Valuation-indiﬀerent weighted indexing for Euro Zone government bonds

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Valuation-indifferent weighted indexing for Euro Zone

government bonds

S.P. Moes August 23, 2011

Supervisor: Prof. Dr. T.K. Dijkstra (University of Groningen) Supervisor: Drs. W.H. de Vries RBA (Kempen Capital Management)

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Master’s Thesis Econometrics, Operations Research and Actuarial Studies Supervisor: Prof. Dr. T.K. Dijkstra (University of Groningen)

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Valuation-indifferent weighted indexing for Euro Zone

government bonds

S.P. Moes

August 23, 2011

Abstract

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Glossary of terms

To increase the readability of this paper, we present a glossary of terms.

Bid-ask spread: The difference in bid- and ask prices for securities (Fontanills, Gentile, and Cawood, 2001).

Bps: Basis points. One basis point is equal to 0.01% (Vernimmen, Quiry, Dallocchio, Le Fur, and Salvi, 2009).

Bullet bond: Bond which redeems at, and only at, the predetermined maturity date (Fabozzi and Fabozzi, 1995).

Cap-weighting: Cap-weighting, short for market-capitalization weighting. The practice of weighting constituents in an index or portfolio based on their market capitalization relative to the total market capitalization of all the constituents (Grinblatt and Titman, 2004).

Credit rating: Rating assigned by rating agencies that assesses the credit worthiness of the debt issue of a government or corporation. Well known rating agencies are Fitch, Moody’s and S&P. Bonds within the highest rating categories are referred to as investment grade bonds and reflect a good score on credit worthiness. Bonds within the lowest rating categories are sometimes referred to as bonds with a junk status (Fabozzi and Fabozzi, 1995). Most credit ratings are by described by characters rather than numbers, see Ap-pendix D for the computation of an average credit rating.

EBITDA: Earnings before interest, taxes, depreciation and amortization (Grinblatt and Titman, 2004).

Euro Zone: Union of 17 European countries that have adopted the euro as their local currency. As of January 1 2011, the Euro Zone is composed of Austria, Belgium, Cyprus, Estonia, Finland, France, Germany, Greece, Ireland, Italy, Luxembourg, Malta, Nether-lands, Portugal, Slovenia, Slovakia and Spain.

Face value: The amount to be repaid to the investor when the bond matures, is called or retired according to sinking fund provisions (Fabozzi and Fabozzi, 1995).

Fixed-rate bond: Bond with a fixed coupon rate, i.e. the bondholder receives a coupon fixed at a predetermined rate. Also known as a straight-coupon bond (Fabozzi and Fabozzi, 1995).

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available for trading on the market (Vernimmen et al., 2009).

Fundamental factors: Factors that disclose information about the economic value of a company or country. Those factors might thereby reveal something about the intrinsic value of a security issued by the corresponding company or country (Fontanills et al., 2001).

Index: In this paper we use the less strict definition of an index, i.e. “any rule-based method of constructing a portfolio” contrary to the more strict definition of an index being “a combination of assets we can all invest in without distorting prices” (Asness, 2006). Intrinsic value: The unobserved value of a security, think of it as the expected discounted present value of future cash flows (Arnott, Hsu, Liu, and Markowitz, 2009).

Noise trading: Contrary to trading on actual information, noise trading is the practice of trading on noise as if it were information (Black, 1986).

Old-age dependency ratio: The old-age dependency ratio is the ratio of citizens aged above 65 years to the part of the population aged 15 to 65 years.

Sinking fund: A sinking fund is an annual obligation of the issuer to set aside an amount of money sufficient to buy back a given percentage of the outstanding bonds (Fabozzi and Fabozzi, 1995).

Valuation-indifferent weighting: The practice of weighting constituents in an index or portfolio based on non-price weighting metrics (Arnott, Hsu, Li, and Shepherd (2010)). Arnott, Hsu, and Moore (2005) used fundamental factors underlying the security, such as book value or sales in the equity market to construct equity indexes. They called this form of indexing Fundamental Indexation R_{. Another example of valuation-indifferent}

weight-ing is equal-weightweight-ing, in which each constituent of a portfolio of index is assigned an equal weight.

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1 Introduction

“If you were... to lend someone a few million dollars, who would you lend it to? A rich man, who had the means to pay it back or a poor one who owed a lot to other people?”(Johnson, 2011).

Many will undoubtedly consider this question to be a rhetorical one, as when it comes to lending a sum of money to someone, one ultimately wants to receive it back. In the example above, the probability of actually doing so is likely to be higher if the loan is made to the rich man than when it is made to the poor and highly indebted person. Hence the choice is not that difficult. But what is it that determines our preference for the former above the latter? What leads us to perceive the credit worthiness of the rich man to be higher than that of the poor one? These questions are not that difficult either, the differences in wealth and indebtness are the indicators for the credit worthiness of these two persons. Likewise, many will prefer a debtor with a high income above one with a low income and not only the size of their outstanding debt influences our perception of their credit worthiness, but also the relation of it to the size of income and spending.

As intuitive as it may sound to prefer the rich man above the poor one, in the con-struction of cap-weighted fixed income indexes the opposite happens. When we consider the market for government debt, cap-weighting implies assigning the largest weights to the governments with the largest outstanding debt, regardless of their income1_{. Hence,}

investing in a cap-weighted index implies lending the largest amount of money to the most indebted government. When governments reach the point of failing to service their debt, the investor is left with the losses. Investors may remember too well the default of Argentina in 2001, and the recent problems of the peripheral countries in the Euro Zone. Being counterintuitive is not the only problem of cap-weighting in the government debt markets. Since the weights in a cap-weighted index are based on the historical issuance of debt, it is unlikely that it will recognize opportunities in new capital markets. Hence a cap-weighted index may rather be called backward-looking than forward-looking as Toloui (2010) points out. In addition, cap-weighting is prone to the so-called ‘buy high, sell low’ bias. The ‘buy high, sell low’ bias arises from the fact that larger weights are assigned to securities that have recently gone up in price and lower weights to securities that have recently experienced a fall in prices. This is the opposite of the traditional investment view to ‘buy low and sell high’.

An alternative way of calculating index weights, valuation-indifferent weighting, is presented as a solution to the problem of selecting the biggest debtors, the ‘buy high, sell

1_{We acknowledge the fact that the comparison between an individual and a government is not complete.}

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low’ bias and the backward-looking nature of traditional indexes. Instead of using market-capitalization to determine the weights in an index, valuation-indifferent weighting implies weighting on non-price weighting metrics (Arnott et al., 2010). In January 2009 PIMCO, a global investment manager, launched three valuation-indifferent weighted bond indexes, in which countries are weighted by their GDP (Toloui, 2010). The GDP weighted indexes have the property that most low-debt countries are assigned larger weights than they have in cap-weighted indexes. In addition, weighting countries on basis of their GDP could be called forward-looking, as the issuing of new debt is generally preceded by rapid economic growth. Lastly, bond prices tend to move inversely to GDP growth and as such GDP weighting solves the ‘buy high, sell low’. The decision by PIMCO to launch valuation-indifferent government bond indexes has not been left unnoticed. In November 2009 Barclays Capital (Toloui, 2010) launched their own breed of valuation-indifferent weighted indexes and more recently Lombard Odier (Milhench, 2010) followed by launching a valuation-indifferent weighted government bond fund. As for the performance of these indexes, Arnott et al. (2010) empirically showed outperformance of a valuation-indifferent weighted index for emerging market government bonds. The index outperformed its cap-weighted benchmark in both bull and bear markets, and in periods with a rising and a falling federal funds rate. This indicates, that valuation-indifferent weighting might be less vulnerable to macro-economic events than the traditional cap-weighting approach.

In addition to the intuitive disadvantage of cap-weighting, the so-called noisy market hypothesis is put forward as an explanation of why valuation-indifferent weighting work. In this framework, market prices of securities are inefficient, in the sense that they do not fully reflect the intrinsic value of the securities, opposed to the well-known efficient market hypothesis. Given this inefficient market, it is suggested by Hsu (2006) that cap-weighting overweights overvalued securities while underweighting the undervalued securities. When pricing errors are not persistent it implies that cap-weighting is a sub-optimal invest-ing strategy. However, not everyone is convinced by the reasoninvest-ing of the noisy market hypothesis as an explanation for the observed outperformance of valuation-indifferent in-dexes. Perold (2007) refutes the statement, that in a market with price inefficiencies the cap-weighted index is sub-optimal. Others, such as Asness (2006) argue that valuation-indifferent weighting is merely value investing in disguise.

1.1 Aim of the paper

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gov-ernment bond indexes compared to a market-capitalization weighted benchmark. In order to do so both a theoretical and empirical study will be performed.

First, we assess whether the noisy market hypothesis is a valid explanation for the observed outperformance of valuation-indifferent weighted indexes. To do so, we examine the arguments put forth by the advocates of valuation-indifferent weighting, for example, Treynor (2005) and Hsu (2006) and the opponents, such as Asness (2006) and Perold (2007). Based on this examination, we present our own view on the noisy market hypoth-esis and its consequences for the performance of cap-weighting indexes.

Subsequently, we initiate our empirical research concerning valuation-indifferent Euro Zone government bond indexes, with the construction of a GDP weighted Euro Zone government bond index. We use backtests to compare its performance to a cap-weighted benchmark. The next step is to examine additional macro-economic factors that can be used to construct a valuation-indifferent weighted index. As in the illustrative example of the rich and the poor man, we focus on factors that are able to gauge the ability of the debtor to meet his obligations. We use these additional factors to construct new valuation-indifferent weighted Euro Zone government bond indexes. In the end, we backtest these indexes to examine their performance relative to a cap-weighted benchmark.

1.2 Research question

We pose the following main research question:

What is the advantage of valuation-indifferent weighted Euro Zone govern-ment bond indexes over a cap-weighted benchmark?

In order to answer this main question, we consider the following questions:

1. What does the noisy market hypothesis entail and how does it imply sub-optimality of cap-weighted indexes in particular markets?

2. How can we explain the empirical evidence of outperformance of valuation-indifferent weighted indexes?

3. How does a GDP weighted Euro Zone government bond index perform compared to a cap-weighted benchmark?

4. Which additional macro-economic factors can be used to construct valuation-indif-ferent weighted Euro Zone government bond indexes?

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1.3 Structure of the paper

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2 Are markets efficient?

According to the capital asset pricing model derived by Sharpe (1964) and Lintner (1965), the cap-weighted portfolio is the optimal investment choice for an investor. The model builds on two pillars, the efficient market hypothesis and the modern portfolio theory.

The efficient market hypothesis was developed by Eugene Fama in the 1960’s. It states that the market price of a security fully reflects all available information concerning this se-curity. Hence ‘prices represent best estimates of intrinsic values.’ (Fama, 1965). The main assumptions underlying the efficient market hypothesis are that the market is frictionless, investors maximize their utility, have rational expectations, are on average correct in their expectations and expectations are adjusted when new information becomes available. We distinguish between three forms of market efficiency: weak-form efficiency, semi-strong form efficiency and strong-form efficiency. Weak-form efficiency refers to efficiency in a market where the only information investors have are the historical security prices or returns. In the case of semi-strong form efficiency, investors have all publicly available information, such as announcements of stock splits or annual reports. When investors, or at least some of them, have inside information in addition to all publicly available information, we speak of strong-form efficiency (Fama, 1970).

A few years earlier Markowitz (1959) posed his modern portfolio theory. This theory states that investors are risk averse and only care about the mean and variance of the return on their investment. Hence an investor, who selects a portfolio at time t − 1 to produce a stochastic return at t, will choose mean-variance efficient portfolios. I.e. a portfolio in which variance is minimized given expected return and expected return is maximized given variance.

In addition to the characteristics Markowitz (1959) bestowed upon investors, Sharpe (1964) and Lintner (1965) assumed that investors all have the same expectations regard-ing the return on assets, which are on average correct and investors are able to borrow and lend at a risk-free rate2. Based on these assumptions Sharpe (1964) and Lintner (1965) presented the result that all investors invest in the cap-weighted market portfolio of all available risky assets (Fama and French, 2004). Although in theory, the portfolio which is mean-variance efficient should incorporate all risky assets, also the ones which are not traded on the market. Common thought is that cap-weighted market indexes are sufficiently representative to be nearly mean-variance efficient.

However, there has been some discussion on one of the pillars on which the capital asset pricing model is resting, i.e. Fama’s view on the relation between market prices of securities and their unobserved intrinsic values. In the 1980’s among others Blume and Stambaugh (1983), Black (1986) and Summers (1986) questioned the validity of the

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efficient market hypothesis. Blume and Stambaugh (1983) discussed the deviation of market prices from their intrinsic value as a result of the bid-ask spread, Black (1986) stated that noise trading, such as speculation and momentum trading, leads to noise in market prices and Summers (1986) proposed an alternative hypothesis, where market prices differ from their intrinsic value by a multiplicative term. Furthermore, (Fama and French, 2004) summarized the results of various researches, which showed the CAPM to be empirically invalid. Since CAPM builds on the assumptions of the efficient market hypothesis, the idea that a cap-weighted index is optimal, might not hold in an inefficient market.

More recently Arnott et al. (2005), Treynor (2005) and Hsu (2006) developed a similar hypothesis to that of Summers (1986). Their hypothesis was coined as the ‘noisy market hypothesis’ by Siegel (2006). It states that market prices are inefficient, i.e. market prices contain noise. A possible explanation of noise in market prices is the influence of speculators and momentum traders (Siegel, 2006). Although bonds have a finite lifetime, predetermined cash flow instances and are more likely to be held for the long run, claiming that prices of government bonds are not subject to noise is not correct. Speculators, momentum traders and market sentiment can have their influence on bond prices as well. This has led Arnott et al. (2009) to assume that the market prices for bonds are different from their intrinsic values by a noise.

2.1 The noisy market hypothesis

In describing a noisy market, we follow the framework of Hsu (2006). Let us first de-fine Pi,t > 0 as the market price of security3 i at some time t, where t ∈ [0, 1, . . . , T ].

Furthermore, let P_i,t∗ > 0 denote the unobserved intrinsic value of security i. Now we assume

Pi,t = Pi,t∗(1 + i,t), ∀i ∈ [1, 2, . . . , N ] (1)

where the i,t are independent of Pj,s∗ ∀i, j, s and t. In addition we assume the i,t are

independently identically distributed (i.i.d.) across assets and time with mean 0 and finite variance σ2, such that i,t > −1 ∀ i and t. We have E[i,t+1i,t] = E[i,t+1]E[i,t]. By

defining the noise in this way, we assign the same probability to a security being overpriced as a security being priced beneath its intrinsic value, as Treynor (2005). By equation (1) we have

E[Pi,t|Pi,t∗] = Pi,t∗. (2)

In the absence of noise, the market price of security i at t is P_i,t∗.

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2.2 Existence of a return drag on a cap-weighted index

Given the assumed noise in prices, Hsu (2006) derives the sub-optimality of cap-weighting. That is, a return drag is said to exist, which is defined as an excess expected return on a so-called valuation-indifferent weighted index over the expected return of a cap-weighted index. Moreover, Hsu (2006) shows that the expected return on the cap-weighted index is even less than the expected return on a cap-weighted index based on intrinsic values. Hsu (2006) shows that the return drag increases with the magnitude of the noise and Arnott and Hsu (2008) and Treynor (2005) claim similar results. The intuition behind their claim is simple. A cap-weighting index overweights overvalued securities, i.e. securities with a market price above their intrinsic value. On the other hand, it assigns underweight to undervalued securities, i.e. securities with a market price below their intrinsic value. In a valuation-indifferent weighted index this bias is avoided, as the securities are weighted independent of their market prices.

To asses whether our noisy market really implies a return drag and the sub-optimality of the cap-weighted index, we use the framework of Section 2.1 and express the one period return Ri,t+1 on security i as

1 + Ri,t+1 = Pi,t+1 Pi,t (3) = P ∗ i,t+1 P_i,t∗ 1 + i,t+1 1 + i,t . (4) We let 1 + R∗_i,t+1= P ∗ i,t+1

P_i,t∗ , i.e. the return on intrinsic value and rewrite equation (4) as

Ri,t+1= (1 + R∗i,t+1)

1 + i,t+1

1 + i,t

− 1. (5)

Now, we have4

E[Ri,t+1] ≥ E[R∗i,t+1], (6)

i.e. the expected real return on a security is always at least equal to the return on intrinsic value. Furthermore, we have

E[Ri,t+1] ≈ ER∗i,t+1 + σ2 1 + ER∗i,t+1 , (7)

this result is equivalent to what Hsu (2006) presented and gives us an indication of the size of the difference between the expected returns.

Let us now assume, without loss of generality, that Si, the number of shares outstanding

for security i satisfies

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Si = 1, ∀i ∈ [1, 2, . . . , N ]. (8)

Hence, the weight of security i in a cap-weighted index at t is given by

wi,t = Pi,t PN k=1Pk,t (9) = P ∗ i,t(1 + i,t) PN k=1Pk,t∗ (1 + k,t) . (10)

Furthermore, let us define

w∗_i,t= P ∗ i,t PN k=1Pk,t∗ , (11)

which is the cap-weight for security i based on intrinsic value rather than the market price. Now, the expected return on the cap-weighted index is given by5

E[RP,t+1] = E " _N X i=1 wi,tRi,t+1 # (12) ≥ E " _N X i=1 w∗_i,tR_i,t+1∗ # (13) = ER∗ P,t+1 . (14)

I.e. inequality (14) shows that the expected return on the cap-weighted index is greater than or equal to the expected return of a cap-weighted index based on intrinsic values, denoted by EhR∗_P,t+1i. Furthermore, we have

E [RP,t+1] ≈ ER∗P,t+1 + ∆, (15)

i.e. the expected return on the cap-weighted index is approximately equal to the expected return on a cap-weighted index based on intrinsic value plus a positive term denoted here by ∆6. In Appendix A.2.2 we show that ∆ satisfies

1 Nσ 2_{(1 + E}_R∗ P,t+1) ≤ ∆ ≤ max k∈N(w ∗ k,t)σ2(1 + ER ∗ P,t+1). (16)

We observe that ∆ is strictly positive and increases with the magnitude of the noise. The result that the expected return of the cap-weighted index is not less than the expected

5_{See Appendix A.2.1 and A.2.2 for the derivations of inequality (14) and approximation (15) and (16).} 6_{In a personal communication, Hsu agreed that the expected return on the cap-weighted portfolio is}

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return on a cap-weighted index based on intrinsic value is in accordance with the findings of Cao (2007).

The consequence for the return drag that Hsu (2006) showed becomes clear if we derive the return on the valuation-indifferent weighted index. Hsu (2006) assumes that we are able to construct weights ˆwi,t such that

ˆ

wi,t = w∗i,t(1 + γi,t) ∀i ∈ [1, 2, . . . , N ] (17)

where γi,t is i.i.d. with mean 0, finite variance such that γi,t > −1 ∀ i and t and it is

uncorrelated with other random variables. I.e. equation (17) implies ˆwi,t is an unbiased

estimator of w∗_i,t. The expected return of an index constructed with weights ˆwi,t satisfies

Eh ˆRP,t+1 i = E "_N X i=1 ˆ wi,tRi,t+1 # (18) ≥ ER∗ P,t+1 , (19)

as a result of inequality (6). In addition we have by approximation (7) that the expected return on the valuation-indifferent index can be approximated by

Eh ˆRP,t+1

i

≈ ER∗

P,t+1 + σ2 1 + ER∗P,t+1 , (20)

When we plug in the inequality (16) in approximation (15) and compare it to approxi-mation (20), we see that the approxiapproxi-mation of the return drag on the cap-weighted index is bounded between (1 − maxk∈Nw∗k,t)σ2

1 + EhR∗_P,t+1i and σ2 N −1_N 1 + EhR∗_P,t+1i. However, the advantage of the valuation-indifferent weighted index stands and falls with the assumption of ˆwi,t being an unbiased estimator of w∗i,t. Finding an unbiased estimator

of w∗_i,t may be hard to achieve in practice. Similarly, the sub-optimality of cap-weighting that Treynor (2005) derives holds only if we are able to find a better approximation of the intrinsic value than the market price.

The non-existence of a return drag on cap-weighting is also argued by Perold (2007). He stresses the fact that while the intrinsic value of a security is unobserved, investors do observe the market price. Hence if we compute the expected return on a security, we need to compute the expected return conditional on the market price Pi,t by means of Bayes

rule. Under the assumption that the prior distribution of intrinsic value is log-uniformly distributed and that noise is independent of intrinsic value P_i,t∗, the expected return on security i conditional on the market price is given by

E [Ri,t+1|Pi,t] = (1 + R∗i,t+1)E

1 1 + i,t

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i.e. the expected conditional return of security i is independent of its market capitalization and hence cap-weighting does not entail a return drag. However, as pointed out in by Dijkstra (2011), the assumption of the prior distribution of intrinsic value being log-uniformly distributed is essential for the result to hold true.

Arnott and Hsu (2008) consider an alternative specification of the noise, such that it is mean-reverting. We assess whether a return drag exists if we assume the noise to be mean-reverting.

Lemma 1. We assume

i,t= ρi,t−1+ δi,t, (22)

where δi,t are i.i.d. with mean 0 and finite variance σ2, such that i,t > −1 ∀ i and t.

Furthermore 0 ≤ ρ < 1 and as a result E [i,t] = 0.

The formulation of the error as an AR(1) process has been used extensively in the literature, by among others Summers (1986) and Campbell and Kyle (1993). It implies that the expected market price of a security is equal to its intrinsic value. Furthermore, we see that when a security is overpriced at t − 1, it is likely to be overpriced at t. Note that for ρ = 0 we are in the framework of Hsu (2006). Under Lemma 1 the expected one period return on security i is given byfootnoteSee Appendix B.1.1 and B.1.2 for the derivations of inequality (23)approximations (24).

E [Ri,t+1] ≥ ERi,t+1∗ , (23) furthermore E [Ri,t+1] ≈ ER∗i,t+1 + σ2 1 + ρ(1 + ER ∗ i,t+1), (24)

From these results we derive the expected one period return on the valuation-indifferent weighted index which satisfies

Eh ˆRP,t+1 i = E "_N X i=1 ˆ wi,tRi,t+1 # (25) ≥ ER∗ P,t+1 . (26)

We can approximate the expected return by

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As is shown in Appendix B.2 and B.3 it implies that the approximation of the return drag on the cap-weighted index is bounded between (1 − maxk∈N wk,t∗ ) σ

2 1−ρ(1 + E h R_P,t+1∗ i) and N −1 N σ2 1−ρ(1 + E h

R∗_P,t+1i). Where we have that the lower bound is at least equal to zero. We noted that finding an unbiased estimator of the true weight is hard in prac-tice. Therefore we consider the case of equal-weighting, which is by definition valuation-indifferent weighting. In the case of equal-weighting, the weight of a security is given by

1

N. The return of an equal-weighted index satisfies

¯ RP,t+1 = N X i=1 ¯ wi,tRi,t+1 (28) ≥ E_¯ R∗_P,t+1 , (29) where ¯R_t+1∗ is the average return on the intrinsic value of all the securities. On the other hand, we have7 E_¯ RP,t+1 ≈ E_¯ R∗_t+1 + σ 2 1 + ρ 1 + E _¯ R∗_t+1 . (30)

We observe that the expected return of an equal-weighted index increases with the magni-tude of noise and is at least equal to the expectation of the average of all securities’ return on intrinsic value. However, compared to a cap-weighted index, the equal-weighted index tends to overweight securities with a low market price, while it underweights securities with a high market price.

2.3 Valuation-indifferent weighting: empirical results

Let us turn our attention to the empirical evidence of outperformance of valuation-indifferent weighted indexes.

Arnott et al. (2005) were the first to give an empirical report on valuation-indifferent weighted indexes. They examined the performance of valuation-indifferent weighted in-dexes for the U.S. stock market. Index weights were determined by six fundamental factors, i.e. book value, cash flow, sales, revenues, dividends and employment. For each factor, U.S. companies were ranked according to this specific factor and the top 1000 of these companies were included in the particular factor index. Subsequently, the score on the specific factor relative to the total score of these 1000 companies determined the weight. In addition to the six valuation-indifferent weighted indexes that were constructed in this way, two other indexes were created. For the first, a composite, Arnott et al. (2005) used four of the six factors, excluding revenues and employment. The reason to exclude these

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factors was because revenues and sales are very similar concepts and employment figures are not available for all companies. The weights each company would have had in the book value, cash flow, sales and dividends indexes were combined in equal proportions. In the case of a non-dividend paying company, the average of the remaining three index weights were used. Again, the top 1000 companies where chosen to be included in the index. The second additional index that was created was a cap-weighted one, with the same constituents as the composite index. Since the only difference between this index and the composite was the weighting scheme, the performance of valuation-indifferent weighting relative to cap-weighting could be measured. During the period 1962-2004, the valuation-indifferent weighted indexes of Arnott et al. (2005) had on average an annual outperformance of 2.15% compared to the cap-weighted benchmark. Further, Arnott et al. (2005) showed that when turnover costs are taken into account, these turnover costs must have had exceeded on average 16.04% to eliminate the excess returns. Moreover, the out-performance was robust across time, and additionally, across phases of the business cycle, across bull and bear markets and across periods of rising and declining interest rates.

Hsu and Campollo (2006) reproduced the studies of Arnott et al. (2005) and of Tamura and Shimizu (2005), who found similar outperformance of valuation-indifferent weighted indexes for the EAFE countries8. Other studies presenting outperformance of valuation-indifferent weighted indexes for stock markets are presented in among others Hemminki and Puttonen (2008) and Houwer and Plantinga (2009). While Hsu and Campollo (2006) argued that this outperformance is due solely to the return drag the noisy market implies on the cap-weighted index, Arnott et al. (2005) were more careful by stating that they remain agnostic of the true driver of the observed excess return. They suggested that superior market index construction, the noise in prices, additional exposure to distress risk or a combination of these three is responsible.

The observations of outperformance of valuation-indifferent weighted indexes are not limited to the equity market. Hsu, Li, and Kalesnik (2010) carried the practice of valuation-indifferent weighting onwards to the U.S. and international listed real estate markets. As weight-determining factors they used sales, dividends, book values and cash flow. For the period 1984-April 2009, Hsu et al. (2010) reported an annual outperformance of 3.96% of the valuation-indifferent weighted index for the U.S. compared to its bench-mark, while the global valuation-indifferent index excluding the U.S. offered an annual outperformance of 2.9 % outperformance in the period 1994-April 2009 versus its bench-mark. In addition, the indexes added ‘substantial large value’ in environments subject to deflation (Hsu et al., 2010). Van der Padt and Van Beek (2010) constructed a valuation-indifferent weighted index for the global real estate market. To ensure that it was possible to replicate the index, only the 150 companies with the highest free float market

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ization were included. The companies were equally weighted on EBITDA, rental income and dividend. The index outperformed a cap-weighted benchmark by 2.28% a year in the period 1988-2009. Compared to the GPR 250 Global9 Van der Padt and Van Beek (2010) measured an annual 4.0% outperformance during the period 1990-2009. The research of Van der Padt and Van Beek (2010) extended the work by Vaessen (2007), who constructed an index for the European real estate market. Vaessen (2007) reported outperformance of 1.6% a year in the period 1989-2007 for an index weighted on rental income, EBITDA and dividend with 100 constituents selected on free float market capitalization relative to a cap-weighted benchmark.

Regarding the nature of our own research, the results of Arnott et al. (2010) are more compelling. Instead of applying the valuation-indifferent weighting theory to stock markets, Arnott et al. (2010) constructed valuation-indifferent weighted indexes for the fixed income market, i.e. investment-grade corporate bonds, high-yield corporate bonds and emerging market bonds. Weights for the corporate bonds were determined based on assets, dividends, cash flow and sales. These valuation-indifferent weighted corporate bond indexes, consisting of 500 companies, were shown to outperform the benchmark before costs in the period January 1997-December 2009 by 2.60% and 0.42% a year respectively. Furthermore, the investment-grade corporate bond index showed an underperformance of around 0.10% in bull markets and in markets with a rising federal funds rate, while gaining an excess return of around 2% in bear markets and in markets with a falling federal funds rate. However, the high-yield corporate bond index outperformed in all mentioned markets with 0.6% to 3.5%. Meanwhile, the PIMCO Global Advantage Bond index, is reported to have outperformed the Barclays mainstream Global Aggregate index by 7.7% in the period February 2009 to March 2011 (Johnson, 2011). Similar results were found for the emerging market government bond index constructed by Arnott et al. (2010). This index showed an annual outperformance 1.43% versus the benchmark in the period January 1997-December 2009 and outperformed by 0.4% to 4.5% in bull markets, bear markets and markets with a rising or falling federal funds rate. In constructing the emerging market government bond index, Arnott et al. (2010) could not use book value or cash flow as weight-determining factors for obvious reasons. Instead population, square root of land area, GDP and energy consumption were used as a measure for the current and potential footprint of a country in the world economy. Whereas GDP is the most widely used measure for the size of an economy, population is used as a measure for the available labor in a country10. Further, land area is a crude proxy for available resources,

9

The GPR 250 Global is an index composed of the 250 largest property securities in the world when measured by market capitalization (Van der Padt and Van Beek, 2010).

10_{Note that the size of the working age population would be a better measure, however, this number is}

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where the square root is taken to prevent assigning extreme weights to large but sparsely populated countries like Russia and small, densely populated countries like Hong Kong. Finally, the energy consumption is used to measure the energy needed to produce goods and services. Security selection is done by weighting securities by the face value of the debt, i.e. issue size (Arnott et al., 2010).

2.4 Valuation-indifferent weighting: drawbacks and critique

Not everyone shares the view that ‘we are on the verge of a revolution’ (Siegel, 2006), when it comes to valuation-indifferent weighting. As we have seen, Perold (2007) and Kaplan (2008) have their doubts about the reasoning that the noisy market hypothesis leads to a return drag for cap-weighted portfolios, as is shown by Hsu (2006) and Treynor (2005).

Apart from the purely mathematical discussion on the noisy market hypothesis and its effect on the expected returns of valuation-indifferent weighted indexes and cap-weighted indexes, there are critics that aim at other aspects of valuation-indifferent weighting. Blitz and Swinkels (2008), Bogle and Malkiel (2006) and Asness (2006) view indifferent weighting as being merely value investing. When for instance, a valuation-indifferent weighting index for equity is based on the fundamental factor book value, the weight difference relative to a cap-weighted index is proportional to the book-to-price ratio of the stock relative to the market. Value stocks, those with a high book-to-price ratio, are overweighted, while growth stocks, those with a low book-to-price ratio, are underweighted. Stocks whose book-to-price ratio matches that of the market have the same weight in valuation-indifferent index as they would have in a cap-weighted index. Hence, according to (Asness, 2006), valuation-indifferent weighted indexing ‘is precisely and solely a very clear and definable value tilt away from cap-weighted indexing’. Blitz and Swinkels (2008) speak of valuation-indifferent weighted indexes as being a ‘new breed of value indexes’. Furthermore, they point out the characteristics of valuation-indifferent weighted indexing which are those of an active strategy. First of all, by definition, not every investor can follow the valuation-indifferent weighted strategy, as is the case for the cap-weighted alternative. For all underweight positions in a security relative to its cap-weight, there must be overweight position of the same size in that particular security. Secondly, valuation-indifferent weighting is not a buy-and-hold strategy, since changes in security prices, unrelated to a change in fundamental factors, create a difference between the actual index weights and the valuation-indifferent weights. Hence, weights have to be rebalanced at some time. Finally, the construction of a valuation-indifferent weighted index implies making some subjective choices, for instance the choice of non-price metrics. Bogle and Malkiel (2006) add that higher turnover costs and additional taxes are due to the necessity to rebalance the valuation-indifferent weighted indexes.

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passive strategy that every investor can follow, i.e. if every investor holds a cap-weighted portfolio, the sum of all these portfolios is the market itself, and liquidity might not prove a big problem. While if every investor could hold a valuation-indifferent weighted portfolio, the sum of these portfolios would not fit the supply of the market. Hence, liquidity and market access need to be screened to ensure that it is possible to replicate a valuation-indifferent weighted government bond index (Toloui, 2010). However, there is room for discussion on the other points mentioned by the critics of valuation-indifferent weighted indexes. First of all, the proponents of valuation-indifferent weighting do not entirely disagree with the view that their concept resembles a value strategy. In the Fundamental Index R _Newsletter11 _{of August 2007, it is stated that compared to cap-weighting, most}

valuation-indifferent weighted index strategies of Research Affiliates LLC have some degree of a value tilt. However, this value tilt is argued to be dynamic, i.e. the value bias of the index varies over time. Furthermore, a value strategy typically excludes growth stocks, while a valuation-indifferent index includes those, albeit by assigning them a relatively low weight. Finally, we at least partly enfeeble the worries of Bogle and Malkiel (2006). That is, we recall the empirical results discussed in the previous section that show outperformance of valuation-indifferent weighted indexes relative to their cap-weighted counterparts, even after considering higher turnover costs.

All in all, given the empirical evidence on the success of valuation-indifferent weighting in different markets, we are led to belief that a valuation-indifferent weighted index can be an attractive alternative to cap-weighted indexes in the government debt market, and more specific, the Euro Zone government debt market.

11

Note that the Fundamental Index R

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3 Existing GDP weighted government bond indexes

From January 2009 to July 2010, PIMCO launched three valuation-indifferent weighted bond indexes, i.e. the PIMCO Global Advantage Bond Index, the Global Advantage Gov-ernment Bond Index and the European Advantage GovGov-ernment Bond Index. The country weights in these PIMCO indexes are determined by GDP, i.e. countries are weighted by their economic size rather than the size of the outstanding debt. Barclays Capital followed a similar approach in constructing among others their Barclays Capital Global Treasury GDP Weighted Index and GDP is but one of ten macro-economic indicators that are used to determine the weights in the Lombard Odier Fundamental Weight Driven Funds. In Section 2 we discussed the theoretical advantage of valuation-indifferent weighted indexes, the empirical evidence of outperformance of those indexes and the possible disadvantages of this weighting scheme, such as higher turnover costs. Irrespective of these advantages and disadvantages, we construct our own GDP weighted Euro Zone government bond index. Before we start with setting up construction rules, we compare those that led to the PIMCO Global Advantage Government Bond Index (PIMCO, 2010b), the PIMCO European Advantage Government Bond Index (PIMCO, 2010a) and the Barclays Capital Global Treasury GDP Weighted Index (Barclays Capital, 2009).

3.1 PIMCO Global Advantage Government Bond Index

The first step in the construction of the PIMCO Global Advantage Bond Index is to divide the world into five economic regions. These five regions are the United States, the Euro Zone, Japan, other industrialized countries12 _{and emerging markets}13_{. These regions are}

assigned weights based upon GDP relative to the GDP of the global economy, where GDP is measured by the average GDP of the last five years. GDP data is provided by the International Monetary Fund and regional weights are recalculated on a yearly basis. New weights are published on September 15 and become effective on October 31.

Within these regions, countries need to meet certain requirements in order to be in-cluded in the index. These are:

• The size of the bond market: For industrialized countries the size of the bond market has to be leastwise 5 billion USD, whereas emerging markets have to have a size of at least 10 billion USD.

• Credit rating: Countries have to be rated by at least one of the three rating agencies Standard & Poor’s, Moody’s or Fitch. The average of these ratings has to be at

12

Australia, Canada, Danmark, New Zealand, Norway, Sweden, Switzerland and the United Kingdom.

13

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least investment grade14.

• The number of bonds in issue: Industrialized countries have to have leastwise five bonds in issue, while emerging markets are required to have at least three bonds in issue.

The following additional rules are set for emerging markets:

• Investability: Markets with lack of liquidity or extensive capital controls and access restrictions are excluded from the index, to assure that the index is investable. • Domestic currency: The emerging markets must have a local currency which is

unequal to the currency of the U.S., Japan, Euro Zone or the other industrialized countries.

Country weights are recalculated annually, and rebalanced on a quarterly basis. I.e. once a year, target country weights are recalculated, while each quarter actual weights are rebalanced towards these target weights. However, on a monthly basis, countries that fall below investment grade are excluded from the index. Note that this implies a recalculation of the country weights for the remaining countries.

The last step in the process is the security selection, which starts by selecting only fixed-rate, non-callable bullet bonds or sinking funds with a remaining maturity of at least 12 months. The minimum issue size naturally depends on the currency in which the bonds are nominated. For bonds denominated in euros the minimum size is 2 billion, as well as for bonds denominated in pounds sterling or USD. The target number of bonds to be selected per country depends on the size of the market, for markets of less than 500 billion USD the maximum number of bonds is 15, while for markets equal to or larger than 500 billion USD the maximum number of bonds is 50. Before making a selection, the bonds are classified according to their remaining maturity in maturity groups:

• Group 1: 1-3 years • Group 2: 3-5 years • Group 3: 5-10 years • Group 4: 10-15 years • Group 5: 15-20 years

• Group 6: more than 20 years

14_{Since the credit ratings of the three agencies are quoted in characters rather than numbers, taking the}

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where for markets smaller than 500 billion USD, the maturity groups 1 and 2 are merged into one group.

Subsequently the cap-weight of each maturity group is calculated, i.e. the market value of all the bonds within a maturity group divided by the market value of all maturity groups combined. These cap-weights are then multiplied by the target number of bonds for the country to determine the number of bonds within each maturity group. For example, when the target number of bonds for country A is 50 and the cap-weight of maturity group 4 is 10%, then the number of bonds to be selected from maturty group 4 is five. In selecting bonds within a maturity group, large bonds and bonds with an original maturity close to the lower bound of the maturity group are preferred.

3.2 PIMCO European Advantage Government Bond Index

As our research focuses on constructing a valuation-indifferent index for the Euro Zone, we are especially interested in the construction rules for the PIMCO European Advantage Index, as this index covers the same region as the index we construct. The rules of the Eu-ropean Advantage Index differ only slightly from those of the PIMCO Global Advantage Index described above. Naturally, the step of dividing the world into economic regions is skipped. Hence the first step is selecting the countries that meet the requirement described above and assign them weights based on their share of Euro Zone GDP. Furthermore, the securities included in the index must have a minimum par amount of 2 billion euro out-standing, similar to what is required in the global index. By July 2010 the index included Austria, Belgium, Finland, France, Germany, Ireland, Italy, Netherlands, Portugal and Spain. This implies that Cyprus, Greece, Luxembourg and Malta are excluded based on one of the rules described above. Whereas Cyprus, Luxembourg and Malta are excluded because of their small size, Greece is ignored because of their credit rating. The credit rating of Greece was downgraded to non-investment grade Ba1 by Moody’s in June 2010, similar to an earlier downgrade by S&P (Gongloff, 2010).

3.3 Barclays Capital Global Treasury GDP Weighted Index

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Figure 1: Barclays Capital Treasury Bond GDP Weighted Index Regions (Barclays Capi-tal, 2009).

The first step in constructing the index is to assign GDP weights to these ten country blocs. In calculating the GDP weights of these country blocs, only the GDP of coun-tries with meaningful representation - at least 0.25% of total market capitalization of the country bloc - in the index are used, in order to prevent an overallocation to countries or country blocs which lack a certain liquidity of their capital markets. This requirement can be compared with that of the PIMCO indexes, in which markets need to have a certain size. As a measure for GDP, a weighted three year average is used. This average consists of 1/2 times the GDP figure of last year, 1/3 times the GDP figure of two years ago and 1/6 times the GDP figure of three years ago. In this way, the average reflects the most recent developments, while incorporating some historical perspective. The data on GDP is provided semi-annually by the World Economic Outlook (WEO) of the Interna-tional Monetary Fund (IMF). Furthermore, the country bloc weights are rebalanced on a monthly basis towards their GDP weights.

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the lowest rating is used.

Instead of the extensive security selection procedure of the PIMCO Advantage Bond indexes, the securities in the Barclays Capital GDP Weighted Treasury Bond Index are selected by cap-weighting. All included securities must be fixed-rate with a remaining time to maturity of at least 12 months and have a certain minimum issue size. In the case of Euro Zone countries, this minimum issue size is 300 million euro.

3.4 Comparison of the existing indexes

We have seen that although the indexes of PIMCO and Barclays Capital are all GDP weighted indexes, they do differ in some respect. Since we are interested in constructing an Euro Zone index, we are not concerned with the discussion on the difference in clas-sifying economic regions. We naturally arrive at the choice of a measure for GDP. Both PIMCO and Barclays Capital use historical data of the IMF, PIMCO uses a five-year equally weighted average, while Barclays Capital uses a three-year weighted average in which recent GDP values receive a higher weight. While the former approach provides more stable weights at the cost of being able to adjust quickly in the case of strong GDP growth or decline, the latter approach seems to capture recent GDP developments better. However, during the last few years we have seen more than once that downward adjust-ments had to be made to published GDP figures, for example for Greece. Hence, to assess the impact of the choice for one of the two GDP averages on performance of the GDP weighted index, we use both weighting methods to construct a GDP weighted Euro Zone government bond index and compare them.

An investment-grade credit rating is a requirement for both indexes of PIMCO and the index of Barclays Capital. The result for a Euro Zone index is that Greece government bonds were excluded at least after the downgrade to non-investment grade of Moody’s in June 2010.

Requirements regarding the size of the bond market, the size of the issue and remaining time to maturity are used to ensure the liquidity of the indexes. Sarig and Warga (1989) argue that over time, an increasing number of bonds of a particular issue are absorbed into investor’s portfolios, with the result that bonds of that issue become illiquid. Hence, remaining maturity is a proxy for the liquidity of bonds. The reasoning of size issue as a proxy for liquidity stems from the intuition that the size of an issue is positively correlated with the amount of trades.

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4 Our GDP weighted Euro Zone government bond indexes

Now that we have discussed the GDP weighted index of PIMCO and Barclays Capital, we turn to the construction of Euro Zone GDP weighted government bond indexes. The first index, referred to as GDP weighted index I, uses country weights that are determined by the three-year weighted average of GDP as used in the Barclays Capital Global Treasury GDP Weighted Index. The second index’ country weights are determined by a five-year equally weighted average of GDP. We refer to this index as GDP weighted index II. In Section 4.1 we consider the construction rules for the indexes, while we present the empirical research on both indexes in Section 4.2. All computations on the indexes are performed in MATLAB.

4.1 Construction rules

We start with devising our own construction rules for a GDP weighted Euro Zone govern-ment bond index. We need rules for the countries to be included as well as rules for which securities are selected within each country.

4.1.1 Country selection

Naturally, we only select countries from the Euro Zone to be included in the index. Before we determine the weights of each country in the index, we exclude any country that does not meet one or more of the following requirements:

• Country has to have the euro as their accounting currency for at least 10 years as of January 1 2011. This requirement ensures that we have enough historical data for each country, such that our conclusions regarding the performance of the GDP weighted index is representative.

• The share of the country’s GDP has to be at least 1% of the total GDP of the Euro Zone. We impose this requirement to exclude countries that have less liquid capital markets.

The first requirement forces us to exclude Cyprus, Estonia, Malta, Slovenia and Slovakia from our index. As for the GDP figures, we use IMF data accessed through Thomson Reuters Datastream. The averages we compute are based on historical figures only, we do not include any forecasted figures. The IMF data leads us to exclude Luxembourg from our index because it does not meet the requirement to have a share of 1% of the total GDP in the Euro Zone.

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January 1, 1999. Country weights are recalculated annually after the release of the IMF World Economic Outlook report in October. New weights are applied after market closure on the last trading day before January 1. Note that Greece introduced the euro as their accounting currency on January 1, 2001. Hence, Greece enters the index in 2001. We saw that the country weights of the GDP weighted indexes of PIMCO and Barclays were rebalanced at least once a year back to their target GDP weights. The argument is that market fluctuations cause the country weights to drift in the direction of cap-weights, so rebalancing is necessary to capture the effect of the alternative weighting measure. However, Arnott et al. (2005) noted that an increase in the frequency of rebalancing increases turnover, without delivering a reasonable return advantage. Therefore, Arnott et al. (2005) reweighted their equity indexes once a year. For our GDP weighted indexes we choose to rebalance annually, since the differences in returns for the GDP weighted index I are not materially significant if rebalanced semi-annually, quarterly or monthly. 4.1.2 Security selection

The next step is to set up rules for the security selection per country in the index. These rules need to ensure that our index is liquid and transparent, such that the index can be replicated. I.e. it is possible to invest in a product that replicates the index and its returns. As we have seen in Section 2.3, Arnott et al. (2010) weight the securities on issue size, while the weight of securities in the indexes of PIMCO and Barclays Capital are determined at least partly by market capitalization. One of the advantages of weighting securities on market capitalization in our index is that we can use existing country indexes to create our GDP weighted Euro Zone indexes. While in the case of issue size weighting we have to construct indexes for each country separately before we can construct the GDP weighted indexes. The main disadvantage of using market capitalization as the determination of the weights of the securities in the index is the reason why we examine GDP weighted indexes, i.e. the noisy market hypothesis and its potential effect on a cap-weighted index. Since the use of existing country indexes greatly reduces the complexity of the con-struction of our GDP weighted indexes, we assess by means of a case study whether issue size weighting offers significantly different returns from cap-weighting. We construct both an issue size and market capitalization index for Finland. We choose Finland, because the number of issues of debt during the period from January 1 1999 to January 1 2011 is rela-tively small, which makes the construction of the index relarela-tively easy. We recognize that the index by no means completely captures the differences or lack of differences between the two weighting schemes. However, it does give us insight in index construction on a security level and it might give us an idea of the differences between the two weighting schemes.

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in euros with a remaining time to maturity of at least 12 months. Furthermore, we only include bonds with an issue size of at least 2 billion euro. These rules are set on basis of the rules of the indexes discussed in Section 3 and the arguments that issue size and time to maturity are proxies for liquidity (Sarig and Warga, 1989). Each month, at the end of last trading day, the characteristics of all bonds in the index are evaluated. When bonds from a particular issue fail to meet the requirements on issue size or time to maturity they are excluded from the index. On the other hand, bonds from new issues that do meet the requirements are included in the index. This implies that index weights are recalculated and rebalanced monthly. In the period January 1 1999 to January 1 2011, Finland government bonds from 19 different bond issues qualified to be included in the index.

Coupon payments received during the month are assumed to accrue interest at the Euribor 1 month interest rate. The total proceeds are reinvested in the index at the next rebalancing of the index weights. We use the last bid prices per trading day to price the bonds. If a price is not available at some date, we use the most recent price available at that specific date.

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Jan/9990 Jan/00 Jan/01 Jan/02 Jan/03 Jan/04 Jan/05 Jan/06 Jan/07 Jan/08 Jan/09 Jan/10 100 110 120 130 140 150 160 170

Issue size weighted index Cap−weighted index

(a) Total return (%).

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 0 1 2 3 4 5 6 7

(b) Annualized returns (%) until December 31 2010, various starting years with January 1 as start date. 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 0 2 4 6 8 10 12 14 16

(c) Difference in annualized returns (bps).

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Jan/99 Jan/00 Jan/01 Jan/02 Jan/03 Jan/04 Jan/05 Jan/06 Jan/07 Jan/08 Jan/09 Jan/10 −20 −15 −10 −5 0 5 10

(a) Difference in monthly returns (bps).

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 0 5 10 15 20 25 30

(b) Turnover (%).

Figure 3: Issue size weighted versus cap-weighted Finland index.

Figure 3(a) presents a graph of the difference in monthly returns between the two indexes. Most noticeable are the figures for the period 1999 to 2003, with differences ranging from around minus 20 to plus 10 bps. Four cases of a large difference in returns occur in March, which might indicate that the differences are caused by annual coupon payments on one of the constituents in that month. When we consider annual turnover15, we observe in the graph in Figure 3(b) that the issue size weighted index on average implies a higher turnover rate.

To find an explanation of the large monthly differences observed in the graph in Figure 3(a), we examine the contributions of the individual issues to the monthly return of the indexes in the months March, September 1999, March 2000, March 2001 and March 2002,

15

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i.e. the months that showed a difference of more than 10 bps. The results are found in Table 116

We observe that the difference between the two indexes are mainly caused by the issue denoted by E. This issue is characterized by a high coupon rate, i.e. 9.5%, which is paid out in March. From the results in Table 1 we deduce that this issue is underweighted in the issue size weighted index relative to the cap-weighted index in the months we highlighted here. When we look at the other issues, we observe that in general high-coupon issues are underweighted in the issue size weighted index and low-coupon issues are underweighted. As the high-coupon issues fell out of the index in 2003, the difference between coupon rates of the constituents declined and hence the difference in monthly returns between the issue size weighted and cap-weighted index became smaller.

In order to say something about the sign of the difference between the monthly re-turns of the issue size weighted index and the cap-weighted index, we perform a test on the difference of means. Our null hypothesis is that there is no difference, against the alternative that monthly returns on the issue size weighted index are larger than those on the cap-weighted index. Before we can perform a one-sample t-test on the differences in monthly returns, it is important to test the differences for autocorrelation. We test for autocorrelation by means of a Ljung-Box Q test, i.e. the test statistic is given by

Q = n(n + 2) h X k=1 ˆ ρ2_k n − k, (31)

where n is the number of observations, ˆρk the sample autocorrelation at lag k and h is

the number of lags included in the test. We have that n = 144 and we let h = 36, i.e. a lag of at most 3 years. The choice for h = 36 is based on the size of the sample auto-correlation coefficients. We have that Q = 163.29 which gives a p-value of 0. Hence, at a significance level of 0.05, we reject the null hypothesis that the observations are inde-pendently distributed against the alternative that the observations are not indeinde-pendently distributed.

Since we suspect autocorrelation in the difference of returns, we have to adjust the standard one-sample t-test. I.e. when ignoring positive autocorrelation, the p-value of the t-test is underestimated. Hence we use the t-test proposed by Albers (1978) which is given by τ = √ n ¯d q s2_d(1 + 2Ph k=1ρˆk) , (32)

where ¯d is the mean of the monthly differences and s2_d is the variance of the monthly differences. Our t-test gives a p-value of 0.87, hence we do not reject the null hypothesis

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of no difference in monthly returns against the alternative of the issue size weighted index having higher returns than the cap-weighted index.

The fact that our Finland indexes include only a limited number of constituents raises the question whether the comparison between the issue size weighted index and the cap-weighted index is representative. Therefore we examined the difference between the two weighting schemes for Belgium as well17. The results are similar to those of the Finland indexes, although the differences in returns are somewhat larger and the turnover rates for both indexes are lower. While we keep in mind the advantages and disadvantages of issue-size weighted indexes, i.e. higher returns versus higher turnover, we choose to use existing cap-weighted country indexes to construct our GDP weighted indexes. The indexes we use are the EFFAS/Bloomberg indexes that include all securities with a remaining maturity larger than 12 months. As a benchmark for our GDP weighted indexes we weight the individual country indexes on the basis of their cap-weight.

4.2 Performance of the GDP weighted indexes

Figure 4 shows the performance of the two GDP weighted Euro Zone indexes compared to the cap-weighted benchmark. I.e. the GDP weighted index I, which is the index based on the weighted three year average of GDP and the GDP weighted index II, in which constituents are weighted based on their five year equally weighted average of GDP. In the graph in Figure 4(a) we observe that the GDP weighted indexes match the cap-weighted benchmark closely. However, they do show a higher total return. While the total return of the cap-weighted benchmark was 67.58% on December 31 2011, the GDP weighted indexes delivered a total return of 69.19% and 69.28% for GDP weighted index I and II respectively.

The differences in total returns between the two GDP weighted indexes and the cap-weighted benchmark remained relatively small until 2008. As the financial crisis began to take form in Europe, the difference began to grow larger, as is seen in Figure 4(b) and Table 2. Figure 4(b) shows the development of the total returns on the GDP weighted indexes relative to that of the cap-weighted benchmark in the period from January 1 2008 to December 31 2010. We see that most of the outperformance of the GDP weighted indexes is generated in the year 2010, while 2009 proved to be a relative good year for the cap-weighted benchmark. Table 2 shows the annual returns on the three indexes over the whole sample period. Again we see a clear distinction between the years before and after January 1 2008. Only in 2001 the difference in returns surpassed the level of 10 bps in the period until 2008, while from 2008 to 2010 the difference was at least 36 bps. Overall, the average annual outperformance of the GDP weighted indexes was around 8 bps.

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Jan/9990 Jan/00 Jan/01 Jan/02 Jan/03 Jan/04 Jan/05 Jan/06 Jan/07 Jan/08 Jan/09 Jan/10 Jan/11 100 110 120 130 140 150 160 170 180 GDP weighted index I GDP weighted index II Cap−weighted benchmark

(a) Total return (%).

Jan/08 Jan/09 Jan/10 Jan/11

−20 0 20 40 60 80 100 120 140 160 GDP weighted index I GDP weighted index II

(b) Relative cumulative performance (%): January 1 2008 - January 1 2011.

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To explain the difference in returns of indexes we examine the contribution of each country weight to the annual performances of 1999 to 2001 and 2008 to 2011 of the cap-weighted benchmark and the GDP weighted index II. in Table 3. A comparison of the GDP weighted index I relative to the cap-weighted benchmark is omitted, since the two GDP-indexes show similar results, with the GDP weighted index II showing slightly better returns. While the first three years showed little difference in the total return of the indexes, the contributions show that on a country level there existed some notable differences. Belgium, Germany and Italy deserve our attention in that respect. In 1999 Germany showed a negative return for both indices, allbeit that the GDP weighted index II lost 24 bps more than the cap-weighted benchmark. This implies a heavier weight for Germany in the GDP weighted index II. In 2000 Germany still received an overweight and with Germany offering positive returns, the GDP weighted index profited and earned 81 bps more on its Germany holding than the cap-weighted benchmark. The difference in returns on Belgium and Italy on the other hand signal a relative underweight for these countries in the GDP weighted index. In Table 2 we observed that the GDP weighted indexes lost 14 bps relative to the cap-weighted benchmark in 2001. From Table 3 we deduce that this underperformance is largely due to the underweight in Belgium and Italy.

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In our examination of the index above, we have considered only one investment horizon, that of an investor who starts to invest in the indexes at January 1 1999 and takes the profits at January 1 2010. In Figure 5 we present two graphs for various investment horizons. The graph in Figure 5(a) shows the annualized returns of the GDP weighted index II and the cap-weighted benchmark for various starting years and January 1 as starting date. Like in the case of the Finland indexes, January 1 2008 was the best moment to invest in the Euro Zone indexes. The differences between the annualized returns are shown in the graph in Figure 5(b). We see that regardless of the starting year, the GDP weighted index II offered an excess return at December 31 2010.

1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 0 1 2 3 4 5 6 GDP weighted index II Cap−weighted index

(a) Annualized returns (%) until December 31 2010, various starting years with January 1 as start date. 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 0 20 40 60 80 100 120

(b) Difference in annualized returns (bps).

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Lastly, the graph of monthly excess returns of the GDP weighted index II in Figure 6 shows that the largest part of the outperformance was realized in 2008 and 2010.

Jan/99 Jan/00 Jan/01 Jan/02 Jan/03 Jan/04 Jan/05 Jan/06 Jan/07 Jan/08 Jan/09 Jan/10 −20 −10 0 10 20 30 40 50

Figure 6: Monthly excess returns (bps) of GDP weighted index II over cap-weighted benchmark.

Given our observation that the GDP weighted indexes showed an outperformance over the last three years of our sample period, while staying close to the benchmark in the less volatile years, we hold the belief that valuation-indifferent weighting is a good alternative to the practice of cap-weighting in government bond indexes. In the next subsection we examine which factors, in addition to GDP, could be used to weight the constituents in a valuation-indifferent weighted index.

4.3 Additional factors to determine index weights

From the study on GDP weighted indexes for the Euro Zone, we have seen that valuation-indifferent weighting might add value compared to cap-weighting. While the excess return of the GDP weighted index was negligible during the years 1999 to 2008, it changed dramatically from the years onwards. The main advantage was that the GDP weighted indexes had an underweight in Greece and an overweight in Germany. An underweight for Greece implies that their share of total Euro Zone debt exceeded their share of total Euro Zone GDP. However, although Greece had a public debt to GDP ratio of more than 100% for years now, it does not hold that only countries that do exceed this threshold are underweighted in the GDP index. Austria for example, has a share of total Euro Zone public debt that exceeds its share of total Euro Zone GDP, without having a huge public debt burden, as measured by the public debt to GDP ratio.

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index that tracks the unemployment and inflation rates. The factors we use in our index need to be indifferent to the market price of debt in order to profit from the return drag on cap-weighted indexes showed in Section 2.2.

When it comes to the credibility of a debtor, it is essentially about the sustainability of its existing debt. One of the most common measures of the sustainability of public debt is the ratio of public debt to GDP, or the evolution of this ratio (IMF, 2000). The measure is quite intuitive, as you might be more eager to lend money to someone whose debt is only a fraction of his income, than to someone whose debt is as big or even bigger than his income, regardless of the total size of the income. I.e. when we compare the economies of Greece and Finland, we observe that Greece has had a larger GDP than Finland since 2000. However, in 2008 the debt to GDP ratio of Finland is about 50 % against over 140 % for Greece. According to this measure, buying Greek government bonds seems less attractive than when considering the GDP figures only. Reinhart and Rogoff (2010) examined the relation between GDP growth and the level of the public debt to GDP ratio. They showed that for twenty developed countries18, a public debt to GDP ratio above 90% was associated with notably lower levels of growth over the period 1946-2009. A drawback of the public debt to GDP ratio as a measure of the ability of a country to finance its debt, is that some governments may not be able to raise revenues from GDP to finance its spending and service of debt (Roubini, 2001; INTOSAI, 2010)19. Hence for these countries, the public debt to GDP ratio might overestimate their credibility. However, despite its shortcomings, it is recognized to be the most important indicator for the indebtness of a country. As such, the debt to GDP ratio is part of the Maastricht criteria, a set of monetary and fiscal rules the Euro Zone members committed themselves to. According to these criteria, the ratio should be below 60% (Buiter, Corsetti, and Roubini, 1993). Given its importance and informative nature, we use the debt to GDP ratio as a factor for our valuation-indifferent indexes. Since debt in the debt to GDP ratio is measured on basis of the face value of debt (IMF, 2000), using this factor to compute index weights does not expose us to the possible return drag caused by noise in prices.

Cecchetti, Mohanty, and Zampolli (2010) point out that a high debt to GDP ratio alone does not cause investors to doubt the credibility of a country. After World War II the United States reached a level of 121%, while the United Kingdom peaked at levels of around 300% without experiencing a default. The key to managing these high debt levels is to achieve a budget balance such that the debt level is stabilized, or even better, that the debt level is decreased. The Maastricht criteria describe the collective aim of

18

Australia, Austria, Belgium, Canada, Denmark, Finland, France, Germany, Greece, Ireland, Italy, Japan, Netherlands, New Zealand, Norway, Portugal, Spain, Sweden, the United Kingdom and the United States.

Valuation-indiﬀerent weighted indexing for Euro Zone government bonds

Valuation-indifferent weighted indexing for Euro Zone

government bonds

Valuation-indifferent weighted indexing for Euro Zone

government bonds

Glossary of terms

Contents

1

Introduction

2

Are markets efficient?

3

Existing GDP weighted government bond indexes

4

Our GDP weighted Euro Zone government bond indexes