1.00001.00001.60001.5000Sum ofreturns
20.42%
Total returns
88
Currency Overlay
I have used extreme exchange rate moves to illustrate the effect. Siegel’s paradox is causedby the inappropriate mixing of multiplicative and additive environments. Geometricallylinking the returns (rather than adding them) from each currency’s perspective over timewill eliminate the positive sum of returns, but not the positive sum across base currencies(i.e. the right-hand column). However, taking logs of the returns will allow the eliminationof errors both over time and across base-currency perspectives. The table below illustrates:US perspective $/£RatePeriod 0Period 1Period 2Period 3Period 4
1.50001.00001.00001.60001.5000Sum ofreturns
UK perspective £/$
Logreturns
Rate
Logreturns
Totalreturns
−0.40550.00000.4700−0.0645
0.66671.00001.00000.62500.6667
0.40550.0000−0.47000.0645
0.00000.00000.00000.0000
0.0000
0.0000
0.0000
5.1.4 Normally distributed period returnsWhy don’t we instead take data that we know (from our assumptions) to be normal, namelythe log returns?In this case we apply exactly the same SD algebra, but this time to the log return series. Themonthly SD is 0.0336, and the annualised SD is 0.11655. But what units is 0.11655 in? It is inlogs17 , and we do not easily think in logs. So we could convert it back into a percentage by thereverse of the route taken above, namely exp(0.11655) − 1 = 12.36%. So by this calculationthe volatility of this very simple data set is a full 0.5% higher than the conventionally acceptedmethod. Which is right?Box 5.4
Fat tails
‘Fat tails’ is an expression used in ﬁnancial markets when ‘rare’ events appear to happenmore often than ‘expected’. The ‘expected’ likelihood of large movements in ﬁnancialmarkets is most commonly derived from the normal distribution, and the appearance ofextreme events more frequently than the normal distribution would predict is evidencethat the normal distribution may be an inappropriate distribution to use for price changeprediction. Why should this be so?The normal distribution is computationally very convenient, and is also commonly observed in many ‘natural’ processes (see Box 3.6). However, price movements in currencyand other markets are not ‘natural processes’, they are the result of the behaviour of numbersof market participants.17In fact, of course, ‘log returns’ are ‘continuous time’ returns. Using these routinely is very difﬁcult (since the real world ofinvestment is measured in discrete time, not continuous time), and would mean that an annual 10% return in continuous time (i.e.‘0.1’ expressed in logs) has an asset value of 100 becoming 110.52 is a year’s time, rather than 110. Not intuitive, and indeed not theindustry standard.
Theory of Currency Hedging of International Portfolios
89
Students of market behaviour will explain that very large market movements can occurwhen there is a positive feedback from market prices to participants (falling markets sell;rising markets buy), reinforcing the price movement. While this behaviour is unstable(i.e. it may not always be present), it may appear strongly at times of market stress.Long-Term Capital Management, a large hedge fund relying on sophisticated statisticalarbitrage, was brought to its knees by its reliance on classical assumptions (i.e. normaldistributions) even though two Nobel Laureate economists on its Board of Directors wouldhave ensured that the maths applied to these distributions would have been right. Applying afat-tailed (rather than normal) distribution would have lowered their (statistical) conﬁdencein their processes, and thereby probably saved them from near collapse.The answer is that the log approach is right. I have already demonstrated the mathematicaland statistical basis for preferring this approach, but we can demonstrate it with a Monte Carloanalysis.5.1.5 A simple testAccording to the normal distribution function, an observation drawn from a normal distributionfalls outside 1.96 × SD of the distribution 2.5% of the time. This is the test:1. We generate a long series of random returns from a lognormal random walk generator. Thegenerator will be genuinely random, with a constant SD and mean return of zero (to reﬂectthe fact that we will be dealing with currency surprise). The random number generatorhas a monthly volatility of 0.040345798, which is that monthly log volatility which wouldproduce a 15% annualised volatility if correctly calculated via logs. The series is 12 000monthly returns.2. Measure the annualised volatility the conventional way, and measure the number of yearsthe annual returns fall outside 1.96 × the measured SD. Record this as a percentage of yearstested.3. Repeat step 2), but measuring the annualised volatility using logs as above, and applyingthe 1.96 test on the log returns, not the percentage returns.4. Repeat both processes 100 times.5. Record the average of the percentage occurrence for both methods (2 and 3).The results of the test are given in Table 5.3. They clearly show that the conventional volatilitymeasure (and annualising methodology) is seriously skewed, and that the log volatility measureproduces answers almost exactly in line with theoretical expectation.Table 5.3 Monte Carlo test results – 100 runs, each run 1000 years
Conventional percentagevolatility measurementLog volatilitymeasurement
Percentage below1.96 × SD
Percentage above1.96 × SD
1.08%
4.13%
2.49%
2.48%
90
Currency Overlay
5.1.6 Relevance to currency hedgingWhy is this so important in a currency context? All of the above is relevant to the theoryof currency hedging because we will be examining the impact of currency exposure, and itsremoval, by reference to historical volatility. Unless we can be conﬁdent in our volatilitycalculations, and our annualising processes, which will all be annualised from high-frequency(monthly or daily) data, then we run the risk that the results will be tainted.There is another feature of volatility calculations which are unique to currency calculations.The reader may think that the result above, namely that negative returns are under-represented(or over-predicted) in the conventional percentage volatility calculation, is not a problem, orperhaps that it is a ‘nice’ problem to have. But currency returns do not have a natural ‘wayup’. I can represent the monthly currency surprise for, say, $/£ as either the value of the poundversus the dollar, or the value of the dollar versus the pound. This means that we can reversethe return series at will, depending on the base currency perspective, and that one person’spositive return is another’s negative.This brings out the fundamental inconsistency of the conventional volatility measurement,the same exchange rate series can generate returns whose distribution depends on the ‘way up’the exchange rate is represented. So extreme performance can at the same time be both underand over-represented, depending on this choice. The log approach completely eliminates theseerrors.I will avoid all of these measurement difﬁculties because throughout this book I will measurevolatility using the log method. I will, however, always convert any resulting logs back topercentages, so as to remain consistent with the return conventions in investment management.If at any stage we use logs in algebra, I will represent them using x.xxxx notation. Similarly, Iwill always represent return percentages as x.xx%.One ﬁnal note of caution. Combining period returns of the various investment classes thatmake up a portfolio requires that we add weighted percentage returns together, including thecontribution from hedging of hedged assets or currency overlay. We cannot add log returns sincethis is, in effect, multiplicative combination.18 However much we may wish to the contrary,a rise in equities by 10% will not make the simultaneous return on bonds increase by 1%!However, we will continue to treat combined series (including whole portfolios) as if they arelognormal, and measure them as such.With this mathematical and measurement background in place, we can move on to thesubstantive issues in currency and international investment.
5.2 THE ‘FREE LUNCH’The expression ‘free lunch’ in currency hedging was coined by Perold and Schulman19 in1988. What they were referring to was their observation that when currency hedging is appliedto international portfolios, the volatility of the historical hedged return series goes down,but (excluding any bid/offer spreads in the hedging contracts) the return remains constant.This is similar to the volatility effect seen from diversifying a portfolio by adding additional,uncorrelated, equal-return stocks. Any portfolio whose expected volatility can be reduced18The only relationship that is multiplicative is the effect of spot exchange rate movements on asset returns. Since spot rates arenot investable stand-alone, this is not a particularly useful observation.19Perold, A. F. and Schulman, E. C. ‘The free lunch in currency hedging: implications for investment policy and performancestandards’, Financial Analysts Journal May/June (1988) pp. 45–52.
Theory of Currency Hedging of International Portfolios
91
without reducing the expected return is not lying on the ‘efﬁcient frontier’, or can be said tobe an ‘inefﬁcient portfolio’.Let me start the explanation of this phenomenon by looking at the theory of diversiﬁcation.
5.2.1 Which way up?Please note that I will refer in this chapter to currency returns expressed as the return of theforeign currency with respect to the base currency. So this means that if we have a USD basecurrency, and we are looking at the behaviour of Japanese equity returns, positive currencyreturns mean the yen is going up against the dollar, negative returns mean the yen is goingdown against the dollar. From a mathematical point of view, this will mean representing theyen/$ exchange rate as dollars per yen, or ‘upside down’ from the conventional measurementof yen per dollar. I will refer to correlation between currency returns as positive if equityreturns move with currency returns, and negative if equity returns move opposite to currencyreturns.Note that if world equity markets are highly correlated, then if we see negative correlationfrom one perspective (say German equity and currency from a US perspective), then we arehighly likely to see positive correlation from the mirror image relationship (i.e. US equityand currency from a German perspective). This implies that while it is possible for currencyhedging not to be risk-reducing from one base currency perspective, this is likely to make thecase for hedging stronger from the counterparty currency perspective.
5.2.2 Adding ‘moving parts’In conventional investment decision-making, the addition of more ‘moving parts’ (individualsecurities, sectors, countries, asset classes) is seen as naturally adding diversiﬁcation. It is alsogenerally understood that the effect of diversiﬁcation is diluted if the new moving parts arepositively correlated with the existing moving parts. But as long as the underlying volatilityof the new moving parts is not materially higher than that of the existing ones, then the overallvolatility of the portfolio will at worst remain constant, and at best fall substantially.The mechanics of diversiﬁcation in underlying assets are straightforward. If an investorholds one stock (or index) and wishes to diversify into another, he sells (say) half his holdingin the original stock and buys the new stock with the proceeds. Each stock’s contribution tothe overall volatility of the portfolio is then only half that of each stock on its own. Thereis some reasonably straightforward maths associated with diversiﬁcation. If a second stock(or index) is introduced into a one-stock portfolio, and the volatility of the two stocks is thesame (at, say, 16% p.a.), and the stocks are held at 50% weight each in the portfolio, then thevolatility of the resulting portfolio will depend on the extent of correlation of the two stocksas shown in Figure 5.1. The maths of this effect can be computed from a statistical theoremthat determines the volatility (standard deviation) of the sum of distributions A (returns fromStock 1 as a percentage of the whole portfolio) and B (returns from Stock 2 as a percentage ofthe whole portfolio) as follows:SD(A + B) =

Var(A) + Var(B) + 2Cov(AB)
(5.1)
92
Currency Overlay20%
Adding stocksreduces risk
15%
Volatility
Vol = 16.0% / Root 2 (no. stocks)= 11.31%
10%
Portfolio vol.Stock 1 vol.Stock 2 vol.
5%
0%−1.0
−0.5
0.0
0.5
1.0
Correlation coefficient
Figure 5.1 Portfolio volatility versus stock correlation coefﬁcient – equally weighted two-stock portfolio, both stocks 16% volatility
whereVar(A) =
n

 2AA2 −n(n − 1)
SD (A) or Volatility (A) =
Cov(AB) =

Var(A)
 n A − A Bi − B i=1 in
(5.2)(5.3)
(5.4)
A and B are the series of returns from each stock as a percentage of the whole portfolioand n = number of observations (i.e. months of returns).Covariance can only be calculated on identical sample sizes, so in the context of asset returns,they are computed for the length of the shorter return series if return series are of differentlengths. To derive the diversifying value√ of adding an uncorrelated stock (i.e. portfolio volatilitybeing reduced from 16% to 16%/ 2 = 11.3%), we can substitute the numerical values intoequation (5.1) above. Note that the volatility of both A and B as a percentage of the wholeportfolio is 16%/2 = 8%, because each stock is 50% of the portfolio. The lack of correlationassumption means that Cov(AB) = 0. Portfolio volatility = SD(A+B) = 8%2 + 8%2 + (2 × 0%)= 2(8%)2√= ( 2 × 8%)2√= 2 × 8%√= 16%/ 2
Theory of Currency Hedging of International Portfolios
93
For n stocks and for the generalised non-zero correlation case:Portfolio volatility = SD(A + B + C + · · · + n) nnn[Cov(i j)], i = j=  [Vol(i)]2 +i=A
i=A j=A
where Vol(i) = SD(asset i) × weight of asset i in the portfolio.Returning to the two-stock model example, if the two stocks in question have the sameexpected return, then the argument for adding stocks to create diversiﬁcation is indisputable.Even if all the stocks (indices) are highly correlated with each other, the volatility of theportfolio can still be reduced. We will return to this graph later when we consider internationaldiversiﬁcation, and equity index correlation.5.2.3 Currency exposure is differentSuppose an investor holds one international stock (or index) as his entire portfolio, and nocurrency exposure. He will hold the stock hedged back into his base currency,20 and currencyﬂuctuations will have no effect on the performance of his portfolio. Suppose he is then encouraged by the logic above to diversify by holding currency exposure. He removes the hedge onhis stock, and he has two ‘moving parts’, not one. However, he does not have to sell any ofthis original stock to do this, and so all of the currency volatility he accepts is in addition tothe existing stock volatility.In mathematical terms, this alters one key component: the SD of each asset in this exampleis not weighted by its proportion in the portfolio because both of the weights are 100%. Wecan produce a graph similar to that of Figure 5.1, but this time continuing to hold one (international) stock and adding currency exposure. Again I have assumed that both the internationalstock volatility and the currency volatility are 16%. Figure 5.2 illustrates the impact of thestock:currency correlation coefﬁcient on portfolio volatility.The maths of this effect closely follows the maths above. Let’s repeat the worked exampleabove, but with the new asset and currency mix. From the basic two-stock formula:SD(A + B) = Var(A) + Var(B) + 2Cov(AB)Substituting values:Portfolio volatility = SD(A+B)

16%2 + 16%2 + (2 × 0%)= 2(16%2 )√= ( 2 × 16%)2√= 2 × 16%
=
It is clear from this analysis that unless there is strong negative correlation between the stockprice and the foreign currency, the volatility of the portfolio will go up when currency exposureis added. The break-even for the negative correlation required for currency to reduce volatilityis −0.5 in this example. This value is not ﬁxed, it depends on the relative volatilities of the20I will use hedge throughout this chapter to mean rolling one-month forward contracts to sell the foreign currency in which theforeign asset is denominated. The amount of the hedge is adjusted monthly to accommodate changes to the value of the assets. This isa passive hedge, no attempt is made to analyse the success or otherwise of active currency managers.
94
Currency Overlay35%30%
Volatility
25%
Vol = 16.0% x Root 2 =22.63%Volatility of portfolio in this case is onlyreduced if negative correlation is morenegative than −0.5Addingcurrenciesincreases risk
20%15%
Portfolio vol.10%
Stock 1 vol.Currency vol.
5%0%−1.0
−0.5
0.0
0.5
1.0
Correlation coefficient
Figure 5.2 Portfolio volatility versus stock : currency correlation coefﬁcient – one stock/one currencyportfolio, both 16% volatility
stock and currency concerned. We will return to this later. If there is either no correlation,or alternatively positive correlation, then the volatility of the portfolio must go up – this is amathematical certainty.It is worth remembering that the sequence of events in international investment (particularlyequity investment) is usually the reverse of this. The ﬁrst decision is to invest abroad, thisbrings both international equity and currency exposure (i.e. 22.63% volatility in Figure 5.2).The second (and usually secondary) decision is to hedge (remove) the currency exposure,reducing the equity portfolio volatility in this example to 16%.5.2.4 Correlation of asset classesOne of the most apparently compelling arguments for retaining currency exposure in international investments is that the unhedged foreign indices (i.e. with currency exposure) almostalways have a lower correlation with domestic equities than hedged foreign indices (i.e. without currency exposure). This observation implies that currencies bring diversiﬁcation. I wantto take the elements of this argument to pieces very carefully, and I will demonstrate that itimplies no such thing.I start with the theoretical case. Let’s make the following assumptions. A fund is investedonly in domestic equities (Index 1) with volatility 16%. On offer are unhedged internationalequities (Index 2), also with volatility 16%. Currency surprise has a volatility of 11%. Theexpected returns of both equity indices are the same, and the expected return of currency is zero.There is no correlation between currency and hedged international equities, and 0.5 positivecorrelation between domestic and unhedged international equities. What do the correlationcurves look like, and what is the optimum investment strategy?
Theory of Currency Hedging of International Portfolios
95
Firstly, we can calculate the volatility of hedged equities. Currency surprise methodologytells us that unhedged returns = hedged returns + currency surprise. Statistical theory tells usthe variance of the sum of two distributions is as follows:Var(A + B) = Var(A) + Var(B) + 2Cov(AB)
(5.5)
so in this context:Variance (unhedged) = Variance (hedged) + Variance (currency)+ 2 Covariance (hedged: currency)Covariance = 0 because correlation currency : hedged asset= 0 (by assumption)Therefore:Volatility (hedged) =

Variance (unhedged) − Variance (currency)
since log(1 + 16%) = 0.1484 and log (1 + 11%) = 0.1044. Then:log[1 + Volatility (hedged)]21 = 0.14842 − 0.10442 = 0.10553And so:Volatility (hedged) = exp(0.10553) − 1 = 11.13%So the volatility of the hedged foreign index is lower than the unhedged at 11.13%, comparedto 16% for the unhedged equity. We had already found this in the example above, although theuse of logs is more accurate.The correlation coefﬁcient, R, is a measure of the correlation between the two series. Itsvalue can only lie between +1 (perfect correlation) and −1 (perfect negative correlation). Pairsof series with a correlation coefﬁcient of zero are said to be ‘statistically independent’. R, thecorrelation coefﬁcient between Index 1 and Index 2, is deﬁned as:R = Cov (Index 1: Index 2)/[Vol(Index 1 ) × Vol(Index 2)]For domestic : unhedged equities we know (from our assumptions) that the correlation is 0.5.From this information we can calculate the correlation of domestic : hedged equities.Hedged equities = unhedged equities − currencyThe correlation coefﬁcient R for domestic : hedged equities is:R = Cov(domestic : hedged) / [Vol(domestic) × Vol(hedged)]A little maths will show that as long as Cov(hedged : currency) = 0 (which it is because wehave assumed R = 0), then:Cov(domestic : hedged) = Cov(domestic : unhedged)21I am adding log returns in this equation. Adding log returns means multiplying percentage returns. As discussed elsewhere in thisbook, this is a complex question when it comes to currency. Unhedged returns are a multiplicative relationship between local returnsand spot returns, but hedging eliminates currency exposure at the additive level, leaving the multiplicative residual term unhedged. Theshorter the rebalancing period, and/or the lower the two constituent volatilities, the smaller this residual term. The effect is material inthis context of correlation calculation.
96
Currency Overlay
We can now calculate Cov(domestic : unhedged):Cov(domestic : unhedged) = 0.5 × Vol(domestic) × Vol(unhedged)= 0.5 × 0.1484 × 0.1484= 0.011014Therefore:R = Cov(domestic : hedged) / [Vol(domestic) × Vol(hedged)]= Cov(domestic : unhedged) / [Vol(domestic) × Vol(hedged)]= 0.011014/(0.1484 × 0.10553)= 0.70(We do not need any conversion back to percentages because this is a ratio.) So we candemonstrate that under these assumptions the apparent correlation between domestic equitiesand international equities rises from 0.50 to 0.70 when international equities are hedged.However, this is only because the volatility of hedged international equities is lower, and forno other reason. Just to recap this point: correlation coefﬁcients are calculated as the covarianceof the two series divided by the product of their respective volatilities. The covariance ofdomestic equities and international equities is the same (on the above assumptions) whether ornot they are hedged or unhedged. But hedged international equities have lower volatility thanunhedged equities, therefore the denominator of the correlation calculation will be lower andthe correlation coefﬁcient will be higher. So it is only the (desirable) reduction in volatilitythrough hedging international equities that increases the correlation coefﬁcient. This, then, isa spurious basis on which to claim that currencies bring diversiﬁcation.Figure 5.3 shows the trade-off from hedging between portfolio volatility and correlation(between domestic and international) as the hedge ratio rises (i.e. as currency is progressively14.00%
Full currency exposure(0% hedge ratio)
Portfolio volatility
13.80%13.60%13.40%13.20%
No currency exposure(100% hedge ratio)
13.00%12.80%12.60%0.50
0.52
0.54
0.56
0.58
0.60
0.62
0.64
0.66
0.68
0.70
Domestic:international correlation coefficient
Figure 5.3 A spurious effect-scattergram of portfolio volatility and domestic : international correlationversus currency exposure
Theory of Currency Hedging of International Portfolios
97
removed from the portfolio). It shows a portfolio consisting of 50% domestic and 50% international, and it uses the assumptions above.This graph emphasises the apparently counterintuitive relationship between domestic : international correlation and portfolio volatility. In this example all of the increase in correlationis accounted for by the reduction in volatility of the international assets as they are hedged. Sorising domestic:international correlation coefﬁcients are a ‘good thing’ in this context!
5.3 HEDGING AND THE EFFICIENT FRONTIERThe analysis and graphs above are not commonly found in the investment literature, theyare peculiar to currency. However, it is possible to look at the impact of hedging in a moreconventional framework, that of the mean-variance optimiser. There is, however, a technicalissue which has made this analysis difﬁcult and slightly inﬂexible – most proprietary optimisersare designed to operate with assets which ‘use up’ capital and have a positive expected return.Passive currency overlay does neither – it requires no capital, and it has an expected returnof zero. Active overlay may have a positive expected return, but it also uses up no capital.However, modern optimising software, particularly if it uses iterative rather than Markowitzmaths, is capable of incorporating currency overlay as well as other ‘non-funded’ activity, likefutures-based equitisation and tactical asset allocation.Box 5.5
Mean-variance optimisation
This technique was ﬁrst developed by Nobel Laureate Harry Markowitz, and developed bymany other economists and ﬁnancial engineers subsequently. His thesis was that investorswill by nature seek higher rather than lower returns, and take less rather than more risk,which he chose to measure by the standard deviation (or volatility) of returns. He showed themathematics by which investors could maximise the expected return of a portfolio subjectto a given risk, or minimise the expected risk of a portfolio subject to a given return. Hedemonstrated that the correlation of returns between asset classes was a key input into thesolution.Portfolios created by this method, and for which no increase in return was possiblewithout an increase in risk (and no reduction in risk was possible without a reduction inreturn), were known as ‘efﬁcient portfolios’ (a subset of Pareto optimality).Modern software can compute optimums for large numbers of asset classes very easily,which was not the case when Markowitz wrote his paper. See also Box 5.6.
5.3.1 Constructing an optimiser including currencyThere are three key variables that need to be established to conduct a mean-variance optimisation. For each asset class that is going to be included, you need:
r Expected returnr Expected volatilityr Correlations with the other asset classesThe choice of these values is the subject of intense academic and practitioner debate, and spacedoes not allow us to join that debate in any detail here. However, to give a ﬂavour of some of
98
Currency Overlay
the issues, it is crucial to the assumptions whether or not the whole exercise is conducted in‘real’ (i.e. index-linked) terms, or in nominal terms. If it is the latter, then a critical assumptionof expected future inﬂation needs to be made. In addition, is the analysis going to be conductedas returns and volatility relative to liabilities, or in current terms? If the former, then thediscounting methodology of the liabilities becomes the driver of the risk-neutral portfolio.Let me give an example. If a pension fund has made a fully inﬂation-proofed promise to itsemployees, and it discounts its liabilities at the risk-free index-linked bond rate, then investingin matching-maturity sovereign bonds is essentially risk-free – i.e. relative volatility is zero.However, if the fund discounts its liabilities using the arcane equity-linked methods embeddedin the UK’s Minimum Funding Rate (MFR) calculation, then index-linked bonds will be farfrom risk-free. So in this case the chosen liability discount rate is the driver of the whole assetallocation process.For the example below, I have chosen to deﬁne expected return and volatility as risk relativeto the discount rate embodied in the new UK accounting standard, FRS17. This requires UKcompanies to discount their liabilities (and publish the resulting present value in their reportand accounts – see FRS17 section (p. 21) for more details). In Table 5.4 I set out these estimatesfor a UK investor.Table 5.4 Assumed asset characteristics
Domestic equitiesInternational equities(unhedged)Passive currency overlayBondsPrivate equityActive overlay
Expected return overliabilities
Expected volatility
2.00%
16%
2.00%0.00%0.00%5.00%0.75%
16%10%2%30%6%
I have put in passive and active overlay, rather than the more conventional ‘hedged international equities’, so that the optimiser is not necessarily constrained to 0–100% hedge ratios.I have assumed 0.75% value added p.a. for active overlay, and zero for passive. We can varythese values later if required (particularly to test sensitivity to transactions costs). The volatilities for the overlays have been taken from 20 years’ history of ‘World ex-UK’ portfolios. Theactive overlay is Record Currency Management’s proprietary process – my guess is that otherproprietary processes would produce similar numbers. The lower volatility for active overlayis explained by a lower average hedge ratio (around 50% vs. 100% for passive hedging).To complete the information needed to run an optimiser, we need a correlation matrix. InTable 5.5 I have used some educated guesses based on 20-year historical averages betweenthe classes. Private equity does not have good marked-to-market information, so these guessesare more guessed than educated! The reader may have good reason to question some of theassumptions; this is to be expected. The principle at stake here is not the assumptions, but themethodology by which assumptions are turned into an asset allocation.5.3.2 Optimiser methodologyMarkowitz’s genius was in devising not only the principles of portfolio optimisation, but alsothe mathematical solution. However, in this era of high-speed data processing, we do not
Theory of Currency Hedging of International Portfolios
99
Table 5.5 Correlation matrix
Domestic equitiesInternational equitiesPassive currency overlayBonds