be revised downward
from the current expectation of 1.473 Sf/$. If the spot FX rate remains at 1.50
Sf/$, we know from the “FX rate form” of the UIRP condition (equation (3-1a)),
the new expected future FX rate is 1.50 Sf/$[1.085/1.10] = 1.48 Sf/$,
representing a lower FX value of the Swiss franc than the previous
expectation, 1.473 Sf/$. Because the interest rate increase is due to a revision
of inflation expectations, the impact of the interest rate change is on the
rationally forecasted FX rate for the future, not on the current spot FX rate.
Note: in high-inflation countries, the spot FX value of the currency can
depreciate relatively rapidly between time-0 and time-1, but this dynamic is
different than the one we are looking at here, which is the impact of an interest
rate change.
Assume that the spot FX rate for the British pound is currently 1.60 $/£. Assume initially
that the 1-year US dollar interest rate is 5% and that the 1-year sterling rate is 10%. Now
7
let the 1-year US dollar interest rate stay at 5% and assume that the 1-year sterling
interest rate instantaneously jumps to 12%. 1. Use the “FX rate form” of the UIRP
condition in equation (3-1a) to determine what FX rate change occurs, assuming that
global investors do NOT change their expectation of the equilibrium future dollar/pound
FX rate a year from now. 2. If the change in the sterling interest rate is due to revised
inflation expectations, what short-term change occurs in the spot FX rate? Answers: 1.
Given the UIRP condition, the original equilibrium expected FX rate is (1.60 $/£)(1.05/1.10) =
1.527 $/£. If the sterling interest rate suddenly jumps to 12%, then the new time-0 equilibrium
spot FX rate is (1.527 $/£)/[(1.05/1.12) = 1.629 $/£. This answer means that the equilibrium spot
value of the pound is higher (1.629 $/£, compared to 1.60 $/£). This spot FX change takes place
“instantaneously.” Note that since the predicted future value of the pound 1 year from now is
unchanged and is still 1.527 $/£, then over the next year, the pound is still predicted to
depreciate to 1.527 $/£. However, at the new spot FX rate of 1.629 $/£, the pound is now
predicted to depreciate by a greater amount between time 0 and time 1. The predicted gradual
depreciation of the pound is consistent with the fact that the interest rate on sterling deposits is
higher than on US dollar deposits. 2. If the sudden rise in the sterling interest rate is due to a
revision of inflation expectations, then the current spot FX rate theoretically is unchanged, but
the expected future FX value of the British pound is revised downward to (1.60 $/£)(1.05/1.12) =
1.50 $/£.
At this point, we pause to point out why the popular “% form” of the
UIRP condition cannot be used to analyze the impact of an interest rate
change. The reason is that there is no way to express where the impact takes
place, in the spot FX rate, the expected future FX rate, or some combination.
Thus the “FX rate form” of the UIRP condition in equation (3-1a) must be
employed.
But there is only so much that such a limited model can
accomplish in addressing such a complex issue. For example, interest rate
increases can simultaneously affect other economic variables operating
behind the scenes of those in the UIRP relationship. In some cases, the
8
interest rate increase could slow the economy (perhaps by design of the
monetary authorities). If so, investors may revise their expected future FX
value of the currency downward. This result could in turn affect the spot FX
rate. If the perceived impact on the economy and the expected future
currency value are very strong, the increased interest rate could lead to a
decline in both the expected future FX value AND in the current spot FX
value. This kind of scenario occurred in the early 1990s in Canada. The Bank
of Canada was raising short-term interest rates to defend the Canadian dollar,
but “the market” perceived the interest rate hikes as being so negative for the
Canadian economy, that the spot FX value of the Canadian dollar actually fell
in response.
In another scenario, the rise in US interest rates in 1994 was
coupled with a decline in the value of the US dollar. The reason turned out to
be that foreign investors in long-term US bonds, as interest rates rose and
bond values fell, decided to get out of the US bond market. Of course, their
sale of US dollars into other currencies caused the spot FX value of the US
dollar to plunge. In turn, the depreciation of the US dollar was further reason
for squeamish foreign investors to pull out.
Recently, the FX value of the euro fell when the ECB (European
Central Bank) announced it would raise short-term interest, in expectation that
economic growth would be further slowed. It was cited that the interest rate
rise would not bolster the sagging euro as global investors already have large
euro-denominated holdings, and thus may not respond much to the interest
rate rise.
The analysis of this section suggests that one’s forecast of interest
rate changes may be a factor that would be useful in forecasting FX changes.
Thus in this chapter two factors have been identified that may belong in a
forecasting model for FX. The two factors are 1) the interest rate differential,
and 2) forecasted interest rate changes.
Note, however, the basic “circularity” of the theory and practice of
9
the UIRP condition. The theory says that UIRP will result because of the
equilibrium future FX forecasts by FX market participants, while FX
forecasters often use the interest rate differential (UIRP) in their forecasts. If
FX market participants only used the UIRP condition to forecast future FX
rates, then the condition would be empty, as it is based on the presumption
that forecasts are made on other fundamental economic factors.
10
Top Financial News
Wed, 30 Aug 2000, 6:45am EDT
Euro Falls to Record Against Yen; Rate Outlook
Threatens Growth
By Candace Carpenter
London, Aug. 30 (Bloomberg) -- The euro tumbled to its lowest ever against the yen
and was just shy of a record against the dollar on concern an expected interest
rate increase by the European Central Bank tomorrow may hinder economic
growth.
``The ECB will go for 50 basis points tomorrow, which is good if you go for rate
differentials; the problem is that it stifles growth,'' said Lorenzo Gallenga, who
oversees about 5 billion pounds ($7.22 billion) at Newton Fund Management.
The euro could drop to 88 yen and surpass the May 19 low of 88.50 U.S. cents
in coming days, he said.
Europe's currency fell as low as 94.09 yen, down from 94.78 late yesterday. It's lost 2.6
percent this week. It declined to 88.82 U.S. cents from 89.22. The euro has
depreciated against the dollar for 14 of its 19 months in existence.
The British pound, meantime, slid to a seven-year low against the dollar of $1.4465 on
speculation the Bank of England won't push up its benchmark interest rate
again, following four increases in the past year.
The euro's descent accelerated after traders triggered pre- set, loss-limiting orders to
sell the currency at the 89.10-cent level, said Eric Robin, head European foreign
exchange sales at Lehman Brothers International. That contributed to losses
against the yen, he said.
Credit Suisse Group's proposed acquisition of U.S.-based Donaldson, Lufkin &
Jenrette Inc. for $13.4 billion in cash and stock is also weighing on the euro
and Swiss franc, Robin said. The company will need to buy dollars, selling
euros and possibly Swiss francs to finance the cash component of the deal, he
said.
``Corporate flows are heading into the U.S. and that seems to be the main factor'' for
the dollar's strength against European currencies, said John Parker, who helps
oversee 1.8 billion pounds ($2.61 billion) at Pavilion Asset Management in
Brighton.
ECB Thursday
Meantime, ECB policy-makers are expected to boost the benchmark interest rate from
11
4.25 percent tomorrow, according to 39 of 42 analysts surveyed by Bloomberg
News yesterday. Eighteen of those 42 economists forecast a 25 basis-point
rise, and 21 predicted a half-point move.
A rate increase, aimed at capping price gains, would be the central bank's sixth since
November. Inflation among the region's 11 nations sped passed the bank's 2
percent target for a second month in July.
While the prospect of rising interest rates tends to support a currency as deposit rates
offer better returns, higher borrowing costs can also make it hard companies
to finance their investments, crimping profits and economic growth.
``The ECB is in a Catch-22 situation,'' said Mike Moran, an economist at Standard
Chartered Bank. ``If it tries to be more aggressive, the growth picture may turn
sour, which is the last thing the euro needs.'' It will probably chart a new low
against the dollar today or tomorrow, he said.
Lagging Growth
The euro's 23 percent decline against the euro, and 29 percent loss against the yen,
since its debut on Jan. 4, 1999, stems mostly from the strength of the U.S.
economy, analysts said. Record U.S. growth continues to lure foreign investors
to dollar- based financial assets, especially as there's little evidence of
accelerating inflation.
The Federal Reserve left key rates on hold at 6.5 percent this month, a nine-year high.
``The economic performance of Europe is seriously lagging the U.S.,'' said Stephen
Hannah, director of research at IBJ International.
In Japan, the yen held steady near a one-month high against the dollar after Japan
said industrial production unexpectedly fell 0.7 percent in July, following a 1.9
percent gain in June. That ran against economists' forecasts for a 0.2 percent
rise, helping stall the yen's rally as stocks declined.
The yen was recently at 106.18 to the dollar, the strongest since July 5, though little
changed from 106.21 late yesterday. The currency's rally to 106 faltered after
the production figures, Tokyo traders said, but it remains the best performing
major currency against the dollar in the past month.
12
SIEGEL’S PARADOX
Recall that he percentage change in the value of a currency must be
computed with FX rates expressed with the currency as the “denominator.” To
review, if the spot FX value of the pound at time 0 is X0$/£ = 1.60 $/£, and if the
spot FX value of the pound appreciates to 2.00 $/£, then the percentage
change in the spot FX value of the pound is (2.00 $/£ – 1.60 $/£)/(1.60 $/£) =
0.25, or 25%. In general, the percentage change in the spot FX value of the
pound over the period from time 0 to time N is XN$/£/X0$/£ - 1.
Can we say that the US dollar correspondingly depreciated by
25% relative to the pound? The answer is: approximately, but not exactly.
Considering the percentage change in the value of the US dollar relative to
the pound requires the use FX rates from the viewpoint of the US dollar as the
“denominator” currency. From this viewpoint, the spot FX rate changes from
X0£/$ = 1/X0$/£ = 1/(1.60 $/£) = 0.625 £/$ to XN£/$ = 1/(2.00 $/£) = 0.50 £/$. The
percentage change in the spot FX value of the US dollar is (0.50 £/$ - 0.625
£/$)/(0.625 £/$) = (0.50 £/$)/(0.625 £/$) – 1 = – 20%, a 20% depreciation of
the US dollar. Although the pound appreciates by 25% relative to the US
dollar, the US dollar depreciates by 20% relative to the pound. This
mathematical artifact is the fundamental basis of what is known as Siegel’s
Paradox.
Assume the spot FX rate for the Swiss franc goes from XSf/$ = 1.50 Sf/$ to XSf/$ = 1.25 Sf/$.
1) Find the percentage change in the value of the Swiss franc, and state whether the
change is an appreciation or a depreciation of the Swiss franc. 2) Find the percentage
change in the value of the US dollar relative to the Swiss franc, and state whether this
change is an appreciation or depreciation of the US dollar. Answers: 1) The FX quotes, in
conventional European terms, must be reciprocated to find the percentage change in the value
of the Swiss franc. Performing this reciprocation directly in the percentage change expression,
we have [1/(1.25 Sf/$)]/[1/(1.50 Sf/$)] - 1 = 0.20, or a 20% appreciation in the value of the Swiss
13
franc (relative to the US dollar). 2) From the point of view of the Swiss franc as the “pricing”
currency, the FX rate has changed from 1.50 Sf/$ to 1.25 Sf/$. The percentage change in the
value of the US dollar is (1.25 Sf/$)/(1.50 Sf/$) - 1 = - 0.1667, or minus 16.67%. Thus, the value
of the US dollar depreciated by 16.67% relative to the Swiss franc, while we know from the prior
example that the value of the Swiss franc appreciated by 20% relative to the US dollar.
Let the notation x$/€ (lower case x) represent the percentage
change in value of the euro (relative to the US dollar). There is an equation
that accurately relates the percentage FX changes from the two different
currency perspectives. The equation is shown below with the US dollar/euro
FX rate used as the representative currencies.
[1 + x$/€][1 + x€/$] = 1
(3-2)
Note that equation (3-2) can also be restated as [1 + x$/€] = 1/[1
+ x€/$], or equivalently, [1 + x€/$] = 1/[1 + x$/€].
Apply equation (3-2) to verify the answers to the previous problem, where the value of the
Swiss franc appreciated by 20% and the value of the US dollar depreciated by 16.67%.
Answer: [1 + x$/Sf][1 + xSf/$] = [1 + 0.20][1 – 0.1667] = 1.
While equation (3-2) is valid for percentage changes over any
horizon, it will be notationally convenient to henceforth regard x$/€ as an
annualized percentage change.
UNCERTAINTY, EXPECTATIONS, AND SIEGEL’S PARADOX
While today’s spot FX rate is observable and given, the spot FX rate for future
times is unknown at the present. For uncertain future spot FX rates, we will
14
often use the concept of expected future FX rates and expected FX rate
changes. [It is easier to discuss the concept of an expectation than to actually
form useful FX forecasts!] Here we’ll make a conceptual point about FX
expectations that relates to Siegel’s Paradox.
Despite the fact that X1$/€ will always be equal to 1/X1€/$, the
expected future FX spot rate, E(X1$/€), will NOT be equal to 1/E(X1€/$). In
words, if we know for certain that the spot FX value of the euro will be 1.20
$/€ a year from now, then the spot FX value of the US dollar a year from
now will for certain be 1/(1.20 $/€) = 0.833 €/$. But if 1.20 $/€ is only the
expected spot FX value of the euro a year from now, then the expected spot
FX value of the US dollar cannot be 1/(1.20 $/€) = 0.833 €/$!!
To see this point, assume there two equally-likely possible
outcomes for the future spot FX rate for a year from now: 0.80 $/€ (≡ 1.25
€/$) and 1.60 $/€ (≡ 0.625 €/$). The expectation of the spot FX value of the
euro a year from now is thus 0.50(0.80 $/€) + 0.50(1.60 $/€) = 1.20 $/€. That
is, the euro is expected to appreciate in FX value by 20% over the next year,
since 1.20/1 – 1 = 0.20, or 20%. At the same time, the expected spot FX
value of the US dollar for a year from now is 0.50(1.25 €/$) + 0.50(0.625
€/$) = 0.9375 €/$. That is, the US dollar is expected to depreciate by 6.25%
over the next year, since (0.9375 – 1)/1 = - 0.0625, or – 6.25%.
Thus we see that, 1/E(X1$/€) = 1/(1.20 $/€) = 0.833 €/$ is NOT
equal E(X1€/$), which was computed directly to be 0.9375 €/$. Also, E(x$/€) 
- E(x€/$), since 20%  – (- 6.25%). Moreover, [1 + E(x$/€)][1 + E(x€/$)]  1,
since (1.20)(1 - 0.0625) = 1.125, not 1. Thus, while equation (3-2) always
holds for any given FX change, it does not mathematically apply for
expectations of those changes.
Taken together, all of these mathematical surprises are
implications of Siegel’s Paradox. In the example, Siegel’s Paradox is that
even though the expected FX value of the euro is 1.20 $/€, the expected FX
value of the US dollar is NOT 1/1.20 $/€ = 0.833 €/$, since the expected FX
15
value of the US dollar is 0.9375 €/$. Siegel’s Paradox is also that the FX
value of the euro (relative to the US dollar) is expected to appreciate by
20%, while at the same time the FX value of the US dollar (relative to the
euro) is expected to depreciate by 6.25%.
By reversing the currency perspective, the UIRP condition implies
that E(XN*Sf/$) = 1/E(XN*$/Sf), or equivalently that 1 + E(x*Sf/$) = 1/[1 + E(x*$/Sf)].
However, we know these conditions cannot hold because of Siegel’s
Paradox.1
Assume that the spot FX value of the euro is currently 0.90 $/€. A year from now there is a
50% chance that the spot FX value of the euro will be 0.75 $/€ and a 50% chance that the
value that the spot FX value of the euro will be 1.10 $/€. What is the expected spot FX
value of the euro a year from now and the expected percentage change in the FX value of
the euro between now and a year from now? What is the expected spot FX value of the
US dollar a year from now and the expected percentage change in the value of the US
dollar? Answers: E(X1$/€) = 0.50(0.75 $/€) + 0.50(1.10 $/€) = 0.925 $/€; E(x$/€) = (0.925
$/€)/(0.90 $/€) – 1 = 0.0277, or 2.77%; E(X1€/$) = 0.50[1/0.75 $/€)] + 0.50[1/(1.10 $/€)] = 1.121
€/$; Since X0$/€ = 0.90 $/€, and thus X0€/$ = 1.111 €/$, E(x€/$) = (1.121 €/$)/(1.111 €/$) – 1 =
0.009 = 0.9%. Note that Siegel’s Paradox in this case has BOTH the euro expected to
appreciate relative to the US dollar (by 2.77%) AND the US dollar expected to appreciate relative
to the euro (by 0.9%)!!
FORWARD FX RATES AND SIEGEL’S PARADOX
Assume that the current 1-year forward FX rate in the market is 1.6605 Sf/$. A
person with Swiss francs can buy 1 US dollar for 1.6605 Swiss francs, for
delivery a year from now. Alternatively, we can say that a person with US
dollars can buy Swiss francs forward, and pay 1/(1.6605 Sf/$) = 0.6022 $/Sf,
Siegel’s Paradox was introduced in Jeremy Siegel, “Risk, Interest Rates,
and the Forward Exchange,” Quarterly Journal of Economics, February, 1975.
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or 0.6022 US dollars per Swiss franc. That is, we can always say that FN$/Sf =
1/FNSf/$. Thus, forward FX rates naturally reciprocate just like spot FX rates.
Some have theorized that forward FX rate is equal to the market’s
expected spot FX rate. Given Siegel’s Paradox, which reveals that
expectations do not reciprocate, we see an immediate problem with this
theory. Suppose that the forward FX rate were equal to the expected spot FX
rate from the direction of $/Sf. In that case, the forward FX rate expressed in
Sf/$ cannot equal the expected FX rate expressed in Sf/$. Since FN$/Sf =
1/FNSf/$, but E(XN$/Sf)  1/E(XNSf/$), the forward FX rate cannot be equal to the
expected spot FX rate from both currency directions! Since the choice of
currency direction viewpoint is arbitrary, Siegel’s Paradox is a problem for the
theory that forward FX rates represent expected spot FX rates.
SUMMARY
Whether actual forward FX rates are useful predictions of future FX spot rates
is an issue that was discussed. A problem is that the uncovered interest parity
condition does not empirically “fit” real world data. Despite that, the UIRP
condition is useful in thinking about the impact of interest rates on FX rates.
Glossary
Asset Market Theory: Changes in interest rates are reflected in spot FX rates.
Fisher Theory: Changes in interest rates are reflected in changes in expected
future FX rates.
Siegel’s Paradox: The mathematical result, and its implications, that the %
change in the FX value of currency A relative to currency B is not equal to the
negative of the & change in the FX value of currency B relative to currency A.
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Discussion Questions
1.
Compare and contrast the covered interest rate parity (CIRP) condition
with the uncovered interest rate parity (UIRP) condition.
2.
Assume that the international law of one price holds initially in the goods
market and that the UIRP condition holds. Explain why a short-run
increase in the interest rate for a currency, other things equal, can
result in the currency being overvalued from the perspective of the
goods market. Use this reasoning to explain why countries with high
economic growth may have trade deficits.
3.
Explain the difference in the reaction of the FX market between an
interest rate change driven by a change in inflation expectation and an
interest rate change driven by increased asset returns but no inflation
change.
Problems
1.
Assume that the spot FX rate for the British pound is 1.60 $/£, the 1-year
US dollar interest rate is currently 8% and the 1-year sterling interest rate
is currently 5%. Assume that the 1-year US dollar interest rate suddenly
rises to 10%, and all else stays the same. Use the UIRP condition to
determine the new spot value of the pound, if the increase in the US
interest rate is a) driven by the expectation of higher US inflation, and b)
driven by higher US short-term asset returns.
2.
If the spot yen/dollar FX rate changes from 125 ¥/$ at time 0 to 133.33
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¥/$ at time 1, what is the percentage change in the value of the yen?
3.
The time-0 FX rate between the Swiss franc and the US dollar is 1.50
Sf/$. The time-1 FX rate is 1.75 Sf/$. What is the percentage change in
the spot FX value of the Swiss franc (relative to the US dollar)? What is
the percentage change in the spot FX value of the US dollar (relative to
the Swiss franc)?
4.
Assume that the FX value of the euro is currently 0.90 $/€. A year from
now there is a 50% chance that the spot FX value of the euro will be
0.625 $/€ and a 50% chance that the spot FX value of the euro will be
1.25 $/€. What is the expected spot FX value of the euro a year from
now. What is the expected percentage change in the spot FX value of
the euro? What is the expected spot FX value of the US dollar and the
expected percentage change in the spot FX value of the US dollar?
Answers to Problems
1. a) 1.60 $/£; b) 1.571 $/£.
2. The yen depreciates by 6.25%.
3. xSf/$ = 0.1667, or 16.67%, x$/Sf = - 0.1428, or – 14.28%.
4. E(X1$/€) = 0.9375 $/€; E(x$/€) = 4.167%; E(X1€/$) = 1.20 €/$; E(x€/$) =
8.01%.
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