Table 1:
Descriptive Statistics for the US Nominal Returns Data (1900-2011)
| Stocks | Bonds | Bills | |
|---|---|---|---|
| Geometric Mean (%) | 9.30 | 5.00 | 3.90 |
| Arithmetic Mean (%) | 11.27 | 5.38 | 3.95 |
| Median (%) | 13.78 | 3.67 | 4.07 |
| Maximum (%) | 57.09 | 40.36 | 14.71 |
| Minimum (%) | -43.54 | -14.90 | -0.02 |
| Standard Deviation (%) | 19.97 | 8.87 | 2.82 |
| Skewness | -0.37 | 1.22 | 0.73 |
| Kurtosis | 2.84 | 5.40 | 4.02 |
| Jarque-Bera Statistic | 2.62 | 54.55* | 14.72* |
1 Note: This table presents the summary statistics of nominal annual returns data for US stocks, bonds, and bills from 1900 to 2011. Geometric and arithmetic mean, standard deviation, median, maximum, and minimum are expressed in percentage. Skewness, Kurtosis, and Jarque-Bera statistic are numbers.
Table 3:
Key Assumptions
| Variable | Assumption |
|---|---|
| Starting Balance | $0 |
| Starting Salary | $25,000 |
| Salary Growth Rate | 4% p.a. |
| Contribution Rate | 9% p.a. |
| Starting Age | 25 |
| Retirement Age | 65 |
Table 4:
Investment Strategies
| Investment Strategy Symbol | Description | |
|---|---|---|
| Lifestyle strategy | 100 stocks 100 bonds Balanced | 100% stocks allocation 100% bonds allocation 60% stocks and 40% bonds allocation |
| LC90 | Modelled after Vanguard Target Retirement Funds. Start with 90% stocks, 10% bonds allocation for the first 20 years, and then decrease the stock weights 2% annually in a linear pattern to reach an allocation of 50% stocks and 50% bonds at retirement. | |
| LC95 | Modelled after Principle Funds. Start with 95% stocks, 5% bonds allocation, and experience a gradual decline 1% annually away from stocks to reach an allocation of 56% stocks and 49% bonds at retirement. | |
| DLC2020 | Start with 100% stocks for the first 20 years. At the end of 20 years, if the target is achieved, the assets are switched to a portfolio comprising of 80% stocks and 20% fixed income; otherwise, remain invested in 100% stocks. This process is also carried out for the next 10 years. In the last 10 years, the assets are switched to a portfolio of 60% stocks and 40% fixed income if the target is achieved. Otherwise, remain invested in 100% stocks. | |
| DLC3010 | Similar to above, except that it starts with 100% stocks for the first 30 years. This process is carried out for the next 5 years. In the last 5 years, the assets are switched to a portfolio of 60% stocks and 40% fixed income if the target is achieved. Otherwise, remain invested in 100% stocks. | |
| Leveraged life cycle strategy | LLC100 | Start with leverage at 2:1 and fully invested in stocks. When the current savings exceed 20% of discount lifetime savings; investors start to deleverage as savings continue to rise, until the current savings hit the 50% stock allocation. After that, investors maintain the market exposure. Risk-free rate plus 100 basis points is used as the margin rate. |
| LLC200 | Similar to above, except that, instead of using risk-free rate plus 100 basis points as the margin rate, it uses a risk-free rate plus 200 basis points as the margin rate. |
Table 5:
Distribution Parameters of Retirement Wealth Ratio (RWR)
| Investment Strategy | Mean | Median | 75th Percentile | 25th Percentile |
|---|---|---|---|---|
| PANEL A (leveraged life cycle versus lifestyle investment strategies) | ||||
| LLC100 | 12.74 | 10.71 | 17.46 | 9.10 |
| LLC200 | 12.54 | 10.57 | 17.11 | 9.00 |
| Balanced | 11.13 | 9.82 | 14.54 | 7.72 |
| 100 Stocks | 17.26 | 17.87 | 21.20 | 13.72 |
| 100 Bonds | 5.53 | 3.76 | 8.12 | 2.91 |
| LLC200 - Balanced | 1.41 | 0.75 | 2.57 | 1.28 |
| PANEL B (leveraged life cycle versus life cycle investment strategies) | ||||
| LLC100 | 12.74 | 10.71 | 17.46 | 9.10 |
| LLC200 | 12.54 | 10.57 | 17.11 | 9.00 |
| LC90 | 13.33 | 12.76 | 17.24 | 9.72 |
| LC95 | 12.66 | 11.32 | 16.63 | 9.54 |
| LLC200 - LC90 | -0.79 | -2.19 | -0.13 | -0.72 |
| LLC200 - LC95 | -0.12 | -0.75 | 0.48 | -0.54 |
| PANEL C (leveraged life cycle versus dynamic life cycle investment strategies) | ||||
| LLC100 | 12.74 | 10.71 | 17.46 | 9.10 |
| LLC200 | 12.54 | 10.57 | 17.11 | 9.00 |
| DLC3010 | 16.96 | 18.32 | 20.49 | 14.98 |
| DLC2020 | 16.39 | 17.69 | 20.45 | 13.20 |
| LLC200 – DLC3010 | -4.42 | -7.75 | -3.38 | -5.98 |
| LLC200 – DLC2020 | -3.85 | -7.72 | -3.34 | -4.20 |
1 Note: This table reports the distribution of RWR for different investment strategies by using historical simulation method. Mean, median, 25th percentile and 75th percentile are reported respectively. Specifically, Panel A reports the comparison results of leveraged life cycle and lifestyle strategies. Panel B reports the comparison results of leveraged life cycle and conventional life cycle strategies. Panel C reports the comparison results of leveraged and dynamic strategies. The details of each strategy are illustrated in Table 4.
Table 6:
Value at Risk (VaR) and Expected Shortfall (ES) Estimates for Different Investment Strategies
| Investment strategy | VaR at different confidence levels | ES at 95% confidence level | ||
|---|---|---|---|---|
| 99% | 95% | 90% | ||
| PANEL A (leveraged life cycle versus lifestyle investment strategies) | ||||
| LLC100 | 5.29 | 6.12 | 6.72 | 5.50 |
| LLC200 | 5.18 | 6.00 | 6.59 | 5.39 |
| 100 Stocks | 4.80 | 6.04 | 7.60 | 5.17 |
| 100 Bonds | 2.38 | 2.57 | 2.74 | 2.41 |
| Balanced | 5.03 | 5.78 | 6.20 | 2.56 |
| LLC200 -- Balanced | 0.15 | 0.22 | 0.39 | 2.83 |
| PANEL B (leveraged life cycle versus life cycle investment strategies) | ||||
| LLC100 | 5.29 | 6.12 | 6.72 | 5.50 |
| LLC200 | 5.18 | 6.00 | 6.59 | 5.39 |
| LC90 | 5.55 | 6.14 | 6.52 | 5.78 |
| LC95 | 5.36 | 6.16 | 6.65 | 5.6 |
| LLC200 – LC90 | -0.37 | -0.14 | -0.07 | -0.39 |
| LLC200 – LC95 | -0.18 | -0.16 | -0.06 | -0.21 |
| PANEL C (leveraged life cycle versus dynamic life cycle investment strategies) | ||||
| LLC100 | 5.29 | 6.12 | 6.72 | 5.50 |
| LLC200 | 5.18 | 6.00 | 6.59 | 5.39 |
| DLC3010 | 4.80 | 6.04 | 7.60 | 5.17 |
| DLC2020 | 4.80 | 6.23 | 7.78 | 5.21 |
| LLC200 – DLC3010 | 0.38 | -0.04 | -1.01 | 0.22 |
| LLC200 – DLC2020 | 0.38 | -0.23 | -1.19 | 0.18 |
1 Note: This table reports the VaR and ES estimates of RWRs for different investment strategies by using historical simulation method. VaR at 99%, 95% 90% confidence levels, and ES at 95% confidence level are reported respectively. Specifically, Panel A compares the VaR and ES leveraged life cycle and lifestyle strategies; Panel B compares the VaR and ES of leveraged life cycle and conventional life cycle strategies; Panel C compares the VaR and ES of leveraged life cycle and dynamic life cycle strategies. The details of each strategy are illustrated in Table 4.
Table 7:
Distribution Parameters of Retirement Wealth Ratio (RWR)
| Investment strategy | Mean | Median | 75th Percentile | 25th Percentile |
|---|---|---|---|---|
| PANEL A (leveraged life cycle versus lifestyle investment strategies) | ||||
| LLC100 | 12.41 | 10.50 | 15.38 | 7.16 |
| LLC200 | 12.22 | 10.31 | 15.11 | 7.09 |
| 100 Stocks | 19.70 | 13.02 | 23.52 | 7.35 |
| 100 Bonds | 4.86 | 4.53 | 5.62 | 3.74 |
| Balanced | 10.81 | 9.29 | 13.28 | 6.65 |
| LLC200 - Balanced | 1.41 | 1.02 | 1.83 | 0.44 |
| PANEL B (leveraged life cycle versus life cycle investment strategies) | ||||
| LLC100 | 12.41 | 10.50 | 15.38 | 7.16 |
| LLC200 | 12.22 | 10.31 | 15.11 | 7.09 |
| LC90 | 13.61 | 10.69 | 16.68 | 7.03 |
| LC95 | 12.71 | 10.28 | 15.62 | 6.96 |
| LLC200 – LC90 | -1.39 | -0.38 | -1.57 | 0.06 |
| LLC200 – LC95 | -0.49 | 0.03 | -0.51 | 0.13 |
| PANEL C (leveraged life cycle versus dynamic life cycle investment strategies) | ||||
| LLC100 | 12.41 | 10.50 | 15.38 | 7.16 |
| LLC200 | 12.22 | 10.31 | 15.11 | 7.09 |
| DLC3010 | 18.31 | 13.23 | 22.69 | 7.54 |
| DLC2020 | 15.91 | 13.20 | 20.18 | 7.73 |
| LLC200 – DLC3010 | -6.09 | -2.92 | -7.58 | -0.45 |
| LLC200 – DLC2020 | -3.69 | -2.89 | -5.07 | -0.64 |
1 Note: This table reports the distribution of RWRs for different investment strategies by using bootstrap simulation method. Mean, median, 25th percentile and 75th percentile are reported respectively. Specifically, Panel A reports the comparison results of leveraged life cycle and lifestyle strategies. Panel B reports the comparison results of leveraged life cycle and conventional life cycle strategies. Panel C reports the comparison results of leveraged and dynamic strategies. The details of each strategy are illustrated in Table 4.
Table 8:
Value at Risk (VaR) and Expected Shortfall (ES) Estimates for Different Investment Strategies
| Investment strategy | VaR at different confidence levels | ES at 95% confidence level | ||
|---|---|---|---|---|
| 99% | 95% | 90% | ||
| PANEL A (leveraged life cycle versus lifestyle investment strategies) | ||||
| LLC100 | 3.03 | 4.28 | 5.16 | 3.52 |
| LLC200 | 2.98 | 4.28 | 5.19 | 3.50 |
| 100 Stocks | 2.02 | 3.36 | 4.50 | 2.53 |
| 100 Bonds | 2.44 | 2.87 | 3.16 | 2.60 |
| Balanced | 2.94 | 4.01 | 4.85 | 3.35 |
| LLC200 - Balanced | 0.04 | 0.27 | 0.34 | 0.15 |
| PANEL B (leveraged life cycle versus life cycle investment strategies) | ||||
| LLC100 | 3.03 | 4.28 | 5.16 | 3.52 |
| LLC200 | 2.98 | 4.28 | 5.19 | 3.50 |
| LC90 | 2.82 | 4.02 | 4.92 | 3.29 |
| LC95 | 2.89 | 4.06 | 5.00 | 3.35 |
| LLC200 – LC90 | 0.16 | 0.26 | 0.27 | 0.21 |
| LLC200 – LC95 | 0.09 | 0.22 | 0.19 | 0.15 |
| PANEL C (leveraged life cycle versus dynamic life cycle investment strategies) | ||||
| LLC100 | 3.03 | 4.28 | 5.16 | 3.52 |
| LLC200 | 2.98 | 4.28 | 5.19 | 3.50 |
| DLC3010 | 2.01 | 3.38 | 4.54 | 2.54 |
| DLC2020 | 2.04 | 3.45 | 4.64 | 2.58 |
| LLC200 – DLC3010 | 0.97 | 0.90 | 0.65 | 0.96 |
| LLC200 – DLC2020 | 0.94 | 0.83 | 0.55 | 0.92 |
1 Note: This table reports the VaR and ES estimates of RWRs for different investment strategies by using bootstrap simulation method. VaR at 99%, 95% 90% confidence levels, and ES at 95% confidence level are reported respectively. Specifically, Panel A compares the VaR and ES leveraged life cycle and lifestyle strategies; Panel B compares the VaR and ES of leveraged life cycle and conventional life cycle strategies; Panel C compares the VaR and ES of leveraged life cycle and dynamic life cycle strategies.