
‘Euro’ used in its old meaning here – not the single currency but the ‘black’ or ‘uncontrolled’ foreign market in domestic interest
rates.15
In 1982 (and again in 1983) the markets attacked the FRF/DEM parity in ERM. The French defended with intervention anddomestic interest rate rises (to about 18%). However interest rates in the ‘Euro-franc’ (uncontrolled francs owned offshore) rose to peakovernight interest rates of 3000%. The French lost the ﬁght and devalued 8% (one day’s worth of 3000% interest rate differential!).
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Currency Overlay
4.4.1.1 Controlling domestic currencies – why do it?Let us begin by listing the main policy objectives that governments claim to target by exchangecontrols on their own currency:1. To prevent or deter capital outﬂows (and sometimes inﬂows)2. To ‘protect’ the exchange rate (i.e. to keep it high, except see 4 below)3. To ration imports (protect domestic producers or control the trade balance)4. To encourage exports (i.e. to depress the exchange rate)5. To isolate the economy from (undesirable) external market factorsNot all countries aim to achieve all these objectives (2 and 4 are mutually incompatible).However, exchange controls will not be maintained by a government unless at least one of theabove are publicly espoused. There may, in addition, be hidden motives that no governmentwill publicly admit to. Three candidates stand out:1. To deter emigration/immigration (mainly the former)2. To levy hidden ‘export taxes’3. To control foreign currency reserves and the (corrupt) sales of import licencesThe motivation behind most of these objectives is self-evident. Once a government has gonedown the exchange/price control route, how do they typically execute them?4.4.1.2 Physical controlsPhysical controls (commonly called exchange controls) can be applied to residents (preventingthem acquiring foreign currencies) and to non-residents (preventing them acquiring the localcurrency, and maybe preventing them buying foreign currencies once they have acquired thelocal currency). The level and mechanisms of control vary widely across countries.Whatever the mechanics of controls, and they are too numerous to mention here, the intentionof the government is the same: to limit by law the ability of residents to sell their own currencyin favour of a foreign currency.4.4.1.3 Price interventionIn the absence of physical controls, or in concert with them, governments have another option. They can choose to use their considerable resources to buy their own currency when itfalls below a reference price of their choice, or sell it when it rises above. The detailed waythey can approach this intervention was laid out in the previous section. If their resourcesare sufﬁcient, this can be a powerful tool; ﬂexible and adaptable, unlike exchange controls.The resources in question are not just the inevitable ﬁnancial clout wielded by every government, but something much more tangible: foreign currency (and gold) reserves. High levelsof reserves, expressed both in absolute terms and as a percentage of that country’s GDP, cangive a government effective control over the ﬂoor at which its own currency will trade. Orcan it? I will argue later that the size of reserves is only one factor when we come to evaluate the ability of a government to defend, or otherwise manipulate, its currency exchangerate.
Foreign Exchange Market – History and Structure
Box 4.6
77
Exchange controls
Both active and passive management of currency risk may be difﬁcult or impossible if thetarget country maintains exchange controls.Exchange controls’ purposes are various, and there is probably a gap between the declaredpurposes (protecting the exchange rate, protecting the reserves, preventing speculation)and the actual purposes (retaining control over import/export resource allocation, keepingimported goods cheap for the urban/elite population, hidden tax on exports, corrupt use ofimport/currency licences, supply of foreign currency to be stolen and sent to Swiss bankaccounts).Exchange controls come in a variety of forms. The most stringent types (e.g. PR China)do not permit foreigners or residents to buy or sell spot or forward currency without apermit. All transactions have ultimately to be routed through the central bank. The nextlevel down of stringency (e.g. South Africa) controls residents’ ability to buy or sell thecurrency, but allows foreigners to exchange offshore currency between themselves, andallows direct investment ﬂows in and dividends/earnings out. The lightest types of controlinvolve controls mainly on forward contracts (to prevent ‘speculation’), and on residents.For those countries with exchange controls but substantial capital markets and trade,offshore commercial banks have developed a contract called the Non-Deliverable Forward(NDF). NDFs are priced and traded like forward contracts as bilateral arrangements, butthe mutual delivery of the currencies is not contemplated – instead a cash payment in anuncontrolled currency is made to compensate for the change between the contract rate andthe actual spot rate at maturity. NDFs tend to be illiquid and therefore expensive to trade.They are unlikely to be suitable as part of a currency overlay portfolio.Where there are substantive exchange controls or highly illiquid forward markets in acountry in a hedged index, most currency overlay managers will argue that the costs ofhedging outweigh the beneﬁts, and leave that exposure unhedged.
4.4.1.4 Conditions for currency market liquidityWe have seen that the FX market developed as something of a ‘black market’ in London. Itdid so because exchanging one currency for another, particularly if neither of the transactionparticipants was a resident of either of the countries of the currencies of the transaction, wasbest carried out outside the legal grasp of the ‘country owner’ of the currency. This tendencyof the FX markets to ﬂee taxation and regulation created, by a process of natural selection, aremarkably elegant, liquid, low-cost and reliable market. Once, and this only ﬁnally occurred inthe mid- to late 1990s, OECD (i.e. developed) countries recognised that they could not controlthe nature and manner of currency exchange, they ﬁnally abandoned all efforts at doing so.Most developing economies have yet to take this step; indeed, many retain legal (exchangecontrol) powers over their own currency which make even offshore trading between offshoreparties difﬁcult or impossible.For countries that have abandoned exchange controls completely, FX market liquidity isdependent on the scale of the country’s economy, the country’s external trade as a percentageof GDP, the extent of overseas ownership of the country’s assets or debt, and the extent ofownership by the country’s residents of overseas assets or debt. With a large, open economy,
78
Currency Overlay
with no exchange controls, an FX market and related hedging products will naturally developwithout any ofﬁcial assistance. Australia is a good example of a country whose increasingopenness, and abandonment of exchange controls, transformed the FX market into the modern,low-cost, ﬂexible market that it is today. For developing countries that have not yet taken thestep of abandoning exchange controls, the extent of liquidity depends on the detailed natureof the exchange controls, and the extent of fulﬁlment of the other conditions above.4.4.1.5 Symptoms of controlled currency economiesGiven the myriad of factors affecting economic performance across sovereign states, it is veryhard to directly isolate, at least by examining only the economic data, the impact of exchangecontrols on an economy. The following are my list of key areas of importance and their impact:(a) Net capital outﬂows. Governments want to ‘keep capital in the country’. A simple enoughidea, but not easily achieved with exchange controls. Here’s why. There is ample evidencethat the existence of, or threat of, exchange controls strongly inhibits inward capital ﬂows,because foreign investors are frightened that their money will be ‘locked up’. Since netcapital outﬂows are the sum of both inward and outward ﬂows, the outward would have tobe reduced for the balance just to remain stable. Capital outﬂows are likely to be inhibited tosome extent by exchange controls, however, they are unlike capital inﬂows in that physicalinvestments (land, buildings, plant) cannot be unbolted and carried off. Portfolio investmentis in theory more vulnerable to capital ﬂight, but if a foreign investor wants to sell equityor bond holdings, he must ﬁnd a buyer. His desire to sell may push the price down, but ifthe buyer is also foreign, then there remains a foreign owner, and if the buyer is local, thencapital ownership is actually repatriated. So ‘capital ﬂight’ is not always what it seems.(b) ‘Protecting’ the exchange rate. As long as the controls are draconian enough, a countrymay choose its own exchange rate and maintain that rate indeﬁnitely. If, as is almostalways the case, that rate is higher than the ‘free market’ rate, then an unofﬁcial or ‘black’market will emerge. At its most extreme, the formal exchange rate might effectively existin name only, with virtually every currency deal being done at ‘special’ rates. Whethermaintaining appearances like this constitutes success is a moot point; it certainly hassevere, often devastating consequences in the real economy.(c) Rationing imports. A universal characteristic of exchange controls is that they restrict theability of residents to acquire foreign currency. This rationing can take many forms, but itwill always give the government power to allocate import/currency licences as they choose.Such choices are by deﬁnition motivated by political considerations, and regrettably butnaturally are often accompanied by corruption in the form of personal payments forlicences. There is no denying that imports are controlled by this process, but the same effectcould be achieved by simple price rationing with free access to foreign exchange at an openmarket price. The advantage of the latter is that the motivation which determines the scaleand mix of imports is entirely economic, not political. Rationing imports is usually, andexplicitly, a device to promote domestic employment, and to (slightly circularly) ‘protectthe exchange rate’. They ﬁt comfortably with the gamut of state intervention in a state-runeconomy; they do not in a market-price-led economy.(d) Monetary isolation. At its extreme, monetary isolation is born of economic xenophobiasuch as held sway in China and the former Soviet Union until the beginning of the 1990s.By rigidly preventing access to roubles or yuan by foreigners, and preventing access to
Foreign Exchange Market – History and Structure
Box 4.7
79
Exchange controls – UK case study
Amongst the most bizarre, and possibly cynical, of the exchange control regimes in recenthistory is that of the UK, home to the currently dominant FX market location, London.Between the Second World War and 1979, i.e. some 40 years, the UK maintained remarkably strict exchange controls on its citizens. UK resident companies had to receivepermission from the Bank of England to conduct foreign exchange transactions on currentaccount (i.e. for imports and exports); UK resident investment institutions (i.e. pensionfunds and life insurance companies) had to buy and sell their non-UK investment throughan arrangement known as the ‘dollar pool’ – which was a ﬁxed pool of external currency,and which traded at an exchange rate premium, reﬂecting the excess demand for dollarinvestments. UK individuals even had to take their passports to the bank to get foreignexchange for spending abroad. Amazingly, the annual limit was £50 a year, raised only inthe 1970s to £200 a year.All this time, London was carving out its position as the pre-eminent location for theforeign exchange market. Non-residents were completely free from exchange controls; theywere reassured by the highly creditworthy UK government, effective but informal bankingregulation (by the Bank of England), transparent and fair legal structure, skilled and educatedworkforce, and convenient time zone. It is hard to think of another democratic jurisdictionwhich so penalised its own population, while so effectively promoting the operation of afree market right in its midst.dollars, yen and Deutschmarks by residents, the authorities effectively cut off theirdomestic economies from the outside world. Inﬂation rates and exchange rates, andin fact all prices, were set by government planners to fulﬁl political objectives. Incentrally planned economies like these, exporting is not an economic activity measured(or measurable) by proﬁt or loss, but by the needs of the planners to acquire hard (i.e.uncontrolled) currencies to spend on imports to keep the economy functioning. The caseagainst monetary isolation is not that it cannot be achieved – it certainly was in thesecountries for decades – but that it does not allow international price signals to ﬁlterthrough to the domestic economy. And the poor historical growth record of the centrallyplanned economies in the 1950–90 period is a testament to the effects of this isolation.4.4.2 TaxationMost markets around the world are taxed. The taxes fall on transactions (stamp duty), onthe proﬁts made by participants (capital gains taxes), and on the income from investments(withholding tax/income tax/ACT/dividend taxes).The global foreign exchange market is, broadly speaking, not taxed. This is not the benevolence of governments, rather it is a reﬂection on their inability to do so. For the free-marketeer,the inability of national governments to reach the global foreign exchange market is a wonderful thing. The very mobility of the markets, and their ability to migrate away from difﬁcultor taxing environments to liberal, welcoming ones is as close to a process of natural selectionboth of the location and of the market structure as one is likely to get.As we have seen above, the locations in which the global foreign exchange market hasﬂourished are those which are welcoming, legally and politically secure, and with a regulatory
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Currency Overlay
‘light touch’, without being casual about the stability of the banking system. Whenever ajurisdiction has imposed transaction or withholding taxes, or excessive regulation or control,the foreign exchange market has promptly moved to alternative locations.4.4.2.1 Tobin taxIn 1978, James Tobin, an American economist (and later a Nobel Laureate), made the suggestion16 that all foreign exchange transactions worldwide should be taxed at a low rate (say0.1% per deal) to ‘throw some sand in the wheels of speculation’. Whatever the merits orotherwise of his arguments, the fatal ﬂaw in his proposal is its sheer unworkability – it wouldbe impossible to get all the possible jurisdictions in the world to agree to such a tax, and itsenforcement would spill over to derivatives and futures, and then to domestic money markets.Even if it were possible to raise the tax, the division of the proceeds (which would be a greatdeal smaller than its proponents imagine) would be a further major obstacle, with competingclaims coming from the owners of the currencies in question, the jurisdiction of the transaction,and all the other jurisdictions without major market places.Interestingly, the Tobin tax has more recently attracted a signiﬁcant lobby following. Thereare a number of campaigning websites, including a number of anti-poverty charities, devotedto putting pressure on politicians to impose the tax.4.4.3 Financial regulationRegulation of the markets under the jurisdiction of each of the major foreign exchange centresis a different matter to taxation. All governments, but particularly business-friendly ones, wishto preserve the integrity of FX market-makers, and to protect their domestic banking systemsagainst any knock-on effects of problems in the foreign exchange market, and vice versa. Thishas meant that in almost all the major centres of the FX market, banking regulations with respectto foreign exchange (capital requirements, risk management processes) have been consistentlytightened over the years. This systemic regulation is regarded by investors as positive up to apoint – excessive regulation, particularly that which seeks to regulate areas that are not seenas a risk, is seen as negative. Most national regulatory environments of the major centres havenot sought to provide major consumer protection. This is largely because private investors areonly indirect users of the market, and then generally on a small scale, and to fund overseastravel and expenditure. Foreign exchange ‘mis-selling’ would in normal circumstances have aminimal effect on individual consumers’ wealth, so governments have wisely decided to leavewell alone.4.4.3.1 Argentina – a case studyThere is, however, a recent case, that of Argentina, in which consumers were a very majorcasualty of a currency debacle.Argentina had established a peg between the Argentinian peso and the US dollar. The peg wasestablished after many years of endemic inﬂation and devaluation, and was intended to providea base of stability and external discipline. The peg was underwritten with a currency board – inwhich central bank peso liabilities were matched one-for-one with US dollar monetary assets16
Tobin, J., ‘A proposal for international monetary reform’, Eastern Economic Journal 4 (1978) 153–159.
Foreign Exchange Market – History and Structure
81
held at the central bank. The government made much of the immutability of this arrangement,and the superior security gained from the currency board underwriting, and the population,who (rather surprisingly) believed the government, went out and borrowed money (mainlyin the form of mortgages) in US dollars. They did so because US dollar interest rates weresigniﬁcantly lower than peso rates.This alone should have rung alarm bells. Interest rates between two currencies that areirrefutably pegged will converge to (close to) zero, otherwise there are risk-free arbitrageproﬁts to be made. If interest rates demanded by savers were higher than US dollar rates, itcould have been regarded as compensation for the exchange rate risk. Alternatively, if pesointerest rates were higher than US rates because Argentina was pursuing its own monetaryagenda, then that should also ring alarm bells. How can two countries with very differentmonetary policies maintain a constant exchange rate? History is littered with examples of theanswer – they can’t in the long-term.Unfortunately, the Argentinian government found itself, in January 2002, unable to defendthe peg any longer, and chose to end the convertibility of the peso, blocking residents’ dollardeposits, ultimately forcing their conversion to pesos. The abandonment of the peg left the peso(by mid-2002) at about 30% of its former value against the dollar, and millions of Argentiniansowing massively more than they had borrowed, with no commensurate increase in the value oftheir assets. The knock-on effect of the devaluation has had a catastrophic effect on consumerspending and business conﬁdence, and also brought many banks to their knees, sending theeconomy into a ‘post-peg devaluation winter’.The Argentinian experience is a warning, if one were needed, of the dangers in attempting tocontrol exchange rates while maintaining a broadly market-based economy. Fortunately, mostof the currencies subject to overlay are not controlled in this way.
5Theory of Currency Hedging ofInternational PortfoliosThis chapter will deal with the theory of currency hedging in international portfolios or, putsimply, why hedging is a good idea, other things being equal. Of course, other things are neverequal, but that issue will be dealt with extensively in the succeeding chapters which look at thepracticalities of currency hedging. As a precursor to this discussion, I will visit the issue of measuring risk and return under the lognormal assumption (see Chapter 3). The chapter will thencover in some detail those arguments, both theoretical and practical, that can be brought to bear.
5.1 LOGNORMAL RANDOM WALK RETURNSI will adopt an assumption that runs through most modern investment theory, namely that thebehaviour of currency exchange rates over time (like that of equity and bond returns) follows alognormal random path. This is to say that if we take the natural log of the change in currencyrates expressed as a percentage over a unit time period (say one day), then these are normallydistributed, with an observable mean and standard deviation. This amounts to saying that therandom linkage from one time period to the next is multiplicative, not additive. This assumptionis the same as that made in the Black–Scholes model; assumptions which I expended someenergy on refuting in an earlier chapter. However, the sharp-eyed reader will note that thelognormal assumption did not bear the brunt of my previous attack – in fact it remained largelyunscathed. Later in the book, I will be looking in more detail at the lognormal random walkassumption, and while its random walk credentials will come under attack, its lognormalityassumptions will not.Percentage changes in currency rates I will call currency returns. However, these are calculated not by reference to spot exchange rates, but by reference to currency surprise (seeChapter 3). In summary, currency returns are the difference between the forward rate at theend of the prior period, and the spot rate at the end of the current period, expressed as a percentage of the spot rate at the end of the prior period. The name ‘currency returns’ is somethingof a misnomer, it derives from the algebra of conventional assets (equities, bonds) and impliesthat such returns are independently investible. As discussed in the currency surprise section,they are not investible without a ‘carrier’ asset.5.1.1 MeasurementThe lognormality assumption for ﬁnancial markets, including currency markets, is very fundamental. We need to reﬂect this fundamental nature of the markets by using accurate measurements for the key variables – return and risk – which match the assumptions we are making.The two matching measures are ‘annual return’, in which returns are measured at an annualfrequency and compounded at any other frequency, and ‘annualised volatility’, in which thestandard deviation of the unbiased normal distribution of the return series is measured.
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Currency Overlay
5.1.2 ReturnsThere is no debate in investment circles about annual return methodology. When we say thatthe return of an investment is 10%, we mean that an asset value of 100 at one year end becomes110 at the next year end.
Box 5.1
Convexity
Convexity is a term that turns up a lot in investing. At its most general, it means that ‘X cango up more than it can go down’, or ‘if I invest $100, I can only lose $100, but I can makean unlimited amount’. Graphically, it can be represented something like this:Value of $100 after 10 yearsat various compound rates of return700600
Value $
500400
Initial investment3002001000−20%
−16%
−12%
−8%
−4%
0%
4%
8%
12%
16%
20%
Annual return
Convexity also turns up a lot in currency analysis. The most obvious feature is that if Isell a foreign currency forward, then I can lose a potentially unlimited amount of money,whereas if I buy a foreign currency forward, I can only lose the cost of the purchase.Interestingly, this statement is only true if I express my gains and losses in my homecurrency. If I express them in the foreign currency, then the converse is true; a forward saleof the foreign currency can be restated as a forward purchase of the home currency, withunlimited gains and limited losses expressed in the foreign currency.A further source of convexity can arise from the way in which currencies are expressed.If I am dollar-based, then expressing the sensitivity of my pounds sterling investments usinga $/£ exchange rate (i.e. the value of the foreign currency (£)) produces linear valuations.However, expressing my yen investments using a yen/$ exchange rate (i.e. the reciprocalof the value of the foreign currency (yen)) produces convex valuation curves.Convexity arises out of the expression of multiplicative (or logarithmic) processes overtime (i.e. investment returns, currency movements) in absolute asset values or currencyamounts. It is important because while investment theory teaches us to think and analysein annualised return terms (i.e. multiplicatively), we actually consume absolute cash inour everyday lives, and our utility functions are more likely to be ‘wealth-based’ than‘return-based’.
Theory of Currency Hedging of International Portfolios
85
We link period returns geometrically, so that they compound. To annualise compound returnsfor periods other than one year, we use the following formula:AR = [(TRt,t+n + 1)( f /n) ] − 1whereAR = annualised return (%)TRt,t+n = total compound return (%) (% of value at end-period t)between the end of period t and the end of period t + n (i.e. n-periods’ return)f = frequency of period in a year (e.g. monthly = 12, quarterly = 4, etc.)Compound returns are simply geometrically linked period returns, as follows:TRt,t+n = [(1 + PRt+1 )(1 + PRt+2 )(1 + PRt+3 )(1 + PRt+4 ) · · · (1 + PRt+n )] − 1where PRt = return for period t expressed in % of value at t − 1.This methodology is fully consistent with lognormal returns, and no errors (not even secondorder) arise as long as these are the consistent deﬁnitions and methodology. However, notethat ‘annual return’ is not the same thing as the rate of return in continuous-time maths. SeeAppendix 2 for details.Box 5.2
Central Limit Theorem
The Central Limit Theorem (CLT) is a statistical law that is widely (if unknowingly) used ininvestment management analysis. In essence it states that if a number of observations of anindependent random variable are taken, then the standard deviation of the resulting samplemean is the standard deviation of the underlying population divided by the square root ofthe number of observations. Furthermore, even if the underlying distribution is non-normal(even strongly so), the distribution of the sample mean tends to normality as the number ofobservations in the sample grows. For most independent variables, a sample size of 10 isgetting pretty normal.The CLT is the basis for all conversion of standard deviations across different frequenciesof data. The logic is as follows (note that returns below are the natural log of (1+percentagemonthly return)) (see Appendix 2).Suppose we measure returns monthly. To calculate annual returns we sum the monthlyreturns. This represents the sum (i.e. 12 × mean) of a sample of 12 observations. Thestandard deviation of the average monthly return in this sample under the CLT will be(Monthly SD)/(Sq. Root 12). But we are interested in the annual sum, not the monthlymean. This is 12 times as large as the monthly mean, so we have to multiply the expressionabove by 12:Annual SD = (Monthly SD)/(Sq. Root 12) × 12 = (Monthly SD) × (Sq. Root 12)Hence the special case of annualising volatilities requires us to multiply by the root of thenumber of observations in a year.While the Central Limit Theorem is very effective at converting any old distribution intonormal, it only works if the variables in the sample are independent. Therefore it remainsa strong theoretical possibility that currency and other markets are not normal at one-yearhorizon if the constituent variables (daily, weekly, monthly returns) are serially correlatedin some way.
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Currency Overlay
A worked example of return methodology is given in Table 5.1, taking one year’s worth ofmonthly return data. The reader might like to satisfy himself that the annual return (which inthis case is the same as the annualised and the total return) is 24.02%.Table 5.1 Example asset returnsPeriod
% Returns
123456789101112
2.4142%4.9287%1.3992%2.5029%−1.4179%6.6249%5.5299%4.4657%2.9206%0.1610%−3.5356%−3.6404%
5.1.3 VolatilityI have slightly laboured the return point to make a much more telling one about volatility. I givea full explanation of the following argument in Appendix 2, but the summary is as follows.The returns above are not normally distributed. They are not because we have assumed that thegenerator of period returns is a lognormal process. Percentages are not logs. We can convertthe series in Table 5.1 into a log series by taking the natural log of each (1 + period return%). This gives us the series in Table 5.2. Just to be clear, these values are the logs of ‘oneplus the percentage returns’ – that is they are the ‘linking’ values for each period of the asset.Because they are logs, they are additive, and because they are logs (and we assume a lognormaldistribution), they are normally distributed. This is the series we need to measure for volatility,not the percentage returns series.Table 5.2 Log asset returnsPeriod
Log returns
123456789101112Sum of logs
0.0238550.0481100.0138950.024721−0.0142810.0641470.0538240.0436880.0287880.001609−0.035997−0.0370830.215276
Theory of Currency Hedging of International Portfolios
87
To conﬁrm the additive nature of these log returns, the reader should take the exponential (or‘anti-log’) of the sum (0.215276), and he will ﬁnd that it equals 1.2402. Take away 1 and youhave the annual percentage return of 24.02% – an illustration that the return methodologies(log and percentage) are (exactly) the same.What about the volatility calculation, though? The conventional volatility (standard deviation) calculation for the return series is (in words) to take the root of the [(sum of the squaresof the percentage returns’ differences from the mean)/n − 1], or in algebra:
SD =
 n (ri − r )2 i=1n−1
whereri = percentage return for period iri = mean percentage return (from sample)This calculation produces a volatility at the horizon of the period’s frequency, so in this case itis the monthly volatility. If we want to annualise it, we use the Central Limit Theorem, whichtells us that if the√ period returns are statistically independent from each other, the volatility atany horizon is [(desired horizon)/(period horizon)] measured in consistent time units (days,months, years). As an example,annualising monthly volatility would mean multiplying the√above formula result by (12/1) = 3.464. See Appendix 2 for fuller details on using theCentral Limit Theorem for annualising.So with this information and a calculator, we can calculate the monthly volatility of theabove series as 3.40%, and the annualised volatility as 11.79% (=3.40% × 3.464). However,this will be taking data that we know to be non-normal (percentage returns). Non-normal datais likely to give us unpredictable and unexpected results, and is not amenable to the series ofstatistical tests that makes normal distributions so helpful – and, surprisingly, the Central LimitTheorem is not going to help us ‘make the distribution normal’ (again see Appendix 2).Box 5.3
Siegel’s paradox
Siegel’s paradox was proposed by Professor Jeremy Siegel in 1972. His basic propositionis that when exchange rates move, the returns when viewed from both bases currencyperspectives sum to greater than 0:
Period 0Period 1Period 2Period 3Period 4
US perspective $/£
UK perspective £/$
Rate
Returns
Rate
Returns
−33.33%0.00%60.00%−6.25%
0.66671.00001.00000.62500.6667
50.00%0.00%−37.50%6.67%
16.67%0.00%22.50%0.42%
19.17%
39.58%
1.5000