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Strategic Asset Allocation: Determining the Optimal Portfolio with Ten Asset Classes Essay

Strategic Asset Allocation: Determining the Optimal Portfolio with Ten Asset Classes Niels Bekkers Mars The Netherlands Ronald Q. Doeswijk* Robeco The Netherlands Trevin W. Lam Rabobank The Netherlands October 2009 Abstract This study explores which asset classes add value to a traditional portfolio of stocks, bonds and cash. Next, we determine the optimal weights of all asset classes in the optimal portfolio. This study adds to the literature by distinguishing ten different investment categories simultaneously in a mean-variance analysis as well as a market portfolio approach.

We also demonstrate how to combine these two methods. Our results suggest that real estate, commodities and high yield add most value to the traditional asset mix. A study with such a broad coverage of asset classes has not been conducted before, not in the context of determining capital market expectations and performing a mean-variance analysis, neither in assessing the global market portfolio. JEL classification: G11, G12 Key words: strategic asset allocation, capital market expectations, mean-variance analysis, optimal portfolio, global market portfolio.

This study has benefited from the support and practical comments provided by Jeroen Beimer, Leon Cornelissen, Lex Hoogduin, Menno Meekel, Leon Muller, Laurens Swinkels and Pim van Vliet. Special thanks go to Jeroen Blokland and Rolf Hermans for many extensive and valuable discussions. We thank Peter Hobbs for providing the detailed segmentation of the global real estate market that supplemented his research paper. Last, but not least, we thank Frank de Jong for his constructive comments and useful suggestions during this study. * Corresponding author, email: r. [email protected] com, telephone: +31 10 2242855. Electronic copy available at: http://ssrn. com/abstract=1368689 1 Introduction Most previous academic studies agree on the importance of strategic asset allocation as a determinant for investment returns. In their frequently cited paper, Brinson, Hood and Beebower (1986) claim that 93. 6% of performance variation can be explained by strategic asset allocation decisions. This result implies that strategic asset allocation is far more important than market timing and security selection.

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Most asset allocation studies focus on the implications of adding one or two asset classes to a traditional asset mix of stocks, bonds and cash to conclude whether and to what extent an asset class should be included to the strategic portfolio, see for example Erb and Harvey (2006) and Lamm (1998). However, because of omitting asset classes this partial analysis can lead to sub-optimal portfolios. This is surprising, as pension funds and other institutions have been strategically shifting substantial parts of their investment portfolio towards non-traditional assets such as real estate, commodities, hedge funds and private equity.

The goal of this study is to explore which asset classes add value to a traditional asset mix and to determine the optimal weights of all asset classes in the optimal portfolio. This study adds to the literature by distinguishing ten different investment categories simultaneously in a mean-variance analysis as well as a market portfolio approach. We also demonstrate how to combine these two methods. Next to the traditional three asset classes stocks, government bonds and cash we include private equity, real estate, hedge funds, commodities, high yield, credits and inflation linked bonds.

A study with such a broad coverage of asset classes has not been conducted before, not in the context of determining capital market expectations and performing a mean-variance analysis, neither in assessing the global market portfolio. The second step in portfolio management, i. e. market timing and security selection are tactical decisions. These are beyond the scope of this study. In short, this study suggests that adding real estate, commodities and high yield to the traditional asset mix delivers the most efficiency improving value for investors.

Next, we show that the proportion of non-traditional asset classes appearing in the market portfolio is relatively small. In the remainder of this study we conduct an empirical and literature analysis to establish long-run capital market expectations for each asset class, which we subsequently use in a mean-variance analysis. Then, we provide an assessment of the global market portfolio. Finally, we show how the mean-variance and market portfolio approaches can be combined to determine optimal portfolios. 1 Electronic copy available at: http://ssrn. om/abstract=1368689 2 Methodology and data Methodology Markowitz (1952, 1956) pioneered the development of a quantitative method that takes the diversification benefits of portfolio allocation into account. Modern portfolio theory is the result of his work on portfolio optimization. Ideally, in a mean-variance optimization model, the complete investment opportunity set, i. e. all assets, should be considered simultaneously. However, in practice, most investors distinguish between different asset classes within their portfolio-allocation frameworks.

This two-stage model is generally applied by institutional investors, resulting in a top-down allocation strategy. In the first part of our analysis, we view the process of asset allocation as a four-step exercise like Bodie, Kane and Marcus (2005). It consists of choosing the asset classes under consideration, moving forward to establishing capital market expectations, followed by deriving the efficient frontier until finding the optimal asset mix. In the second part of our analysis, we assess the global market portfolio.

Finally, we show how the mean-variance and market-neutral portfolio approaches can be combined to determine optimal portfolios. We take the perspective of an asset-only investor in search of the optimal portfolio. An asset-only investor does not take liabilities into account. The investment horizon is one year and the opportunity set consists of ten asset classes. The investor pursues wealth maximization and no other particular investment goals are considered. We solve the asset-allocation problem using a mean-variance optimization based on excess returns.

The goal is to maximise the Sharpe ratio (risk-adjusted return) of the portfolio, bounded by the restriction that the exposure to any risky asset class is greater than or equal to zero and that the sum of the weights adds up to one. The focus is on the relative allocation to risky assets in the optimal portfolio, in stead of the allocation to cash. The weight of cash is a function of the investor’s level of risk aversion. For the expected risk premia we use geometric returns with intervals of 0. 25%. The interval for the standard deviations is 1% and for correlations 0. 1.

In our opinion, more precise estimates might have an appearance of exactness which we want to prevent. We do not take management fees into consideration, except for private equity and hedge funds as for these asset classes the management fees are rather high relative to the expected risk premia of the asset class. Other asset classes have significantly lower fees compared to their risk premia. They are therefore of minor importance, especially after taking the uncertainty of our estimates into account. We estimate risk premiums by 2 subtracting geometric returns from each other.

Hereby, our estimated geometric returns as well as the risk premiums both are round numbers. In the mean-variance analysis, we use arithmetic excess returns. Geometric returns are not suitable in a mean-variance framework. The weighted average of geometric returns does not equal the geometric return of a simulated portfolio with the same composition. The observed difference can be explained by the diversification benefits of the portfolio allocation. We derive the arithmetic returns from the geometric returns and the volatility. Data We primarily focus on US data in the empirical analysis.

The choice for this market is backed by two arguments. First, the US market offers the longest data series for almost all asset classes. This makes a historical comparison more meaningful. For instance, the high yield bond market has long been solely a US capital market phenomenon. Secondly, using US data avoids the geographical mismatch in global data. A global index for the relatively new asset class of inflation linked bonds is biased towards the US, French and UK markets, while a global stock index is decently spread over numerous countries.

We use total return indices in US dollars. Asset classes like real estate and private equity are represented in both listed and non-listed indices, while hedge funds are only covered by non-listed indices. Non-listed real estate and private equity indices are appraisal based, which may cause a smoothing effect in assumed risk of the asset class. This bias arises because the appraisals will not take place frequently. However, interpolating returns causes an underestimation of risk.

Also, changes in prices will not be immediately reflected in appraisal values until there is sufficient evidence for an adjustment. Statistical procedures to mitigate these data problems exist, but there is no guarantee that these methods produce accurate measures of true holding-period returns, see Froot (1995). As these smoothing effects can lead to an underestimation of risk, this study avoids non-listed datasets and instead adopts listed indices for real estate and private equity. The quality of return data of listed indices is assumed to be higher as they are based on transaction prices.

Ibbotson (2006) supports this approach and states “Although all investors may not yet agree that direct commercial real estate investments and indirect commercial real investments (REITs) provide the same risk-reward exposure to commercial real estate, a growing body of research indicates that investment returns from the two markets are either the same or nearly so. ”. For hedge funds we will use a fund of fund index that we unsmooth with Geltner (1991, 1993) techniques. Fung and Hsieh (2000) describe the important role of funds of hedge funds as a proxy for the market portfolio of hedge funds.

Appendix A and B contain our data sources. In appendix A we discuss our capital market expectations and in appendix B we derive the market portfolio from a variety of data sources. 3 3 Empirical results Capital market expectations We estimate risk premia for all asset classes based on previous reported studies, our own empirical analyses of data series and on the basic idea that risk should be rewarded. Obviously, estimates like these inevitably are subjective as the academic literature only provides limited studies into the statistical characteristics of asset classes.

Moreover, there is generally no consensus among academics and we lack long term data for most asset classes. Our results should therefore be treated with care, especially since mean-variance analysis is known for its corner solutions, being highly sensitive in terms of its input parameters. In this study we proceed with the risk premia and standard deviations as shown in Table 1. Appendix A contains the reasoning for these estimates and for the correlation matrix. [INSERT TABLE 1] Mean-variance analysis

Table 2 shows the optimal portfolio based on the mean-variance analysis and its descriptive statistics for a traditional portfolio with stocks and bonds as well as a portfolio with all assets. On top of the traditional asset classes of stocks and bonds, this analysis suggests that it is attractive for an investor to add real estate, commodities and high yield. The Sharpe ratio increases from 0. 346 to 0. 396. The allocation to real estate is quite high. To bring this into perspective, we would suggest that the proposed portfolio weight is overdone.

When one, for example, would be willing to perceive utilities as a separate asset class, it is likely that it also would get a significant allocation as this sector also has a low correlation to the general stock market. Table 2 also illustrates that mean-variance analysis tends towards corner solutions as it neglects credits which has characteristics comparable with bonds. However, with these parameters it prefers bonds in the optimal portfolio. [INSERT TABLE 2] Figure 1 shows the benefits of diversification into non-traditional asset classes.

In the volatility range of 7% to 20% the diversification benefits vary between 0. 40% and 0. 93%. This additional return is economically significant. For example, at a volatility of 10% the additional return is 0. 56%. The efficient 4 frontier of a portfolio with stocks, bonds and the three asset classes real estate, commodities and high yield comes close to the efficient frontier of an all asset portfolio. By adding these three asset classes, an investor almost captures the complete diversification benefit. [INSERT FIGURE 1] For various reasons not all investors use cash to (un)leverage their investment portfolio.

Therefore, it is interesting to observe the composition of efficient portfolios in a world without the risk free rate. Figure 2 shows the asset allocation on the efficient frontier in an all asset portfolio starting from a minimum variance allocation towards a risky portfolio. It maximizes the expected excess return constrained by a given volatility. [INSERT FIGURE 2] In the least risky asset allocation, an investor allocates 77. 7% of the portfolio towards fixed-income assets. Next to bonds and stocks, real estate and commodities receive a significant allocation in portfolios with a volatility in the range of 7. %-12. 5%. High yield is also present in most of the portfolios in this range. For riskier portfolios, private equity shows up and, in the end, it ousts bonds, real estate, commodities and stocks (in this order). For defensive investors, inflation linked bonds and hedge funds enter the portfolio. We have tested the sensitivity of the mean-variance analysis to the input parameters. Table 3 shows the impact on the optimal portfolio of an increase and a decrease in the expected volatility of an asset by a fifth, all other things being equal.

Note that a change in volatility affects both the arithmetic return and the covariance matrix. Again, this table demonstrates the sensitivity of a mean-variance analysis to the input parameters. An increase in expected volatility leads to a lower allocation to that asset class. High yield even vanishes completely from the optimal portfolio. It is noteworthy that commodities are hardly affected by a higher standard deviation. A decrease in volatility mostly leads to a higher allocation, with the exception of hedge funds and commodities.

Commodities, despite their expected zero risk premium, add value due to the strong diversification benefit. In this analysis, they appear to be insensitive to a change in their expected volatility. Credits and bonds are quite similar asset classes and, in a mean-variance context, the optimal portfolio tends to incline towards one or the other. [INSERT TABLE 3] In short, the mean-variance analysis suggests that adding real estate, commodities and high yield to the traditional asset mix of stocks and bonds creates most value for investors. Basically, adding these 5 three asset classes comes close to an all asset portfolio.

Private equity is somewhat similar to stocks, but shows up in riskier portfolios, moving along the efficient frontier. This part of the efficient frontier is interesting for investors in search of high returns without leveraging the market portfolio. Hedge funds as a group do not add value. Obviously, when investors attribute alpha to a particular hedge fund, it changes the case for that fund. This also applies to private equity. Credits and bonds are quite similar asset classes and in a mean-variance context the optimal portfolio tends to tilt to one or another.

Inflation linked bonds do not show up in our mean-variance analysis. The inflation risk premium and the high correlation with bonds prevent an allocation towards this asset class in that setting. However, for defensive investors who primarily seek protection against inflation this asset class can be very interesting. Market portfolio Both academics and practitioners agree that the mean-variance analysis is extremely sensitive to small changes and errors in the assumptions. We therefore take another approach to the asset allocation problem, in which we estimate the weights of the asset classes in the market portfolio.

The composition of the observed market portfolio embodies the aggregate return, risk and correlation expectations of all market participants and is by definition the optimal portfolio. In practice however, borrowing is restricted for most investors and at the same time borrowing rates usually exceed lending rates. The result is that the market portfolio is possibly no longer the common optimal portfolio for all investors, because some might choose risky portfolios on the efficient frontier beyond the point where no money is allocated to the risk free rate.

In addition, an investor’s specific situation could also lead to a different portfolio. Despite this limitation, the relative market capitalization of asset classes provides valuable guidance for the asset allocation problem. In this setting, the market-neutral weight for a particular asset class is its market value relative to the world’s total market value of all asset classes. Figure 3 shows the global market portfolio based on a variety of data sources. Appendix B provides details about the market portfolio and its dynamics for the period 2006-2008.

The asset classes stocks and investment grade bonds (government bonds and credits) represent more than 85% of the market for these years. At the end of 2008 we estimate this number at 88. 8%. This means that the size of the average remaining asset class is less than 2. 0%. Based on this analysis, we conclude that the proportion of non-traditional asset classes appearing in the market portfolio is relatively small. [INSERT FIGURE 3] 6 Combination of market portfolio and mean-variance analysis The mean-variance analysis can be combined with the market portfolio.

Here, we choose to take the market portfolio as a starting point which we subsequently optimize with turnover and tracking error constraints. We choose to take the market portfolio as a starting point, as it embodies the aggregate return, risk and correlation expectations of all market participants without the disadvantage of delivering the corner solutions of the mean-variance analysis. Table 4 shows the optimal portfolios with different tracking error constraints and a maximum turnover of 25% (single count) relative to the market portfolio.

In other words, in this example we limit ourselves to finding optimized portfolios with portfolio weights that do not differ more than 25% from the market portfolio, calculated as the sum of the absolute difference between the market portfolio and the optimized portfolio for each asset class. Focusing on the 0. 25% tracking error constraint, it appears that the analysis recommends especially adding real estate, commodities and high yield, and removing hedge funds and inflation-linked bonds. This is logical, as the results from the meanvariance analysis are applied in this market-portfolio-adjustment process.

There is a 12. 5% shift in portfolio weights. Obviously, less constraints result in a higher risk premium and a higher Sharpe ratio, until we end up with the theoretically optimal portfolio from the mean-variance analysis. Within this methodology, investors must determine their own individual constraints, while the market portfolio and the portfolio optimized by mean-variance are considered as the boundaries for the asset classes. [INSERT TABLE 4] 4 Summary and conclusions Our mean-variance analysis suggests that real estate, commodities and high yield add most value to the traditional asset mix of stocks, bonds and cash.

Basically, adding these three asset classes comes close to an all asset portfolio. The portfolio with all assets shows a diversification benefit along the efficient frontier that varies between 0. 40% and 0. 93% in the volatility range of 7% to 20%. That is an economically significant extra return for free. Another approach to the asset allocation problem is assessing the weights of the asset classes in the market portfolio. Based on this analysis we conclude that the proportion of non-traditional asset classes appearing in the market portfolio is relatively small. 7 One can combine the mean-variance analysis with the market portfolio.

Within this methodology, investors must determine their own individual constraints, while the market portfolio and the portfolio optimized by mean-variance are considered as the boundaries for the asset classes. 8 References Altman, E. I. , 1998, “The Anatomy of the High Yield Bond Market: After Two Decades of ActivityImplications for Europe”, research paper. Amin, G. , and H. Kat, 2005, “Welcome to the Dark Side: Hedge Fund Attrition and Survivorship Bias 1994-2001”, Journal of Alternative Investments, vol. 6, no. 2, 57-73. Bodie, Z. , A. Kane and A. Marcus, 2005, “Investments”, McGraw-Hill, sixth dition. Brinson, G. P. , L. R. Hood and G. L. Beebower, 1986, “Determinants of Portfolio Performance”, Financial Analysts Journal, vol. 42, no. 4, p. 39-44. Dimson, E. , P. Marsh and M. Staunton, 2003, “Global Evidence on the Equity Risk Premium”, Journal of Applied Corporate Finance, vol. 15, no. 4, p. 27-38. Dimson, E. , P. Marsh and M. Staunton, 2009, “Global Investment Returns Sourcebook 2009”, Credit Suisse. Doyle, E. , J. Hill and I. Jack, 2007, “Growth in Commodity Investment: Risks and Challenges for Commodity Market Participants”, study from the FSA Markets Infrastructure Department.

Elton, E. J. , M. J. , Gruber and D. M. C. Agrawal, 2001, “Explaining the Rate Spread on Corporate Bonds”, Journal of Finance, vol. 56, no. 1, 247-277. Erb, C. B. , and C. R. Harvey, 2006, “The Tactical and Strategic Value of Commodity Futures”, Financial Analysts Journal, vol. 62, no. 2, p. 69-97. Fama, E. F. , and K. R. French, 2002, “The Equity Premium”, Journal of Finance, vol. 57, no. 2, p. 637660. Froot, K. A. , 1995, “Hedging Portfolios with Real Assets”, Journal of Portfolio Management, vol. 21, no. 4, p. 60-77. Fugazza, C. , M. Guidolin and G.

Nicodano, 2007, “Investing for the Long-Run in European Real Estate”, Journal of Real Estate Finance and Economics, vol. 34, no. 1, p. 35-80. 9 Fung, W. , and D. A. Hsieh, 2000, “Performance Characteristics of Hedge Funds and Commodity Funds: Natural vs. Spurious Biases”, Journal of Financial and Quantitative Analysis, vol. 35, no. 3, p. 291-308. Geltner, D. M. , 1991, “Smoothing in Appraisal-Based Returns”, Journal of Real Estate Finance and Economics, vol. 4, no. 3, 327-345. Geltner, D. M. , 1993, “Estimating Market Values from Appraised Values without Assuming an Efficient Market”, Journal of Real Estate Research, vol. 8, no. , 325-345. Gorton, G. , and K. G. Rouwenhorst, 2006, “Facts and Fantasies about Commodity Futures”, Financial Analysts Journal, vol. 62, no. 2, p. 47-68. Graham, J. R. , and C. R. Harvey, 2008, “The Equity Risk Premium in 2008: Evidence from the Global CFO Outlook Survey”, working paper. Graybill, F. A. , 1983, “Matrices with Applications in Statistics”, Wadsworth International Group, California. Grishchenko, O. V. , and J. Huang, 2008, “Inflation Risk Premium: Evidence from the TIPS Market”, working paper. Hammond, P. B. , 1999, “Using Inflation-Indexed Securities for Retirement Savings and Income: The TIAA-CREF Experience”, in J.

Brynjolfsson and F. J. Fabozzi, Handbook of Inflation Indexed Bonds, New Hope. Hordijk, A. C. , and C. Ahlqvist, 2004, “European Market Dimensions: An Inventory of the Investable Market in 11 European Countries”, The Compendium of Real Estate Papers 2. Ibbotson, 2007, “Private Equity and Strategic Asset Allocation”, research paper. Ibbotson, 2006, “Commercial Real Estate: The Role of Global Listed Real Estate Equities in a Strategic Asset Allocation”, research paper. Jong, F. de, and J. Driessen, 2005, “Liquidity Risk Premia in Corporate Bond Markets”, working paper. 10 Kaplan, P. D. , and S.

L. Lummer, 1998, “GSCI Collateralized Futures as a Hedging and Diversification Tool for Institutional Portfolios: An Update”, Journal of Investing, vol. 7, no. 4, p. 11-17. Kaplan, S. N. , and A. Schoar, 2005, “Private Equity Performance: Returns, Persistence, and Capital Flows”, Journal of Finance, vol. 60, no. 4, p. 1791-1823. Lamm, R. M. , 1998, “Asset Allocation Implications of Inflation Protection Securities”, Journal of Portfolio Management, vol. 24, no. 4, p. 8-13. Li, L. , 2002, “Macroeconomic Factors and the Correlation of Stock and Bond Returns”, working paper. Liang, Y. , and N. M.

Gordon, 2003, “A Bird’s Eye View of Global Real Estate Markets”, Prudential Real Estate Investors, research paper. Lummer, S. L. , and L. B. Siegel, 1993, “GSCI Collateralized Futures: A Hedging and Diversification Tool for Institutional Portfolios”, Journal of Investing, vol. 2, no. 2, 75-82. Markowitz, H. M. , 1952, “Portfolio Selection”, Journal of Finance, vol. 7, no. 1, p. 77-91. Markowitz, H. M. , 1956, “The Optimization of a Quadratic Function Subject to Linear Constraints”, Naval Research Logistics Quarterly, vol. 3, no. 1-2, 111-133. Masters, M. W. , 2008, Testimony Before the US Senate.

McKinsey Global Institute, 2007, “The New Power Brokers: How Oil, Asia, Hedge Funds, and Private Equity are Shaping Global Capital Markets”, research paper. McKinsey Global Institute, 2008, “Mapping the Global Capital Markets – Fourth Annual Report”, research paper. Norman, E. J. , G. S. Sirmans and J. D. Benjamin, 1995, “The Historical Environment of Real Estate Returns”, The Journal of Real Estate Portfolio Management, vol. 1, no. 1, p. 1-25. Ong, S. E. , and M. Ranasinghe, 2000, “Portfolio Variance and Correlation Matrices”, Journal of Real Estate Portfolio, vol. 6, no. 1, p. 1-6. Phalippou L. 2007, “Investing in Private Equity Funds: A Survey”, working paper. 11 Phalippou, L. , and O. Gottschalg, 2007, “The Performance of Private Equity Funds”, Review of Financial Studies, forthcoming. Posthuma, N. , and P. J. van der Sluis, 2003, “A Reality Check on Hedge Fund Returns”, working paper. Hobbs, P. , and H. Chin, 2007, “The Future Size of the Global Real Estate Market”, RREEF research paper. Swensen, D. F. , 2000, “Pioneering Portfolio Management”, Free Press, New York. Welch, I. , 2008, “The Consensus Estimate for the Equity Premium by Academic Financial Economists in December 2007”, working paper.

Wilshire, 2008, “Report on State Retirement Systems: Funding Levels and Asset Allocation”, research paper. 12 Appendix A: Capital market expectations Risk premia for stocks and bonds are well documented and long term data series extending over 100 years are available. We will therefore start with the risk premia for stocks and bonds. Then, we derive the risk premia of other asset classes by comparing historical performance data and consulting the literature. In order to estimate volatilities and correlations, we rely more on our own historical data, due to a lack of broad coverage in the literature.

Below, we discuss returns and standard deviations for each asset class. Afterwards, we estimate correlations among all asset classes. Stocks Extensive research on the equity-risk premium has been conducted in recent years. Fama and French (2002) use a dividend discount model to estimate an arithmetic risk premium of 3. 54% over the period 1872-2000 for US stocks, while the realized risk premium for this period is 5. 57%. In the period 19512000, the observed difference is even larger. They conclude that the high 1951-2000 returns are the result of low expected future performance.

However, the US was one of the most successful stock markets in the twentieth century, so a global perspective is important to correct this bias. Dimson, Marsh and Staunton (2009) use historical equity risk premia for seventeen countries over the period 1900-2008. They conclude that their equity risk premia are lower than frequently cited in the literature, due to a longer timeframe and a global perspective. Table A. 1 provides an overview of historical risk premia and volatilities. TABLE A. 1 OVERVIEW OF HISTORICAL RISK AND RETURN CHARACTERISTICS FOR STOCKS SOURCE COUNTRY RISK ST. DEV. ANN. ST. DEV.

PREMIUM OF MONTHLY ON CASH RETURNS MSCI US US 3. 1% 18. 4% 15. 4% MSCI WORLD WORLD 3. 0% 18. 8% 14. 8% FAMA AND FRENCH (2002)* US US US US UK WORLD 3. 9% 2. 5% 6. 0% 5. 2% 4. 2% 4. 4% 18. 5% 19. 6% 16. 7% 20. 2% 21. 8% 17. 3% 22. 0% N. A. N. A. N. A. N. A. N. A. N. A. 16. 0% DATA 1970-2008 1970-2008 1872-2000 1872-1950 1951-2000 1900-2008 1900-2008 1900-2008 1970-2008 DIMSON, MARSH AND STAUNTON (2009) ST. DEV. OF MSCI WORLD IN EURO’S * STANDARD DEVIATION OF THE RISK PREMIUM INSTEAD OF THE STANDARD DEVIATION OF THE NOMINAL RETURN. WE DERIVE GEOMETRIC DATA BY USING THE EQUATION RG = RA – 0. 5*VARIANCE

Both Fama and French (2002) and Dimson, Marsh and Staunton (2003, 2009) find that the historical equity premium was significantly higher in the second half of the twentieth century than it was in the first half. Dimson, Marsh and Staunton (2009) expect a lower equity premium in the range of 3. 0%- 13 3. 5% going forward. In this study we use an equity risk premium of 4. 75%. This is slightly above the average of countries in a long timeframe and corresponds well with consensus estimates among finance professors as documented by Welch (2008) and among CFOs as reported by Graham & Harvey (2008).

The other estimate we need is stock market volatility. Dimson, Marsh and Staunton (2009) find a standard deviation of 17. 3% for global equity during the 109 year period 1900-2008. Over the period 1970-2008 the global MSCI index had a volatility of 18. 8% and 22. 0% expressed in dollars and euros respectively. We average these last two figures and estimate the volatility of stocks at 20%. 1 Government bonds Dimson Marsh and Staunton (2009) also evaluate the risk premium of bonds over cash. Their data point to a lower risk premium than the Barclays Government Indices which have been available since 1973, see Table A. . The last decades have been extremely good for government bonds. We use a geometric risk premium of 0. 75% for government bonds over cash, in line with the long term historical average from Dimson, Marsh and Staunton (2009). TABLE A. 2 OVERVIEW OF HISTORICAL RISK AND RETURN CHARACTERISTICS FOR GOVERNMENT BONDS SOURCE COUNTRY RISK ST. DEV. ANN. ST. DEV. PREMIUM OF MONTHLY ON CASH RETURNS BARCLAYS TREASURIES US 2. 2% 6. 5% 5. 4% US 3. 6% 6. 3% 5. 0% US 3. 0% 5. 5% 4. 8% DIMSON, MARSH AND STAUNTON (2009) US UK WORLD 1. 2% 0. 4% 0. 8% 8. 3% 11. 9% 8. 6% N. A. N. A. N. A. DATA 973-2008 1984-2008 1999-2008 1900-2008 1900-2008 1900-2008

The volatility of bonds has been significantly lower in

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