Econ 426 Note 6.0: Floating Nominal Exchange
Rates
Gregor Smith
Queen’s University
January 2016
Currency Names and Data
For those (like me) with poor pronunciation, use ISO 4217 codes.
e.g. CAD, USD, GBP, RMB, JPY, EUR, CHF
Recent history is at the Pacific Exchange Rate Service
Volume
Daily trading volume for Canadian institutions is about $100B, or
$24T per year, which is roughly 14 times the flow of GDP.
From the BIS Survey, world daily volume is about $3T daily.
These enormous values suggest that more than trade in goods and
services is involved.
Yet swings in the prices have large consequences for income and
employment in certain sectors and countries.
We’ll see that economists have often tried to explain these swings
with monetary factors, albeit with limited success.
Incidence and Notation
A diverse group of countries floats, as we’ll see in FT figures
2.2.–2.4.
The current rate is called the spot rate, from the spot market.
We’ll usually denote it S and measure it in local currency.
For many currencies there also is a forward market and price F .
That price is known today but the actual transaction occurs say 3
months in the future.
Who would use a forward contract?
Covered Interest Parity
At a common maturity and absent controls or taxes, CIP:
1 + i = (1 + i ∗ )
F
S
See FT figures 2.8 and 2.9. CIP seems to hold.
In logarithms:
i = i∗ + f − s
Uncovered Interest Parity
1 + it = (1 + it∗ )
Et St+1
St
See FT figure 2.10.
In logarithms:
it = it∗ + Et st+1 − st
where Et (∆st+1 ) is the expected rate of depreciation of the
domestic currency.
UIP is an hypothesis. With CIP you can see that it implies:
Et St+1 = Ft (sometimes called unbiasedness). See section 7 for
evidence.
Explaining or Predicting S
The exchange rate is the relative price of two monies, so see
money demands: M d = PY and M s = M d in each country.
Next, PPP:
SP ∗
P
(so this may be at best a long-run theory).
1=
Thus in growth rates (differences of logs):
∆s = ∆m − ∆m∗ − (∆y − ∆y ∗ )
See FT figures 3.7–3.9. This simple theory may work for high
inflation rates.
More General Money Demand
This more complex but realistic case is like FT 4.4 but using some
mathematics to as a short-cut to their combined approach.
1. Money Demand:
M d = PY exp(−αi)
or in logs
m = p + y − αi
and similarly in the foreign (starred) country.
2. PPP
S=
P
P∗
or in logs
s = p − p∗
3. UIP in logs:
i − i ∗ = Et st+1 − st
Exercise: Combine these three building blocks to find the
fundamental equation of the monetary model of the exchange rate.
st =
α
1
[mt − mt∗ − (yt − yt∗ )] +
Et st+1
1+α
1+α
Digression on Automobiles
Why does the Civic cost more than the Focus?
Half the difference is due to features (‘fundamentals’) labeled X
and half is due to expected, resale value.
Pt = Xt0.5 [Et Pt+1 ]0.5
and doubling both doubles the price.
In logs:
1
1
pt = xt + Et pt+1
2
2
These two factors matter for any asset, from art to real estate.
Solving Forwards
But what explains Et pt+1 ? A similar mix:
1
1
pt+1 = xt+1 + Et+1 pt+2
2
2
and so by substitution and using the E operator carefully:
1
1
1
pt = xt + Et xt+1 + Et pt+2
2
4
4
Repeat until the remainder becomes small, which will be true for
automobiles.
Forward-Looking Asset Price
With some patience:
∞
X
1
1
1
(0.5)j+1 Et xt+j
pt = xt + Et xt+1 + Et xt+2 + ... =
2
4
8
j=0
which is a present-discounted value model. Future fundamentals
matter since they affect the resale value.
But how do we forecast future fundamentals?
Suppose cars depreciate deterministically:
xt+1 = 0.9xt
Then
Et xt+j = 0.9j xt
Exercise: Write pt as a function of xt .
Forecasting Fundamentals
More generally, suppose that
xt = ρxt−1 + t
where t is white noise.
See if you can show that the solution is:
pt =
1
xt
2−ρ
It makes sense that slower depreciation leads to a higher price.
End of Digression
Monetary Model
The fundamental is xt = mt − mt∗ − (yt − yt∗ ), and
st =
1
α
xt +
Et st+1
1+α
1+α
or we know that this means:
∞
st =
1 X α j
Et xt+j
1+α
1+α
j=0
(as long as the remainder term vanishes).
Economic Lessons
Relative monetary policy matters.
The future can be anticipated (’discounted’) so the currency value
may not react to an event.
Or there can be a big reaction to a surprise that affects future
policy.
Solution
Suppose that we find fundamentals follow our first-order
autoregression:
xt = µ + ρxt−1 + t
now including an intercept term as in econometrics.
Solving forwards can be tricky and time-consuming so we can
instead use guess-and-verify. Guess the solution is:
st = k0 + k1 xt
where k0 and k1 are unknown coefficients.
Solution (continued)
Remember that:
st =
1
α
xt +
Et st+1
1+α
1+α
so using our guess:
k0 + k1 xt =
α
1
xt +
[k0 + k1 (µ + ρxt )]
1+α
1+α
This is a bit challenging but you can see that:
k1 =
1
1 + α − αρ
and
k0 = αµk1
so the theory predicts what happens when we regress st on xt .
Let us see what happens if ρ = 1 so xt follows a random walk.
Then the solution is:
st = αµ + xt
Thus st also follows a random walk (as we can confirm).
This is a resilient empirical model, even though we are not always
sure of the identity of xt .
Empirical Evidence
Imagine running a regression like this:
st = β0 + β1 st−1 + β2 xt
We often find β̂1 is significantly different from 0 and near 1, while
β̂2 is insignificantly different from 0 (even after looping over
databases).
Equivalently, a forecasting tournament finds st predicts st+1 better
than xt+1 does. It is hard to improve on the random walk model
even knowing future fundamentals.
There are large swings in exchange rates that have serious
consequences yet are hard to explain (other than with large
inflation differences), so ...
Speculative Bubbles?
Logic: The term ‘bubble’ often is used informally to refer to a
high price level.
Formally a rational speculative bubble occurs when the price rises
(or at least is expected to rise) and this appreciation constitutes a
return.
What is the minimum rate at which the price must rise?
Bubbles are similar to Ponzi schemes too.
Math: Remember the remainder term? It turns out that our
fundamental equation has multiple solutions if that term is not
zero. In particular, adding a term like
1+α
1+α t
bt =
bt−1 =
α
α
gives a solution that still satisfies the equation.
e.g. In our original automobile example α = 1 and so
pt = 0.91xt + 2t
will also be a solution, if cars can last forever (so there will always
be someone to sell to).
History: An oft-cited example is the Dutch tulipmania of 1677.
Other examples are the South Sea bubble and the Mississippi
bubble.
Perhaps there are recent examples.
Pitfall: A pitfall in testing for speculative bubbles is the possibility
of an ‘apparent’ bubble: bubbly price behavior even when no
bubble is present, simply because we have misunderstood the
market’s forecast of future fundamentals.
e.g. Imagine a biotech stock price that rises sharply as information
spreads about the firm’s promising research. Then suppose the
research fails, and the price falls back. A statistician looking at the
price and earnings data years later may falsely conclude a bubble
was present.
A similar concept sometimes is invoked in current markets: a ‘peso
problem’. This is an extreme event in the fundamentals (relative
monetary policy, say) that occurs very rarely. It affects expected
future fundamentals, and so the exchange rate, but may not be
observed in the historical sample of xt .
Currency Transactions Tax
A CTT is also called a Tobin tax. See the North-South Institute
links.
What are two purposes of a CTT?
What are two likely problems with implementing a CTT?
Key Lessons
1. Monetary factors matter, at least with large, persistent
differences in monetary policy.
2. Expected future policy matters, so events can be ‘discounted’ in
currency prices.
3. We’ve learned how to solve and test asset-pricing models in
general.
4. But our record of explaining floating exchange rates is fairly
poor.
5. Be sceptical of forecasts of floating exchange rates: the current
level usually is the best forecast, and changes generally are
unpredictable.