# Defined Contribution Pension Plans: Who Has Seen the Risk?

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## Abstract

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## 1. Introduction

Sophisticated employers should choose their plan defaults carefully, since these defaults will strongly influence the retirement preparation of their employees. Policymakers should also recognize the role of defaults, since policymakers can facilitate, with laws and regulations, the socially optimal use of defaults.(Choi et al. 2002, p. 104)

- With typical glide path or constant proportion strategies, there is an unacceptably large probability of shortfall in terms of meeting the target final wealth goal.
- Even with an optimal dynamic QS asset allocation strategy, there is still a fairly high probability of shortfall. This shortfall probability can be reduced to what we view as a reasonable level by increasing the total contribution rate or reducing the replacement ratio, compared to the base case. Another possibility is to substitute an equal-weighted equity index for the value-weighted index, but this may not work in practice due to higher costs associated with equal-weighted indexes, which are not recognized in our model.

## 2. Formulation

#### 2.1. Deterministic Glide Paths

#### 2.2. Adaptive Strategies

- (i)
- withdraw cash ${c}_{i}={W}_{i}^{-}+{q}_{i}-\left(\right)open="("\; close=")">{W}^{*}{e}^{-r(T-{t}_{i})}-{Q}_{i}$ from the portfolio; and
- (ii)
- invest the remainder $\left(\right)$ in the risk-free asset.

## 3. Data and Parameter Estimates

#### Robustness to Parameter Estimation

## 4. Base Case Scenario

- Constant proportion, i.e., $p=const$.
- Linear glide path, as in Equation (5).
- Time consistent QS optimal strategy, as described in Section 2.2. Recall that this strategy is also multi-period pre-commitment MV optimal.

- When we change input parameters (e.g., invest in different assets, allow ${L}_{max}>1$, etc.), we may need to recompute the expected wealth target $E\left[{W}_{T}\right]={W}_{d}$, the equity weight for the constant proportion strategy, the glide path parameters $({p}_{max},{p}_{min})$, and the quadratic wealth target ${W}^{*}$ (along with the associated optimal control) in order to meet this target.
- The quadratic wealth target ${W}^{*}$ exceeds the target expected real terminal wealth ${W}_{d}=E\left[{W}_{T}\right]$. This is because the QS optimal strategy will de-risk if ${W}^{*}$ is attainable by investing only in the risk-free asset, so there is not much chance of exceeding this quadratic target by a significant amount. This implies that the average terminal wealth, factoring in paths where the accumulated savings does not ever reach ${W}^{*}$, must be lower than ${W}^{*}$.

## 5. Alternative Assumptions

#### 5.1. Effect of Contribution Fraction

#### 5.2. Effect of Salary Escalation Rate

#### 5.3. Effect of Leverage

#### 5.4. Long-Term Bond Index

#### 5.5. Equal-Weighted Equity Index

#### 5.6. Effect of Replacement Ratio

#### 5.7. Summary Regarding Alternative Assumptions

- varying the accumulation fraction ${F}_{c}$, i.e., the investor saves ${F}_{c}{I}_{0}{e}^{{\mu}_{I}{t}_{i}}$ at each rebalancing date;
- varying the real salary escalation rate ${\mu}_{I}$;
- use of leverage;
- alternative bond index: use of a 10 year T-bond index instead of a 3 month T-bill index;
- alternative stock index: use of an equal-weighted equity index instead of a value-weighted index; and
- varying the replacement ratio R.

## 6. Conclusions

- reducing the final salary target replacement ratio ($40\%$ or less);
- increasing the total (employee and employer) contribution rate to $25\%$ per year;
- using alternative stock investment indices, such as an equal-weighted index. The backtests of an equal-weighted index perform well, but it is not clear that this will persist in the future. In addition, we have not factored in the additional costs of this type of index.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1. | Statistics from the Thinking Ahead Institute’s “Global Pension Assets Study 2018”. Available online: www.thinkingaheadinstitute.org/-/media/Pdf/TAI/Research-Ideas/GPAS-2018.pdf. |

2. | In other words, the employee must explicitly decide to opt out of a TDF if she desires a different asset allocation strategy. |

3. | Although our focus in this article is on the US setting, we note that TDFs are now being marketed in various parts of Europe, in part due to regulatory developments (Pielichata, 2018). Major US vendors such as Vanguard and Fidelity have launched TDFs in Canada in recent years. In addition, some life-cycle products such as Time Pension which has been popular in Denmark are similar in some respects to TDFs. |

4. | See, for example, Hedesström et al. (2004) (Sweden), O’Connell (2009) (New Zealand), and Dobrescu et al. (2018) (Australia). |

5. | An obvious extension would be a GARCH/stochastic volatility model. However, Ma and Forsyth (2016) document that mean-reverting stochastic volatility effects for Heston-type stochastic volatility models are negligible for long-term investors. Since multivariate GARCH/stochastic volatility models are typically mean-reverting, this suggests that stochastic volatility may be unimportant for long-term investors under these models as well, but this has not been proven. |

6. | Recall the time consistent QS strategy has the same controls as the pre-commitment MV strategy. |

7. | As discussed below, in the case of an optimal QS strategy, the investor may also withdraw cash from the portfolio at an action time. |

8. | Since the investor rebalances her portfolio discretely, insolvency could also occur if ${L}_{max}>1$ in the special case of the model where jumps are ruled out ($\lambda =0$), i.e., the value of the risky asset follows geometric Brownian motion. |

9. | More precisely, suppose that insolvency occurs at time t, i.e., ${S}_{t}+{B}_{t}<0$. Letting ${t}^{+}$ be the instant after t, then ${B}_{{t}^{+}}={S}_{t}+{B}_{t}$ and ${S}_{{t}^{+}}=0$. |

10. | For example, we can exogenously specify ${p}_{min}$ and find the value of ${p}_{max}$ which generates the desired expected terminal wealth via Newton iteration. Alternatively, we can exogenously set ${p}_{max}$ and numerically find the appropriate value of ${p}_{min}$. |

11. | If problem (7) is not convex, there may be solutions to problem (8) that are not solutions to problem (7). However, these spurious solutions can be eliminated in a straightforward way (Dang et al., 2016; Tse et al., 2014). |

12. | More precisely, our calculations are based on data from Historical Indexes, ©2015 Center for Research in Security Prices (CSRP), The University of Chicago Booth School of Business. Wharton Research Data Services (WRDS) was used in preparing this work. This service and the data available thereon constitute valuable intellectual property and trade secrets of WRDS and/or its third-party suppliers. |

13. | As noted by Bloom et al. (2014), this rate has been used by the US Congressional Budget Office in its long-term projections. |

14. | This is a bit more aggressive in terms of taking on equity market risk than the strategy considered by Bengen (1994) which involved equal weights between the equity and bond markets. Keep in mind that here we are investing in a 3-month T-bill index, whereas Bengen used intermediate maturity Treasury bonds which offer somewhat higher average returns. |

15. | In other words, if the size of a block extends past the end of the sample in 2015:12, the return data resumes at the start of the sample in 1926:1 for the duration of the block. |

16. | In other words, the QS optimal strategy will appear riskier than the constant proportion or glide path strategies according to tail risk measures such as value-at-risk or conditional value-at-risk, provided that the risk measure is calculated using sufficiently low cumulative probabilities. |

17. | Of course, the equity allocation for the constant proportion and glide path cases is fixed in advance, being at most time-dependent and not varying at all in response to realized returns. |

**Figure 3.**Base case scenario results in the historical market. Input data provided in Table 1 and Table 2. Results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12 with expected blocksize $\widehat{b}=2$ years. (

**a**) cumulative distributions of real terminal wealth for various strategies. Wealth units: thousands of dollars; surplus cash included for the QS optimal case; (

**b**) mean and standard deviation of the fraction allocated to the equity market for the QS optimal strategy.

**Figure 4.**Cumulative distributions of real terminal wealth for various contribution fractions ${F}_{c}$ for the QS optimal strategy. Wealth units: thousands of dollars. Input data provided in Table 1 and Table 2, except as noted. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12 with expected blocksize $\widehat{b}=2$ years; surplus cash included.

**Figure 5.**Cumulative distributions of normalized real terminal wealth for various salary escalation rates ${\mu}_{I}$ for the QS optimal strategy. Input data provided in Table 1 and Table 2, except as noted. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12 with expected blocksize $\widehat{b}=2$ years; surplus cash included.

**Figure 6.**Results for cases allowing and excluding leverage. Input data provided in Table 1 and Table 2, except as noted. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12 with expected blocksize $\widehat{b}=2$ years. (

**a**) cumulative distribution function of real terminal wealth for cases allowing leverage (${L}_{max}=1.5$) and excluding it (${L}_{max}=1$). Wealth units: thousands of dollars; surplus cash included; (

**b**) mean and standard deviation of the fraction allocated to the equity market for the case where leverage (${L}_{max}=1.5$) is allowed.

**Figure 7.**Cumulative distribution of real terminal wealth for QS optimal, glide path, and constant proportion strategies when the 10-year T-bond index is used. Wealth units: thousands of dollars. Input data provided in Table 1 and Table 2, except as noted; surplus cash included for the QS optimal strategy. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12. Expected blocksize $\widehat{b}=2$ years.

**Figure 8.**Cumulative distribution of real terminal wealth when the equal-weighted equity index is used. Wealth units: thousands of dollars. Input data provided in Table 1 and Table 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12 with expected blocksize $\widehat{b}=2$ years; surplus cash flow included for the QS optimal strategy. (

**a**) synthetic market; (

**b**) historical market.

**Figure 9.**Cumulative distributions of real terminal wealth with different salary replacement ratios R for the QS optimal strategy. Input data provided in Table 1 and Table 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using historical data from 1926:1 to 2015:12 with expected blocksize $\widehat{b}=2$ years; surplus cash included. (

**a**) synthetic market;

**b**) historical market.

**Table 1.**Annualized parameter estimates based on real monthly data from 1926:1 to 2015:12. These values originally appeared in Forsyth and Vetzal (2019) and are reproduced here for convenience. Parameters for the equity market indexes were estimated using the threshold technique of Cont and Mancini (2011). The average returns for the bond indexes were calculated as $log\left[B\right(T)/B(0\left)\right]/T$, where $B\left(t\right)$ denotes the index level at the time t.

Equity Market Index | $\mathit{\mu}$ | $\mathit{\sigma}$ | $\mathit{\lambda}$ | ${\mathit{p}}_{\mathit{up}}$ | ${\mathit{\eta}}_{1}$ | ${\mathit{\eta}}_{2}$ |

Value-weighted | 0.0889 | 0.1477 | 0.3222 | 0.2759 | 4.4273 | 5.2613 |

Equal-weighted | 0.1183 | 0.1663 | 0.4000 | 0.3333 | 3.6912 | 4.5409 |

Bond Market Index | Average Return | |||||

3-month T-bill | 0.00827 | |||||

10-year T-bond | 0.02160 |

**Table 2.**Data for base case scenario. Cash is injected into the portfolio at times $t=0,1,\cdots ,29$. Market parameters for the equity and bond indexes are provided in Table 1.

Initial salary ${I}_{0}$ | $50,000 |

Salary escalation rate ${\mu}_{I}$ | 0.0127 (Bloom et al., 2014) |

Contribution fraction ${F}_{c}$ | 0.20 |

Accumulation period T | 30 years |

Safe withdrawal rate ${w}_{r}$ | 0.04 |

Equity index | Value-weighted |

Bond index | 3-month T-bill |

Investment strategies | Constant proportion, glide path, QS optimal |

Rebalancing interval | 1 year |

Maximum leverage indicator ${L}_{max}$ | 1.0 |

If insolvent | Trading stops |

**Table 3.**Base case scenario results. Wealth units: thousands of dollars. Input data provided in Table 1 and Table 2. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12.

Strategy | Expected Value | Standard Deviation | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<700)$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<800)$ | Expected Surplus Cash |
---|---|---|---|---|---|

Synthetic Market | |||||

Constant proportion | 915 | 519 | 0.39 | 0.51 | NA |

Glide path | 915 | 519 | 0.39 | 0.51 | NA |

QS optimal | 915 | 244 | 0.19 | 0.24 | 21 |

Historical Market (Expected blocksize $\widehat{b}=1$ year) | |||||

Constant proportion | 876 | 402 | 0.38 | 0.51 | NA |

Glide path | 872 | 398 | 0.39 | 0.52 | NA |

QS optimal | 904 | 232 | 0.18 | 0.24 | 26 |

Historical Market (Expected blocksize $\widehat{b}=2$ years) | |||||

Constant proportion | 869 | 376 | 0.38 | 0.51 | NA |

Glide path | 866 | 372 | 0.39 | 0.52 | NA |

QS optimal | 911 | 221 | 0.17 | 0.23 | 31 |

Historical Market (Expected blocksize $\widehat{b}=5$ years) | |||||

Constant proportion | 862 | 349 | 0.37 | 0.50 | NA |

Glide path | 861 | 347 | 0.38 | 0.51 | NA |

QS optimal | 924 | 213 | 0.16 | 0.21 | 38 |

**Table 4.**Base case scenario results for the QS optimal strategy with random variation of market parameters $\mu $ and $\sigma $, Wealth units: thousands of dollars. Input data provided in Table 1 and Table 2, except as noted. Monte Carlo simulations with $(\mu ,\sigma )$ drawn from a uniform distribution with the indicated limits along each of 160,000 paths.

Market Parameters | Expected Value | Standard Deviation | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<700)$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<800)$ | Expected Surplus Cash |
---|---|---|---|---|---|

Synthetic market | 915 | 244 | 0.19 | 0.24 | 21 |

$\mu \in [0.04889,0.01289]$ | 916 | 245 | 0.19 | 0.24 | 21 |

$\sigma \in [0.1077,0.1877]$ | 915 | 245 | 0.19 | 0.24 | 21 |

**Table 5.**Effect of varying contribution fraction ${F}_{c}$ for the QS optimal strategy. Wealth units: thousands of dollars. Input data provided in Table 1 and Table 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12.

${\mathit{F}}_{\mathit{c}}$ | Expected Value | Standard Deviation | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<700)$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<800)$ | Expected Surplus Cash |
---|---|---|---|---|---|

Synthetic Market | |||||

0.15 | 915 | 440 | 0.36 | 0.42 | 18 |

0.20 | 915 | 244 | 0.19 | 0.24 | 21 |

0.25 | 915 | 150 | 0.09 | 0.13 | 21 |

Historical Market (Expected blocksize $\widehat{b}=2$ years) | |||||

0.15 | 916 | 380 | 0.34 | 0.41 | 12 |

0.20 | 911 | 221 | 0.17 | 0.23 | 31 |

0.25 | 909 | 126 | 0.07 | 0.14 | 67 |

**Table 6.**Effect of varying salary escalation rate ${\mu}_{I}$ for the QS optimal strategy. Units for ${W}_{d}$: thousands of dollars. Remaining wealth values are normalized by ${W}_{d}$ for each case. Input data provided in Table 1 and Table 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12.

${\mathit{\mu}}_{\mathit{I}}$ | Wealth Target ${\mathit{W}}_{\mathit{d}}$ | Expected Value $({\mathit{W}}_{\mathit{T}}/{\mathit{W}}_{\mathit{d}})$ | Standard Deviation $({\mathit{W}}_{\mathit{T}}/{\mathit{W}}_{\mathit{d}})$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}/{\mathit{W}}_{\mathit{d}}<0.8)$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}/{\mathit{W}}_{\mathit{d}}<0.9)$ | Expected Surplus Cash $(/{\mathit{W}}_{\mathit{d}})$ |
---|---|---|---|---|---|---|

Synthetic Market | ||||||

0.0175 | 1056 | 1.0 | 0.31 | 0.25 | 0.30 | 0.02 |

0.0127 | 915 | 1.0 | 0.27 | 0.20 | 0.25 | 0.02 |

0.0075 | 783 | 1.0 | 0.22 | 0.15 | 0.21 | 0.02 |

Historical Market (Expected blocksize $\widehat{b}=2$ years) | ||||||

0.0175 | 1056 | 0.99 | 0.29 | 0.23 | 0.29 | 0.03 |

0.0127 | 915 | 0.97 | 0.24 | 0.18 | 0.25 | 0.03 |

0.0075 | 783 | 0.99 | 0.19 | 0.14 | 0.21 | 0.04 |

**Table 7.**Effect of varying maximum leverage indicator ${L}_{max}$ for the QS optimal strategy. Wealth units: thousands of dollars. Input data provided in Table 1 and Table 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12.

${\mathit{L}}_{max}$ | Expected Value | Standard Deviation | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<700)$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<800)$ | Expected Surplus Cash |
---|---|---|---|---|---|

Synthetic Market | |||||

1.0 | 915 | 244 | 0.19 | 0.24 | 21 |

1.5 | 915 | 205 | 0.12 | 0.17 | 24 |

Historical Market (Expected blocksize $\widehat{b}=2$ years) | |||||

1.0 | 911 | 221 | 0.17 | 0.23 | 31 |

1.5 | 904 | 186 | 0.11 | 0.18 | 46 |

**Table 8.**Results for the optimal QS strategy when the 10-year T-bond index is used instead of the 3-month T-bill index. Wealth units: thousands of dollars. Input data provided in Table 1 and Table 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12.

Strategy | Expected Value | Standard Deviation | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<700)$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<800)$ | Expected Surplus Cash |
---|---|---|---|---|---|

Synthetic Market | |||||

Constant proportion | 915 | 437 | 0.34 | 0.47 | NA |

Glide path | 915 | 438 | 0.34 | 0.48 | NA |

QS optimal | 915 | 222 | 0.16 | 0.21 | 19 |

Historical Market (Expected blocksize $\widehat{b}=2$ years) | |||||

Constant proportion | 900 | 374 | 0.34 | 0.47 | NA |

Glide path | 897 | 376 | 0.34 | 0.48 | NA |

QS optimal | 881 | 201 | 0.17 | 0.26 | 67 |

**Table 9.**Results when the equal-weighted equity index is used instead of the value-weighted equity index. Wealth units: thousands of dollars. Input data provided in Table 1 and Table 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12.

Strategy | Expected Value | Standard Deviation | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<700)$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}<800)$ | Expected Surplus Cash |
---|---|---|---|---|---|

Synthetic Market | |||||

Constant proportion | 915 | 546 | 0.38 | 0.51 | NA |

Glide path | 915 | 553 | 0.39 | 0.52 | NA |

QS optimal | 915 | 185 | 0.11 | 0.16 | 44 |

Historical Market (Expected blocksize $\widehat{b}=1$ year) | |||||

Constant proportion | 837 | 327 | 0.39 | 0.54 | NA |

Glide path | 831 | 319 | 0.39 | 0.55 | NA |

QS optimal | 904 | 162 | 0.10 | 0.16 | 49 |

Historical Market (Expected blocksize $\widehat{b}=2$ years) | |||||

Constant proportion | 827 | 293 | 0.38 | .54 | NA |

Glide path | 820 | 283 | 0.38 | 0.55 | NA |

QS optimal | 915 | 139 | 0.07 | 0.13 | 51 |

Historical Market (Expected blocksize $\widehat{b}=5$ years) | |||||

Constant proportion | 815 | 248 | 0.36 | 0.53 | NA |

Glide path | 808 | 242 | 0.37 | 0.55 | NA |

QS optimal | 932 | 115 | 0.04 | 0.10 | 54 |

**Table 10.**QS optimal results with varying salary replacement ratios R. Units for ${W}_{d}$: thousands of dollars. Remaining wealth values are normalized by ${W}_{d}$ for each case. Input data provided in Table 1 and Table 2, except as noted. Synthetic market results computed using Monte Carlo simulations with 160,000 sample paths. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12.

R | Wealth Target ${\mathit{W}}_{\mathit{d}}$ | Expected Value $({\mathit{W}}_{\mathit{T}}/{\mathit{W}}_{\mathit{d}})$ | Standard Deviation $({\mathit{W}}_{\mathit{T}}/{\mathit{W}}_{\mathit{d}})$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}/{\mathit{W}}_{\mathit{d}}<0.8)$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}/{\mathit{W}}_{\mathit{d}}<0.9)$ | Expected Surplus Cash $(/{\mathit{W}}_{\mathit{d}})$ |
---|---|---|---|---|---|---|

Synthetic Market | ||||||

0.4 | 732 | 1.0 | 0.16 | 0.10 | 0.15 | 0.02 |

0.5 | 915 | 1.0 | 0.27 | 0.20 | 0.25 | 0.02 |

0.6 | 1098 | 1.0 | 0.37 | 0.31 | 0.36 | 0.02 |

Historical Market (Expected blocksize $\widehat{b}=2$ years) | ||||||

0.4 | 732 | .99 | 0.15 | 0.09 | 0.17 | 0.0 |

0.5 | 915 | .97 | 0.24 | 0.18 | 0.25 | 0.03 |

0.6 | 1098 | 1.0 | 0.35 | 0.30 | 0.35 | 0.02 |

**Table 11.**Comparison of shortfall probabilities. Results are normalized by ${W}_{d}$ for each case. Input data provided in Table 1 and Table 2, except as noted. Historical market results based on 10,000 bootstrap resampled paths using data from 1926:1 to 2015:12 with expected blocksize $\widehat{b}=2$ years.

Case | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}/{\mathit{W}}_{\mathit{d}}<0.8)$ | $\mathit{Pr}({\mathit{W}}_{\mathit{T}}/{\mathit{W}}_{\mathit{d}}<0.9)$ |
---|---|---|

Base Case, Table 2 | ||

Constant proportion | 0.43 | 0.54 |

Glide path | 0.43 | 0.54 |

QS optimal | 0.18 | 0.25 |

QS Optimal | ||

Contribution Fraction | ||

${F}_{c}=0.25$ | 0.09 | 0.17 |

${F}_{c}=0.15$ | 0.37 | 0.43 |

Salary Escalation Rate | ||

${\mu}_{I}=0.0175$ | 0.23 | 0.29 |

${\mu}_{I}=0.0075$ | 0.14 | 0.21 |

Leverage | ||

${L}_{max}=1.5$ | 0.13 | 0.19 |

Alternative Bond Index | ||

10-year T-bond | 0.19 | 0.28 |

Alternative Stock Index | ||

Equal-weighted | 0.09 | 0.16 |

Replacement Ratio | ||

$R=0.4$ | 0.09 | 0.17 |

$R=0.6$ | 0.30 | 0.35 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Forsyth, P.A.; Vetzal, K.R.
Defined Contribution Pension Plans: Who Has Seen the Risk? *J. Risk Financial Manag.* **2019**, *12*, 70.
https://doi.org/10.3390/jrfm12020070

**AMA Style**

Forsyth PA, Vetzal KR.
Defined Contribution Pension Plans: Who Has Seen the Risk? *Journal of Risk and Financial Management*. 2019; 12(2):70.
https://doi.org/10.3390/jrfm12020070

**Chicago/Turabian Style**

Forsyth, Peter A., and Kenneth R. Vetzal.
2019. "Defined Contribution Pension Plans: Who Has Seen the Risk?" *Journal of Risk and Financial Management* 12, no. 2: 70.
https://doi.org/10.3390/jrfm12020070