Trading behavior of the Dutch Insurance companies in view of sovereign downgrades

(1)

Trading behavior of the Dutch Insurance companies in

view of sovereign downgrades

(2)

University of Groningen

(3)

1 Introduction

From 2009 on, investors became aware of the Sovereign debt crisis due to rising private and government debt levels around the world and downgrades of government debts. In particular, before 2009 the debt crisis was mainly associated with emerging and developing economies hence it was a new phenomena for developed countries (Reinhart, 2010).

With the advent of the debt crisis, for several European countries such as Portugal, Greece, Spain, Ireland and Italy the sovereign credit ratings dropped to their historically lowest levels 1_{. At the start of 2006, Standard & Poor's (S&P) rated Italy, Ireland, Greece, Spain and} Portugal with AA−_{, AAA, A, AAA, AA}− _{credit ratings but at the end of 2012, listed in the} same order, as only BBB+_{, BBB}+_{, SD, BBB}− _{and BB.}

The debt crisis has led to revisiting the discussion on how investors react to the sovereign credit rating downgrade announcements. Afonso, Furceri and Gomes (2012) use daily data on bond yields and Credit Default Swap (CDS) spreads for the European Union (EU) countries and nd evidence that government bond yields react to changes in rating categories and to outlooks. These eects are more prominent in the case of a negative announcement. They also nd rating announcement spill-over eects from lower rated countries to higher rated countries. Curiously, they nd that countries that have been downgraded less than six months ago face higher spreads than countries with the same rating which have not been downgraded within the last six months.

1_{S&P is one of the three largest rating agencies which assigns the credit worthiness grades to bonds,}

(6)

Hite and Warga (1997) research how the changes in corporate credit ratings impact bond prices of bonds listed on NYSE. They study bond prices 12 months before and 12 months after the rating change. They nd evidence that downgraded rms do reveal a signicant announcement eect in the announcement month and in the preannouncement period. Moreover, they show that the strongest reaction is when a bond moves to a non-investment grade. Comparing the impact of Moody's versus S&P credit rating announcements, they conclude that both rating agencies play a similar role in terms of the information they provide to the public debt markets. Hand et al. (1992) report asymmetric results with respect to rating change downgrades and upgrades. They observe signicantly negative average excess bond and stock returns for downgrades, but weaker positive average excess bond returns for upgrades. They also report that for downgrades, the average excess bond returns are stronger for below investment grade bonds than for investment grade bonds.

In general, the previous literature analyze the response of investors to downgrades based on observed bond price eects, yield spreads, returns and CDS spreads. For example in the context of examining change in returns, the standard event study methodology is to link the rating event to abnormal returns. Thereafter, it is established if there is a systematic relation between the abnormal returns and the credit rating downgrade announcements in a specied time window (Hite and Warga, 1997). Alternatively, Afonso, Furceri and Gomes (2012) construct an event study to examine change in average bond yield spreads over Germany during occurrence of a credit rating event.

The dataset used in this study allows for a more rigorous analysis. This strictly condential dataset is provided by De Nederlandsche Bank (DNB) 2 _{and has detailed information on the} trade ows and positions in bonds held by the Dutch insurance companies. To measure the reaction to a downgrade, instead of one return observation at time t, the dataset contains up

(7)

to 4800 transaction observations on dierent countries, depending on the number of insurance companies participating in the sample at that time period and trading the bonds of some particular country. This allows for a more precise estimation of the downgrade eect on the insurer's behavior.

In order to estimate the eect of a downgrade, I follow two approaches. The rst approach is to examine directly the impact of a downgrade announcement on the trade ows of insurance companies. For this purpose, the Trade Flow model (TF) is developed, relating the trade ows to the credit rating announcements, nancial and macroeconomic variables in the panel dataset context. The second approach, is to examine how the Dutch insurance companies react to downgrades as compared to the overall market reaction. This is examined using the Structural demand model (SD).

The idea of the Structural demand model was introduced by Friedman (1977) and was originally intended to model the implied equation for the long term interest rates. In this model the impact of a credit event on asset demand is not considered. There are two important dierences between the dataset of Friedman (1977) and the one used in this paper which necessitates adaptation of the original model. First, the original dataset used by Friedman (1977) contains the ow-of-funds data on only six investment categories such as long term corporate bonds, holdings of residential mortgages etc. Due to the broad asset aggregation, Friedman (1977) was able to model desired equilibrium holding of these assets as a function of returns on alternative investment categories. In the new setting, given the large choice of bonds issued by various countries, this is not anymore possible. Instead of modeling the equilibrium holding, the Linear Homogeneous Selection model (LHS) proposed by Brandt et al. (2009) is used in this study.

(8)

funds, life insurance companies, other insurance companies etc. Given this level of aggregation, Friedman (1977) modeled demand of each investor category separately. In this paper, there is information on an individual investor level and estimation is performed simultaneously for all investors accounting for time invariant, unobserved individual eects in the panel data setting. The nding of this paper is that the credit rating announcements do provide new information to the Dutch insurance companies. In particular, in response to downgrades, investors reduce their net investments in bonds of a downgraded country. There was no evidence that credit rating upgrades would increase these investments. For the net trade ows, the strongest reaction to downgrades is within BBB bond rating category. The role of economic variables in explaining net investment ows turned out to be rather limited.

The remainder of this paper is devoted to the analysis of the impact of credit rating downgrades on the trading behavior. It is organized as follows, Section 2 discusses literature relevant for the model construction such as explanatory variables and methodological approaches. In Section 3, I introduce notation used in this paper. In Section 4 exogenous variables of this paper are discussed in detail. Thereafter, in Section 5 two methodological approaches to estimate the credit rating downgrade eect on investment positions will be discussed. Section 6 describes the data for empirical estimation and estimation results. Finally, Section 7 concludes.

2 Background

In this Section I shall review relevant literature on the variables used to explain the investment ows and on alternative methodological approaches used to examine the eect of downgrades on the investment ows. There are several references in the literature regarding variables explaining the investment ows.

(9)

investment ow of securities is positively correlated with the market return and the previous weeks' returns. Depending on either there is a positive or a negative correlation between the investment ows and returns, they classify investors as belonging to the positive or negative feedback traders.

Robin and Bierlaire (2012) use ve stock indicators to model an investor's buying, selling or holding decision. These indicators are provided by an actual investment bank. The calculations of these ve indicators are condential. However it is known that they capture quality, sentiment, technique, value and price of a stock. The Quality indicator is said to measure the fundamental quality of a company and is based on specic nancial and economic data. The Sentiment indicator is a measure based on a combination of estimates such as next year earning estimates. The Technique indicator is again a combination of indicators that analyze the companies activities on the market by identifying a chart pattern of prices, such as the momentum reversal. This momentum is dened as the dierence between two prices of the same stock for a chosen time horizon. The Value indicator is an objective measure of a company and a classic valuation metric from which it is derived could be, for example, price to earnings ratio. Finally, The Price indicator characterizes the price of a stock associated with the company. Robin and Bierlaire (2012) conclude that according to investors, these indicators constitute the main information for making decisions.

(10)

productivity shocks against Germany. However this variable turns out not to be of relevance. In order to account for persistence of spreads, lagged spreads are used. To represent macro fundamentals, they use growth rate of industrial production as an indicator for change in credit worthiness.

Catren et al. (2006) examine which news leads investors to trade in FX markets. Among other return components, they examine the link between macroeconomic fundamentals and exchange rate returns. The macroeconomic news variables included in the investigation were CPI, purchasing managers' index, GDP, industrial production, unemployment rate, retail trade, trade balance, business condence and ECB interest rate announcements. Their nding is that the osetting impact on the various return components can blur the eect of macroeconomic data releases on aggregate FX excess returns.

Event study methodology is the standard practice to examine the impact of downgrades on investors' reaction. For example, Hand et al. (1992), Hite and Warga (1997) and Afonso, Furceri and Gomes (2012) relate rating events to abnormal dierences in the yields and CDS. They compare model generated movements based on times when no rating event takes place with actual movements. The returns generated by the model are dened by regressing returns on the market portfolio returns. The regression coecients for the model generated returns are calculated based on periods free of rating events. Thereafter average abnormal bond returns during, for example, a period of 30 days before and after a rating event are examined to see if rating changes exert a sustainable impact on bond returns.

(11)

changed signicantly over time. The turning point is the onset of the debt crisis. The conclusion is that since summer 2007 the movements in macro and scal fundamentals explain movements in spreads well. Moreover, since 2009, markets are pricing risks which were not considered relevant before and within the crisis.

In this paper I shall follow the regression approach to model the trading behavior of investors in reaction to sovereign downgrades. Further details on the particular kind of methodology will be discussed next.

3 Notation

Before proceeding with describing each model separately, common notation shared by both models needs to be introduced.

Consider at time t for t = 0, ..., T an investor j for j = 1, .., k who has wealth Wj t. This wealth is invested into bonds of dierent countries i for i = 1, ..., n. Denote the invested amount in country's i bonds at time t by yj

i,t then W j t = ι0ny j t. Let a n vector w j t keep a record on each country's weight in the portfolio. The ith entry of this vector is

wj_i,t = y j i,t

W_tj. (1)

By time t the country i has issued total debt which is denoted by di,t. All countries together have issued dt =Pn_i=1di,t. Given these denitions it is possible to introduce so called market weights for each country wM

i,t

wM_i,t = di,t dt

. (2)

(12)

sells same country's bonds during the same month. Let bj

i,t+1 and s j

i,t+1 denote bought and sold values of bonds. The buys of bonds can be nanced by selling some other bonds or by using additional cash ow. Let Fj

t+1 denote additional cash available to investor j at time t + 1. At any time, all of it will be invested in bonds so that F_t+1j =Pn

i=1(b j i,t+1− s j i,t+1)from this value Fj

i,t+1 is additionally invested in the country's i bonds.

In practice, by time t one country has issued many types of bonds which are available on the market thus there does not exist just a single bond. Assume for the moment that at time t the country i has issued s bonds for s ∈ Z. Let yj

is,t denote how much the investor j has

invested in the sth bond issued by the country i. This value will change over time due to price changes and due to additionally invested cash. The cumulative change of invested value in all bonds issued by the country i is

∆yj_i,t+1=X s∈Z

(y_ij_s_,t+1 − y_ij_s_,t). (3)

In particular, denote the change in value due to price mutations by ∆˜yj

i,t+1 then

∆˜y_i,t+1j =X s∈Z

(yj_i_s_,t+1− yj_i_s_,t) − F_i,t+1j . (4) Let Pis,t denote a price of the sth bond. From time period t to t + 1 the return on this bond

is given by r_i_s_,t+1 = _P is,t+1 P_i_s_,t − 1 . (5)

At this point it is convenient to use a weighted average to aggregate these multiple returns for each country separately. The weighted average return of the country i is investor specic since it depends on the combination of bonds and on invested amount in the country i the investor j has chosen. Let rj

(13)

received by the investor j. In total an investor can invest in n countries. Let a n vector rj t+1 collects an investor specic returns from each of them. Given this information, the return on the portfolio, rp;j t+1, can be calculated as r_t+1p;j = n X i=1 wj_i,trj_i,t+1= w0_tjrj_t+1. (6)

At time t + 1, an investment in country's i bonds has changed in value. As discussed before, the prices of bonds will change from time period t to t + 1. Let yH;j

i,t+1 denote the invested value in the country's i bonds by the investor j at the start of the period t + 1

yH;j_i,t+1= (1 + rj_i,t+1)yj_i,t. (7)

Over time, the relation between yH;j

i,t+1 and y j i,t+1 is

y_i,t+1j = yH;j_i,t+1+ bj_i,t+1− sj_i,t+1. (8)

A n-vector φj

t+1represents the net bought amount of dierent countries' bonds by the investor j. This is also the dependent variable of the TF model. If the investor has previously invested in the country i, then yj

i,t(1 + r j

i,t+1) 6= 0 and the entry i of the net investment ow vector is given by φj_i,t+1= b j i,t+1− s j i,t+1

y_i,tj (1 + r_i,t+1j ) for i = 1, .., n j = 1, .., k. (9) In case there was no previous investment, net investment ow is dened as

φj_i,t+1= b

j

i,t+1− s j i,t+1

y_i,tj (1 + r_i,t+1j ) + bj_i,t+1− sj_i,t+1 for i = 1, .., n j = 1, .., k. (10) To see the range of φj

(14)

implies that φj

i,t+1 = 1. The second situation occurs when the investor decides to close the position so that sj i,t+1= y j i,t(1+r j i,t+1)and b j

i,t+1= 0, in this case (9) implies that φ j

i,t+1 = −1. On basis of this analysis it follows that φj

i,t+1∈ [−1, 1].

Overall in the market, there are k investors who can invest in n countries. Bond trades are based on news regarding country specic variables observed by all insurers or on insurer specic variables. I refer to each news characteristic by l for l = 1, ..., p. These explanatory variables are collected in a nk × p matrix Zt. Denote the sth row of this matrix by zts,• and the lth column by zt

•,l. A typical entry of this matrix is represented by (Zt)s,l. At this point, Table 1 lists names, notation, source, expected sign and a model for each explanatory variable. More details on how they are constructed and what is their role in models will be explained in Section 4.

Finally, there is additional notation since both models need to be adapted to the panel data setting. At this point I will focus on notation, further particulars of panel data models will be provided in Section 5.

There are two relevant aspects which are considered in the panel data context. First, depending on the properties of a model, there will be either the Fixed eects or the Random eects present. In this dataset, the panel data unit is an individual j holding bonds of the country j. Denote this individual, time invariant country-insurer eect by cj

i. Second, let j

i,t represent stochastic part of a model. Potentially these errors are cross sectionally dependent. Since the panel data unit is the insurer j holding bonds of the country i, there are two sources of possible cross-sectional dependence. Let fi,t ∼ i.i.d.(0, 1) be a common factor for all insurers having invested in a country i and dene a parameter γj

i ∼ i.i.d(0, σγ2) which measures the impact of this common country factor on the investment of the insurer j. Let bj

t ∼ i.i.d.(0, 1) be a common factor across dierent investments of the same insurer, then ηj

(15)

Table 1: Explanatory variables

Variable Notation Source Sign Model

Relative strength index at t + 1 (6 months) RSI_i,t+1j Constructed/DNB + the TF and the SD Lagged bond value change ∆˜y_i,t−1j

y_i,t−1H;j Constructed/DNB + the TF and the SD

at t versus total value (for t > 0)

Additional cash ows at t + 1 F_t+1j DNB + the TF and the SD Credit Default Swaps, 10Y, seniority CDS10i,t Datastream − the TF and the SD

The Manufacturing P M Ii,t Markit + the TF and the SD

Purchasing managers Index

The Manufactured production Index M anP rodXi,t Eurostat + the TF and the SD

Unemployment rates U rateit Eurostat − the TF and the SD

The Composite leading indicator CLIi,t OECD + the TF and the SD

A downgrade from AAA class at t + 1 DB;AAA_i,t+1 S&P − the TF A downgrade (upgrade) D_i,t+1B;AA(U_i,t+1B;AA) S&P −(+) the TF from class AA at t + 1

A downgrade (upgrade) DB;A_i,t+1 (U_i,t+1B;A) S&P −(+) the TF from class A at t + 1

A downgrade (upgrade) DB;BBB_i,t+1 (U_i,t+1B;BBB) S&P −(+) the TF from class BBB at t + 1

A downgrade within AA at t + 1 DW,AA_i,t+1 S&P − the TF A downgrade within A at t + 1 D_i,t+1W,A S&P − the TF A downgrade within BBB at t + 1 D_i,t+1W,BBB S&P − the TF A downgrade within BB at t + 1 DW,BB_i,t+1 S&P − the TF A downgrade from class AAA LAAA;B_i,t+1+b Constructed/ S&P − the SD at t + 1 for b ≥ 0 periods ago

A downgrade from class AA LAA;B_i,t+1+b Constructed/ S&P − the SD at t + 1 for b ≥ 0 periods ago

A downgrade from class A LA;B_i,t+1+b Constructed/ S&P − the SD at t + 1 for b ≥ 0 periods ago

there is no cross-sectional dependence, then the error term reduces to independent identically distributed vj

i,t ∼ [0, σv2].

This completes the introduction of the relevant notation shared by the TF and the Structural Demand models. In Section 4 the explanatory variables of Table 1 are discussed in detail.

4 Exogenous variables

(16)

for purposes of the Structural demand Model, explanatory variables will be transformed before using them in the model. More details on how this transformation is performed is postponed until Section 6.

One could agree that an important aspect to explain the net investment ows is information on the past bond price changes. In order to incorporate bond price changes in a model, I choose to use a technical indicator which is known to traders as the Relative Strength Index (RSI). The 6 months RSI is constructed using monthly, observed value mutations due to pure price changes of bonds. The value mutations are observed in the dataset and for the same country can be dierent across dierent insurers. This is so since dierent insurers hold dierent combinations of individual bonds issued by the same country. The index is constructed in several steps. First the indicator function for positive I+,j

i,t+1 and negative value changes I −,j i,t+1 is dened I_i,t+1+,j =        1 if ∆˜y_i,t+1j > 0 0 else I_i,t+1−,j =        1 if ∆˜y_i,t+1j < 0 0 else.

In the next step the ratio of average value over 6 month of positive versus the average value over the 6 months of negative price changes is calculated to obtain the Relative strength measure, Sj i,t+1 S_i,t+1j = P5 l=0I +,j i,t+1−l∆˜y j i,t+1−l P5 l=0I −,j i,t+1∆˜y j i,t+1−l . (11)

This quantity is used to calculate the RSI which is measured on the scale from 0 to 100, RSI_i,t+1j = 100 − 100

1 + S_i,t+1j . (12)

(17)

To control for dierences in countries' credit risk, I use Credit Default Swap rates for 10 years (CDS). A CDS is a nancial agreement which compensates a buyer of a contract in the event of a credit event. The higher the rates, the lower is the credit quality and more likely that a credit event occurs. Hence one would expect a negative relation between the demanded value of bonds and CDS rates.

To control for dierences in economic performance which also captures a dimension of credit risk and expected return, I tried several forward looking economic indicators. These include the Purchasing Manufactures' Index, the Composite Leading Indicator and unemployment rates. The PMI by Markit has a global coverage, including U.S., Japan, Canada etc. It also covers the largest European economies however it omits Northern Europe countries, Swiss and number of smaller European countries.

The main variable of interest is the credit rating downgrade indicator. 3 _{One can distinguish} two types of downgrades. The rst one is a downgrade which can lead to changes in an investment grade for a country. In one investment grade belong all ratings which contain either AAA or AA or A or BBB letters as part of their notation. The second type of downgrade occurs within the same investment grade. This means that after a downgrade, rating stays in the same group which is either AA+_{, AA, AA}− _{or A}+_{, A, A}− _{or BBB}+_{, BBB, BBB}−_. Denote by DB;g

i,t+1 for g ∈ {AAA, AA, A, BBB} downgrades which lead to the change of an investment grade. For example the downgrade from the AAA to AA rating is denoted by

3_{To keep it concise I refer here to downgrades only, however downgrade can be replaced with an upgrade}

(18)

D_i,t+1B;AAA 4. The downgrade is dened by D_i,t+1B;AAA = D_i,t+1B RAAA_i,t where D_i,t+1B =               

1 if a downgrade of the country i at time t + 1 changes its investment grade

0 else RAAA_i,t =               

1 if the country i at time t is rated AAA

0 else.

Let DW

i,t+1 denote the downgrades which occur within the same investment grade. These downgrades are dened in the analogue way to DB

i,t+1.

Another separate indicator function is dened for an upgrade of a credit rating. For example, an upgrade from AA to AAA is denoted by UB;AA

i,t+1 = Ui,t+1B RAAi,t , where

U_i,t+1B =       

1 if an upgrade of the country i at time t + 1 changes its investment grade 0 else.

As will become clear in Section 5.2.1, the dependent variable of the LHS model is not an investment ow variable but rather an investment stock variable. This implies that the stock adjustment resulting from a credit event should persist over several periods. In order to dene the variable measuring the eect of a downgrade in the LHS model, consider a situation when b periods ago for b = 0, .., B there has been a downgrade of a AAA rated bond to AA class. Let the step function LAAA

t represent the downgrading eect starting from the announcement month and persisting as long as there is no other downgrade or an upgrade

LAAA;B_i,t+1+b = D_i,t+1+bB;AAA b Y

j=1

(1 − D_i,t+1+b−jB )(1 − U_i,t+1+b−jB ) for b 6= 0

LAAA;B_i,t+1+b = D_i,t+1B;AAA for b = 0. (13)

This concludes the discussion of exogenous variables. In Section 5, models to detect the

4_{All downgrades in the dataset have the same order. For example a rating is reviewed from A+ to A}

(19)

impact of a credit event on the trading behavior are presented.

5 Models

I distinguish two possible ways to measure the impact of a Sovereign credit rating downgrade on investors trading behavior. The rst way relates directly trade ows to arrivals of new information and is called the TF model. The new information regards the economic performance, credit rating changes and government bond returns of dierent countries. The second way is given by the Structural demand model within a general framework provided by Friedman (1977). It relates the trading behavior to portfolio management rules and imposes restrictions on the form of variables entering the demand model. To consider this in more detail, the portfolio management rules are described by two models, the Linear Homogeneous Selection model (LHS) and the Marginal adjustment model (MA).

For the purposes of this paper I shall modify a variant of the LHS model dened by Brandt et al. (2009) and introduce two variants of the Marginal adjustment model. One version describes trades with and other without transaction costs. The model with transaction costs is a modied version of the model by Brandt et al. (2009).

The LHS model denes the selection of the optimal portfolio weights from an investor's point of view. In the framework of this paper, the model is adjusted to reect the perspective of the researcher who does not know the exact form of the LHS model and works with a hypothetical model.

(20)

combining it with the LHS model results in the Structural demand regression model. This model describes the trading behavior relative to the overall market.

The common underlying assumption for both models is that there are no feedback eects. This is due to the fact that only a small subset of the global asset demand, as represented by the Dutch insurance companies, is modeled. Hence I assume that there is no market power of combined Dutch insurance companies to shift the right hand side variables.

5.1 The Trade ow model (TF)

The objective of the Trades Flow model is to examine how the traded quantities of bonds change when news of a credit rating downgrade arrive. The model to explain the net investment ows of a bond i owned by an investor j consists of deterministic and stochastic parts

φj_i,t+1 = zt+1_ij,•θ + uj_i,t+1, (14)

uj_i,t+1 = cj_i + j_i,t+1 (15) j_i,t+1 = γ_ijfi,t+1+ ηjib j t+1+ v j i,t+1. (16)

The denition of the net buys variable implies that changes in relatively small positions are perceived equally important to changes in large positions. Since bonds of countries with lower credit ratings have larger downgrade probabilities but also they represent smaller proportions in portfolios, the current standardization allows to preserve the strength of a downgrade eect on bond trades in the dataset.

An individual, time invariant country-insurer eect cj

i is considered to be a random variable and represents unobserved heterogeneity between panel data units. For example, cj

(21)

technical indicators and economic data. The investor j might be successful at market timing of the country i and less so when trades bonds of some other country. Moreover, there might be dierences between investors in a market timing ability.

If cj

i is correlated with the observed regressors ztij,•, as can be the case for example with the market timing ability, then cj

i is treated as the xed eects. In this case E(c j

i|ztij,•) 6= 0.5 Note that the origin of the name xed eects comes from the earlier treatment of these eects as parameters (Cameron and Trivedi, 2005) but in fact it is a random variable.

The TF model examines the changes in the investment ow variable. Another possibility is to examine how the investment stock responds to credit event announcements. The object of the next Section is the Structural demand model to give some insight in this topic.

5.2 The Structural demand model (SD)

5.2.1 The Linear Homogeneous Selection Model (LHS)

Consider the investor j who selects portfolio weights in order to maximize utility. Conditional expected utility of holding a bond portfolio is a function of the portfolio's returns. Denote it by (u(rp;j

t+1)). When deciding on portfolio proportions, an investor does not know the future returns. The best she can do is to maximize the expected utility, given the current information. The optimal investment weights dened at t solve

w∗;j_t ≡ arg max wj_t E

t[u(rp;jt+1)] = arg max wj_t E t[u(w 0j tr j t+1)]. (17)

Consider n countries which have issued bonds. The optimal portfolio weights can be parameterized as a function of p bond characteristics observed at time t and represented by a nk × p matrix Zt. Let a p vector θ contain xed parameters. Then wt= f (Zt; θ). The

5_{Note that the second underlying assumption for the FE model is that for a stochastic part v}j

i,t∼ [0, σ 2 v]

(22)

most simple policy specication for the portfolio weights has a linear form wt = wMt +

1

nZtθ. (18)

Portfolio weights should always sum up to one and short positions in bonds are not allowed hence ι0_w

t = 1, ι0wtM = 1, wt ≥ 0. Consequently the deviations of the optimal portfolio weights from the market benchmark weights should sum up to zero

1 nι

0

Ztθ = 0. (19)

More details on how to ensure that (19) holds are provided in Appendix A.

Given a parametrization for the optimal portfolio weights, an investor solves the utility maximization problem (17). Instead of choosing from n asset classes at each point in time, there is a reduced dimensionality problem to choose a p vector with the optimal parameters, e.g., one parameter for each bond characteristic. This parameter vector applies to all bonds at all times. The coecients θ are estimated by maximizing the corresponding sample analogue of (17) max θ 1 T T −1 X t=0 u( n X i=1

[w_i,tM + zt_i,•θ]r_i,t+1j ). (20)

The form of the utility function is usually chosen to represent a constant relative risk aversion (CRRA)

u(r_t+1p;j ) = (1 + r p;j t+1)1−γ

1 − γ , (21)

(23)

weights as a function of the asset's characteristics. The solution to the optimization problem is θ∗ _{and the optimal investment weights are}

w_it∗;j = w_i,tM + zt_l,•θ∗. (22)

Brandt et al. (2009) point out that tests about parameters address the question if a given characteristic is related to the moments of returns in such a way that an investor would nd it optimal to deviate from the benchmark portfolio weights. Brandt et al. (2009) note that this is not the same as testing if the given characteristic is cross-sectionally related to the conditional moments of bond returns. There are two reasons for that. First, the benchmark portfolio weights may already reect an exposure to this characteristic and it is not optimal to change it. Second, the characteristic may be correlated with rst and second moments in an osetting way such that the conditionally optimal portfolio weights are independent of the characteristic. Having dened the LHS model, next step is to relate this to the MA model to obtain the trade in bonds. However, before doing so, I shall briey discuss the necessary modications of the presented LHS model.

5.2.2 The modied version of the LHS

(24)

The modied LHS model is w_it∗;j = w_i,tM + ˆzt_i,•θ∗₁+ cj_i + j_it (23) j_i,t = γ_ijfi,t+ ηjib j t + v j i,t. (24)

For the unobserved individual eect cj

i, it is tested if H0 : E(cji|wi,tM, ˆztl,•) = 0 holds. If it cannot be rejected then the model has the Random eects otherwise the model has the Fixed eects. This is formally tested with a Hausman test.

This concludes the discussion of the necessary modications to the LHS model. The next section will introduce the MA model in the absence of transaction costs.

5.2.3 The Marginal adjustment model (MA) without transaction costs

The MA model combined with the LHS model imply the constrained demand equation for bonds. This will complete the model called the Structural demand model.

At the very start of the investment period, an investor has invested available funds optimally. However, at time t + 1 new information arrives and the optimal fund allocation changes. A n vector with new optimal weights w∗;j

t+1 using the LHS model is determined and the investor j has to decide on how to adjust the portfolio by trading. At this point, assume that there are no transaction costs thus at any point in time, the investor will adjust to hold the optimal portfolio. At the start of the period t + 1 the investor j has invested in the country's i bonds yH;j

i,t+1 but wants to have y∗;j

i,t+1. The adjustment takes place through trades

(25)

At any point in time, the stock of country's i bonds owned by the investor j can be expressed as a proportion of the total wealth. Then by using (23) it is possible to obtain the relation between the portfolio weights and the LHS model variables

bj_i,t+1− sj_i,t+1+ y_i,t+1H;j Wt+1

= wM_i,t+1+ ˆzt+1_i,• θ∗+ cj_i + j_i,t+1. (26)

The model (26) is the Structural demand equation which is simplied by discarding transaction costs. It relates one entry of a vector containing portfolio weights to the LHS model variables. Of particular interest is the eect of a downgrade on the portfolio weights. This coecient will indicate if on average, ceteris paribus, an investor acts as the market in general or deviates from it. Part of a country's downgrade eect will be captured by the market weights variable wM

i,t+1, since the price decline resulting from a downgrade would register in decline of the market weights. This type of portfolio weights decline is also known as passive portfolio rebalancing which does not involve active trading.

It turns out that introduction of transaction costs in the model complicates estimation substantially. In Appendix B, I follow the stock adjustment mechanism proposed by Brandt et al. (2009) to obtain the MA model if there are transaction costs. This model however is not used for the estimation due to its complicated nature. In Section 5.2.4 I will discuss a preliminary version of model which accounts for presence of transaction costs.

5.2.4 The Marginal adjustment model with transaction costs

(26)

Friedman (1977) proposes another version for the MA model in case there are transaction costs. To illustrate it, I shall use the notation of the current paper. Consider as an example the bonds of the country i = 1 and the investor j who owns them. As new information arrives, the necessary adjustment in stock is

∆yj_1,t+1= n X i=1 θ1,i(w ∗;j i,t+1W j t − y j i,t) + w ∗;j 1,t+1∆W j t+1. (27)

This adjustment depends not only on deviation from the equilibrium of the country i = 1 but also from deviations of other portfolio investments. In addition, the model distinguishes between investments due to new cash ows and due to reallocation of the existing stock. The drawback of this model is that the number of parameters which need to be estimated becomes very large if there are many dierent investment opportunities. However, the model of Friedman (1977) has two important features. First, an error correction mechanism underlies the asset allocation. Second, the necessary correction is relative to disequilibrium in other investment positions.

Consider a variant of the Error correction model

∆wj_i,t+1= ∆wM_i,t+1+ ∆ˆzt+1;j_i,• φ + κ(w_i,tj − wM i,t − ˆz

t;j i,•θ

∗

) + j_i,t+1. (28)

In this specication, the adjustment of an investment position depends only on its own disequilibrium in the past. At this point, this specication leaves out an aspect of relative adjustment, however it reduces the number of parameters.

In the long run an insurer j selects weights for country's i bonds according to the equilibrium relation so that wj

i,t− wMi,t−ˆz t;j

(27)

can be rewritten in levels as

wj_i,t+1 = w_i,t+1M + wM_i,tξ + ˆz_i,•t+1;jφ + ˆzt;j_i,•ζ + cj_i + j_i,t+1 (29) j_i,t+1 = γ_ijfi,t+1+ η j ib j t+1+ v j i,t+1. (30)

In the next Section, the Trade Flow and the Structural demand model without transaction costs will be applied on the Dutch insurance companies.

6 Empirical application

6.1 Data

(28)

types of bonds, issued by 80 dierent countries. Approximately 90% of these investments are in the European countries. The data frequency is monthly, covering period from April 2006 until September 2012. Appendix D summarizes invested proportions in government bonds in relation to assigned credit ratings.

In total there are 765 399 lines of observations on an individual bond level. Each line of observation contains at least one non zero entry on either a starting value, or on bought or sold volumes, or on price mutations or on an ending position of a bond at time t. Over the whole data period, the total number of trades is 43 262. Since the data frequency is monthly, it is possible that at each given month in one line of observations, both buy and sell transactions will be observed. In this case buy and sell transactions would overlap at time t. In the data set there are 16 197 non overlapping monthly bond purchase observations, 20 491 non overlapping bond sale observations and 6574 overlapping buy and sell observations.Descriptive statistics for buy and sell transactions at an individual bond level are shown in Table 2.

Table 2: Descriptive statistics by insurer groups on an individual bond level ( average bought and sold values are in Euros )

Type Nb of buys Nb of sells Avg. buy Avg. sell Std buy Std sell Non-life insurers 9 463 11 036 6055.38 5297.23 12 118.43 11 221.6

Life insurer 13 308 16 029 15 801.54 12 772. 18 31 076.3 28 316.05

(29)

(30)

Figure 1: Evolution of average bond buy and sell values for countries downgraded over the period of the last four months (the upper panel) and for not downgraded countries in the last four periods (the lower panel)

6.2 Results

6.2.1 The Trade Flow Model

(31)

economic and nancial variables of each country listed in Table 1 6_.

As mentioned in Section 5, the general specication of errors in Model (14) considers a possibly of the cross-sectional dependence.7 _{In order to treat this problem, time varying common shocks} across dierent insurers occurring within the same country are removed with Quarter-Country dummy variables. Similarly, time varying shocks across bonds of dierent countries held by the same insurer are removed with Month-Insurer variables. All together this results in roughly 3300 month-insurer dummy variables and 2000 quarter-country dummies some of which were dropped due to multicollinearity. 8 _{After this correction, the resulting model residual is called} v_i,tj .

The nal item before discussing the estimation results is the choice of the Fixed eects (FE) versus the Random eects (RE) model, representing time invariant, unobserved eects in the panel data. According to Hsiao (2003), the FE model is used when investigator makes inferences conditional on the eects that are in the sample and the RE model for making unconditional or marginal inferences with respect to a population. In this case, the relevant population on which inferences are to be made is fully represented in the dataset, hence the xed eect model would be an appropriate choice.

To test for the FE versus RE model formally, one can examine the Hausman test statistic. Denote the coecient estimates from the FE model with ˆθF E and the estimates from the RE

6_{Summary statistics on explanatory variables is available on request.}

7_{The formal test for the cross-sectional dependence in panel data is called Pesaran's CD test (Pesaran,}

2004) and the routine is programed in additional package supported by STATA. In this setting it was not possible to apply the command as STATA reported the problem that the dataset is highly unbalanced and there are not enough observations to test for the cross sectional dependence. When I applied the test to a data subset of most actively traded countries' bonds with many transaction observations such as Germany, the same error report resulted.

8_{I chose to specify quarter-country dummies instead of month-country dummies, since otherwise the}

(32)

model with ˆθRE. The Hausman test statistic is dened as

(ˆθF E − ˆθRE)0(Var(ˆθF E) −Var(ˆθRE))−1(ˆθF E − ˆθRE) (31)

and is asymptotically χ2 distributed with degrees of freedom equal to the rank(Var(ˆθ) F E − Var(ˆθRE)). For this dataset, the statistic was not well dened since the matrix (Var(ˆθF E) − Var(ˆθRE))was reported to be a negative denite.

The test of overidentifying restrictions is an alternative for the Hausman test but did not allow to include dummy variables in the model specication, however the test statistic was strongly signicant and in favor of the FE model. In Table (3) I reported the estimates of the FE and RE models, note that for the downgrade indicators, the dierences in FE versus RE model coecients are not large.

In Table 3, net investment ows are described with four model variations. Model (1) has RE and insurer and quarter-country dummies included. Model (2) has FE and month-insurer and quarter-country dummies included. Model (3) has RE, Model (4) has FE and in both cases only month-insurer dummies are included. Model (2) is the best specication in view of the cross-sectional dependence and explained reasons for preferring FE over RE. This model is the main focus of the further discussion.

The estimation results of Model (2) show that there are only three signicant variables. The rst one is the cash ow (Fj

t) which has a positive eect on the net bond purchases. The second one is the past price performance variable ∆˜yi,t−1j

yH;j_i,t−1 which has a negative relation to the net bond purchases. The sign would suggest that investors follow a contrarian investment strategy9_{. The third relevant eect is a downgrades within BBB category which trigger 17%} 9_{De Haan and Kakes (2011) uses information on investments in broad, aggregated asset categories to}

(33)

Table 3: The Trade Flow Model

(1) (2) (3) (4)

NIF RE NIF FE NIF RE NIF FE

F_tj y_i,tH;j 5.066*** 1.265*** 0.945*** 0.480*** (1.633) (0.0828) (0.0954) (0.00627) RSI_i,tj -0.000512 -0.000808 -0.000442 -0.000618* (0.000786) (0.000730) (0.000340) (0.000358) ∆˜y_i,t−1j yH;j i,t−1 -1.412** -1.848** . . (0.715) (0.911) . . CDS10i,t -0.00123 -0.00123 -0.000400* -0.000635** (0.000755) (0.000782) (0.000220) (0.000273) U ratei,t 0.0311 0.0325 0.00635** 0.0134* (0.0194) (0.0207) (0.00323) (0.00742) CLIi,t 0.0550 0.0569 -0.000930 -0.00168 (0.0515) (0.0506) (0.00190) (0.00711) DB;AAA_i,t -0.633 -0.650 -0.673 -0.707 (0.556) (0.563) (0.561) (0.588) DB;AA_i,t -0.393 -0.389 -0.452 -0.468 (0.329) (0.322) (0.399) (0.407) DB;A_i,t -0.534 -0.547 -0.576 -0.627 (0.453) (0.444) (0.472) (0.496) DB;BBB_i,t -0.337 -0.326 0.111 0.0836 (0.419) (0.410) (0.108) (0.1000) DW ;AA_i,t -0.0912 -0.0870 -0.139*** -0.109*** (0.0740) (0.0724) (0.0465) (0.0398) DW ;A_i,t 0.719 0.725 0.979 0.939 (0.790) (0.787) (0.852) (0.843) DW ;BBB_i,t -0.143* -0.173** . . (0.0739) (0.0860) . . DW ;BB_i,t 0.0346 -0.0432 0.230 0.261 (0.136) (0.138) (0.180) (0.161) U_i,tB;A -0.0605 -0.0562 -0.0763 -0.0131 (0.0861) (0.0867) (0.0535) (0.0385) U_i,tB;BBB -0.115 -0.0835 -0.0287 -0.0707 (0.114) (0.105) (0.0196) (0.0873) Month*Insurer Yes Yes Yes Yes Quarter*Country Yes Yes No No Observations 21407 21407 20566 20566

R-squared . 0.156 . 0.096

Note: This model is dened for a observation at time t for t>0. The model is estimated with standard errors clustered at the country-insurer level. Note that the initial version of Models (1) and (2) included:DB;BB

i,t , D

B;B i,t , U

B;AA i,t

which were omitted due to multicollinearity. When estimating version (3) and (4) additional control variables included were ∆˜yj

i,t, ManP rodXi,t and

P M Ii,t. *, ** and *** denote signicance at the 10%, 5% and 1% level,

(34)

sales of owned amount for downgraded bonds.

One would expect that the credit rating downgrade resulting in the change of an investment grade has the strongest eect on trades. This was not supported from the side of data since no relevant eects for these kind of downgrades were found. In Models (3) and (4) which control only for Month-insurer dummies, the relevant reaction takes place within downgrades of AA category.

The role of economic variables did not have any relevant explanatory power in the framework of reviewed models. However, the riskiness of investments as measured by CDS10i,t had expected negative eect on net bond purchases when Quarter-Country dummies were omitted from the model.

6.2.2 The Structural Demand model

The Trade ow model measured the impact of a country's credit rating downgrade on the trade ows. In this section I shall examine what is the impact of a credit rating event on the accumulated stock of bonds. The accumulated stock is measured relative to the total wealth of an investor.

There are four variants of the SD model, see Table 4. First, they dier in the treatment of the unobserved, time invariant eect. I include estimates from the xed and the random eects models since the result of a Hausman test for xed versus random eects could not be interpreted for this dataset. The reason is the same as the one explained in case of the TF model. However, since there is a close similarity between the xed eect and the random eect coecients for downgrade variables, this does not constitute a major problem.10

10_{Magnitudes of downgrading eects were very similar in the Models (2) and (4) but less so in the Models}

(35)

Second, model will become a dynamic panel data model if the lagged dependent variable is included11_{. In methodological section, I discussed two versions of the Structural demand model} which made the distinction between the model with and without transaction costs. Due to transaction costs, portfolio weights can be away from the equilibrium, which leads to the path dependence. The implication of the path dependence is that the portfolio weights one chooses today are aected by the deviation from the equilibrium in the past. In order to accounts for this, one can specify a dynamic panel data model which includes the autoregressive term. 12 In all models, to control for time varying, unobserved factors resulting in possible cross sectional dependence, Month-Insurer dummies were included. There are some dierences between the models. In particular, Model (1) is estimated with the random eects and with the lagged dependent variable to account for the path dependence. Model (2) is estimated with the random eects and without the lagged dependent variable. Model (3) is estimated with the xed eects and with the lagged dependent variable which is included for the same reason as in Model (2). Finally, Model (4) is estimated with the xed eects and without the lagged dependent variable.

In general all four models indicate negative, relevant eects of downgrades when adjusting the stock of bonds. These eects are relevant for all included credit grades. The most dierences in coecient magnitudes of downgrading eects are due to the inclusion of the lagged dependent variable. I shall discuss each model in greater detail next.

In Table 4, results of Model (1) show that the eects of credit rating downgrades are relevant

11_{When estimating the dynamic xed eects model which results from the lagged dependent variable or}

when model has a residual autoregression, coecients suer from the Nickell bias (Nickell, 1981). However, as pointed out by Beck and Katz (2004), Nickel bias is less or equal to 2% once T = 20.

12_{If the path dependence is not the case, then Achen (2001) shows that if errors have autocorrelation and}

(36)

in determining portfolio weights for downgrades resulting in bond rating of AA, A, BBB class. However, these eects have economically small magnitude. The largest eect of a downgrade is when bonds enter BBB category followed by A category. The ordering makes sense since the lower the category, the higher the credit risk and larger reaction to this event.

Variables representing the macro fundamentals' do not have a relevant role in explaining variation in weights. Increase in nancial variable CDS10i,t has expected negative eect on portfolio weights. Note that the market weights variable does not have any relevant eect on portfolio weights. This makes sense since both the lagged dependent and the market weights variables are highly persistent hence its eect is absorbed by the dynamic term.

The conclusions from Model (2) are qualitatively similar to those of Model (1). The downgrading eects are still relevant in explaining portfolio weights. In magnitude they are relatively similar across all three rating classes. The relevant macro fundamentals explaining portfolio weights are standardized P MIi,t and CLIi,t variables with expected positive sign. Peculiar results are that the standardized ManP rodXi,t has a negative eect and CDSi,t have a positive eect which contradicts the expected signs for these eects.

Model (3) and (4) imply the same conclusion that downgrading eects are relevant. What diers is the ordering of the magnitude. In both models the strongest eect is for bonds downgraded from AAA category followed by downgrades from A category. In both models the CLIi,t variable is relevant in explaining portfolio weights and has the expected positive sign. Less clear result of Model (4) is that the standardized ManP rodXi,t has a negative eect on the portfolio allocation.

(37)

Table 4: The Structural Demand Model

(1) (2) (3) (4)

RE &lag RE & no lag FE &lag FE & no lag

wj_i,t−1 0.965*** 0.905***

(0.00557) (0.0152) wM

i,t 0.0486*** 1.382*** 0.382*** 1.915***

(0.00718) (0.147) (0.0607) (0.423) Std_ManP rodXi,t 0.000498 -0.00760* -0.00107 -0.00932**

(0.000358) (0.00388) (0.000810) (0.00388) Std_Uratei,t -0.000128 -0.00618 -0.000852 -0.00689

(0.000190) (0.00401) (0.000714) (0.00419) Std_CLIi,t 9.86e-05 0.0359*** 0.00321** 0.0363***

(0.000377) (0.00658) (0.00145) (0.00682) Std_P MIi,t -0.000191 0.00296** 0.000250 0.00302** (0.000178) (0.00137) (0.000280) (0.00136) Std_CDS10i,t -0.00112*** 0.00955*** 2.25e-05 0.0102*** (0.000195) (0.00338) (0.000673) (0.00356) C_LAAA;B_i,t+b -0.00369*** -0.0403*** -0.00825*** -0.0432*** (0.000765) (0.00828) (0.00145) (0.00822) C_LAA;B_i,t+b -0.00800*** -0.0367*** -0.00568*** -0.0332*** (0.000867) (0.0117) (0.00163) (0.0121) C_LA;B_i,t+b -0.00868*** -0.0462*** -0.00765*** -0.0406*** (0.00269) (0.00858) (0.00248) (0.00834) Constant -0.00124*** -0.0254*** -0.0203*** -0.0469 (0.000279) (0.00927) (0.00466) (0.0372) Observations 22697 23143 22697 23143 R2 _. _. _0.817 _0.081

Month*Insurer Yes Yes Yes Yes Observations 22697 23143 20566 20566

R-squared . 0.156 . 0.096

(38)

7 Conclusion

The objective of this paper was to model and estimate the reaction of the Dutch insurance companies towards sovereign downgrades. To get an insight in this area, the unbalanced panel data set with the Dutch insurers' trading information was used. The data indicated that the sovereign downgrade does provide a new information to investors, decreasing the net bought value of bonds. There was no evidence that the same holds for upgrades since no relevant trading reactions were detected. These ndings are in line with Hand et al. (1992) who document a stronger reaction to downgrades as compared to upgrades.

The downgrades were classied with respect to the credit rating classes. This was done to see if the eect of downgrade depends on the investment rating category an asset has before the credit event. The Trade Flow model with the xed eects and with the dummy variables included to account for the cross-sectional dependence, indicated dierences. According to results, a negative reaction towards downgrades is within BBB rating category only.

(39)

(40)

References

Achen, C. (2001). Why lagged dependent variables can supress the explanatory power of other independent variables, Working paper pp. 142.

Afonso, A., Arghyrou, G. and Kotonikas, A. (2012). The determinants of sovereign bond yield spreads in the emu, Working Paper 36 ISSN N 0874-4548: 140.

Afonso, A., Furceri, D. and Gomes, P. (2012). Sovereign credit ratings and nancial markets linkages: application to european data, Journal of International Money and Finance 31: 606 638.

Beck, N. and Katz, J. (2004). Time-series-cross-section issues:dynamics, Working paper pp. 1 25.

Brandt, M., Santa-Clara, P. and Valkanov, R. (2009). Parametric portfolio policies: Exploiting charcteristics in the cross-section of equity returns, The Review of Financial Studies 22: 34113447.

Cameron, A. and Trivedi, P. (2005). Microeconometrics, Vol. 5(1).

Catren, O., Osbat, C. and Sydow, M. (2006). What drives investors' behaviour in dierent fx market segments? a var-based return decomposition analysis, ECB Working Paper 706: 1 36.

De Haan, J. and Kakes, J. (2011). Momentum or contrarian investment strategies: Evidence from dutch institutional investors, Journal of Banking & Finance 35: 22452251.

Friedman, B. (1977). Financial ow variables and the short-run determination of long-term interest rates, Journal of Political Economy 85: 661689.

(41)

Hite, G. and Warga, A. (1997). The eect of bond-rating changes on bond price performance, Journal of Financial Economics 53: 3551.

Hsiao, C. (2003). Analysis of Panel Data, Cambridge Universiy Press.

Kamesaka, A., Nofsinger, J. and Hidetaka, K. (2003). Investment patterns and performance of investor groups in japan, Pacic-Basin Finance Journal 11: 122.

Nickell, S. (1981). Biases in dynamic models with xed eects, Econometrica 49(6): 1417 1426.

Pesaran, M. (2004). General diagnostic tests for cross section dependence in panels, Cambridge Working Papers in Economics No. 0435.

Reinhart, C. (2010). This time is dierent chartbook: country histories on debt default, and nancial crisis, NBER Working Paper 15815 .

(42)

Appendix A: Centralized and standardized variables for

the SD model

In order to ensure that the constraint (19) holds, instead of directly using bond characteristics as inputs for the optimal portfolio weights, Brandt et al. (2009) rst standardize these characteristics across all stocks at time t to have a zero mean and optionally a unit standard deviation. If variable is transformed in a way that it has a zero mean only, I refer to it as a centralized variable. If in addition a variable is transformed to have a unit standard deviation, I refer to it as a standardized variable.

To formalize the notion of a centralized and standardized variable which in this case is a bond characteristic, I need to introduce some additional notation. Dene a sample average of a bond characteristic l by 1

nι 0

zt_•,l for l = 1, .., p. Dene a sample standard deviation of a bond characteristic l by \vart(zl) = _n−11 Pn_i=1((Zt)i,l− \µt(zl))2. Then a typical element of a centralized variable is (Zt)i,l− 1 nι 0 zt_•,l (32)

and a typical element of a standardized explanatory variable is given by (Zt)i,l− 1_nι0zt•,l

q \ vart(zl)

. (33)

(43)

Appendix B: The MA model with transaction costs and

without intermediate cash ows

Version of the MA model with transaction costs but without intermediate cash ows was originally proposed by Brandt et al. (2009). Due to presence of transaction costs, the investor j will not always choose to re-balance weights to match the optimal allocation perfectly. A n vector with transaction costs for the investor j is denoted by vj

t. These costs depend on the portfolio's bond turnover and reduce portfolio returns.

In each period the turnover of the portfolio is the sum of absolute portfolio weight changes from one period to the next. The changes in weights which are due to dierences in the bond portfolio's relative returns are not included. The cost adjusted portfolio returns are calculated according to

˜

rj_p,t+1= w0trt+1− v0t|wt− wt−1|. (34)

Since prices of bonds included in the portfolio change from time t to time t + 1, also their portfolio weights have changed. If an investor just keeps on holding what she already has in the portfolio, these new proportions are a n vector wH;j

t+1 with the ith entry given by

wH;j_i,t+1= 1 + r j i,t+1 1 + rp;j_t+1 w j i,t. (35)

If this portfolio is suciently close to the optimal portfolio, then it is better not to trade any further, which implies that at time t + 1

wj_t+1 = wH;j_t+1 if (w∗;j_t+1− wH;j_t+1)0(w∗;j_t+1− wH;j_t+1) ≤ k2, (36)

(44)

If the "hold" portfolio is far from the optimal portfolio, then trading will continue until the boundary of no trade region is reached

wj_t+1 = αtwt+1H;j+ (1 − αt+1)w ∗;j t+1 if (w ∗;j t+1− w H;j t+1) 0 (w∗;j_t+1− wH;j_t+1) > k2 (37) and solving (w∗;j t+1− wt+1)0(w ∗;j t+1− wt+1) = k2 implies that αt+1 = k (w∗;j_t+1−wH;j_t+1)0_(w∗;j t+1−w H;j t+1) 1 2.

At this point I will not map the Marginal adjustment model to this trading behavior, since the model does not contain intermediate cash ows. I will proceed next with discussing necessary modications when intermediate cash ows are included. Thereafter I link the MA model to the trading behavior.

Appendix C: The MA model with transaction costs and

intermediate cash ows

Assume for the moment that Fj

t ≥ 0. For positive cash ows I shall assume that

F_tj ≤ X

w_i,t∗j>wj;H_i,t

(w_i,t∗j − w_i,tj )W_tj, (38)

so that the allocation of Fj

t is well dened. The available money will be invested to get closer to the optimum and the ith entry of a n vector with new weights, wY ;j

t , is wY ;j_t,i =          Ftj W_tj w∗j_i,t−wH;j_i,t P w∗;j i,t−w H;j i,t >0 (w∗;j_i,t−wH;j_i,t )+ W_t−1j W_tj w H;j i,t if w ∗;j i,t > w H;j i,t w_i,tH;j else. (39)

Now consider a situation when Ft < 0, in which case there is a money withdrawal. The investor would sell those bonds which are above their optimal levels. For bond sales to be well dened, I assume that Fj

t ≤ P w_i,tH;j>w_i,t∗;j(w H;j i,t − w ∗;j i,t)W j

(45)

weights are wY ;j_t,i =          Ftj W_tj w_i,tH;j−w_i,t∗;j P wH;j i,t −w ∗;j i,t>0 (wH;j_i,t −w∗;j_i,t)+ W_t−1j W_tj w H;j i,t if w H;j i,t > w ∗;j i,t w_i,tH;j else. (40) If it holds that (w∗;j t − w Y ;j t )0(w ∗;j t − w Y ;j

t ) ≤ k2 then no further adjustment is necessary and wj_t = wY ;j. If this is not the case, then there is a further adjustment and the new weights at the start of period t will be a combination

w_t,ij = αtwi,tY ;j+ (1 − αt)w ∗;j

i,t. (41)

To dene αt, solve for the non trade boundary condition (NT) (w ∗;j t − w j t)0(w ∗;j t − w j t) = k2. From (41) it follows that wj

i,t− w ∗;j i,t = αt(w Y ;j i,t − w ∗;j i,t), so that αt= k (wY ;j_t −w∗;j_t )0_(wY ;j t −w ∗;j t ) 1 2.

The corresponding trade in bonds eliminates the dierence between amount the investor wants to have and her current amount

bj_i,t− sj_i,t = yi,t − yi,tH;j. (42)

Denote the situation when non trade boundary is violated (w∗

t − wt)0(wt∗ − wt) > k2 with ¬NT and else with NT. Normalizing (42) by wealth, results in demand equation

bj_i,t− sj_i,t+ yH;j_i,t

(46)

If Fj

(47)

Trading behavior of the Dutch Insurance companies in view of sovereign downgrades