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Germany show that (i) systematic (non-diversifiable) risk as measured by bet
maturity (i.e. as n increases, b(") increases), (ii) all p(")sare less than unity a
sum of the two estimated coefficients on the E&+, and E,R?, in equation (1
unity as suggested by the theory. The above results hold when White's (1980
for heteroscedasticity is implemented (although no test or correction for high
correlation appears to have been undertaken). For the U K and USA the esti
p(")s is also statistically significant and rises with term to maturity n. Thu
consider a portfolio model which explicitly includes a risk premium, whi
on term to maturity, we find that expected holding period yields do have a
pattern. This is in contrast to the ex-post average HPY, which for the U K and U
exhibit systematic behaviour. But this is perfectly consistent, since the theory
expected returns at a point in time, conditional on other variables and not fo
sample averages of actual returns. For five- and 10-year bonds for the USA th
#1(5) = 0.71, p(lo) 0.92 and hence the risk premium for the USA is about
that for Germany.
The above evidence suggests that holding period returns are broadly in
with the zero-beta CAPM. Hence expected HPYs on long bonds depend on th
of such bonds where the latter varies with the term to maturity. Chapter 17
CAPM model to allow for a time varying term premium (as well as one that
term to maturity), that is we consider the case where T = T ( n ,z,).
For spot yields on three- and six-month bills (i.e. pure discount bonds) the U
tends not to support the EH. For US spot yields on longer-term bonds (i.e. n
the spread tends to be an unbiased predictor for future changes in shor
thus supporting the EH at the long end of the maturity spectrum. For the U
test is broadly supported by the data, even at the short end.
For longer maturity coupon paying bonds, regression and variance bounds
on the yield to maturity, the holding period yield and bond prices tend to
EH on US data. On UK data variance bounds tests on HPYs and long ra
supportive of the EH yet the latter does not appear to be grossly at odds w
There is some evidence that the (zero-beta) CAPM may provide a usefu
holding period yields.
Coupon paying bonds with long maturities are rather like stocks (except th
payment is known and largely free of default risk). We therefore have som
paradox in that long-term bonds broadly conform to the EMH whereas stocks
both types of asset are traded in competitive markets. Similarly, on the evid
us, short maturity bonds (bills) do not conform to the EMH based on the US
tentative hypothesis to explain the different results for long-term bonds and st
noise traders are more active in the latter than in the former market. It may
case that the greater volatility in short rates relative to long rates noted in S
may imply that agents perceive that taking positions at the short end is fairl
hence a time varying risk premium could explain the failure of the EH and t
the short end. The modelling of time varying risk premia is examined in Cha

In this appendix we briefly examine (i) the conditions under which the long rate is
and (ii) the use of forward rates in testing hypotheses about the term structure.

Is the Long Rate a Martingale?
In some of the early empirical work on the term structure researchers focused on whe
rate behaved as a martingale. This evidence will not be examined in detail but the basic
important early strand in the term structure literature will be set out. To investigate th
152,) =
under which long-term interest rates exhibit a martingale property, that is E,(R,+I
the EH applied to spot yields:

where a term premium TI"' on the n-period bond has been included. Leading (1) one-pe
the n-period rate at t 1 is given by:
+ -[Er+lrr+n - rrl
+ [T!;: - T 9
= n-'[A + B + C +D]

where A = RE forecast error: qf+l = rf+l - E f r f + l ,B = revisions to forecasts: w!yl =
Err,+,, C = Ef+lrf+n r, and D = change in the term premium: Ti:), - TI"'.
There is no guurantee that 'C'is either a constant or zero. However, for large
C/n may be small relative to the sum of the other terms, although this is not guarant
speaking we need to assume a transversality condition, such that C / n + 0. If the te
is constant and the term 'C' is small (i.e. approaches zero) then we can write (3) as

where pf+l is the weighted sum of error terms {qf+l, w:yl) in ( 3 ) and hence Ef(pf
follows immediately from (4) that under these conditions Rj"' is a martingale, that
RI"' = 0.
According to (3) the only reason we get a change in R(") between t and t 1 is:

a non-zero forecast error for the short rate at c 1 (term 'A'),
revisions to expectations between t and t 1, about short rates in one or more fu
that is term B (e.g. 'news' of a credible counterinflationary policy might imply a
rf+j in future periods and hence a decrease in some or all of the (Er+lrf+j - Ef

a change in the agent's views about Ef+lrf+n(i.e. term 'C'),
a change in the agent's perceptions of the term premium required on long-term
term 'D).

The first three items in the above list involve the arrival of new information or 'news' an
the monetary authorities cannot systematically influence long rate. At best, the authorit
cause 'surprise' changes in the long rate by altering expectations of future short ra
instituting a tough anti-inflation package). If the authorities could systematically influe
premium then it would have some leverage on long rates. However, it is difficult to
authorities might accomplish this objective by, for example, open market operations. He
cited implication of the above analysis is that under the EH RE the authorities canno
yield curve.
Empirical tests of the martingale hypothesis are based on (4) and usually involve a r
AR,':: on informatiod dated at time c or earlier Q. In general AR!:: is found to depend
in $2, although the explanatory power is low (e.g. Pesando (1983) using US data). How
is close to a martingale process this is consistent with the evidence that long rates ten
conform to the EH (with a time invariant term premium).
structure. (The concepts used will also be useful in analysing the forward foreign exch
in Part 4.) The methodology employed is shown by using simple illustrative example
follows the notation in Fama (1984).
Consider the purchase of a two-period (zero coupon) bond. Such a purchase autom
you in to a fixed interest rate in the second period. At time ˜t™ the investor know
one-year spot interest rate, and R(2, t ) the two-year spot rate (expressed at an annual
he can calculate the implicit interest rate he is receiving between years 1 and 2. Deno
+ +
as the forward rate applicable for years t y to t n. Then the forward rate between
and t 2 is given by:

+ R(2, ?)I2 = [l + R(1, t>][l + F(2, 1 : t ) ]
F(2, 1 : t ) is the implicit forward rate you ˜lock in™ when purchasing a two-year
ranging (1):
+ +
F ( 2 , l : t ) = [(l R(2, t)I2/(1 R(1, t))] - 1

For example, if you purchase a two-year bond at R(2, t) = 10 percent per annum a
9 percent per annum then you must have ˜locked in™ to a forward rate of 11.009 percen
in the second year. Note that (4)is an ˜identity™, there is no economic or market behavio
If we let lower case letters denote continuously compounded rates (i.e. r(n, t) = ln[
then (1) and (2) can be expressed as linear relationships and

f(2, 1 : t ) = 2r(2, t ) - r(1, t )
For a three-period horizon we have

+ 2f(3,1 : t )
3r(3, t ) = r(1, t )
+ 1 to t + 3 and hence a
where f(3, 1 : t) is the implicit forward rate for the years t
investment horizon of 3 - 1 = 2 years. Note also that

+f(3,2 :t )
343, t ) = 2r(2, t )
Equations (4) and ( 5 ) can be used to calculate the implicit forward rates f(3, 1 : t
from the observed spot rates. Implicit forward rates can be calculated for any horiz
appropriate recursive formulae if data on spot rates are available. From the above e
following recursive equation is seen to hold:

We now return to the two-period case, and examine variants of the expectations hypo
in terms of forward rates. If the EH plus a risk premium T(2, t) holds then

Comparing (3) and (7) we see that

Hence the implicit forward rate is equal to the market™s expectation of the return on
bond, starting one period from now, plus the term premium. If T(2, t) = 0 then (8)
with the pure expectations hypothesis. The PEH therefore implies that the implicit fo
an unbiased predictor of the expected future spot rate. Subtracting r(1, t) from both sid
Tests of the PEH (i.e. T = 0), or the EH (T = constant) or the LPH (i.e. T = T"),
assume a time invariant term premium, are usually based on regressions of the form:
+ 1) - r(1, t ) = a + P[f(2,1 : t ) - 4,r ) ] +
r(1, t qr+1

where we expect a = -T and fi = 1. Thus under the pure expectations hypothesis w
forward premium to be an unbiased predictor of change in the future spot rate (i.e
a = 0). Under a constant term premium we also expect that no other variables dated
are statistically significant in (10). Note, however, that if T is non-constant and correla
forward premium, then OLS on (10) yields inconsistent (biased) estimates. Similar
(10) apply to implicit forward rates over different horizons.
Mishkin (1988) examines equations like (10) for short horizons, namely for two-
horizons, using data on US Treasury bills over the period 1959-1982. He finds t
month ahead forecasts = 0.40 (se = 0.11) and R2 = 0.11 but as the horizon is exten
r(1, t + m ) - r(1, t ) , m = 3 , 4 , 5 , 6 ) the forward premium generally has little or no pred
(i.e. $ * 0 statistically). In all cases Mishkin finds he can reject Ho:# 1 and this
with the regression tests reported in the main text, which tend to reject the expectation
on US data at the very short end of the maturity spectrum. Fama (1987) presents resul
horizons using (approximate) spot rates on US Treasury bills from one to five years'
finds that one can reject H o : p = 0 for horizons greater than one year, although in m
also finds H o : p # 1 is rejected. Hence the EH (with a constant term premium) is
invalid. However, the forecast power of regressions like (10) improve as the horizon is
For example, for four-year changes in one-year spot rates r(1, t 4) - r(1, t ) the forw
explains 0.48 percent of the variability in the dependent variable (see Figure 10.A1
not reject H o : /3 = 1. The above results using forward rates are consistent with those
this chapter, namely that at the very short end (less than four years) the EH perform
improves somewhat at longer maturities. For further evidence on these types of tests
of the term premium to forecast inflation, see inter alios Mishkin (1988, 1990) and F


Figure 10A1 Four-year Change in Spot Rate (Solid Line) and Forecast Change (D
using the Forward-Spot Spread. Source: Fama and Bliss (1987). Reproduced by perm
American Economic Association
equation (10) will not yield an estimate of = 1 in the sample.
The presence of the Peso problem implies that the forward rate is a biased
E r r ( l ,t 1) within the sample period examined. The Peso problem may be viewed
finite sample problem or a ˜missing variable™. If we had a long enough data set it is (ju
able that over the whole sample, positive and negative ˜Peso effects™ would cancel. A
one could also argue that we obtain the result / # 1 because the ˜missing variable™ in th
equation (10) is a term premium which might vary over time as the perceivedfuture b
the monetary authorities alters. If the variability in the term premium is correlated with
premium, then the OLS estimate of is biased.

1. This is most easily demonstrated using continuously compounded spot ra
We have:

The logarithmic HPY is

The null hypothesis is that the HPY is equal to the risk-free (three-month

= (1/4)rt
Using (3) and (4):
2R, - rr+l = rf
Equation (5) is equivalent to (10.4) in the text under the null of t
similar equation to (5) can be derived using simple interest rates (i.e.
Pj6™)/(Pi6), = 4(M - P,˜3™)/P!3™ we use the approximation (1 r,+1/4
This is left as a simple exercise for the reader.

Most basic texts on financial markets and portfolio theory provide an intro
alternative measures of bond returns together with theories and evidence o
structure. More practitioner based are Fabozzi (1993) and Cooke and Rowe
the USA and for the UK, Bank of England (1993). At a more technical level, a
of tests of the term structure can be found in Section I1 of Shiller (1989), p
Chapters 12, 13 and 15, and Melino (1988).
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The Foreign Exchange Market

The behaviour of the exchange rate particularly for small open economies tha
a substantial amount of international trade has been at the centre of macr
policy debates for many years. There is no doubt that economists™ views abo
exchange rate system to adopt have changed over the years, partly because ne
has accumulated as the system has moved through various exchange rate reg
worthwhile briefly outlining the main issues.
After the Second World War the Bretton Woods arrangement of ˜fixed but
exchange rates™ applied to most major currencies. As capital flows were smal
subject to government restrictions, the emphasis was on price competitiveness
that had faster rates of inflation than their trading partners were initially allowed
from the International Monetary Fund (IMF) to finance their trade deficit. If a ˜fu
disequilibrium™ in the trade account developed then after consultation the defi
was allowed to fix its exchange rate at a new lower parity. After a devaluatio
would also usually insist on a set of austerity measures, such as cuts in public e
to ensure that real resources (i.e. labour and capital) were available to switch
growth and import substitution. The system worked relatively well for a numb
and succeeded in avoiding the re-emergence of the use of tariffs and quotas tha
a feature of 1930s protectionism.
The US dollar was the anchor currency of the Bretton Woods system and the
initially linked to gold at a fixed price of $35 per ounce. The system began to c
strain in the middle of the 1960s. Deficit countries could not persuade surplus c
mitigate the competitiveness problem by a revaluation of the surplus countries
There was an asymmetric adjustment process which invariably meant the defi
had to devalue. The possibility of a large step devaluation allowed speculat
way bet™ and encouraged speculative attacks on those countries that were pe
have poor current account imbalances even if it could be reasonably argued
imbalances were temporary. The USA ran large current account deficits which
the amount of dollars held by third countries. (The US extracted seniorage by th
Eventually, the amount of externally held dollars exceeded the value of gold in
when valued at the ˜official price™ of $35 an ounce. At the official price, free co
of dollars into gold became impossible. A two-tier gold market developed (w
market price of gold very much higher than the official price) and eventually co
the Bretton Woods system, and floated their currencies.
In part, the switch to a floating exchange rate regime had been influenced b
economists. They argued that control of the domestic money supply woul
desired inflation and exchange rate path. In addition, stabilising speculation
agents would ensure that large persistent swings in the real exchange rate an
price competitiveness could be avoided by an announced credible monetary poli
in the form of money supply targets). Some of these monetary models of exc
determination will be evaluated in Chapter 13.
Towards the end of the 1970s a seminal paper by Dornbusch (1976) sho
FOREX dealers are rational, yet goods prices are ˜sticky™, then exchange rate ov
could occur. Hence a contractionary monetary policy could result in a lo
competitiveness over a substantial period with obvious deflationary consequen
trade, output and employment. Although in long-run equilibrium the econo
move to full employment and lower inflation, the loss of output in the transi
could be more substantial in the Dornbusch model than in earlier (non-rational
models, which assume that prices are ˜flexible™.
The volatile movement in nominal and real exchange rates in the 1970s led
to consider a move back towards more managed exchange rates which was
reflected in the workings of the Exchange Rate Mechanism (ERM) from the e
European countries that joined the ERM agreed to try and keep their bilatera
rates within announced bands around a central parity. The bands could be either
percent) or narrow ( 2.5 percent). The Deutschmark (DM) became the ancho
In part the ERM was a device to replace national monetary targets with Germa
policy, as a means to combat inflation. Faced with a fixed exchange rate again
a high inflation country has a clear signal that it must quickly reduce its rate of
that pertaining in Germany. Otherwise, unemployment would ensue in the hig
country which would then provide a ˜painful mechanism™ for reducing inf
ERM has a facility for countries to realign their (central) exchange rates in th
fundamental misalignment. However, when a currency hits the bottom of its ba
of a random speculative attack, all the Central Banks in the system may try a
the weak currency by coordinated intervention on the FOREX market.
The perceived success of the ERM in reducing inflation and exchange rate v
the 1980s led the G10 countries to consider a policy of coordinated interventi
Plaza and Louvre accords) to mitigate ˜adverse™ persistent movements in their o
cies. The latter was epitomised by the ˜inexorable™ rise of the US dollar in
which seemed to be totally unrelated to changes in economic fundamentals. Rec
economists have suggested a more formal arrangement for currency zones an
bands for the major currencies, along the lines of the rules in the ERM.
Very recently the ERM itself has come under considerable strain. Increas
mobility and the removal of all exchange controls in the ERM countries facilita
ulative attack on the Italian lira, sterling and the franc around 16 September 19
as Black Wednesday). Sterling and the lira left the ERM and allowed their cu
monetary union in Europe are complex but one is undoubtedly the desire t
the problem of floating or quasi-managed exchange rates. One of the main t
part of the book is to examine why there is such confusion and widespread d
the desirability of floating exchange rates. It is something of a paradox that
are usually in favour of ˜the unfettered market™ in setting ˜prices™ but in the
exchange rate, perhaps the key ˜price™ in the economy, there are such diverge
This section of the book provides the analytic tools and ideas which will
reader to understand why there is a wide diversity of views on what drives th
rate. We discuss the following topics:

the interrelationships between covered and uncovered interest parity, purcha

parity and real interest rates
testing for efficiency in the spot and forward markets and whether market

are rational when setting FOREX ˜prices™
whether economic fundamentals drive the exchange rate and under what

exchange rate overshooting might occur
the impact of ˜news™ on the volatility of exchange rates, the so-called Pe

and the influence of noise traders in the FOREX market
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Basic Arbitrage Relationships in
FOREX Market
This chapter outlines the basic concepts needed to analyse behaviour in t
market. These concepts are first dealt with sequentially, in isolation, before dis
inter-relationship between them.

There are two main types of ˜deal™ on the foreign exchange (FOREX) market.
the ˜spot™ rate, which is the exchange rate quoted for immediate delivery of th
to the buyer (actually, delivery is two working days later). The second is t
rate, which is the guaranteed price agreed today at which the buyer will take
currency at some future period. For most major currencies, the highly traded
are for one to six months hence and, in exceptional circumstances, three to
ahead. The market-makers in the FOREX market are mainly the large banks.
The relationship between spot and forward rates can be derived as follows. A
a UK corporate treasurer has a sum of money, LA, which he can invest in the
USA for one year, at which time the returns must be paid to his firm™s UK sh
Assume the forward transaction is riskless. Therefore, for the treasurer to be
as to where the money is invested it has to be the case that returns from inve
UK equal the returns in sterling from investing in the USA. The return from
in the UK will be A ( l r) where r is the UK rate of interest. The return
from investing in the USA can be evaluated using the spot exchange rate S (E
forward exchange rate F for one year ahead. Converting the A pounds into d
give us A / S dollars which will increase to (A/S) (1 r*) dollars in one year™s
is the US rate of interest. If the forward rate for delivery in one year™s time
then the UK corporate treasurer can ˜lock in™ an exchange rate today and re
+ r)F in one year™s time. (We ignore default risk.) Equalis
certainty, f(A/S)(l
we have:
A ( l r ) = (A/S)(l r*)F

which becomes
--- + r
where r is measured as a decimal. The above equations represent the ˜cove
parity™ (CIP) condition which is an equilibrium condition based on riskless a
CIP doesn™t hold then there are forces which will restore equilibrium. For e
r > r* and f = s then US residents would purchase UK securities pushing
up and interest rates down. US residents would also have to buy sterling sp
dollars forward, hence spot sterling would appreciate (i.e. s falls) and f woul
tending to restore equality in (11.3). In fact, because the transaction is riskles

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