For bonds this is likely to create a better hedge than theend-prior-month value methodology, although it does create another level of complexity thatreduces transparency. For equities, a sophistication like this would be unlikely to be sufﬁcientlymaterial (since yields are so low) to warrant the additional complexity.One index provider (JPM) discounts daily valuations of monthly forward contracts.
7.5 WORKED EXAMPLESIn the following pages, I have set out worked examples of the algebra for the three main typesof benchmark discussed in this chapter. Each column has algebra showing how the numberspresented are calculated. Each cell value has a column reference, and a t subscript. The use oft means the current period; the use of t − 1 means the prior period value, and so on. If I omitthe t subscript, then I am referring to the current period values.
7.5.1 Asset plus currency overlay methodologyI have chosen a ﬁctitious two-asset/two-currency international portfolio. We have a EUR basecurrency investor, and the two target currencies are USD and JPY. So, in lay terms, a Eurozoneinvestor has invested in only two assets, one in the US and one in Japan. Thinking of the assetsas stock indices makes this a more intuitive benchmark (although of course they could be anytype of asset).
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We are required to calculate hedged returns for a benchmark for this portfolio. We are giventhe values of the assets at the end of each period, and the relevant exchange rates. Table 7.2shows the portfolio assumptions.Table 7.2 Portfolio information (EUR base)JPY asset
USD asset
End-period JPY End-period End-period End-period USD End-period End-periodasset valuespot rate forward rateasset valuespot rate forward rate(JPYm)JPY/EURJPY/EUR(USDm)USD/EUR USD/EURPeriod endABCDEF0123
1000.001021.00931.20950.30
115.7013115.9451118.3738117.4127
115.368240115.610410118.033330117.066370
15.0015.1014.8414.61
0.90110.93430.98760.9804
0.8999400.9330600.9862800.979070
Table 7.3 shows the calculation for a fully (100%) hedged benchmark. In the ‘total return’row, the calculation uses geometric linking for unhedged return and hedged return. I havenoted that we do not use geometric linking for the ‘contribution from hedging’ since we cannotreinvest in stand-alone currency overlay, which is the implicit assumption of geometric linking.Table 7.3 Asset plus currency overlay benchmark (100% BHR)
Column Formula:Period end0123Total return
UnhedgedEUR asset valuebenchmarkat spot ratesreturn(A/B) + (D/E) (Gt -Gt−1 )/Gt−1GH25.2924.9722.8923.00
(1.27%)(8.31%)0.45%(9.07%)
Benchmarkcontributionfrom hedging[BHR{(At−1 /Ct−1 −At−1 /Bt ) + (Dt−1 /Ft−1 −Dt−1 /Et )}]/Gt−1J
100% hedgedbenchmark returnH+JK
2.59%4.41%(0.58%)6.31%∗
1.32%(3.90%)(0.13%)(2.76%)
∗ 6.31% = (2.76%) − (9.07%). The geometrically linked return of ‘contribution from hedging’ is 6.50%, but this is
not investable.
Table 7.4 shows the relevant additional columns to calculate a 50% hedged benchmark,and Table 7.5 the same working for an unhedged benchmark. I have included the 0% hedgebenchmark working for completeness – but of course the contribution from hedging is zero,and the benchmark return equals the unhedged return.Finally, I have added two tables (Tables 7.6 and 7.7) which are not strictly a benchmarkcalculation. They show how an actual overlay portfolio’s returns relative to benchmark wouldbe calculated under this methodology. The marked-to-market valuations (MMVs) of an actualportfolio at the end of each month are illustrated in Table 7.6, but they are not derivablefrom information presented here – they come from the (assumed) valuation of the individualmanager’s currency positions. I show how overlay performance could be calculated from thisinformation.
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Table 7.4 Asset plus currency overlay benchmark (50% BHR)
Column formula:Period end
Benchmark contributionfrom hedging[BHR{(A t−1 /Ct−1 − At−1 /Bt ) +(Dt−1 /Ft−1 − Dt−1 /Et )}]/Gt−1L
50% hedgedbenchmark returnH+LM
1.30%2.20%(0.29%)3.14%∗
0.03%(6.11%)0.16%(5.93%)
0123Total return
∗ Total resulting from the subtraction or addition of other totals, not geometricallylinked.
Table 7.5 Asset plus currency overlay benchmark (0% BHR)
Column formula:Period end
Benchmark contributionfrom hedging[BHR{(At−1 /Ct−1 − At−1 /Bt ) +(Dt−1 /Ft−1 − Dt−1 /Et )}]/Gt−1N
0% hedgedbenchmark returnH+NP
0.00%0.00%0.00%0.00%
(1.27%)(8.31%)0.45%(9.07%)
0123Total return
Table 7.6 Additional portfolio information – actual overlay portfolioMMV and cash ﬂow (50% BHR)
Period end0123
End-period EURvaluation of actualoverlay (MMV)Q
Cash ﬂow fromoverlay in period(EUR)R
Cum cash ﬂow EURS
0.0000.2820.937(0.024)
0.0000.0000.0000.891
0.0000.0000.0000.891
7.5.2 Embedded currency plus currency overlay methodologyKeeping the same portfolio information, Table 7.8 sets out the workings for an ‘embeddedcurrency plus currency overlay’ benchmark. I have not included a total return row, sincenone of these returns contains the ‘carrier’ asset that allows geometric linking. I have alsoonly included a 50% BHR calculation – the reader can work through other hedge ratios ifrequired.
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Table 7.7 Actual portfolio and performance calculation (50% BHR)
Column formula:Period end0123Total return
% Return fromoverlay[(Qt − Qt−1 ) + (St − St−1 )]/Gt−1T1.12%2.62%(0.31%)
Portfolio return∗∗H+T
Relative performanceof overlay portfolio vs.benchmarkU−M
U
V
(0.16%)(5.69%)0.14%(5.70%)
(0.18%)0.42%(0.02%)0.23%∗∗
∗ Total resulting from the subtraction or addition of other totals, not geometrically linked.∗∗ Column U is not the actual portfolio return. It is the benchmark unhedged asset return (column H) plus the actual
currency overlay return. The actual unhedged asset return would not normally be used in currency overlay benchmarkcalculations.
Table 7.8 Embedded currency plus currency overlay benchmark (50% hedged)
Column formula:Period end0123
Embedded currency(= currency surprise)−{(At−1 /Ct−1 − At−1 /Bt ) +(Dt−1 /Ft−1 − Dt−1 /Et )}/Gt−1W
Benchmarkcontribution fromhedging[BHR{(At−1 /Ct−1 − At−1 /Bt ) +(Dt−1 /Ft−1 − Dt−1 /Et )}]/Gt−1X
50% hedgedreturnW+XY
(2.59%)(4.41%)0.58%
1.30%2.20%(0.29%)
(1.30%)(2.20%)0.29%
In essence, this benchmark clearly shows that a 50% BHR eliminates half of the embeddedcurrency exposure in an international portfolio. This is much clearer under this methodologythan under the ‘asset plus currency overlay’ methodology in Section 7.5.1.7.5.3 Currency overlay onlyThis is the easiest methodology to calculate, and to conceptualise. Only the overlay return isrecorded in the benchmark – there are no elements from the underlying portfolio. Table 7.9shows the calculations for a 50% hedged benchmark, along with actual overlay returns and relative performance. None of the values in this table is investable back into the return series fromwhich they emerge, and so none is geometrically linked. Total or annualised returns will be approximations – as long as this is recognised, the method of totalling or annualising is not critical.
7.6 TRACKING ERRORThis section brings together this chapter and Chapter 6. Tracking error in this context is theannualised standard deviation of the percentage differences (usually monthly) between actual
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Table 7.9 Currency overlay only benchmark (50% BHR)
Column formula:Period end0123
Benchmarkcontribution fromhedgingLZ
% Return fromoverlayTAA
Relativeperformance ofoverlay portfoliovs. benchmarkAA − ZAB
1.30%2.20%(0.29%)
1.12%2.62%(0.31%)
(0.18%)0.42%(0.02%)
returns and benchmark returns. We use logs for all the volatility calculations (see Appendix 2),but percentages for the arithmetic required to calculate the tracking error. Columns V in Table7.7 and AB in Table 7.9 above are examples of the relative return series which generate thetracking error measure.7.6.1 Passive hedgingTracking error arises in passive hedging because the actual overlay portfolio is not identical tothe benchmark portfolio, and is not identically valued. We can list (more or less exhaustively)the possible sources of tracking error:
r Interest rate differential durationr Dealing prices/benchmark prices mismatchr Valuation rates for outstanding forward contractsr Hedge ratio variationsr Rebalancing trades – buffer and timingr Discounting methodology on valuationsWe take each source in turn.7.6.1.1 Interest rate differential durationIf the actual portfolio contains forward contracts with different maturities to the benchmark,then variations both in the absolute size and shape of the term structure of interest rates, andvariations in the term structure, will generate deviations between the actual portfolio and thebenchmark. We can calculate the size of this effect from historical data. Figure 7.5 showsthe tracking error purely from this effect for horizons of 2m, 3m, 6m and 12m (versus a 1mbenchmark), for a three-currency hedge (JPY, EUR, GBP) for a USD-based investor.It is clear from Figure 7.5 that this source of tracking error is relatively important – certainly atcontract maturities over 3m. For example, the 6m basket tracking error is 29 bp. However, unlikemany sources of tracking error, which are the result of continuous small discrepancies, thissource is a direct function of whether or not there are large swings in interest rate differentials,whether the differentials are wide, and whether the term structure of interest differentials issloping rather than ﬂat.
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Currency Overlay0.90%MSCI weighted basket
0.80%
JPYEUR
Tracking error % p.a.
0.70%
GBP
0.60%0.50%0.40%0.30%0.20%0.10%0.00%2m
3m
6m
12m
Original contract maturity (months)
Figure 7.5 Tracking error versus 1m benchmark (USD base, overlapping roll-over, variable contractmaturity, 1983–2002, three currencies)
There were several periods in the 1980s and 1990s when these conditions obtained, but lessso in recent history. If we only use the last ﬁve years’ data (Nov 1997–Nov 2002), we canreproduce Figure 7.5 but with the shorter data, and ﬁnd much lower tracking errors. Figure 7.6illustrates this. In this period, the 6m basket tracking error is 14 bp.The tracking errors in Figures 7.5 and 7.6 are compiled from an overlapping roll-overprocess, rather than a bullet process. An overlapping rolling process is where, for example,in 3-month original maturity hedging, one-third of the exposure matures each month, andthe maturing contract is rolled 3m at the end of each month, rather than 100% of the exposurebeing rolled 3m every three months. For a 6-month original maturity hedge, the contracts aredivided into units of one-sixth of the total hedgeable amounts, and one-sixth is rolled eachmonth, for six months. Likewise for all the other original maturity lengths.This smoothes the effect of the interest rate differential ‘snapshot’ at roll-over, and createsa hedge closer to constant maturity than the very jagged saw-tooth interest rate differentialduration that a ‘bullet’ roll-over creates. It also signiﬁcantly smoothes cash ﬂows.We can chart the two tracking errors (overlapping and bullet), and compare them graphically.Figures 7.7 and 7.8 show the 20-year and 5-year data, respectively. Again, the tracking errorsfor the most recent period (1997–2002) are much smaller.
7.6.1.2 Dealing prices/benchmark prices mismatchEven if there was sufﬁcient information to allow rebalancing and rolling trades to be executed at precisely the time that the feeds for the valuation rates for the benchmark are taken,
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151
0.50%MSCI weighted basket
0.45%
Tracking error % p.a.
JPY
0.40%
EUR
0.35%
GBP
0.30%0.25%0.20%0.15%0.10%0.05%0.00%2m
3m
6m
12m
Original contract maturity (months)
Figure 7.6 Tracking error versus 1m benchmark (USD base, overlapping roll-over, variable contractmaturity, 1997–2002, three currencies)0.80%Overlapping
0.70%
Bullet
Tracking error % p.a.
0.60%0.50%0.40%0.30%0.20%0.10%0.00%2m
3m
6m
12m
Original contract maturity (months)
Figure 7.7 Tracking error versus 1m benchmark – overlapping versus bullet roll-over (USD base,variable contract maturity, 1983–2002, three-currency basket)
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Currency Overlay0.45%Overlapping
0.40%
Bullet
Tracking error % p.a.
0.35%0.30%0.25%0.20%0.15%0.10%0.05%0.00%2m
3m
6m
12m
Original contract maturity (months)
Figure 7.8 Tracking error versus 1m benchmark – overlapping versus bullet roll-over (USD base,variable contract maturity, 1997–2002, three-currency basket)
nevertheless there would still be a difference between the valuation rates, and the actual ratesdealt.There are two elements at work here. If the benchmark is based on mid-prices (as most are),then at each roll, and each rebalance, the actual portfolio will receive slightly less good pricesthan the benchmark. This will generate a very small tracking error, and a small but consistentunderperformance. Secondly, there is a natural (small) variation between the prices used for‘deals’ in the benchmark, and the prices that a particular manager can negotiate for a particularlive deal in the market. Given the efﬁciency and liquidity of the currency market, this is likelyto be a very small (<1 bp) contributor to tracking error.
7.6.1.3 Valuation rates for outstanding forward contractsMost 1-month rolling benchmark calculations use only the end-period spot rate to value (andclose) the maturing hedge. All actual passive portfolios will maintain contracts outstanding atthe end of a reporting period, and they will require forward prices to value them.At the most obvious level, a live 1m maturity passive portfolio will roll contracts at the endof the month. This means that at all times there are contracts outstanding – there is never anytime when there are no contracts outstanding. Outstanding contracts will have to be valued(even if they have only just been executed). Most portfolio valuation reports will value forwardcontracts at ‘dealable’ rates; this will usually be the appropriate (i.e. adverse) bid or offerprices. If mid-prices are used, then this will minimise (but not eliminate) the tracking errorversus benchmark.
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Where forward prices include bid/offer spreads, the side of the market (LHS/RHS) of thespot price off which the forward FX swaps are priced will depend on whether the simultaneousrebalance is an asset rise (sell more foreign currency – so the foreign currency bid price isused) or an asset fall (buy back foreign currency – so the foreign currency offer price is used).This switching of the spot reference price can be responsible for a tracking error of around10 bp p.a. – although since it is only ‘shifting’ of return between one month and the next, itwill cumulate over time to zero.7.6.1.4 Hedge ratio variationsA difference between the benchmark hedge ratio and portfolio hedge ratios for each currencyin the benchmark can occur for a variety of reasons. Among the most common are:
r Not hedging a very small currency allocation for reasons of materialityr Not hedging a currency for reasons of liquidity or pricingr Overhedging a currency because it is acting as a proxy for a currency in the categories abover Changing a hedge mid-month because of, say, an absolute limitation of max 100% hedgeratio
r Holding hedge ratios at levels other than benchmark because of rebalancing buffers or timingThe tracking errors generated by these mismatches are proportional to the size of currencymovements. For a USD-based investor with market cap equity weights, an across-the-board1% hedge ratio mismatch will produce an annualised tracking error of about 10 bp. This makeshedge ratio mismatches quite an important source of tracking error.7.6.1.5 Rebalancing trades – buffer and timingThis tracking error source is dealt with in some detail in Chapter 6. There is a cost/trackingerror trade-off between close adherence to benchmark methodology, and rebalancing costsarising from spot deals. The costs issues were dealt with earlier; the tracking error issues aredealt with here.Rebalancing buffer: Rebalancing buffers are used by passive overlay managers to reducecosts. Table 6.7 to 6.13 set out costs for some key variables under the control of the manager.However, in choosing to use buffers for rebalancing, a manager also accepts a trade-off interms of tracking error.We can calculate the tracking error resulting from the use of buffers. For a US investor,with market cap weights, and a buffer policy expressed as a percentage of the amount of eachcurrency exposure, we can calculate the approximate tracking error as set out in Table 7.10.The maths for this calculation gets quite complicated, so the table has been put together withnumerical analysis and a Monte Carlo simulation.This tracking error table can be represented in conjunction with the costs table. Figure 7.9 isan ‘XY graph’.7 We can see that we have reached a depth of analysis in which we are measuringcosts in fractions of a basis point p.a. The graph indicates that, in spite of the popularity ofbuffers, in fact they contribute very little to cost reduction. In return, they generate relatively7An XY graph is a scattergram which joins the lines between dots. Scattergrams like this can curve in any direction, even‘backwards’ on themselves, round in circles, and the line crossing itself. Figure 7.9 happens to be rather conventional in shape.
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Currency OverlayTable 7.10 Tracking error vsrebalancing buffers% Buffer size
Tracking error (bp)
1%2%3%4%5%6%7%8%9%10%
0.700%
1.54.47.410.914.117.520.123.526.029.0
1%2%
Costs (bp) p.a.
0.60
3%Buffersize
4%0.505%
6%0.40
7%8%9%
0.30
10%
0.200.0
5.0
10.0
15.0
20.0
25.0
30.0
Tracking error (bp) p.a.
Figure 7.9 XY Chart of costs and tracking error versus rebalancing buffer size (monthly rebalancing,nil delay)
wide tracking errors for fractional changes in cost levels. This would mitigate against their usefor any but the most cost-dominated clients.Rebalancing timing: To calculate this we ﬁrst calculate the average size of rebalancing required for a monthly rebalance. The average rebalancing size (3.5%) comes from the ‘1 month’row in Table 6.4. Then we calculate the volatility of currencies, calculated from the observedaverage historical volatilities. This average is the unweighted average of the MSCI weightedcurrency surprise volatility for the four major base currencies 1980–2002, and comes out at10.2%. The formula for the tracking error from delays is then (ignoring logs for simplicity):Tracking error = Annual currency vols × Days delay/Days in month× Average size of rebalance
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155
I have not derived this equation here – the interested reader should either apply standardstatistical rules to derive this, or conﬁrm its accuracy with numerical testing. This equationproduces a tracking error with a root function of the length of the delay, and the resultingtracking error values are shown in Table 7.11.Table 7.11 Tracking error vsrebalancing delayDelay inrebalancing
Tracking error(bp)
1 day2 days3 days4 days5 days6 days7 days8 days9 days10 days
7.410.012.514.516.518.119.820.922.123.2
It is clear from this table that delay in rebalancing can be a quite signiﬁcant source oftracking error. Since there are no dealing-cost consequences from reducing the delay – onlyadministrative effort – currency overlay managers should press clients and their custodianshard to produce timely asset valuations.7.6.1.6 Discounting methodology on valuationsThis is likely to be a very minor source of error, particularly at the current (2003) low levelof interest rates. It is also a completely artiﬁcial one. It will only arise if the benchmarkuses discounted valuations of the MMVs of the overlay programme, and the manager’s orcustodian’s valuation does not, or vice versa. It can be completely eliminated by a commonreporting methodology across service providers.7.6.2 Summary on tracking errorTracking error attribution is like performance attribution for bonds. Each element can beseparately analysed, but their contribution to total tracking error is not a simple sum of parts;rather, it is a calculation based on the standard rule for combining series:Variance (combined) = Variance Series 1 + Variance Series 2 + Variance Series 3+ · · · + Variance Series n + 2Cov(Series 1, Series 2)+ 2Cov(Series 1, Series 3) + · · · + 2Cov(Series n − 1, Series n)andTracking error =

Variance (combined)
So for n tracking error contributors, there are n + [n(n − 1)]/2 [= (n 2 + n)/2] terms in theequation. The ‘difﬁcult’ terms are the covariances – they are not ‘intuitively’ easy to guess or
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hypothesise. They may be negative or positive, depending on a variety of factors. The interestedreader (and he/she would have to be very interested to go into this level of detail!) might liketo conduct their own experiments to determine the covariances of these tracking error series.I would like to conclude this chapter with an approximate combined tracking error table. Forthe purpose of this exercise, I will assume that all the covariance terms are zero – i.e. that noneof the contributors to tracking error are correlated (positively or negatively) with each other.This means that the total tracking error is the root of the sum of the squares of the constituentparts. I have included three types of passive mandate in the table (Table 7.12) to give a ﬂavourof the range of costs possible. I have ‘guessed’ some of the more opaque values, but generallyspeaking heroic efforts to minimise tracking error will be increasingly expensive. Similarly, adetermination to minimise costs will inevitably mean the appearance of more tracking error.Table 7.12 Passive tracking error for three strategiesSource oftracking errorInterest rate differentialValuation ratesHedge ratio variationsRebalancing timingRebalancing bufferMiscellaneousTotal tracking error
Passive strategy1 – benchmarkhugging (bp)
Passive strategy2 – costminimising (bp)
Passive strategy3 – error/costcompromise (bp)
032100311.0
149814.50824.8
105512.50317.8
One point remains: tracking error is a relative concept. There is no compulsion for a clientto adopt one of the published benchmarks as their benchmark. Many will not because thebenchmark asset allocations do not match their chosen strategic allocations. This leaves openthe possibility of designing a benchmark methodology which is lower cost to execute, tomatch in a passive portfolio, than the published benchmarks. This allows the client to reducepassive hedging costs (which is a real gain), while not generating tracking error – which is arelative (and therefore somewhat artiﬁcial) measure. The contents of this and the two precedingchapters should help benchmark designers in this task.
8Overlaying Different Asset ClassesThis chapter develops the theoretical arguments made in Chapter 5 into the real world of actualassets. Currency overlay is most commonly associated with cross-border equity holdings.Accordingly, equities, and equities’ return behaviour, will comprise the bulk of this chapter.There are two reasons why equities have dominated bonds in currency ‘underlay’. The ﬁrst isthat there is much more cross-border equity investment than cross-border bond investment. Theoverall weight of equities in pension funds (the main cross-border investors) is much higherthan the overall weight of equities, and the proportion of that higher weight is also muchhigher. The second reason is a little perverse. It has long been accepted that cross-border bondinvestment does not make economic sense (or at least sense in modern portfolio theory) unlessthe bonds are fully currency hedged. Leaving bonds unhedged increases their volatility fromc. 6% to c. 12%, without increasing their expected return. This simple observation has meantthat almost all international bond mandates are benchmarked against hedged benchmarks.This, in turn, has meant that hedged mandates have been handed out, and that internationalbond managers have been expected to hedge as a matter of routine. This has meant that currencyhedging expertise has been developed in bond management ﬁrms, and the conditions for thedevelopment of stand-alone overlay in this asset class have been absent. Interestingly, however,the style of active currency management adopted by bond managers has been very differentfrom the styles adopted by overlay managers. Bond managers typically take opportunisticdecisions to lift currency hedges in ad hoc decision-making processes. Overlay managers havedeveloped more structured processes.
8.1 EQUITIES8.1.1 Correlation – the historical evidenceThe discussion in Chapter 5 has been largely theoretical. However, it established that thecritical relationship needed for us to be able to determine the effect of hedging on internationalportfolios is the correlation (if any) between currencies and equities. I want to turn now to thehistorical evidence to test this relationship.What relationship to test? We have assumed in all of the above theory that there is nocorrelation between currency returns and equity returns. We need to examine the extent towhich this assumption is satisﬁed, or violated, in practice. It is important to understand whythe correlation between hedged equities and currency surprise is the right one to test.What we are trying to do is to assess whether ‘equity returns’ and ‘currency returns’ arecorrelated. But what are ‘equity returns’? For a domestic market, this is easy: the returns ofthe chosen equity index. For an international market, what do we do about the currency issue?Clearly if we measure the correlation between currencies and unhedged international equities,we are highly likely to get a strong positive correlation. This is because the internationalasset is having the currency return added to it; and then we are measuring the correlationbetween this measure and the foreign currency. If there was not a strong positive correlation, it
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would be indicative of a strong negative correlation between hedged international equities andcurrency.But why have I not mentioned ‘local’ returns in equities? The reason is that local returnsare not investable for anyone other than a local investor. Correlations between currencies andan uninvestable index may be academically interesting, but are not useful in determining assetstrategies. What we need to do is analyse what is actually available to international investors.What is available, and looks very like ‘local’ returns in correlation if not in absolute returns, is‘hedged international’. This is the only way that an investor can access international marketswithout accepting currency risk, and therefore this is the measure we are interested in.We could look at hedging in a slightly different way which might help intuitive understanding. Because, for the reasons above, we expect unhedged equity:currency correlations to bestrongly positive, undertaking hedging (whose returns are the negative of currency surprise)alongside unhedged international is like investing in an asset class negatively correlated withan existing class (good), but with an expected return of zero. ‘Shorting’ equities cannot perform the same function because of the expected negative returns (i.e. negative because of thepositive expected equity risk premium for long equity positions).8.1.2 Correlation evidenceHow do historical currency:hedged asset correlations compare to the assumptions in the aboveexample? Taking the nearly 20-year period Jan 80–Sep 00, I have set out in Figure 8.1 thecorrelations for four major investor base currencies (USD, EUR, JPY, GBP). The measure is thecorrelation coefﬁcient between the hedged returns of the international equity market expressedin the base currency, and the returns of a weighted basket of the relevant currencies.1 So