# Portability, Salary and Asset Price Risk: A Continuous-Time Expected Utility Comparison of DB and DC Pension Plans

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## Abstract

**:**

## 1. Introduction

## 2. Model Setup

#### 2.1. DB Pension Plan

#### 2.2. DC Pension Plan

#### 2.3. Matching the Employee’s Contributions

## 3. Utility-Based Comparison

#### 3.1. Utility Functions and Certainty Equivalents

#### 3.2. Expected Utility Results

**Proposition 3.1**(Expected utilities under the DB plan.). The expected utility for the power utility function is given by:

**Proof.**The proof of the proposition is given in the Appendix. □

## 4. Numerical Analysis

#### 4.1. Means of Comparison

#### 4.1.1. Indifference Job Switching Intensity

Utility | Risk aversion | $\pi =0.4$ | $\pi =0.57$ | $\pi =0.75$ | $\pi =0.9$ |

CRRA | $\gamma =1$ | 0.3423 | 0.3169 | 0.3058 | 0.3088 |

$\gamma =2$ | 0.2540 | 0.2690 | 0.3087 | 0.3673 | |

$\gamma =4$ | 0.0897 | 0.1724 | 0.3052 | 0.4309 | |

LA | ${\eta}_{1}=2.25$ | 0.3068 | 0.2532 | 0.1929 | 0.1349 |

${\eta}_{1}=5$ | 0.2965 | 0.2504 | 0.1890 | 0.1399 | |

DD | ${\eta}_{2}=2.25$ | 0.3526 | 0.3121 | 0.2831 | 0.2958 |

${\eta}_{2}=5$ | 0.3061 | 0.2801 | 0.2821 | 0.3442 |

#### 4.1.2. Further Indifference Parameters

Utility | Risk aversion | ${\pi}^{*}$ | ${\sigma}_{S}^{*}$ |

CRRA | $\gamma =1$ | - | 0.1625 |

$\gamma =2$ | 0.7656 | 0.0999 | |

$\gamma =4$ | 0.7551 | 0.0975 | |

LA | ${\eta}_{1}=2.25$ | 0.3502 | - |

${\eta}_{1}=5$ | 0.3453 | - | |

DD | ${\eta}_{2}=2.25$ | 0.3654 | 0.0945 |

${\eta}_{2}=5$ | {0.1218},{0.9152} | 0.0925 |

**Figure 1.**Values for the indifference switching ${\lambda}^{*}$ for different levels of the salary volatility ${\sigma}_{S}$.

#### 4.1.3. Certainty Equivalents

**Table 3.**Values for the certainty equivalent ratio $\frac{C{E}^{DB}}{C{E}^{DC}}$ for $\lambda =0.25$.

Utility | Risk aversion | $\pi =0.4$ | $\pi =0.57$ | $\pi =0.75$ | $\pi =0.9$ |

CRRA | $\gamma =1$ | 1.1265 | 1.10881 | 1.0748 | 1.0803 |

$\gamma =2$ | 0.8651 | 0.8792 | 0.9322 | 1.0053 | |

$\gamma =4$ | 0.6831 | 0.7716 | 0.9268 | 1.1121 | |

LA | ${\eta}_{1}=2.25$ | 0.9168 | 0.8539 | 0.7873 | 0.7419 |

${\eta}_{1}=5$ | 0.9250 | 0.8647 | 0.7901 | 0.7415 | |

DD | ${\eta}_{2}=2.25$ | 0.9863 | 0.9120 | 0.8645 | 0.9254 |

${\eta}_{2}=5$ | 0.8733 | 0.8151 | 0.8210 | 1.1025 |

#### 4.2. Sensitivity Analysis

**Table 4.**Values of ${\lambda}^{*}$ for a piecewise constant and U-shaped portability loss size with $\beta =\phantom{\rule{3.33333pt}{0ex}}[0.95\phantom{\rule{1.em}{0ex}}0.9\phantom{\rule{1.em}{0ex}}0.99]$ (original $\beta =0.95$).

Utility | Risk aversion | $\pi =0.4$ | $\pi =0.57$ | $\pi =0.75$ | $\pi =0.9$ |

CRRA | $\gamma =1$ | 0.2721 | 0.2533 | 0.2413 | 0.2441 |

$\gamma =2$ | 0.1989 | 0.2089 | 0.2431 | 0.2857 | |

$\gamma =4$ | 0.0631 | 0.1268 | 0.2288 | 0.3374 | |

LA | ${\eta}_{1}=2.25$ | 0.2503 | 0.2032 | 0.1531 | 0.1102 |

${\eta}_{1}=5$ | 0.2501 | 0.2019 | 0.1528 | 0.1145 | |

DD | ${\eta}_{2}=2.25$ | 0.2742 | 0.2447 | 0.2260 | 0.2321 |

${\eta}_{2}=5$ | 0.2360 | 0.2147 | 0.2179 | 0.2583 |

**Table 5.**Values of ${\lambda}^{*}$ for a piecewise constant and decreasing salary trend with ${\mu}_{S}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}[0.0225\phantom{\rule{1.em}{0ex}}0.0175\phantom{\rule{1.em}{0ex}}0.01]$ (original ${\mu}_{S}=0.015$).

Utility | Risk aversion | $\pi =0.4$ | $\pi =0.57$ | $\pi =0.75$ | $\pi =0.9$ |

CRRA | $\gamma =1$ | 0.3500 | 0.3257 | 0.3118 | 0.3148 |

$\gamma =2$ | 0.2601 | 0.2733 | 0.3164 | 0.3692 | |

$\gamma =4$ | 0.0974 | 0.1797 | 0.3068 | 0.4370 | |

LA | ${\eta}_{1}=2.25$ | 0.3247 | 0.2695 | 0.2032 | 0.1506 |

${\eta}_{1}=5$ | 0.3195 | 0.2687 | 0.2021 | 0.1595 | |

DD | ${\eta}_{2}=2.25$ | 0.3773 | 0.3324 | 0.2972 | 0.2917 |

${\eta}_{2}=5$ | 0.3344 | 0.3050 | 0.2895 | 0.3442 |

**Figure 2.**Values for the indifference switching ${\lambda}^{*}$ for different levels of the salary drift ${\mu}_{S}$.

**Table 6.**Values for the certainty equivalent ratio $\frac{C{E}^{DB}}{C{E}^{DC}}$ for a piecewise constant and decreasing job switching intensity $\lambda =[0.3\phantom{\rule{1.em}{0ex}}0.2\phantom{\rule{1.em}{0ex}}0.1]$.

Utility | Risk aversion | $\pi =0.4$ | $\pi =0.57$ | $\pi =0.75$ | $\pi =0.9$ |

CRRA | $\gamma =1$ | 1.1707 | 1.1339 | 1.0990 | 1.1153 |

$\gamma =2$ | 0.8974 | 0.9166 | 0.9735 | 1.0419 | |

$\gamma =4$ | 0.7119 | 0.8085 | 0.9792 | 1.1695 | |

LA | ${\eta}_{1}=2.25$ | 0.9567 | 0.8869 | 0.8227 | 0.7643 |

${\eta}_{1}=5$ | 0.9658 | 0.8984 | 0.8333 | 0.7772 | |

DD | ${\eta}_{2}=2.25$ | 1.044 | 0.9627 | 0.9117 | 0.9358 |

${\eta}_{2}=5$ | 0.9529 | 0.8885 | 0.8829 | 1.1205 |

#### 4.3. Comparative Statics

**Figure 3.**Values for the indifference switching ${\lambda}^{*}$ for different levels of the career length T.

**Figure 4.**Values for the indifference switching ${\lambda}^{*}$ for different levels of the employee’s contribution rate q.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Appendix: Derivation of Proposition 3.1

## Conflicts of Interest

## References

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^{1}According to the Bureau of Labor Statistics Employee Benefits Survey in 1991 only 13% of full-time workers were covered by portability provisions (see Foster (1994) [7] for the different categories of portability provisions).^{2}Our framework excludes the case in which the employee can switch between the DB and DC plan.^{3}That is, we allow for a possibly time-varying, but deterministic intensity.^{4}As already mentioned, we do not model the endogenous choice of job mobility. Neither do we estimate the job intensity.^{5}Accordingly, this shows that the salary risk and the portability risk are interconnected in practice.^{6}In a more realistic setup, as suggested above, the ${Y}_{i}$’s would be stochastic and also directly depend on the salary process S, particularly the trend of the salary ${\mu}_{S}\left(t\right)$. We could also include the trend in a deterministic way to the ${Y}_{i}$’s, but in order to keep the impact of the model parameters clear, we stick to our simple assumption.^{7}We do not require the employers costs to be necessarily the same in the two retirement plans, since the pension plans are compared from the employees perspective, and in practice, the costs the employer bears in the two plans also differ.^{8}Of course, this is just a theoretical assumption here. In reality, employees in DC often bear higher costs, since they need to contribute periodically a fixed rate to the DC account, while the employees in a DB plan often bear less costs, as most of the contributions in the DB plan are variable deficit contributions and are mainly covered by the employer.^{9}In the sequel, we will frequently abbreviate the mean-shortfall utility as LAutility and the mean-downside deviation utility as DD utility.^{10}Similar estimation methods have been used in [23].^{11}This is a so-called hybrid pension plan, which has the main features of DB plans, but with the main difference being that pension benefits are portable.^{12}Note that the independence of ${W}^{S}$ and the Poisson process N immediately implies this independence.

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**MDPI and ACS Style**

Chen, A.; Uzelac, F.
Portability, Salary and Asset Price Risk: A Continuous-Time Expected Utility Comparison of DB and DC Pension Plans. *Risks* **2015**, *3*, 77-102.
https://doi.org/10.3390/risks3010077

**AMA Style**

Chen A, Uzelac F.
Portability, Salary and Asset Price Risk: A Continuous-Time Expected Utility Comparison of DB and DC Pension Plans. *Risks*. 2015; 3(1):77-102.
https://doi.org/10.3390/risks3010077

**Chicago/Turabian Style**

Chen, An, and Filip Uzelac.
2015. "Portability, Salary and Asset Price Risk: A Continuous-Time Expected Utility Comparison of DB and DC Pension Plans" *Risks* 3, no. 1: 77-102.
https://doi.org/10.3390/risks3010077