REGULATORY CAPITAL ON INSURERS ASSET ALLOCATION & TIME HORIZONS OF THEIR GUARANTEES

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1 DAEFI Philippe Trainar May 16, 2006 REGULATORY CAPITAL ON INSURERS ASSET ALLOCATION & TIME HORIZONS OF THEIR GUARANTEES As stressed by recent developments in economic and financial analysis, optimal portfolio should include a sufficient share of stocks for three main reasons: - stocks contribute to diversification of insurers portfolio ; - stocks can better hedge real risks linked to productivity or wage shocks - stocks have a better ability to cover long term liabilities than short term one QIS 2 are taking into account the first reason, i.e. the diversification effect of stocks. But, they ignore the second and the third reason when they are as important as the first one. Indeed, one may reasonably argue that these two characteristics, because of their multidimensional aspects, are difficult to deal with in a standard formula. The hedging potential of stocks is well documented and not questioned by experts. Indeed, no simple formula can measure exactly the hedging potential of stocks with regard to real risks such as productivity or wage risks and internal models are precisely designed for handling such questions. At the same time, it is not sure that a standard formula would be less able to take into account the flavour of this hedging potential than other dimensions of risk that are currently included in QIS 2. One simple way to do that would be to define reference portfolios by line of business and to calculate the capital loading on assets as a proportion of the gap between this reference portfolio and the effective investment portfolio of the company. There are other ways, as we will see afterwards. Until now no convincingly practical or theoretical argument for excluding these solutions from QIS 2 has been given Unlike the hedging potential of stocks, their better ability to cover long term liabilities than short term one is more controversial because it is neither common knowledge of experts nor perceived as a core issue. This note argues this perception is erroneous and that it is desirable to take the ability of stocks to cover long term liabilities into account in QIS 2 for three main reasons : - first of all, the ability of stocks to cover long term liabilities is not well documented and no more questioned, despite the fact that it has remained until now a topic of highly sophisticated economic and finance academics ; - secondly, it is a characteristic of stocks, which induces consequences of first order magnitude and not of second order magnitude as often argued ; - thirdly, taking into account the time horizon of liabilities for calculating the capital loading of assets may be easy to deal with in the standard formula of solvency II. 1

2 1. At the opposite of the efficient market theory, current financial analysis point to time-varying investment opportunities. Traditional mean-variance financial analysis typically focuses on short term expected returns and risk. This focus is valid as long as the term structure of the risk-return trade-off is flat and investment opportunities are constant. Indeed, it is the hypothesis on which the traditional meanvariance analysis of academic and applied finance is based : - assets have constant expected returns which can be estimated from their historical means and which are therefore unconditional ; - the variance of each asset return is proportional to the horizon over which it is held : a ten years variance is ten times larger than a one year variance ; but, annualized variances are independent of the time horizon, and there is a single number that summarizes risks for all holding periods. Traditional analysis thought short-term mean-variance provides inputs that are valid for all meanvariance investors, regardless of their investment horizon. As a consequence, the efficient frontier for the asset allocation is the same at all horizons. This is clearly the option chosen by CEIOPS in its QIS 2. Even in this context of constant investment opportunities as defined by the couple of mean and variance, it has been rigorously demonstrated 1 that, when risks are independent over time, a longer time horizon raises the optimal exposure to risk in the short term only if the absolute risk tolerance, i.e. the inverse of the absolute risk aversion, is convex (and subhomogeneous). Whether the absolute risk tolerance of the regulator is convex and whether it is for all wealth levels is a pure empirical question. Until recently, most models in finance used hyperbolic absolute risk aversion or constant relative risk aversion utility functions, for which time horizon has no effect on the optimal portfolio. However, these functions were used for simplicity reasons rather than for realism or theory. But, more importantly, a large body of research in empirical finance, undertaken since the end of the 1980 s, has established not only that investment opportunities, i.e. variance and returns, change over time but also that changes on bonds and equities are highly persistent. Yet, since the work of Samuelson and Merton 2, financial economists have understood that, in this case, long term investing may differ from short term investing. Long-term investors will care about shocks to investment opportunities and may wish to hedge their exposure giving rise to intertemporal hedging demands for financial assets. This response to time-varying investment opportunities has been labelled strategic asset allocation. This is a very important issue. It has been rigorously established 3 that, when expected returns are time-varying and the term structure of risk is not flat, the efficient frontiers at different horizons do not coincide and the short-term mean-variance analysis can be misleading for investors with 1 For an overview of this argument cf. C. Gollier (2001) : The economics of risk and time, MIT Press and L. Eeckoudt, C. Gollier & H. Schlesinger (2005) : Economic and financial decision under risk, Princeton University Press 2 Cf. P.A. Samuelson (1969) : Lifetime portfolio selection by dynamic stochastic programming, Review of Economics and Statistics, vol. 51, n 3 and C. Merton (1973) : An Intertemporal Capital Asset Pricing Model, Econometrica, vol. 41, n 5 3 For an overview of the most recent developments cf. J.Y. Campbell & L.M. Viceira (2002) : Strategic asset allocation : potfolio choice for long-tem investors, and (2005) : The term structure of the risk-return tradeoff, Financial analysts Journal, april, with its companion paper (2004) : Long-Horizon Mean-Variance analysis : a user guide, Harvard University working paper, and J.Y. Campbell, L.M. Viceira & Y.L. Chan (2001) : A multivariate strategy of asset allocation, NBER, working papers n

3 longer investment horizons. In this case, the short-term mean-variance analysis is able to describe only the short end of the curve. 2. These changing opportunities results mainly from mean reversion of stock returns, which alters significantly long term mean-variance. For long, business experts have pointed the ability of long term investors to capture an excess return thanks to a larger exposure of their portfolio to equity risk. They have made their recommendation, having observed that empirical mean-variance return of bonds and equities is time varying. As pointed by the graph below, for France, over a half century, one can observe that volatility of bonds and equities are falling with the lengthening of time horizon, and that the one of equities is falling more quickly than the one of bonds. Annual volatility 25% 20% 15% 10% Assets short & long term, empirical & theoretical volatility equities : empirical volatility equities : theoretical volatility Bonds : empirical volatility Bonds : theoretical volatility 5% 0% time horizon (years) The empirical volatility is the volatility effectively observed over the corresponding temporal horizon, whereas the theoretical volatility is the volatility for a random Brownian motion with independent returns between t and t+1. A theoretical volatility curve up above an empirical volatility curve shows a mean reversion phenomenon while a theoretical volatility curve below an empirical volatility curve show the opposite, i.e. a cumulative spread phenomenon in comparison with the mean. Furthermore, as illustrated by the graph below, probability of cumulative mean annual return on equities at 96,5 th percentile is growing rapidly, more rapidly than the one of bonds, with the le ngthening of investment horizon, and becomes positive at 25 years horizon. 3

4 15% Real return at 96,5th percentile and mean value (french data since 1945) Real return 10% 5% 0% -5% -10% -15% -20% -25% -30% mean bond return bond return at 96,5th percentile stock return at 96,5th percentile mean stock return Investment duration (years) As a consequence of these behaviours of equity returns, the Sharpe ratio of stocks is growing with investment duration. 3. Econometric estimations specify this mean reversion process and dependent correlations between bonds, stocks and T-bills. But, observed time varying mean-variance presented above are biased. Their calculation uses overlapping annual data in an inconsistent way : different annual data are used several times depending from year position and period length. Furthermore, a very large body of empirical research since the 1980's 4 shows that, contrary to the efficient market hypothesis, equity returns, as well as bond returns, are not following a random walk and are (partially) predictable. The optimal dynamic portfolio strategy is therefore an affine function of the vector of state variables describing investment opportunities, with coefficients that are a function of the investment horizon. Simulating mean-variance of stocks, bonds and T-bills on a multiyear basis, thanks to predictive models, allows bypassing the flaw of empirical multiperiod mean-variance. Studies based on sophisticated Vector Auto Regressions (VAR) on US data find that meanreversion in stock and bond returns decreases the volatility per period of real stock and bond return at long horizons and that this behaviour is more pronounced for stock returns, while reinvestment risk increases the volatility of real T-bill returns. Inflation risk increases the volatility per period of the real return on long term bonds held to maturity, as it appears clearly in the graph below 5. 4 For the most recent empirical estimations cf. Campbell & alii (2002 & 2005). VAR simulation presents the advantage of extracting the autonomous shocks specific to each variable. 5 Cf. idem 4

5 Furthermore, as pointed by the graph below, stocks and bonds exhibit low positive correlation at both ends of the term structure of the risk, but they are highly correlated at intermediate investment horizons. Inflation is the main reason of this shape of the correlation. Indeed, inflation is negatively correlated with bond and stock real return at short term and positively correlated at long horizons Optimal asset allocation of long term insurers is substantially altered by this time varying mean-variance Because of time varying mean-variance of asset returns, the efficient frontier of insurers asset allocation should also be time varying following the variations of the relative variance and mean of equities (or bonds). As time horizon increases, the Sharpe ratio of stocks increases and stocks become relatively more secure, justifying an overweighting of equities in the asset allocation of 6 Cf. idem 5

6 long term insurers. So, as time horizon increases, the minimum variance portfolio is increasingly biased toward stocks, as illustrated by calculations presented in the table below whose figures are extracted from Campbell & alii 7 using volatilities and correlations presented above and applied to a buy-and-hold strategy. The same calculations are made for the tangency portfolio, i.e. the optimal portfolio of bonds and stocks, which is increasingly biased toward stocks as time horizon widens. Even if they are also applied to a buy-and-hold strategy, which is consistent with regulatory concerns, one should nevertheless have in mind that they do not take into account regulatory standards that are required for insurance companies, such as a minimum rate of survival for example. Of cause, the figures below cannot be considered as representative of the tangency portfolio of an insurance company. However, all the figures below point rightly to the fact that the gap between the short term and long term prudent/optimal portfolios is of first order magnitude, and not of second order magnitude as often argued. Minimum variance Tangency portfolio of Risky fund portfolio of portfolio of risky assets risky assets a buy-and-hold investor Horizon Horizon (excl. T-bills) 1 year 25 years 1 year 25 years Stocks Sharpe ratio 0,201 0,407 0,201 0,407 Stocks e 43 % 47 % 72 % 5 years bond 100 % 57 % 53 % 28 % This time bias is the result of the increasing positive correlation between stocks and bonds at intermediate time horizons and the pronounced decrease of the volatility per period of stock return at long investment horizons. This bias is aggravated when one considers the intertemporal insurers' hedging demands for financial assets : stocks appear to better hedge inflation as well as productivity and wealth shocks than other assets 8. Of cause, mean excess return of stocks with regard of mean excess return on bonds plays also a significant role in the calculation. Of cause, one has to remember that the shares of risky assets in a portfolio are proportional to the ratio of their mean excess return to their variance-covariance matrix. More rigorously, they are equal to the ratio of their mean excess return to the product of their variance-covariance matrix with their relative risk aversion [(µ-r)(rs) -1 ] 9. Here we should make due reference to a constraining condition concerning risk aversion for getting bias in favour of stocks in asset allocation : in face of strong mean reversion of stocks return, a longer time horizon induces risk averse investors (or regulators) to take more stocks only if their relative risk aversion is larger than unity 10. From a strict regulatory and supervisory point of view this condition is easy to satisfy because regulatory and supervisory bodies have high relative risk aversion, probably higher than Regulatory standards cannot ignore these behaviours when calculating capital requirements on asset risks. It would be highly questionable if CEIOPS would deliberately choose to ignore these main conclusions of the last twenty years economic and financial analysis, i.e. the facts that volatility and correlations of main asset categories vary over time and that consequently the optimal asset allocation vary with the time horizon of the insurer. Ignoring these conclusion would lead 7 cf. idem 8 For a more detailed examination of the relationships between inflation and stock prices cf. J.Y. Campbell & T. Vuolteenaho (2004) : Inflation illusion and stock prices, American Economic Review, May 9 Cf. C. Gollier et alii (2005) 10 Cf. idem 6

7 regulators to require as much capital for a badly managed short term insurer, that is over-invested in equities for speculative reasons, and a well managed long term insurer, that is as much invested in equities as the former but for hedging reasons. In order to get a good evaluation of asset risk and to discriminate accurately between good and bad asset managers, the regulatory standards and the standard formula, as far as it is concerned, should take explicitly into account the term structure of the risk-return tradeoff of insurers. This term structure of investment is largely, if not exclusively, determined by the term structure of insurance liabilities. It should underlined that what is at stake is not an overall reduction of capital requirement on stocks but a reduction proportionate to the effective risk of stocks with regard to the commitments of the insurer. Moreover, in doing so, the standard formula would be consistent with market pricing of portfolios transfers. There are three elegant solutions for taking into account the time varying level of mean-variance so as to be consistent with QIS 2 architecture : - proportionate capital requirement on assets to the gap between the effective asset structure of insurer and a predefined reference portfolio calculated by line of business on the basis of their respective term structure; - choose equity shocks or equity lading coefficients proportionate to assets volatilities and correlations that are consistent with the term structure of insurance liabilities, having also taken into account automatic contract renewal ; at a first step, observed empirical volatilities and correlations could be used, having in mind that observed volatilities and VAR calculated volatilities do not differ substantially ; one should underline that this rule does not necessitate predicting asset returns even if its accuracy follows directly from the (partial) predictability of stock and bond returns; - calculate a duration of equities 11 and proportionate capital requirement on assets to the duration gap between assets and liabilities ; this solution, which may be conceptually referred to immuzation and which has been supported by the French industry, would be easy to implement and would hugely simplify the current proposition of QIS 2. Most previous conclusions are based on a buy-and-hold strategy which is the way regulators and supervisors are currently thinking. Optimal asset allocation rebalancing strategies in the face of changing investment opportunities is probably more difficult to compute. But, as for buy-andhold investors, asset return predictability can have large effects on the asset allocation decisions of rebalancing investors. Such strategic intertemporal hedging portfolios tilt the total portfolio away from short term mean-variance frontier. 6. The prospective view of the framework does not oppose taking mean reversion of stocks and other assets returns into consideration The framework of the EU Commission proposes a prospective view of solvency. It does not say that solvency should be appreciated instantaneously but that it should be appreciated on a one year prospective horizon : insurers should be able to cover their commitments with their current assets with a probability of 99,5 % at the end of next year. Some people fear that taking into account long term volatilities and returns may make solvency requirements more difficult to satisfy next year. These fears are misplaced because next year solvency will not be controlled on the basis of next end year balance sheet but on basis of the prospective balance sheet at the end of the year following next year, as anticipated at the end of next year. At first glance, the prospective view of the framework seems to live sufficient margin 11 Standard & Poor s estimates equity current duration to be 15 years, near the trough of 14 years at the beginning of the nineties peak (the last peak was 23 years in 1998). 7

8 for taking into account mean reversion behaviour of stocks and bonds returns and for using multiperiod volatilities when calculating capital requirements on assets (as well as liabilities). An encompassing understanding of the framework clearly allows integrating time varying means and variances for the calculation of capital requirements. If this is not the case, one should then question the effective prospective dimension of the framework and ask for its enlargement. 8