Financial Intermediaries’ Asset–Liability Dependency and Low-Interest-Rate Environment: Evidence from EU Life Insurers

This research studies the relationships between the two sides of life insurers' balance sheet and investigates whether and how they changed during recent past years, when European Central Bank monetary policy drove market rates to unprecedented low levels. By using a canonical correlation analysis, we study the internal structure of the links within and between the asset and liability sides of 24 major European Union (EU) life insurers' balance sheets over the 2007– 2015 time horizon.<br><br>We find strong and substantial evidence that life insurers' assets and liabilities have become more independent over time. We argue that the declining trend of market interest rates has contributed to the generalized reduction in the linkage between the asset side and the liability side of EU life insurers, and has made insurance companies more exposed to ALM-related risks relative to the period before the financial crisis.


Introduction
Life insurers' decisions concerning asset and liability management (ALM) depend on a complex set of economic and institutional factors, such as the regulatory constraints and the conditions prevailing in¯nancial markets. From this latter perspective, the issues associated with the decline and the unprecedented low levels of market interest rates pose signi¯cant challenges for both the insurance industry and regulators. EIOPA has paid special attention to the interest rate risk across European life insurers (EIOPA 2016a(EIOPA , 2015, and in 2014 and 2016 it conducted industry-wide stress tests to assess the resilience of the European life insurance industry to a prolonged period of low interest rates (EIOPA 2014(EIOPA , 2016b. Based on EIOPA's last Financial Stability Report (EIOPA 2018), because of the prolonged low interest rate environment, the pro¯tability of the insurance industry has experienced a signi¯cant deterioration and its solvency is at risk. Nevertheless, investigating the impact of interest rates on life insurers' activity is complex since changes in interest rates a®ect the balance sheet, investment income and premium rates, and, more in general, the demand for insurance products (Swiss Re 2012).
The impact of interest rates on a life insurer's balance sheet depends on a possible accounting mismatch and on the extent of the duration mismatch between assets and liabilities. As for the former, should assets be marked to market and liabilities be recorded at book value, a decline in interest rates will cause an increase in the insurer's equity. With regard to the latter, since life insurers' liabilities may have maturities longer than 30 years and, in some markets, assets with the same maturity are either unavailable or illiquid, a decline in interest rates threatens companies' pro¯tability and can even cause solvency issues if rates stay low for long.
As far the pricing of insurance products is concerned, in life insurance, given a certain level of bene¯ts, a decrease in interest rates increases the premiums paid by the policyholders. For example, the annual payments required to obtain the same bene¯ts when interest rates are at 4% are lower than the payments needed if investment returns stand at 2.5%. From a broader perspective, a reduction in interest rates can have di®erent impacts on the demand for life insurance products. On the one hand, declining interest rates can either make these products more expensive or reduce their bene¯ts, like in the case of an annuity, where, due to lower interest rates, the present value of the future lifetime payments rises and, consequently, the corresponding lump sum premium must rise to keep bene¯ts at the same level. On the other hand, a reduction in interest rates can have a positive impact on some savings products providing returns higher than market rates.
Furthermore, a decline in interest rates signi¯cantly a®ect life insurers' pro¯tability since they invest most of their premium income in high-quality bonds and, therefore, investment income is a major source of earnings for them. Nevertheless, even if the sensitivity of life insurance products to changes in interest rates can be very high, income response to a change in interest income is slow because insurers invest only current premiums at current market yields.
Finally, the impact of interest rates on life insurers depends also on policyholders' behavior, which in some circumstances is far from being rational. Some insurance products give to policyholders options whose exercise can a®ect insurers' exposure to interest rate risk, such as the options to extend the term of the policy or to increase the payments or the insured sum originally set in the contract. Therefore, assuming that the embedded interest rate guaranteed by a contract is higher than market rates, policyholders have the incentive to extend the contract beyond the original maturity, or to increase premium payments/the sum insured if the original interest rate is guaranteed.
In this paper, we tackle the issue of the impact of interest rates on life insurers' activity following an integrated-ALM perspective, based on balance sheet data. From our view, life insurance companies take the decisions concerning the assets to buy and the liabilities to sell under a condition of uncertainty and the outcomes of such a decision process involve interactions among the assets, among the liabilities and between the asset and liability sides of the balance sheet. The purpose of this research is to uncover the relationships between life insurers' assets and liabilities and to investigate how these relationships evolved during past recent years, when ECB's monetary policy decisions drove market rates to unexperienced low levels. In particular, by assuming that the structure of a life insurer's assets is a function of the structure of its liabilities, we contribute to prior literature investigating¯nancial institutions' ALM by testing the hypothesis that assets of a life insurer have multiple correlation patterns with its liabilities and surplus categories. We also investigate the nature and the strength of the relationships between the two sides of the balance sheet.
We run our empirical analysis by using a canonical correlation estimation technique. Canonical correlation analysis, introduced by Hotelling (1935), unlike regression analysis, which relates a single-dependent variable to a linear combination of independent variables, correlates linear combinations of two sets of variables. Canonical correlation analysis considers variation within each set of variables and between the two sets, while regression considers only variation within one set of independent variables and between one dependent variable and that set. Therefore, canonical correlation analysis is an appropriate technique for analyzing relationships between the securities bought and sold by an intermediary (and represented as assets and liabilities in the intermediary's balance sheet) because these securities are correlated both within each side of the balance sheet and between the two sides of the balance sheet. More in details, we¯rst assess the relationships within the assets and within the liability of the balance sheet, by separately taking into account each of the two sides of the balance sheet. Then, under a joint perspective, we study the links of asset accounts with liability accounts. Even if the analysis of canonical correlations provides an excellent statistical tool, our results have to be interpreted with caution because the analysis of balance sheet data is not able to comprehensively capture the complexity of the issues investigated here, which is essentially given by the above discussed numerous channels through which low interest rates a®ect life insurers. Therefore, additional qualitative and quantitative evidence could improve our understanding.
This research considers 24 major life insurers from 11 European Union (EU) countries, representing circa 56% of the whole EU insurance industry in terms of total assets (as of December 2015), and whose behavior is observed over the 2007-2015 horizon. As of December 2017, the overall European insurance industry roughly accounts for 30% of the global insurance market and European life premiums (€ 710 billion) represent 32% of 2017 global life premiums (Insurance Europe 2018). We decided to focus on the EU life insurers not only for the seize of the industry, but also because they have traditionally made large use of insurance products with minimum guaranteed rates of return and pro¯t sharing mechanism. In times of low interest rates, this might represent not only a threat for insurers' pro¯tability, but it might also endanger their solvency position and, eventually, due to their relevance and strong interconnections with other¯nancial intermediaries, it might represent a threat for the¯nancial system as a whole. Based on our results, life insurance companies seem to run their business as if they decide their funding policies after identifying good investment opportunities. We¯nd strong and substantial evidence that insurers' assets and liabilities have indeed become more independent over time. We argue that the declining trend of market interest rates over the examined time horizon has contributed to the generalized reduction in the linkage between the asset side and the liability side of EU life insurers, and has made insurance companies more exposed to ALM-related risks relative to the period before the¯nancial crisis broke out.
The rest of the paper is organized as follows: Section 2 provides a review of the literature investigating the exposure of life-insurers' activity to interest rate risk and of the few previous studies using canonical correlation technique to investigate insurers' ALM; in Sec. 3, we present the rationale for using canonical correlation analysis and outline its application to this research; Sec. 4 describes our life insurer sample and the variables we use in our analysis; in Sec. 5 we discuss the main results of our empirical analysis; Sec. 6 provides concluding remarks.

Interest Rates and Life Insurers' Asset and Liability Management: A Literature Review
According to prior research , there are two main channels through which low interest rates a®ect insurers': the \income channel" and the \balance sheet channel". Based on the former, low long-term interest rates make it increasingly hard for insurers to achieve investment returns in excess of guaranteed returns embedded in policies issued in the past. The \balance sheet channel" rests on a valuation e®ect: low yields result in an increase in the values of both assets and liabilities. The rise of the liabilities is higher than that of the assets because the assets invested in¯xed-term instruments are a fraction of the total liabilities. From this perspective, relative to nonlife sectors, life insurers experience a higher exposure to interest rate declines because of the higher duration mismatch between assets and long-term life insurance liabilities. Antolin et al. (2011) have examined the impact of protracted periods of low interest rates on insurance companies. From a¯nancial stability perspective, they point out two major concerns: the¯rst is that insurers may search for higher yields via riskier investments (\gambling for redemption" behavior); the second is the decline in interest rates causing an interest-risk-hedging activity that might further reduce bond yields (this has been con¯rmed for German life insurers by Domanski et al. 2015). From an ALM perspective, if their liabilities are characterized by longer duration than assets, insurers have to deal with the reinvestment risk associated with the fall of interest income due to the rollover into lower-yielding debt both of the coupon payments from¯xed-income instruments and of the principal from maturing debt. Overall, the impact of low interest rates seems to depend on the contribution of investment income to the overall pro¯tability: the greater is the contribution of interest income to an insurer's pro¯tability, the larger is the fall in its pro¯tability when interest rates stay low for a long period. With regard to the second source of concern just mentioned above, based on an empirical analysis of German life insurers, Domanski et al. (2015) actually show that German insurers buy more long date bonds to improve their matching strategy and further push down yield on bonds.
In their analysis of the impact of interest rate levels on the¯nancial performance of a sample of 127 European insurers,  show the complexity of these two channels. In particular, they¯nd that long-term interest rates have a positive impact on insurers' return on assets, with a bigger (and more statistically signi¯cant) impact on small and medium-sized companies and the life and health sectors. At the opposite, the long-term yields does not a®ect large insurers, which may result from their superior diversi¯cation ability, or the property and casualty sector, probably because nonlife insurance is characterized by a short pay-out pattern and by contracts made on a yearly basis with the possibility to increase prices at renewal. The authors also provide a model-based scenario analysis to assess in a forward-looking manner the e®ects of a prolonged period of low interest rates on the solvency and pro¯tability of a representative life insurer active in Germany, France, Italy and the Netherlands. Their results show that: (i) the negative e®ect would be more pronounced in the case of a high volatility in¯nancial returns, and (ii) speci¯c characteristics of the company (duration mismatch, asset allocation, etc.) matter in determining the extent of the impact. Berdin & Gründl (2015) assess and quantify the e®ects of the low interest rate environment on the balance sheet of a representative German life insurer, given the current asset allocation and the outstanding liabilities. In particular, they generate a stochastic term structure of interest rates and stock market returns to simulate investment returns of a stylized life insurance business portfolio in a multi-period setting. To take into account di®erent scenarios and to check the robustness of their results, the authors calibrate di®erent capital market settings and di®erent initial situations of capital endowment. Their results suggest that a prolonged period of low interest rates would markedly a®ect life insurers' solvency, leading to a relatively high cumulative probability of default, especially for less capitalized companies. The German insurance sector is the object of a previous paper by Kablau & Wedow (2012) also, who develop a di®erent approach to assess life insurers' resilience to a protracted period of low interest rates. Based on their results, a relatively large number of German life insurers would not be able to meet the minimum regulatory capital requirements set by Solvency I if the interest rates were to stay su±ciently low for long.
There are two approaches to measure life insurers' exposure to interest rate risk: bottom-up and top-down. The former would use detailed data on assets and liabilities and would allow estimating the interest rate risk of each company or even of each of the products sold by a certain insurer. According to the latter, since stock analysts take into account the product mix and interest rate guarantees of the products sold by the insurers, their stock price re°ects the sensitivity of their liability to interest rate movements. By using a top-down based approach, Hartley et al. (2016) measure the exposure to interest rate risk of U.S. and U.K. insurers through a two-factor model of life insurer stock returns. Their empirical evidence shows that, despite being exposed to similar changes in interest rates, U.S. life insurers experienced an increase in their risk exposure as interest rates strongly decreased in recent years, whereas in U.K. life insurers' risk remained low and roughly at the same level observed prior to the¯nancial crisis. According to the Authors, these di®erences are due to the more widespread use of products that combine guarantees with options for policyholders to adjust their behaviour by U.S. life insurers relative to their U.K. counterparts. Berdin et al. (2017) develop an analytical framework for a forward-looking assessment of insurers' pro¯tability and solvency by modeling the balance sheet of an insurance company involved in both life and nonlife business. The balance sheet is calibrated using country level data to make it representative of the major euro area insurance markets, and then projected forward under stochastic capital markets, stochastic mortality developments and stochastic claims. The research focuses on European markets where the relatively high guarantees and generous pro¯t participation schemes make the insurers largely exposed to reinvestment risks. The results suggest that insurers more exposed to products with¯nancial guarantees display a marked reduction in both pro¯tability and solvency over time. The speci¯c local regulation and the business practices (e.g. the minimum return guarantees and duration mismatches) signi¯cantly a®ect both pro¯tability and solvency. As the business portfolio becomes more diversi¯ed and less concentrated on interest rate sensitive business, both pro¯tability and solvency improve. By assuming a group perspective, if capital redistribution within the group is allowed, a low interest rate environment might also negatively a®ect the solvency position of the nonlife business, generally characterized by a limited exposure to interest rate risk, due to both lower return on assets and, most importantly, to the capital redistribution from the nonlife towards life business.
Prior studies have investigated widely the issues associated with a proper valuation and management of interest rate risk in the insurance sector. In particular, there is a huge literature dealing with the valuation of life insurance policies with a minimum guaranteed rate of return and a pro¯t participation scheme (see, among the others, Grosen & Løchte Jørgensen 2000, Bauer et al. 2006, Zaglauer & Bauer 2008. As concerns asset-liability management and hedging strategies, Lee & Stock (2000) examine the duration and convexity matching strategies to hedge interest rate risk, and Gerstner et al. (2008) developed an integrated asset and liability model where a portfolio of di®erent life insurance policies evolve over time depending both on mortality developments and on the asset return generated by the investment portfolio.
Due to the peculiarities of the insurance business, interest rate risk has been studied jointly with the longevity risk. Mahayni & Steuten (2013) examine the e®ect of stochastic longevity and stochastic interest rates on a portfolio of deferred annuities. By focusing on solvency requirements based on the investment decisions and the associated shortfall probability of the annuity provider, they conclude that the impact of stochastic mortality is low if compared to the impact of stochastic interest rates. Berdin (2016) assesses the e®ect of interest rate risk and longevity risk on the solvency position of a life insurer selling policies with minimum guaranteed rate of return, pro¯t participation and annuitization option at maturity. Following their methodology, an existing banking book of policies and an existing asset allocation calibrated on observed data are projected forward according to di®erent scenario built based on stochastic¯nancial markets and stochastic mortality developments. Among these scenarios, the authors focus on a prolonged period of low interest rates and huge decrease in mortality rates. Even their results show that interest rate risk is de¯nitely the major threat to life insurers, whereas longevity risk can be easily mitigated mainly through product design and prudential pricing.
As the studies using canonical correlation analysis to investigate the relationships between asset and liability accounts of¯nancial intermediaries' balance sheets are concerned, this technique has not been speci¯cally used to study the impact of interest rates on insurers' ALM, and prior research is mostly focused on banks (see Simonson et al. 1983, DeYoung & Wom 2008, among the others). Within the insurance literature, the seminal work of Stowe & Watson (1985) is the¯rst to use a canonical correlation approach to study the relationships between the assets and liabilities of US life insurers' balance sheets. This analysis is important as the structure of asset and liability depends on interactions among the assets, among the liabilities, and between assets and liabilities, on top of the constraints induced by regulation and¯nancial market conditions. Stowe & Watson (1985) work builds on the idea that life insurers solve a portfolio optimization problem when structuring their assets and liabilities. They follow an approach according to which the mean-variance model, based on Markowitz (1952) classic theoretical framework, provides a simple explanation of¯nancial intermediation, where securities positively held can be thought as intermediary's assets and those negatively held as intermediary's liabilities. Instead of focusing on either the asset or the liability side of the balance sheet, this approach allows to incorporate explicitly dependencies between assets bought and liabilities sold by a¯nancial intermediary. Pyle (1971), Francis & Archer (1979) and Francis (1978) use a mean-variance model to study the optimal balance sheet structure of¯nancial intermediaries, where the optimal balance for each asset and each liability is a function of the expected returns on each assets and the expected costs of each liability as well as the covariances between all assets and liabilities. Stowe (1978) applies a similar meanvariance framework to life insurers.

The Canonical Correlation Analysis: Mathematical Framework and Application to this Research
This paragraph provides a description of the canonical correlation technique, developed by Hotelling (1935Hotelling ( , 1936, in order to clarify the rationale for adopting it in our study. Canonical correlation is a multivariate analysis technique describing the relationships between two sets of variables, named criterion variables and predictor variables. In our case, these two sets of variables are, respectively, the asset and liability/capital accounts of an insurer's balance sheet. Let the asset and liability/capital variables be denoted, respectively, by the matrices X and Y . The number of rows of each matrix represents the n insurers of our sample, while the number of columns indicates the di®erent categories of asset (q 1 ) and liability (q 2 ) taken into account in our analysis. Consequently, X is n Â q 1 and Y is n Â q 2 . The variables used are expressed as a proportion of total assets.
The canonical correlation methodology attempts to¯nd linear combinations of X and Y so that the correlation between them is as high as possible. The linear combinations of X and Y are denoted, respectively, by u i and v i : where a i and b i are vectors to be estimated and are, respectively, q 1 Â 1 e q 2 Â 1. We refer to the scalars that constitute the vectors as canonical coe±cients, to the linear combinations of X and Y as canonical variables and to the correlations between the canonical variables as canonical correlation coe±cients. The canonical correlation coe±cients and the canonical coe±cients are obtained by solving the following equations a : where R 11 is the covariance matrix between asset variables; R 22 is the covariance matrix between liability variables; R 12 is the covariance matrix between asset and liability variables, R 21 is its transposed and I is the identity matrix. a For a more detailed description, see Anderson (2003).
Equations (3.2a) and (3.2b) can be rewritten as systems of p linear equations in p unknown coe±cients. These systems of linear equations will have nontrivial solutions only if their determinants are zero.
The largest value of that satis¯es both equations (3.2a) and (3.2b) is the¯rst characteristic root, or, in other words, the¯rst eigenvalue, of the following matrices: Vectors a 1 e b 1 are its corresponding eigenvectors, which constitute the weights (canonical coe±cients) for the linear combinations u 1 and v 1 . The¯rst canonical correlation coe±cient (R 1 ) is the square root of the¯rst characteristic root. In symbols: Particularly, there will be a number p of canonical correlation coe±cients equal to the minimum between q 1 and q 2 . Canonical correlation coe±cients represent the variance shared by linear combinations of assets and liabilities. Each successive canonical correlation coe±cient will be smaller than the last since each successive root will explain less and less of the data. For each canonical correlation coe±cient, we have di®erent pairs of canonical variables. Each pair of canonical variables is uncorrelated with the others.
In order to determine the number of statistically signi¯cant canonical correlation coe±cients, we use the test proposed by Bartlett (1941) which tests the null hypothesis that there is no relationship between the predictor and the criterion variables, or that there are no more than k signi¯cant canonical pairs, where k is equal to zero. When this hypothesis is rejected, k is set equal to 1 and Bartlett's test is performed for this new value until the signi¯cance level is exceeded and the number of statistically signi¯cant canonical pairs is determined.
Since the variables used in our study are expressed as a proportion of total asset the sum of these proportions add to unity, which makes R 11 and R 22 singular. To avoid this singularity, we eliminate one variable from each set. The informational content of the remaining q 1 À 1 and q 2 À 1 variables does not change. Consequently, the number of canonical correlation can be lower than the minimum between q 1 À 1 e q 2 À 1. In symbols we can have: p minðq 1 À 1; q 2 À 1Þ: ð3:6Þ The nature of the relations between asset and liability can be studied by examining the canonical loadings, which are the correlation between the original variables and their own canonical variables. The canonical loadings give a measure of the total amount of variance in the actual data accounted for by the canonical variables; they are the elements of the matrices S 1 and S 2 , with dimension q 1 Â p and q 2 Â p, obtained as follows: where U and V are n Â p matrices whose columns contain the canonical variables obtained by solving (3.2a) and (3.2b). A and B are matrices with dimension, respectively, q 1 Â p and q 2 Â p, whose columns are formed by the eigenvectors of the characteristics roots. Following Cli® & Krus (1976), we rotate simultaneously the canonical loadings using Kaiser (1958) normalized varimax criterion. This simpli¯es the interpretation of the nature of the relationships between the canonical variables without a®ecting the total predictable variance. Each element S jk;1 and S jk;2 of matrices S 1 and S 2 is the correlation coe±cient between the jth asset/liability variable and the kth asset/liability canonical variable (for j ¼ 1; 2; . . . ; q 1 /q 2 and K ¼ 1; 2; . . . ; p). The canonical correlation coe±cients and the canonical loadings can be used to study the nature of the relationship between a speci¯c asset and a speci¯c liability. Its underlying logic is represented in Fig. 1 below and can be explained as follows. If the canonical correlation coe±cient between two canonical variables is large (relation 1), and the canonical loadings for a speci¯c asset q 1 (relation 2a) and a speci¯c liability q 2 (relation 2b) are both large, we can assume that the speci¯c asset q 1 and the speci¯c liability q 2 are interconnected (relation 3). Since the canonical correlation coe±cients represent the variance shared by linear combinations of asset and liability/capital variables and not the variance shared by the original asset and liability accounts, it is possible that a high correlation between only one asset variable and only one liability variable could lead to a very large canonical correlation coe±cient. In order to address this issue and further investigate the links between asset and liability accounts, we calculate the redundancy coe±cients that provide a measure of the average ability of asset (liability) variables, taken as a set, to explain variation in liability (asset) variables taken one at a time. For each canonical correlation coe±cient (k ¼ 1; . . . ; p), the redundancy coe±cients can be obtained as follows: where s 2 jk;1 and s 2 jk;2 are, respectively, the elements of matrices S 1 and S 2 and R 2 is the canonical correlation coe±cient. As shown by Stewart & Love (1968), the sum of the redundancy coe±cients across all the canonical correlation coe±cients represents a measure of the proportion of the variance of asset variables predictable from liability variables (R 1 ) and vice versa (R 2 ). In symbols:  Brislin & Santomero (1991) with regard to the banking sector, we include in our sample only companies more than 10 years old. We perform canonical correlation analysis on these data in each year from 2007 to 2015, where each set of annual calculations is independent from the others. In order to ensure that our¯ndings will re°ect the impact of changes in¯nancial markets on asset-liability dependence while holding (as best possible) insurer management and business strategy constant, the models are estimated only for survivor insurers that appear in the data every year during the sample period.
We subdivide insurer assets into the following eight accounts: cash and deposits with credit institutions, bonds and other¯xed interest securities, shares and other variable interest instruments, assets held to cover linked liabilities, mortgage loans, real estate loans, receivables arising out of insurance/reinsurance operations and other assets. As concerns the liability side, we have taken into consideration the following six accounts: capital, life reserves, provisions for linked liabilities, payables arising out of insurance/reinsurance operations, external borrowing and other liabilities. Since there is not any a priori to de¯ne these accounts prior to applying canonical correlation analysis, we make our decisions based primarily on their nature and maturity characteristics. Each of these accounts is expressed as a percentage of total assets.
The mean balance sheet proportions for this balance sheet breakdown are shown in Table 1. Our sample intermediaries' major investments are in bonds and other xed rate securities (BOND), which increase over the 2007-2015 time horizon from 38.4% to 47.8% of total assets, with a 2.49% compound annual growth rate. The percentage of assets invested in shares and other variable interest securities (SHARE) experience a decline from the 11.8% observed in 2007 to the 5.6% of the 2015 year-end, with a À7.94% compound annual growth rate. Overall, the decreasing interest rates and the raise in uncertainty and stock market volatility can explain these opposite trends. The former might have created incentives to buy¯xed-rate bonds and securities in order to bene¯t from capital gains that insurers might have reasonably perceived as highly likely to occur, whereas the latter might have discouraged risky investments in volatile stock markets and less pro¯table variable interest securities. EU life insurers might have also decided to use their liquidity to support their investments in bonds and¯xed rate securities since cash and deposits with credit institutions (CASH) shows a substantial decrease of 10.14% at a À1.18% compound annual growth rate, being, on average the 4.79% of total assets. Assets held to cover linked liabilities (AHCLL) are an important share of our insurers' balance sheets, representing the 11.8% of their total assets over the entire sample period. Their compound annual growth rate is 0.74%, which is very close to the 0.79% value observed for the correlated item provisions for linked liabilities (PLL) within the liability side. On average, the sum of mortgage and real estate loans (ML and REL, respectively) represent 10.6% of total assets, with an overall 7.98% decrease to the 9.46% of the 2015 year-end from the 10.28% as of December 2007. By focusing on the receivables and payables arising out of insurance/reinsurance operations, EU life insurers seem to be more involved in transferring their risk to other insurers/reinsurers rather than to take risk from other companies. Receivables from insurance/reinsurance operations are, on average, 2.67% of total assets, with a compound annual growth rate of À2.16%, whereas payables represent the 1.73% of total assets, with an even greater negative compound annual growth rate (À3.07%).
On the liability side, the capital endowment (CAPITAL) of our insurance companies has remained quite stable, being equal to 10.7% at the end of 2007 and to 10.1% as of December 2015, and representing, on average, 9.86% of total assets. Prior literature hypothesized a positive relationship between the level of an insurer's capital and its investments in riskier assets such as stocks (Stowe 1978). Therefore, the reduction in shares and variable interest securities, as well as the reduction of lending exposure, might help to explain the substantial stability of EU life insurers'  capital endowment, assuming that they have bought high-quality bonds and¯xed rate securities. Life insurance reserves (LIFERES) represent the major share of the balance sheet liability side, amounting, on average, to 45.7% of total assets, and raising from 43.9% to 47.1% over the period we take into account. Finally, PLL represent another signi¯cant component of EU life insurers' liabilities, being on average equal to 11.78% of their total assets, and experiencing a 7.38% increase from 2007 to 2015. Both of these two latter accounts show a 0.79% compound annual growth rate. Finally, the average share of external borrowing is 2.13% over the entire sample period, with an overall À0.35% reduction and an approximately null compound annual growth rate. Table 2 provides simple correlations between the asset and liability proportions for the cross-section of our sample EU life insurers in each of the major years taken into account. We have highlighted in bold only \strong" correlations, which we de¯ne in ad hoc fashion as correlations greater than 0.30 in absolute value and statistically signi¯cant. By focusing on the relationships between asset and liability accounts, we only¯nd¯ve relationships con¯rmed for all the three years 2007, 2011 and 2015. In particular, EU life insurers systematically show an almost perfect positive correlation between AHCLL and PLL, a positive correlation between the variable CASH and the variable CAPITAL, and,¯nally, a positive correlation between mortgage loans (ML) and LIFERES. Insurance/reinsurance receivables (IRR) positively correlates with CAPITAL and negatively with LIFERES. Among the correlations of the other major asset accounts, the share of bonds and other¯xed rate securities (BOND) positively correlates with the amount of capital, reserves and retained earnings (CAPITAL) only in 2007, whereas the variable SHARE positively correlates with LIFERES in 2007 and 2015, but not in 2011. From simple pair-wise correlation analysis, where we completely ignore movements and co-movements of other assets and liabilities, we move to canonical correlation analysis, which considers simultaneously the economic complexities both within and across the two sides of the balance sheet.

Applying Canonical Correlation Analysis to EU Life Insurance Companies: Analysis and Results
The results of the canonical correlation analysis are discussed in this section. Five canonical variate pairs were derived for our sample insurers, since we have eight variables describing the asset side and six variables for the liability side of the balance sheet, and one was dropped from each group to avoid the singularity issue. b Table 3 displays the canonical correlations referred to the three main years in our data. On average, based on the magnitude and statistical signi¯cance of the canonical correlation coe±cients referred to years 2007, 2011 and 2015, the asset and liability b See Sec. 3 above.
variables exhibit a relatively high degree of collective dependence. For example, the second canonical correlation in the column referred to year 2007 is equal to 0.93, which means that the second factor extracted from the asset accounts data and the second factor extracted from the liability accounts data have a linear correlation of 0.93. Moving down each column, the canonical correlations tend to decline in both explanatory power and statistical signi¯cance. For example, with regard to the 2011 column, the¯rst approximate F-statistic of 5.95 allows us to reject the null hypothesis that all the¯ve canonical correlations coe±cients are zero. Similarly, the second F-statistic of 2.60 rejects the null hypothesis that second, third, fourth, and fth canonical correlations coe±cients are zero. The third F-statistic is also statistically signi¯cant at the 10% con¯dence level, which allows to reject the null hypothesis that third, fourth, and¯fth canonical correlation coe±cients are zero, but the fourth F-value is not: therefore, three or fewer, out of¯ve, canonical pairs are necessary to represent the asset-liability relationship in that particular year. Overall, we¯nd that the number of statistically signi¯cant canonical correlations declines over our sample period. In particular, we observe¯ve statistically signi¯cant canonical correlation coe±cients in 2007, with the coe±cients 0.99, 0.93, 0.83 and 0.65 all signi¯cant at the 1% con¯dence level and the coe±cient 0.55 marginally signi¯cant at the 10%. There are three statistically signi¯cant canonical correlation coefcients in 2011, with the coe±cients 0.99 and 0.89 both signi¯cant at the 1% con¯dence level, and the coe±cient 0.79 at the 10%. Finally, there are only two statistically signi¯cant canonical correlation coe±cients in 2015, 0.99 and 0.94, both signi¯cant at the 1% con¯dence level. Overall, this evidence suggests a declining trend in the strength of the asset-liability relationship within EU life insurers.
Nevertheless, the statistics displayed in Table 3 represent relationships between linear combinations of asset variables and linear combinations of liability variables, and these canonical correlations may or may not indicate systematic relationships between or among the underlying asset and liability variables. Consequently, in order to address this particular issue, and provide support to the inference we make 0.55* 0.06 0.00 À À À 0.00 À À À The F-statistic tests whether there is any association between the p pairs of canonical variables. Note: ***, **, * ¼ signi¯cance level of 1%, 5%, and 10%, respectively, using Bartlett's Chi-square test. through our statistical analysis, we report information for the proportion of variance coe±cients and the redundancy coe±cients in Table 4. By construction, the proportion of variance statistics in the Panel A of Table 4 must sum to 100%, and, in order to correctly read the table, we point out that the larger is the number of the reported signi¯cant loadings, the more interconnected are our insurers' assets and liabilities.
In 2007, about 55.2% (21.45% þ 25.57% þ 8.15%) of the variation in the actual assets data is explained by the asset canonical variables in the¯rst three loadings. Four years later, in 2011, a higher percentage of 86.8% (37.19% þ 49.65%) of the variation in the actual assets is explained by the asset canonical variables in the¯rst two loadings. Finally, in 2015, 50.3% of the variation in the actual assets is explained by the asset canonical variables in the¯rst loading. As far the liability side is concerned, 60.5% (25.96% + 20.70% + 13.81%) of the variation in the actual liabilities data is explained by the liability canonical variables in the¯rst three loadings. In 2011, 79.9% (45.92% + 34.03%) of the variation in the actual liabilities is explained by the liability canonical variables in the¯rst two loadings. At the end of our sample period, 66.7% of the variation in the actual liabilities is explained by the liability canonical variables in the¯rst loading. All else equal, this suggests that the relationships among the various asset and liability accounts become less complex over time, since we observe a reduction in the number of signi¯cant loadings and, accordingly, an increase in the share of the variance explained by the¯rst loadings.
Panel B of Table 4 shows the redundancy coe±cients calculated following the approach described in the previous Sec. 3 (Stewart & Love 1968), where we have pointed out that the sum of these coe±cients across all the canonical correlation coe±cients represents a measure of the proportion of the variance of asset variables predictable from liability variables and vice versa. In other words, redundancy coe±cients are a measure of the average capacity of asset (liability) accounts, taken as a set, to explain variation in liability (asset) accounts taken one at a time. Therefore, they are expected to provide insights about the link of causality working between the two sides of our insurers' balance sheets.
The redundancy coe±cients in the panel B of Table 4 sum to well less than 100% across the loadings for all the years taken into account. In 2007, the liability canonical variables explain 51.45% of the variation in the asset variables, while the asset canonical variables explain 67.4% of the variation in the liability variables. Both the share of the variation of the asset variables explained by the liability variables and that of the variation of the liability variables explained by the asset variables experience a decline in 2011 and 2015. In particular, with regard to 2011, liability variables explain 40.02% of the variation of the asset variables and asset accounts are able to explain 61.89% of the variation of the liability ones, whereas in 2015 these values decrease to 34.19% and 56.99%, respectively.
Overall, we can draw two informal inferences from the results shown in Table 4. First, causation runs more strongly from assets to liabilities than from liabilities to assets since, for all the three years examined, the variation of the liabilities explained by the assets is higher than the variation of the assets explained by the liabilities. In terms of our life insurers' behavior, this seems to suggest that they seek funding and/or determine funding mix only after¯nding investment opportunities rather than being pools of deposits looking for lending opportunities. Second, despite the relatively large size of the redundancy coe±cients, the importance of the¯rst loadings in the calculation of these coe±cients, suggests that a relatively small number of relationships among individual asset and liability accounts drives the strong canonical correlations shown in Table 3.
We look more closely at links between the individual asset and liability accounts in Table 5, which focuses on the individual asset-liability relationships in the  Fornell & Larcker (1980) and Perrault & Spiro (1978) explain the rotation of the canonical loadings. In this study, the canonical loadings are rotated using Kaiser's (1958) normalized varimax criterion which leaves the total predictable variance unchanged. The rotated loadings are given, for each of the three years 2007, 2011 and 2015, in the columns of Panels A, B and C of Table 5, and the interpretations are drawn from these rotated loadings.
Given our results in Table 4, we limit our analysis here to the linkages suggested by the¯rst and the second loadings. Correlations between individual actual asset accounts and their asset canonical variables appear on the left-hand side of the tables, while correlations between individual actual liability accounts and their liability canonical variables appear on the right-hand side of the tables. Based on the logic of Fig. 1,¯nding simultaneous strong canonical loadings for asset and liability accounts implies a strong relationship between the underlying asset and liability variables, because the canonical correlations in both the¯rst and second loadings are empirically large and statistically strong (see Table 3). Following Fornell & Larcker (1980), we use a 0.30 threshold to determine a \strong" relationship between the original variables and the canonical variables.
We¯nd a limited number of strong and economically sensible relationships among the variables in Table 5. In particular, for canonical variable 1R, the dominant relationship is between cash and deposits with credit institutions (CASH) and capital and reserves (CAPITAL), which have strong canonical loadings with the same sign in all the three years taken into account, thus providing a strong evidence that these two balance sheet accounts systematically move up and down together. According to this evidence, insurers with large amounts of capital are better able to hold large portions of their portfolios in cash. Function 2R in all the panels of Table 5 has a positive loading for the AHCLL on the asset side and a positive loading for the PLL on the liability side. Both variables have loadings of nearly unity and all other original variables have relatively low loadings with this canonical variable. This almost perfect matching of these two accounts is entirely expected as insurers have hedged this asset and liability category.

Concluding Remarks
Low interest rates have become a signi¯cant threat to the stability of the life insurance industry, especially in countries where products with relatively high guaranteed returns sold in the past still represent a prominent share of the total portfolio. Given the current market environment and the expected persistence of extremely low interest rate scenario, it is important to investigate and quantify how and to what extent the unprecedented market conditions have a®ected life insurers' behavior in terms of asset and liability management.
Canonical correlation analysis has allowed us to examine and interpret signi¯cant empirical relationships within the asset and liability structures of a group of large life insurers and the cross-balance sheet relationships that we have identi¯ed suggest some interesting considerations, which can shed some light on insurers' decisions under unprecedented market conditions. In particular, contrary to expectations, insurance companies seem to run their business as if they decide their funding policies after identifying good investment opportunities. Overall, low market rates may have a negative impact on companies' business model. We¯nd strong and substantial evidence that insurer assets and liabilities have indeed become more independent over time. We argue that the declining trend of market interest rates over the examined time horizon has contributed to the generalized reduction in the linkage between the asset side and the liability side of EU life insurers, and has made insurance companies more exposed to ALM-related risks than they were in the period before the¯nancial crisis broke out.
Based on our¯ndings, further investigation and a deeper comprehension of the relations between insurer assets and liabilities are crucial from both a regulatory and supervisory perspective since it might help to de¯ne qualitative and quantitative measures of liquidity requirements that are more consistent with insurers' actual behavior, during both benign market conditions and stressed¯nancial markets.
The statistical methodology employed in this paper is completely appropriate for a¯rst examination of the phenomena we are interested in and can supplement the frequently employed regression models in investigating relationships in which the decision outcome involves a set of several variables instead of a single variable. Nevertheless, our¯ndings should be interpreted with caution since further investigation into asset-liability linkages is needed in order to generate more robust insurerlevel evidence and to get a more comprehensive understanding of the di®erent channels through which interest rates a®ect life insurers' activity. One potential development would be to apply canonical correlation analysis to time-series data at the insurer level. Such an approach would generate insurer-speci¯c estimates of canonical correlations and redundancy coe±cients, which could then be regressed on insurer-speci¯c arguments to test a variety of hypotheses.