hedging portfolio as a hedge against changes in investment opportunities. Thus the models of long term buy and hold portfolio is conceptually appealing. The other reason for rebalancing is the asset liability management that depends on demographic structure of plan's participants. This might be solved by dividing whole portfolio into many small portfolios according to the maturities of liabilities that need to be matched and solve the optimization problem for each portfolio individually. Then the stochastic programs for asset liability management can be used as in paper of Dupačová and Polívka (2004). The term structure of the risk-return tradeoff can be included into these programs later on.
In section 5 we were searching for optimal portfolio allocation where we took the horizon effect from previous sections into account. First we constructed global minimum variance portfolio, where we found that for longer investment horizons the weight of T-bills decreases and weight of stocks increases. For 50 years investment horizon the weight of stocks rises to more than 4% in basic model and more than 10% in extended version of the model. Then we were trying to find optimal asset allocation with respect to risk tolerance.
First we minimized the Value at Risk for various investment horizons. We found that with increasing investment horizon the weight of stocks increases dramatically. Then we defined three representative investors according to their risk tolerance and constructed optimal asset allocation for various investment horizons. In the last part of empirical research we treated T-bills as riskless asset and constructed tangency portfolio for different investment horizons. We found again that with increasing horizon the weight of stocks increases and the weight of bonds decreases. The weight of stocks in tangency portfolio exceeded 60%
for investment horizons larger than 25 years. Thus the main conclusion of empirical research of this paper is that institutional investors with long investment horizons such as pension funds should include more stocks into their portfolio even if they are very risk averse.
In the last section of this paper we used the results of empirical research for comparison with the practice of Czech pension funds and their asset allocation. We found that Czech pension funds are not only strongly risk averse, but also optimize their asset allocation as short term investors, which is suboptimal. This finding is not as surprising when we realize that regulations of pension funds in Czech Republic force pension funds to act as a myopic investor. We found that the allocation of stocks is maybe even under the theoretically optimal allocation of global minimum variance portfolio. Therefore the suggestion of this paper is to relax the regulation of minimum investment return for one year and make this regulation for longer period. This would enable Czech pension funds to act as a long term investor and use the advantages of the term structure of risk-return tradeoff. Our research applies only to buy and hold investor, but as a theoretical concept may be appealing. Our suggestion for further research using the similar concept of vector autoregressive model is to use the asset return predictability to simulate the short term rebalancing portfolio as the
sample data from last financial crisis whether our vector autoregressive model has good predictive power and whether it would suggest rebalancing the portfolio in the good direction just before the crisis.
8. List of Figures
Figure 1: Efficient combinations of variance and expected return... 5
Figure 2: Efficient portfolios... 6
Figure 3: Set of efficient portfolios... 7
Figure 4: Efficient frontier... 7
Figure 5: Binomial model of bond pricing... 9
Figure 6: Annualized percent standard deviations of real returns... 12
Figure 7: Correlation of real returns implied by quarterly VAR (1) estimates... 13
Figure 8: Composition of global minimum variance portfolio... 13
Figure 9: Risk in standard mean-variance approach……….. 17
Figure 10: Risk in standard mean-variance approach………... 17
Figure 11: Annualized Percent Standard Deviations of Real Returns………... 28
Figure 12: Correlation of Real Returns ………. 29
Figure 13: Annualized Percent Standard Deviation……….. 33
Figure 14: Correlation of Real Returns………. 34
Figure 15: Annualized Standard Deviation for 3 bonds……… 35
Figure 16: Correlation of T bill Returns and 3 Bonds………... 36
Figure 17: Correlation of Stock Returns and 3 Bonds………... 36
Figure 18: Extension by REIT- Annualized standard deviation………... 39
Figure 19: Extension by REIT- Correlation……….. 40
Figure 20: Extension by REIT- Correlation of REIT with other asset classes…………. 40
Figure 21: Extension by REIT- Annualized standard deviation (significant)…………... 44
Figure 22: Extension by REIT- Correlation (significant)………... 44
Figure 23: Extension by REIT- correlation of REIT with other asset classes (sig.)... 45
Figure 24: Extension by REIT and Hedge funds- Risk of T-bills... 48
Figure 25: Extension by REIT and Hedge funds- Annualized standard deviation... 49
Figure 26: Efficient frontiers... 55
Figure 27: Efficient frontiers (extended version)... 56
Figure 28: Tangency portfolio... 59
Figure 29: Tangency portfolio (extended version)... 60
Figure 30: 40 years charge ratio... 62
Figure 31: Portfolio limits on OECD pension funds` investment in equities, 2007…… 64
Figure 32: Pension fund asset allocation, 2009……… 65
9. List of Tables
Table 1: Optimal allocation to two-period bond... 10
Table 2: VAR estimation results, 1952 Q2 – 2002 Q4... 11
Table 3: Mean and standard deviation... 15
Table 4: Mean and standard deviation ……….. 15
Table 5: VAR estimation results- Coefficients on lagged variables ………. 25
Table 6: VAR estimation results- variance covariance matrix of shocks……….. 26
Table 7: VAR estimation results- Coefficients on lagged variables……….. 30
Table 8: VAR estimation results- Variance, Covariance matrix of shocks……… 31
Table 9: Testing the assumptions……….. 32
Table 10: Extension by REIT- coefficients on lagged variables……….38
Table 11: Extension by REIT- covariance matrix………. 38
Table 12: Extension by REIT- coefficients on lagged variables (significant)………….. 41
Table 13: Extension by REIT- Covariance matrix (significant)……… 42
Table 14: Testing the assumptions (extended model)……… 43
Table 15: Extension by REIT and Hedge funds- coefficients on lagged variables... 46
Table 16: Extension by REIT and Hedge funds- Covariance matrix... 47
Table 17: Coefficients on lagged variables (European data)... 50
Table 18: Global minimum variance portfolio... 52
Table 19: Global minimum variance portfolio (extended version)... 52
Table 20: Minimal Value at Risk portfolio... 54
Table 21: Minimal Value at Risk portfolio (extended version)... 54
Table 22: Optimal asset allocation... 58
Table 23: Optimal asset allocation (extended version)... 58
Table 24: Performance of Czech pension funds: Nominal returns (%)... 67
Table 25: Performance of Czech pension funds: Real returns (%)... 67
Table 26: Asset structure of Czech pension funds... 69
Table 27: Liability structure of Czech pension funds... 70
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