# An Optimal Turkish Private Pension Plan with a Guarantee Feature

## Abstract

**:**

## 1. Introduction

- The participant has the right to choose his/her amount of contribution; however, this amount can not be below the minimum amount of contribution which is determined by the retirement company;
- The participant can select one of the three pension funds that are specified by the law:
- Money market funds including Liquid Government Funds, Liquid Private Sector Funds, Liquid Mixed Funds, other funds, etc;
- Government/Private Sector/International debt instruments funds/precious metals/Index Funds;
- Stocks/Other Funds: Stock funds, international stock funds, foreign country funds, sector funds, flexible funds, mixed funds, etc;

- The participant has the right to change the allocation of his/her own retirement fund among specified funds at most six times in a year;
- The participant has the right to change the pension plan at most four times in a year;
- The participant has the right to change the amount of contribution and has the right to suspend paying contribution. During the period of suspension, his/her account is protected, and, at any convenient time, he/she might continue contribution payments;
- The participant has the right to transfer his/her savings to another company if he/she is already in the system at least for two years;
- The participant has the right to quit the system.

## 2. Model Setting and Maximizing Utility

**Proposition 1.**

**Corollary 2.**

**Remark 1.**

**Special Case of the Model as a classical CPPI Strategy**

**Remark 2.**

**Special Case of the Model as a classical CPDO Strategy**

## 3. Application

- Fund management charge—$2\%$ of the monthly contribution ($0.0645\%$ on a daily basis);
- Total fund investment charges—annually $2.25\%$ of the fund’s total fund value ($0.00617\%$ on a daily basis).

#### 3.1. Comparison of Optimal Strategy with the Strategies Used in Industry

- Domestic government bond based pension fund (GBF): on average include $80\%$ of the government bonds, $15\%$ of other riskless instruments and $5\%$ of risky stocks (totally $95\%$ of riskless assets);
- Stock based pension fund (SBF): on average include $80\%$ of stocks and $20\%$ of other riskless instruments.

- The values of pension funds at maturity, ${V}_{T}$ for the GBF and SBF strategies are simulated;
- ${V}_{T}$ values under optimal strategy are simulated for various ${r}_{min}/\mathrm{year}\phantom{\rule{0.277778em}{0ex}}\in [-1,0.45]$, and the volatilities are calculated;
- Then, the risk equivalent benchmark optimal strategy of the corresponding strategy is determined;
- The internal returns $({r}^{int})$ of the strategies for mean, $E\left({V}_{T}\right)$ and median value of the pension fund at maturity are calculated;
- Finally, the comparison is done.

#### 3.2. Sensitivity of Some Characteristic Features

## 4. Conclusions

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Volatility of the value of the pension funds at maturity under optimal strategy (OS) for various ${r}_{min}$ ($\sigma =0.015$).

**Figure 2.**Volatility of the value of the pension funds at maturity under optimal strategy (OS) for various ${r}_{min}$ ($\sigma =0.025$).

**Figure 3.**Optimal Multiplier versus Value of the Pension Fund for different values of σ when ($\gamma =1.2$).

**Figure 4.**Optimal Strategy versus Value of the Pension Fund for different values of σ when ($\gamma =1.2$).

**Figure 5.**Optimal Multiplier versus Value of the Pension Fund for different values of γ when ($\sigma =\phantom{\rule{3.33333pt}{0ex}}0.015$).

**Figure 6.**Optimal Strategy versus Value of the Pension Fund for different values of γ when ($\sigma =0.015$).

**Table 1.**Internal return of Domestic Government Based Fund(GBF), Stock Based Fund(SBF) and Equivalent Optimal Strategy(OS) based on mean and median final value of the pension funds at maturity (fixed ${r}_{min}$ and $\sigma =0.015$).

Mean | Median | |||||
---|---|---|---|---|---|---|

Portfolio | ${\mathit{\sigma}}_{{\mathit{V}}_{\mathit{T}}}$ | ${\mathit{r}}_{\mathit{min}}$ (Corresponding ${\mathit{F}}_{\mathit{T}}$ ) | ${\mathit{r}}_{\mathit{int}}$ | ${\mathit{r}}_{\mathit{int}}^{\mathit{OS}}$ | ${\mathit{r}}_{\mathit{int}}$ | ${\mathit{r}}_{\mathit{int}}^{\mathit{OS}}$ |

GBF | 550 | 2.0327$\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ and 2.7027$\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ (17,452 and 20,068) | 0.0888 | 0.0888 and 0.0851 | 0.0887 | 0.0880 and 0.0859 |

SBF | 11423 | 3.0227$\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ (21,489) | 0.1166 | 0.0831 | 0.0944 | 0.0848 |

**Table 2.**Internal return of Domestic Government Based Fund(GBF), Stock Based Fund(SBF) and Equivalent Optimal Strategy (OS) based on mean and median value of the pension funds at maturity (fixed ${r}_{min}$ and $\sigma =0.025$).

Mean | Median | |||||
---|---|---|---|---|---|---|

Portfolio | ${\mathit{\sigma}}_{{\mathit{V}}_{\mathit{T}}}$ | ${\mathit{r}}_{\mathit{min}}$ (Corresponding ${\mathit{F}}_{\mathit{T}}$ ) | ${\mathit{r}}_{\mathit{int}}$ | ${\mathit{r}}_{\mathit{int}}^{\mathit{OS}}$ | ${\mathit{r}}_{\mathit{int}}$ | ${\mathit{r}}_{\mathit{int}}^{\mathit{OS}}$ |

GBF | 915 | 1.1727$\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ and 3.3027$\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ (14,692 and 22,834) | 0.0888 | 0.0890 and 0.0850 | 0.0885 | 0.0882 and 0.0858 |

SBF | 23197 | 1.1970$\times \phantom{\rule{0.166667em}{0ex}}{10}^{-4}$ (21,0231) | 0.1168 | 0.1926 | 0.0556 | 0.1807 |

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**MDPI and ACS Style**

İşcanog̃lu-Çekiç, A.
An Optimal Turkish Private Pension Plan with a Guarantee Feature. *Risks* **2016**, *4*, 19.
https://doi.org/10.3390/risks4030019

**AMA Style**

İşcanog̃lu-Çekiç A.
An Optimal Turkish Private Pension Plan with a Guarantee Feature. *Risks*. 2016; 4(3):19.
https://doi.org/10.3390/risks4030019

**Chicago/Turabian Style**

İşcanog̃lu-Çekiç, Ayşegül.
2016. "An Optimal Turkish Private Pension Plan with a Guarantee Feature" *Risks* 4, no. 3: 19.
https://doi.org/10.3390/risks4030019