Investing internationally: currency issues for superannuation funds

Susan Thorp

2003/05

SCHOOL OF ECONOMICS

DISCUSSION PAPER

ISSN 1323-8949 ISBN 0 7334 2026 5

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Investing internationally: currency issues for superannuation funds

Susan Thorp

School of Economics

University of New South Wales

Sydney Australia 2052

Email: s.thorp@unsw.edu.au

March 2003

Abstract:

Australian superannuation funds have increased portfolio allocations to foreign assets, exposing members to currency volatility. Fund managers commonly exclude currency-hedged foreign equities and unhedged foreign bonds from portfolios. Mean-variance optimising allocations are used here to investigate this restriction. For robustness, ex post allocations are retested under Bayesian priors and with forecasts of long-run equity premiums under UIP. Optimal portfolios draw on both excluded asset classes, with optimal hedging ratios around 75 per cent. The hedging restriction and home bias together cost members about 100 basis points at 10 per cent volatility, but most of the cost is due to home bias. By tacitly inducing home bias, these common exclusions from the choice set are costly to members.

Keywords:

SUPERANNUATION; CURRENCY HEDGING; INTERNATIONAL DIVERSIFICATION

Acknowledgements

Many thanks to Geoffrey Kingston, Hazel Bateman, Carol White, Lance Fisher, John Piggott, and members of the Seminar at the Reserve Bank of Australia for their helpful comments. This research is supported in part by the Australian Research Council.

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1. Introduction

Do individuals hold optimal portfolios? Do they do a good job of hedging risks?

These questions, raised by Lewis (1999) in her survey of the debate over equity home

bias, remain to be answered. In the context of Australian Superannuation Guarantee

regulations they become even more significant for their public policy implications: Do

Australian superannuants (or the trustees who act on their behalf) hold optimally

hedged investment portfolios?

It is curious and puzzling that many Australian fund managers have restricted the

opportunity set of their portfolio allocations by excluding two foreign asset classes:

currency-hedged foreign equities and unhedged foreign bonds. This paper contends

that such restrictions are not optimal on mean-variance criteria under reasonable

assumptions, and when combined with restrictions on total offshore holdings,

dramatically reduce returns to members at moderate volatility. Most of these

efficiency gains arise from international diversification rather than widening the

currency-hedging choice set, but four potential foreign asset classes are still better

than two.

In what follows implications of the restriction for a fundamental mean-variance,

single-horizon portfolio allocation model are set out, adopting as closely as possible

the perspective of a member of a defined contribution, balanced growth

superannuation fund, and confining the experiments to tools and data generally

accessible to trustees and managers. The question of portfolio allocation studied here

covers the usual components of the balanced-fund choice set: domestic and foreign (in

3

this case, US) equity indexes, domestic and foreign bond indexes and domestic cash.

This choice set is wider than many studies of currency hedging which focus on either

equities or bonds (see Izan , Jalleh and Ong, 1991, for an Australian example). The

strategic approach, by covering both equities and bonds, is more in the style of Jorion

(1989). No short positions are allowed (apart from those implied by hedging),

consistent with a borrowing-constrained defined contribution fund. All analysis uses

real returns and covariances calculated from end-month data (March 1983 to March

2001). The sensitivity of results to estimation risk is limited by repeating the ex post

experiments over returns adjusted by Bayesian shrinkage in the style of Jorion (1985,

1986), and over returns fixed by a long-run Uncovered Interest Parity (UIP)

assumption.

The analysis demonstrates that optimal portfolio allocations employ both hedged

foreign equities and unhedged foreign bonds for moderate risk tolerances. A 60:40

portfolio for example, has an optimal currency hedge ratio close to 75 per cent.

Further tests indicate that measurable reductions in portfolio volatility arising from

currency hedging alone only emerge when portfolios are mainly invested offshore,

and even then they are small. This finding is consistent with Jorion (1989) and Beggs,

Brooks and Lee (1989). However, exchange rate volatility does contribute to home

bias1 when the portfolio choice set is restricted, and home bias is costly. The

combined effect of the hedging and total offshore restrictions diminishes portfolio

returns by about one per cent per annum at ten per cent volatility.

1 A point made by other commentators, see Battellino (2002, p20):

“There will come a point where superannuation trustees will resist further increases in their exposure to exchange rate fluctuations, and direct their funds managers either to stop increasing the allocation to offshore equities or, more likely, to invest on a hedged basis.”

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This paper looks at currency hedging issues from different perspectives, theoretical

and practical, normative and positive. Empirical background is set out in Section 2.

Section 3 deals with some relevant theory. Section 4 presents the data and summary

statistics, Section 5, the model and optimisation results, and Section 6 concludes.

2. Superannuation industry practice

Since the deregulation of the 1980s, Australian investors have enjoyed growing

participation in global capital markets. Capital flows in both directions have

increased, with an associated net increase in the offshore holdings of superannuation

funds (Battellino, 2002). Of the approximately $520 billion of superannuation assets

reported by the Australian Prudential Regulation Authority (APRA) in June 2002,

nineteen per cent were invested offshore. About one sixth of this was held in bonds

and the remainder in equities.2

2.1 Currency hedging

For the most part foreign equity assets are not hedged against currency risk.

Practitioners’ anecdotal evidence for this fact (Van Eyk 2001, Muysken and Burt

2000) gained formal support from a recent ABS/RBA survey (Reserve Bank 2002). In

general across sectors, Australian investors tended not to hedge their foreign currency

equity assets, regarding the exchange rate exposure as ‘part of the rationale for the

investment decision’3. Some superannuation funds were found to hedge currency risk,

(about 20 per cent of funds’ total foreign currency equity assets). The remaining 80

2 See APRA Superannuation Trends, June 2002 and Battellino (2002, p20).

3 Reserve Bank (2002), p. 57.

5

per cent unhedged, however, represents a substantial exposure to exchange rate

changes, even if such risks are part of the investment rationale, since offshore

investments will at some time be used to fund retirement incomes denominated in

Australian dollars4.

Industry discussion proposes a number of reasons for hedging the currency exposure

of fixed interest assets and not of equities. One argument decomposes portfolios into

‘low risk’ (fixed interest) and ‘growth’ (international equities) products and argues

that controlling volatility in growth products is less important than the potential regret

risk when currency hedging lowers returns. Hedged investment in international bond

markets, by contrast, can be seen as a natural way to extend the limited duration and

liquidity of the domestic fixed interest market at comparable levels of return and risk.

‘Peer risk’ is also cited as a major issue.5 Since international equity fund performance

is measured against an unhedged benchmark, tracking error for a currency-hedging

manager must be larger. Fear of performing badly during depreciations may lead

many fund managers to reject hedging a priori (Van Eyk 2001). Other commentators

suggest a lack of confidence or failure of expertise among fund managers (Dunstan

2001).

2.2 Historical currency gains

Regardless of any reason for these restrictions, one needs to ask whether exchange

rate volatility is a large enough component of portfolios to matter. As a preliminary

4 This practice could be at odds with OECD guidelines on ‘currency matching’. See OECD (2001, p.430).

5 See Muysken & Burt (2000) and Van Eyk (2001).

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guide to the exchange rate ‘contribution’ to nominal superannuation portfolios,

consider Figure 1. Since the mid 1980s superannuation funds have increased their

holdings of foreign assets while concurrently benefiting from a more or less steady

depreciation of the Australian Dollar (AUD) against most major currencies.

Consequently, part of the expansion in holdings of overseas assets can be explained in

terms of favourable currency movements, in conjunction with a reluctance to

rebalance portfolios after positive returns have increased the share of an asset class.

Figure 1 shows an asset class breakdown of superannuation fund holdings allowing

for changes in the value of the Australian dollar. A measure of the possible

contribution of a depreciating currency to growth in overseas assets is made by

comparing the proportion of assets held overseas while adjusting for changes in the

exchange rate. Here it is assumed that all superannuation funds’ overseas portfolios

were held in fixed proportions through the period6 and that they were entirely exposed

to currency movements. The lighter shaded area in ‘overseas assets’ gives an

indication of how much currency movements may have contributed to growth in this

category. In fact, variations in the Australian dollar in the past three years are large

enough to constitute a small asset class on their own. They represent a potentially

important source of portfolio volatility.

6 55% in US dollars, 25% in Euros or Deutschmarks, 10% in UK pounds, and 10% in Japanese yen. Using these weights, an asset-weighted index analogous to the Reserve Bank’s trade-weighted exchange rate index was constructed. This index was then used to ‘deflate’ the Australian dollar amount of total overseas superannuation assets.

7

Figure 1: Total superannuation assets: exchange-rate adjusted

0

100,000

200,000

300,000

400,000

500,000

600,000

Jun-95 Jun-96 Jun-97 Jun-98 Jun-99 Jun-00 Jun-01 Jun-02

Cash and Deposits

Loans and PlacementsInterest Bearing Securities

Equities in Units and Trusts

Land and Buildings

Other Assets

Assets Overseas

Exchange-rate adjustment

Source: APRA Superannuation Trends June 2002, RBA Bulletin.

3. Theory

Given that currency exposure is considerable, it remains to be seen if there are

analytic reasons for excluding currency hedging (or including it in the case of bonds).

Under what circumstances is it better for investors to passively accept the impact of

movements in the exchange rate as part of their portfolios? When, if ever, can hedging

currency risk be a useful improvement for a risk-averse investor? These questions can

be addressed within a unified framework by means of the Sharpe ratio.7

7 On the nature and limitations of this statistic, see for example, Goetzmann et. al. (2002).

8

For any given foreign asset i, the unhedged real rate of return to the domestic

investor over one period is given by:

ππ −++= ** srr ii (1)

and the hedged return by:

iishhrr iih −+=−= * , (2)

where:

ri = real rate of return to asset i for domestic investor;

s = rate of change in the price of foreign currency;

π = inflation rate for domestic consumer; and

i = one period risk-free borrowing and lending rate

(* indicates foreign variable).

Borrowing in foreign currency and lending in domestic currency at the beginning of

the period (or the forward contract equivalent of such a transaction) and reversing the

transaction at the end of the period effects a one-period hedge.

3.1 The case for hedging policy irrelevance

If purchasing power parity (PPP) holds in every period, and consumption bundles are

the same in both countries, then s = π – π* and unhedged real rates of return are

equalised across countries. If the international Fisher relation (UIP) also holds and

one-period interest differentials also describe the inflation differential, hedging has no

impact on the rate of return. The Sharpe ratios for hedged and unhedged investments

in any given offshore asset class will be the same.

9

3.2 The case for full hedging

As an empirical proposition, however, PPP has little support. Deviations from PPP

have been found to be large and persistent. The half-life of deviations from PPP is

estimated at three to five years, and in the short run it is hard to distinguish either the

nominal or the real exchange rate from a random walk.8 This additional volatility in

the exchange rate drives a wedge between real returns to investors in different

countries and contributes to the risk associated with holding foreign assets.

Hedging can reduce volatility where there are deviations from PPP. By equation (2)

the return to the hedged portfolio reduces to:

*** iirr iih −++=+ ππ (3)

The nominal hedged return is equal to the local market nominal return plus the

interest rate differential. Note that the precision of this equation is conditional on

knowing the local currency value of the underlying asset at the end of the period – not

a trivial issue for hedging equity portfolios. But given that this value is known, and

recognising that interest differentials typically show less variability than exchange

rates, hedging is likely to reduce volatility9.

8 See Rogoff (1996) for a survey of PPP literature. Cashin and McDermott (2001) estimate a half-life of 50 months for Australia with a 90% confidence interval of 21months to infinite. Cassie (2001) argues for mean reversion in the AUD post-float.

9 To the extent that (non-zero) returns to the forward contract are correlated with the underlying asset, the variance minimising hedge ratio may not equal one. The minimum-variance hedge ratio is given by the slope coefficient in the regression of the domestic currency foreign asset return on the return to the forward contract. See for example Luenberger (1998) and Solnik (2000). The allocation experiments reported below effectively solve this problem on a portfolio level by allocating shares between hedged and unhedged asset classes to minimise overall variance.

10

Indeed some studies assert that currency hedging provides reduced volatility at no

loss of return: a result which Perold and Schulman (1990) label a ‘free lunch’. This

relies on a number of assumptions about exchange rate behaviour, some of which can

be illustrated in the following simplified framework.

Consider two risky assets, the real exchange rate and a foreign security, each

describing geometric Brownian motions. The process for the real exchange rate is:

SdzSdtdS ss σµ += (4)

where µs is the drift coefficient and σs is the volatility coefficient.

The return to the instant forward contract is assumed non-stochastic:

FdtdF fµ= (5)

For the purposes of illustration assume also that inflation rates are both fixed at zero,

so that π = π* = 0, and that the risky security returns are independent, so that σrs = 0.

IF UIP drives the drift term in the exchange rate process (i.e., µs = i – i*), then the

instantaneous Sharpe ratio (SR) for an unhedged investment in the risky foreign

security is given by:

sr

iiirESRσσ +

−−+=

)()( **

. (6)

By comparison, and assuming Covered Interest Parity, the instantaneous Sharpe ratio

for the hedged investment is given by

r

iiirESRσ

−−+=

)()( **

, (7)

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which has the same numerator as (6), but a smaller denominator. Using ‘ball park’

estimates for these quantities, with r* of 0.08 and σr of 0.15, an interest differential at

0.03, domestic interest rate of 0.05 and σs at 0.1, full currency hedging raises the

Sharpe ratio from 0.24 to 0.4.10

3.2 The case for partial hedging

The ‘free lunch’ argument needs to be qualified in a number of ways. According to

Black(1990) there are net gains to holding exchange rate risk that are not evident in

Perold and Schulman’s framework. He argues that investors will always want to hold

some exposure to the exchange rate in order to exploit the non-linearity of returns

available through Jensen’s inequality11.

Again employing the Sharpe ratio, Black’s argument can be illustrated by using a

process for the real exchange rate that is symmetrical between different ‘home’

currencies. Specifically, let:

dzdtiiSd sσ+−= )(ln * (8)

10 Assuming that risky assets have non-zero covariance changes the scale, but not the direction of this change unless 2cov(s, r) is negative and larger than σs

2 in absolute value. The cov(s,r) also appears in the numerator of both ratios (by Ito’s lemma).

11 The historical puzzle of Siegel’s Paradox was first raised in Siegel (1972), where he deduces that the expected value of the reciprocal of an exchange rate is greater than the reciprocal of the expected value of the exchange rate, so that E(1/s)>1/E(s). If this is so, the forward rate cannot be an unbiased predictor of the future spot rate bilaterally (Kritzman 2000). Hull (2000, p.521) argues that apparent gains disappear when measured against the correct numeraire. Others argue that net increases in welfare are not available by exploiting the non-linearities offered by Siegel’s paradox (Kritzman, 2000 and Roper, 1975).

12

The relationship between (8) and (4) can be demonstrated using Ito’s Lemma:

dzdtSd sss σσµ +−= )21(ln 2 . (9)

Hence,

)(21 *2 iiss −+= σµ , (10)

and the numerator in the Sharpe ratio for unhedged returns should be augmented by

the factor σs2/2.. The numerators in the hedged and unhedged Sharpe ratios will

diverge by this adjustment. Given UIP, the Sharpe ratio for an unhedged investment in

foreign cash is σs/2. Using the ‘ball park’ figures, σs/2 is about 0.05, much less than

for a hedged or an unhedged investment in foreign stocks, yet enough to be taken

seriously.

Any systematic divergence between forward rates and realized spot rates weakens the

‘free lunch’ case. For example, if there exist exchange rate risk premia, time-varying

or otherwise, (see Kritzman, 1993, Solnik and Dumas, 1995, and Hodrick and

Vassalou 2002) returns to hedge contracts may be predictably non-zero12.

Non-zero returns aside, exchange rate volatility also matters as it covaries with

portfolio returns. Exposure to deviations from PPP may allow members to hedge

against domestic inflation risk (Adler and Dumas, 1983), especially to the extent that

they include imported goods in their consumption bundles. Systematic covariance

12 Many studies test alternative currency hedging rules based on different hypotheses about the returns to forward contracts. Eun and Resnick (1997) test various passive and active hedging rules. The results reported here use a passive hedging rule.

13

between exchange rate changes and returns to other assets can be exploited to reduce

portfolio variance.13 The motivation to hedge exchange rate risk arises from the

chance to reduce portfolio volatility once these possibilities have been exhausted. If

portfolio volatility in real terms can be reduced by including hedged assets it seems

reasonable to include both hedged and unhedged categories in the decision set.

If it is agreed that partial hedging is potentially useful, the discussion shifts to the

problem of deciding how much to hedge and how best to implement the hedge. Given

the possibilities for covariance with domestic inflation and other asset returns, it is

unlikely that full hedging, as Perold and Schulman (1990) propose, will be optimal.

The 70 per cent ‘universal’ hedge proposed by Black (1990a, 1990b), depends on an

equilibrium where all investors hold the same portfolio of equities plus some

optimally hedged combination of domestic and foreign cash, and exhibit average risk

tolerance. The universal hedge ratio also depends on successful empirical estimation

of the relevant parameters. Moreover, Black confined attention to real exchange rates

in a zero-inflation environment in which stocks, domestic cash and foreign cash are

the only asset classes. For these reasons, relying on a universal ratio may be both

heroic and arbitrary (Solnik 1998, p. 49).

While disputing seemingly arbitrary ratios, Solnik agrees that the ‘optimal portfolio is

the world market portfolio partly hedged against currency risk’ (Solnik 1998, p. 47).

The degree of hedging is dependent on the covariance relationships between the

exchange rate and component assets of any portfolio.

13 See Lewis(1999) pp. 580ff.

14

The advantages of partial hedging, at least for portfolios with offshore components

of a reasonable size, say above 20 per cent of total, have been documented by a

number of writers. (See Jorion, (1989), Kaplanis and Schaefer (1991), Glen and

Jorion (1993) and Solnik (1998), for example, and Izan et. al. (1991)14 and Beggs et.

al. (1989) for an Australian perspective).

3.4 Currency overlays

The timing of currency hedging decisions in putting together a portfolio is also

important to outcomes. Hedging decisions can be made at the time the portfolio is

constructed or in an overlay after asset shares have been allocated. Industry practice

frequently has been to use currency managers whose specialist work is to make and

maintain overlays, but some researchers argue against separating the two parts of the

management problem. Jorion (1994) and Grinold and Meese (2000) demonstrate that

funds which make their initial allocation over unhedged international assets and then

make a lower level decision about hedging will invest less offshore and hedge less

than funds which make their initial allocation over both hedged and unhedged asset

classes. They argue that the overlay strategy fails to utilise the full possibilities of

investing offshore.

Market practice in Australia has favoured unhedged offshore equity positions and

hedged fixed interest. In the next two sections are set out the data and model used to

14 Izan et. al. compare the performance of fully hedged and unhedged equity portfolios from the perspective of an Australian investor, concluding that hedged portfolios are preferred on the basis of a comparison of Sharpe ratios. The focus here is on deriving a portfolio which is optimally hedged rather than fully hedged, and where equities compete for a place with such ‘safe’ assets as superannuation funds usually hold.

15

test whether the exclusion of these two asset classes is measurably important to a

borrowing-constrained superannuation fund member.

4. Data

The aim here is to mimic the single-horizon portfolio allocation choice offered to a

superannuation fund member, enlarged by including currency-hedged international

equities and unhedged international bonds. In order to maximise the run of data

available and to simplify the calculations, the US acts as proxy for all foreign

holdings.15

4.1 Preliminaries

The test portfolio covers seven asset classes: Australian bonds, Australian equities,

unhedged US bonds and equities, hedged US bonds and equities, and Australian cash.

Where possible, observations were taken on the last working day of each month for

the period March 1983 to March 2001.

Bond returns were calculated from the Datastream Tracker Index (TUSGVAL(RI))

for the US, and the Commonwealth Bank16 bond returns index for Australia. Equity

returns were derived from the Datastream Total Market Indexes for Australia and the

US (TOTMKAU(RI), TOTMKUS(RI)). The Australian index covers 160, and the US

index, 1000 stocks. Bond and equity returns indexes assume all coupons/dividends are

reinvested.

15 Anecdotal evidence suggests more than half of foreign asset holdings have been invested in the United States.

16 This series ran longer than the comparable Datastream index. Data were kindly supplied by AMP

16

Australian cash rates were measured by the yield on 30-day bank-accepted bills for

Australia17, and for the US, the one-month certificate of deposit rate18.

All unhedged US dollar values were translated to AUD values at the relevant

AUD/USD spot rate as reported in the RBA Bulletin Database. Hedged returns were

calculated as a rolling forward contract over a 30-day horizon19. Since the monthly

variation in an equity index can be substantial, the effectiveness of the currency hedge

is partly dependent on a manager’s ability to predict the end-period local-currency

value of the index. One approach would be to hedge the spot value of the underlying

asset at the beginning of the period (in which case 1+α would drop out of (11)

below). Another would be to assume perfect foresight and set 1+α to the exact change

in the local currency index. To avoid both of these extremes, α is set equal to the

mean increase over the sample. The hedged returns for bonds and equity will

therefore include a forecasting error dependent on the true return’s deviation from

mean. The gain or loss on the forward currency position each period using this type of

hedge is:

( ) 11

1 1* S

ii

SαpFC t*t

ttt

−

++

+= ++ 1t (11)

Henderson.

17 RBA Bulletin Database

18 Board of Governors of the Federal Reserve System Database

19 This short-horizon forward transaction is conventional practice according to Muysken and Burt (2000). They estimate that the total cost of this type of currency hedging is 0.2% on an annual basis.

17

sample over theindex in the increasemonthly average the tperiod endat index equitiesor bond US theof value USD

rate (*)for US rate,cash day -30 tperiod endat rate exchangespot positioncurrency forwardon lossor gain

:where

*

==

===+

αt

t

pi

SFC 1t

The sum of FC and the change in the spot value of the index over the period gives the

hedged return in AUD.

Asset returns and covariances were calculated from annualised monthly changes, and

real series were the result of final discounting by the (annualised) percentage change

in the Private Consumption Deflator.20

4.2 Summary statistics and correlations

Table 1 gives descriptive statistics and correlations for each of the real annualised

series. There are at least two observations worth making here. Firstly, hedging is

effective in reducing the volatility of individual series, at little cost in terms of

expected returns. The standard deviation of the hedged US bond series is about half

that of the unhedged series, close in size to the volatility of the domestic bond index.

Hedging lessens volatility for the US equities series also, but not so dramatically. The

standard deviation of the hedged offshore series, however, does compare well with the

same measure for the Australian equities series.

20 The Private Consumption Deflator is published quarterly (ABS Catalogue 5206.0) and monthly values were inferred by linear interpolation of the quarterly levels. Nominal returns were deflated by the annualised percentage change in the interpolated index for the current period.

18

Table 1: Summary Statistics and Correlations for Real Returns

Annual per cent changes:

Australia USA Bonds Equities Cash Bonds hedged Equities hedged Mean 7.67 13.00 5.69 8.69 8.44 15.07 14.82 Median 8.82 15.07 4.98 6.12 8.50 15.24 18.41 Max. 59.39 190.23 16.34 151.77 59.93 183.42 159.86 Min. -46.08 -491.18 -5.69 -112.29 -43.96 -195.76 -270.61 Std. Dev. 5.39 18.58 0.96 12.49 5.52 17.36 15.58 Obs. 216 216 216 216 216 216 216

Correlation Matrix:

Aust. Bonds Equities Cash

US Bonds hedged

US Equities hedged

Aust. Bonds 1.0000 0.4079 0.0427 -0.0388 0.3477 0.1387 0.3093 Equities 1.0000 -0.0095 -0.3000 -0.0651 0.2618 0.5098 Cash 1.0000 -0.0427 0.1299 -0.0065 0.0735 US Bonds 1.0000 0.4956 0.5276 -0.0430 hedged 1.0000 0.2744 0.2616 US Equities 1.0000 0.7843 hedged 1.0000

Secondly, the correlation matrix foreshadows benefits from diversification. Offshore

unhedged bonds are negatively correlated with domestic assets. Hedged US bonds and

Australian equities are negatively correlated. Covariance with cash is negative for

Australian equities, and unhedged US equities. Other pairings show low positive

correlations, especially those relating cash to other assets.

Table 1 data also confirm that hedging increases the correlation of foreign and

domestic assets. This increasing correlation is due to the fact that much exchange rate

volatility is unrelated to variability in domestic assets and can be diversified away

(Jorion 1989, p. 51). The large reductions in volatility for individual series that

hedging has achieved are not so apparent in a portfolio context.

19

Diversifying into foreign markets exploits the risk-reduction possibilities of holding

combinations of domestic and foreign assets.21 Domestic investors have tended to

restrict foreign asset holdings to what appears to be sub-optimal levels. The

correlation matrix reported here gives superficial support to the risk reduction benefits

of diversifying internationally, but the other inputs to the optimisation process, the

mean returns, are imprecisely estimated.22 Uncertainty about the usefulness of sample

moments of historical returns series may be a factor in both home bias and currency

hedging decisions. The results reported in the following section begin with the ex

post, or certainty equivalent, data then conduct the tests again using alternative returns

assumptions as a guide to the robustness of the results.

Section 5 outlines the asset allocation experiments.

5. The Optimal Portfolio

The question of what is the optimal combination of hedged and unhedged assets for

any given level of risk tolerance provides a useful starting point for analysis of the

currency risk in superannuation portfolios. Mean-variance optimisation models begin

with the proposition that investors are concerned about return and risk. Efficient

portfolios are those combinations of assets which offer the highest possible return for

the least variation and efficient frontiers trace out that set of portfolios over a range of

volatilities.

21 The natural international extension of Markowitz’s (1959) modern portfolio theory was first made by Grubel (1968). A series of ex post applications followed including Solnik (1974) and Lessard (1976).

22 See Lewis (1999) for a summary of the debate over whether gains to diversifying offshore are statistically significant.

20

5.1 Setup

The problem is to minimize:

∑=

n

jijiij ww

1,21 σ (12)

subject to:

...., ,2 ,1for 0

1

)()(

1

1

niw

w

rEwrE

i

n

ii

n

ipii

=≥

=

=

∑

∑

=

=

(13)

irEiw

rE

jiσ

i

i

p

ij

asset return to expected )(asset toallocated wealth totalof proportion

return expected portfolio )(

and assetsbetween covariance :where

==

=

=

This problem assumes that investors are interested in portfolio return and variance

per se. Outcomes are consistent with utility maximization for risk averse investors

when asset returns have an elliptical distribution or under the assumption of quadratic

utility.23 This formulation is also myopic and will be consistent with long-horizon

23 See Ingersoll (1987, pp.104ff).

21

utility maximisation only if utility functions are logarithmic or asset returns are

independent and identically distributed over time.24

5.2 Asset allocations using ex post real returns

Consider first a portfolio constrained only in that all asset shares be zero or greater (no

short positions). 25 The optimal shares are shown in Figure 2.26 The minimum variance

frontier is here calculated across twenty evenly spaced observations between the

minimum variance and maximum return portfolios. Portfolio standard deviations

around 10-12 per cent per annum correspond to moderate risk aversion27.

24 For more complete discussion see Campbell and Viceira (2002).

25 The non-negativity constraints have a number of effects on the model when compared with outcomes allowing short selling. The two-fund separation spanning results of the unconstrained model no longer hold, extending to k-fund separation. The frontier can be derived as a linear combination of ‘corner’ portfolios. For additional discussion see Sharpe (1991).

26 All optimisation calculations were made in Matlab.

27 Assuming negative exponential utility functions defined over returns to wealth and normally distributed rates of return, coefficients of relative risk aversion along the frontier range from around 40 at the lower end to around 2 at the more risk tolerant end. See Sharpe (2001) or Cochrane (2001) for the utility theoretic foundations of mean-variance analysis.

22

Figure 2: Composition of Efficient Portfolios

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Australian Equities

US Equities - hedged

US Equities -

US Bonds - unhedged

US Bonds - hedged

Cash

Australian Bonds

Figure 2 shows that more risk-tolerant investors prefer high concentrations of US

equities, topped up with domestic equities. As the investor becomes less risk tolerant,

US bonds are favoured over US equities of both types, with a rapid flight to cash at

the lowest levels of risk tolerance. Notice that contrary to prevailing practice among

many Australian superannuation funds, hedged equities are incorporated in almost all

optimal portfolios, as are unhedged bonds. Proportions held offshore and

hedged/unhedged ratios are presented in Table 2.

23

Table 2: Optimal hedging weights

Hedged to offshore Offshore to portfolio Hedged to portfolio 0.00 0.01 0.00 0.71 0.09 0.07 0.79 0.18 0.14 0.81 0.27 0.22 0.83 0.36 0.30 0.83 0.45 0.37 0.84 0.54 0.45 0.84 0.62 0.53 0.85 0.71 0.60 0.85 0.80 0.68 0.84 0.86 0.72 0.80 0.86 0.69 0.77 0.85 0.65 0.73 0.85 0.62 0.70 0.84 0.59 0.66 0.84 0.55 0.62 0.83 0.52 0.59 0.83 0.49 0.55 0.82 0.45 0.00 1.00 0.00

Results reported in Table 2 support partial hedging of both equities and bonds. The

optimal hedge ratio is quite stable across a range of portfolios and volatilities, ranging

from 55 to 85 per cent of offshore assets for all but the extreme portfolios. For a

portfolio with about 60 per cent allocated to equities, the optimal hedged component

is remarkably close to Black’s ‘universal’ suggestion: around 70-75 per cent.

5.3 Efficient frontiers

These calculations demonstrate firstly that hedging is valuable for both equities and

bonds. They suggest that even during periods of depreciation some proportion of

hedged assets in a portfolio beats none. Secondly it appears that home bias could be

costly to the investor.

24

As a guide to how much such restrictions might cost, consider the results of the same

experiment but where one portfolio is restricted to 25 per cent offshore, comprised of

hedged bonds and unhedged equities (constraints which roughly approximate

prevailing practice) and the third portfolio covers only domestic assets. The efficient

frontiers generated by these three are shown in Figure 3.

As one would expect, over all levels of portfolio return the least-constrained portfolio

dominates, achieving higher levels of expected returns for any volatility. At a

volatility of around 10 per cent per annum, for example, lack of hedging and home

bias can cost may cost 100 basis points in returns.

Figure 3: Efficient frontiers – ex post

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16 18 20

Port

foli

o E

xpec

ted

Ret

urn

% p

.a.

Portfolio Standard Deviation % p.a.

Unconstrained

Constrained

Domestic

25

If we narrow the search however, and consider what the restricted hedging rule alone

costs investors, the gains look much less substantial. In Figure 4, the unconstrained ex

post frontier is mapped against an efficient frontier for which total foreign assets are

not restricted, but unhedged US bonds and hedged US equities are held to zero.

Figure 4: Efficient Frontiers – ex post returns and restricted hedging

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16 18

Unconstrained

Restricted Hedging

Port

foli

o E

xpec

ted

Ret

urn

% p

.a.

Portfolio Standard Deviation % p.a.

Payoffs to relaxing the hedging restriction are insignificant in terms of reduced

volatility for the more conservative portfolios, and are evident, but still small, at the

upper risk range. They amount to a difference of about 4 basis points. (The two

frontiers converge at the extreme of risk because short-selling constraints force 100

per cent of both allocations to unhedged US equities.) This result may explain

industry practice: in a typical balanced portfolio with foreign asset allocations below

26

30 per cent of total, the costs of hedging equities may outweigh the benefits. Even

though optimal portfolios include substantial weightings in the excluded asset classes,

sticking with the ‘hedge bonds not equities’ rule in a domestic-heavy portfolio is not

particularly costly in terms of volatility. And the strategy may protect managers

whose performance is measured against industry benchmarks which entrench the

restriction. Concerns arise where the marginal increases in volatility associated with

unhedged foreign equity holdings weight portfolios towards domestic assets since

home bias dramatically reduces efficiency.

The preliminary evidence in this section is based on historical sample mean rates of

return which are subject to some well-known weaknesses. The next section tests the

robustness of these results using two approaches to the question which rely less on ex

post means.

5.4 Bayes-Stein estimation

Expected returns based on historical means of stock market returns have very large

standard errors so that they are very poor predictors of the ‘true’ mean. Scanning the

historical means and standard deviations in Table 1 confirms this view: the standard

deviations of mean returns are almost as large as the means themselves.

Unfortunately, longer runs of data (and they would need to be very long) are not

likely to reduce this problem when the financial environment that generates them is

also changing. Estimates of the standard deviations of returns, however, are much

more accurate28. The sample covariance matrices are much closer to true population

28 Luenberger (1998, p. 217). For n independent identically distributed observations of returns, the sample variance has itself a variance of 2σ4/(n-1). One year’s observation at a monthly frequency produces reasonable estimates of σ2 on these assumptions.

27

values and therefore better inputs to the optimisation problem than the sample mean

returns.

Given the unreliability of ex post sample means, techniques for improving on these

have been developed. One approach is to shrink the historical estimates back towards

a more reliable ‘prior’ distribution using Bayesian methods. Jorion (1985, 1986) was

one of the first researchers to apply this technique to portfolio allocation problems.

Since then it has been widely applied and modified (see, for example, Izan et.al.

(1991), Black and Litterman (1991), Eun and Resnick (1997) and Connor (1997)).

The Bayes-Stein shrinkage estimator as developed by Jorion recalculates the mean

returns as a weighted average of the sample means and the return to the minimum

variance portfolio. The weights in the Bayes-Stein mean calculation are derived as a

function of the sample size and the sample covariance matrix. The predictive density

function has a mean return vector given by:

0)1( Yww 1YR 1 +−= (14)

where R is the vector of predictive returns, Y1 is the vector of ex post sample means

and Y0 is the mean return for the minimum variance portfolio. When the factor w is

equal to zero, R reduces to the ex post mean vector. When w is equal to 1, the

minimum variance return is obtained. The ex post sample mean vector used above is

therefore a special case of this more general formulation. As the historical sample

becomes larger, R converges towards the ex post mean vector.

The shrinkage factor w is calculated as:

))(2()'()1)(2()1)(2(

00 1YΣ1Y 11

1 YNTTYTNTNw

−−−−+−+−+

=−

(15)

28

where:

N = number of assets to be allocated

T = number of observations

Σ = covariance matrix29

Figure 5 shows the efficient frontiers generated using the Bayes-Stein mean vector

compared with the ex post frontiers. Unconstrained and restricted frontiers are

reported, with the broken lines indicating the corresponding Bayes-Stein adjusted

frontiers. The shrinkage factor has (predictably) scaled down the expected portfolio

return per unit of volatility for both portfolios without changing the tenor of the

results. In fact the optimal hedge ratios (not reported here) are unchanged, as is the

pattern of asset allocation.

29 This unknown covariance matrix is estimated from the sample covariance matrix. See Jorion (1986) for this and the adjusted covariance matrix of the predictive distribution.

29

Figure 5: Efficient frontiers – Bayes-Stein adjusted and ex post

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16 18

ex post

ex post

Bayesian

Bayesian

Port

foli

o ex

pect

ed r

etur

n %

p.a

.

Portfolio standard deviation % p.a.

5.5 Forecasting real returns

Another approach to dealing with the problem of badly estimated means is to take

forecasts based on long-run equilibrium conditions in international capital markets

and use them as inputs to the calculation process. Loosely following the forecast of

long-term returns made by Campbell (2001) and (2002), assuming UIP holds, and

holding net returns to currency hedging to zero, all real equity returns are reset at 5.5

per cent, all bond returns at 3.5 per cent and cash at one per cent. This strategy aims to

remove from optimisation calculations some of the historical biases due to the bull

market of the past twenty years.

30

Optimal portfolio shares using these revised forecast returns and the estimated

correlation matrix are shown in Figure 6 below. The results show portfolios making

use of both hedged and unhedged foreign assets, but more heavily weighted towards

bonds since the new returns assume that the equity premium has narrowed

considerably. Because real returns to each equity class are now assumed to be equal,

the most risky portfolio is evenly shared between them. Again the allocations support

including both hedged equities and unhedged bonds in the choice set.

Figure 6: Optimal Portfolio Shares – forecast real returns

0

0.2

0.4

0.6

0.8

1

1.2

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Australian Equities

US Equities - hedged

US Equities - unhedged

US Bonds - unhedged

US Bonds - hedged

Cash

Australian Bonds

31

Efficient frontiers in the style of section 5.3 also confirm previous conclusions:

restricting offshore allocations and the currency-hedging choice set forces substantial

inefficiencies on the portfolio allocations.

Figure 7: Efficient Frontiers – forecast real returns

1

2

3

4

5

6

0 2 4 6 8 10 12 14 16 18 20

UnconstrainedConstrained

Domestic

Port

foli

o E

xpec

ted

Ret

urn

% p

.a.

Portfolio Standard Deviation % p.a.

Finally, comparing frontiers across only the hedging restriction reinforces earlier

observations that the gains to widening the choice set are small until foreign equities

have a substantial weight. Figure 8 makes the comparison. Deviation between the

frontiers increases rapidly only where nearly all weight is allocated to equities. In this

scenario, allowing currency hedging of foreign equities reduces the portfolio standard

deviation by four per cent for the most risky frontier point. At a standard deviation of

14 percent, hedging alone increases returns by 25 basis points.

32

Figure 8: Efficient Frontiers – forecast returns and restricted hedging

3.0

3.5

4.0

4.5

5.0

5.5

6.0

4 6 8 10 12 14 16 18 20

Port

foli

o E

xpec

ted

Ret

urn

% p

.a.

Portfolio Standard Deviation % p.a.

Unconstrained

Restricted Hedging

Despite the likely estimation risks associated with using sample means in such

optimisations, initial results appear to be robust to both Bayesian adjustment and

alternative forecasts.

6. Summary and Conclusions

Pension fund managers around the world have increasingly recognised the benefits of

diversifying portfolios offshore. Australian superannuation funds similarly are

including a growing share of overseas assets in their holdings, thus making efficient

management of exchange rate risk an important consideration. Survey evidence

demonstrates that many managers restrict their choices by excluding currency–hedged

foreign equities and unhedged foreign bonds from portfolios. This potentially costly

33

restriction has here been tested for mean-variance efficiency, from the perspective of

a borrowing-constrained Australian superannuant holding ‘balanced growth’ assets.

It is shown that contrary to common practice, optimal portfolios incorporate both

currency-hedged equities and unhedged bonds. In fact, frontier portfolios consistent

with moderate risk tolerance have hedge ratios around 75 per cent of total offshore

allocations. On their own, the currency hedging restrictions probably have a small

impact, in the order or 3-4 basis points, but when combined with low offshore

allocations, returns per unit volatility worsen dramatically.

If superannuation investment managers contemplate greater offshore weights without

including at least currency hedged equities as a part of their portfolio choice set, lower

foreign allocations are more likely, aggravating inefficiency.30

30 This parallels a well-known observation by Jorion (1994) and others that the overlay approach to currency hedging tacitly introduces a home bias. Note also that this study has not considered the home-bias inducing impact of Australia’s dividend imputation system.

34

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