# Immunization Strategies for Funding Multiple Inflation-Linked Retirement Income Benefits

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## Abstract

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## 1. Introduction

## 2. Theoretical Framework

#### 2.1. Term Structure Specification

#### 2.2. Interest Rate Risk Measures

#### 2.3. Security Design

## 3. Empirical Study

#### 3.1. Portfolio Design and Testing Methodology

- (i)
- the present value of the asset portfolio matches that of the expected liability cash flow stream, i.e., the immunization portfolios are self-financing;
- (ii)
- For each sub-portfolio, the vector of the M-Absolute measures matches the residual maturity of each liability;
- (iii)
- The sum of the M-Absolute interest rate risk measures has to be equal to the sum of the residual maturities of each liability due.

#### 3.2. Performance Metrics

_{z}stands for the current quarterly rebalancing period for the portfolio and ${Q}^{j}$ are the bond units of bond j in the portfolio. If the j-th bond was not held in the portfolio before a given quarterly rebalancing period, the respective transaction amount is set to zero. The amount to be reinvested includes any coupons received since the last rebalancing period and any amount bought or sold at the rebalancing date. The average turnover measure assesses the frequency and extent to which immunizing portfolios have to be rebalanced (buy/sell trades, coupon, and principal reinvestment trades) in order to match immunization constraints.

#### 3.3. Data

## 4. Results

## 5. Sensitivity Analysis: Zero Interest Rate Scenario

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Disclaimer

## Appendix A. U.S. TIPS Inflation Adjustment

_{v}is the number of days in month v, d is the day of the month v when settlement occurs and v is the month on which settlement takes place.

_{v}day for the first day when the bond’s inflation component starts to accrue (the base day), allows for the calculation of the base index (${I}_{base}$). This way, it is possible to compute a daily Index Ratio to adjust for daily inflation changes in the bond and whenever it is traded, making the inflation accrual steadily over each month instead of adjusting only once a month, when the new figure of the price index is published. The daily adjusted Index Ratio is given by the expression

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Panel A—3-Year Immunization Horizon | |||||

M-Absolute—Sub-Portfolio Horizon | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 0.48% | 0.64% | 45.01% | 1.66% | 28.93 |

2-year | 1.53% | 1.13% | 61.49% | 1.26% | 22.34 |

3-year | 1.94% | 0.59% | 29.29% | 0.86% | 20.01 |

Naive—Sub-portfolio horizon | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 0.09% | 0.30% | 24.10% | 0.86% | 7.84 |

2-year | 0.83% | 0.46% | 29.41% | 0.77% | 27.92 |

3-year | 2.50% | 1.06% | 52.97% | 0.45% | 15.73 |

Panel B—5-year immunization horizon | |||||

M-Absolute—Sub-portfolio horizon | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 0.56% | 0.67% | 45.39% | 1.78% | 29.33 |

2-year | 1.90% | 1.27% | 69.10% | 1.90% | 20.25 |

3-year | 2.52% | 0.69% | 32.72% | 0.74% | 19.38 |

4-year | 5.27% | 1.36% | 54.23% | 0.70% | 14.72 |

5-year | 6.70% | 0.73% | 27.06% | 0.55% | 15.20 |

Naive—Sub-portfolio horizon | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 0.16% | 0.32% | 24.69% | 0.88% | 7.89 |

2-year | 1.09% | 0.53% | 32.06% | 0.64% | 29.97 |

3-year | 3.10% | 1.24% | 58.93% | 0.46% | 15.55 |

4-year | 3.85% | −0.01% | −0.26% | 0.85% | 29.39 |

5-year | 7.08% | 1.10% | 37.63% | 0.72% | 26.45 |

Panel A—3-Year Immunization Horizon | |||||

M-Absolute—Sub-Portfolio Horizon | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 5.98% | 9.03% | 209.29% | 0.61% | 20.49 |

2-year | 9.67% | 15.71% | 270.38% | 0.40% | 13.40 |

3-year | 8.98% | 18.05% | 298.01% | 0.31% | 10.68 |

Naive—Sub-portfolio horizon | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | −0.29% | 2.74% | 76.73% | 1.18% | 21.54 |

2-year | −0.22% | 5.76% | 99.06% | 0.96% | 19.78 |

3-year | −1.08% | 7.84% | 129.41% | 0.40% | 12.64 |

Panel B—5-year immunization horizon | |||||

M-Absolute—Sub-portfolio horizon | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 7.04% | 7.81% | 180.20% | 0.64% | 21.26 |

2-year | 11.38% | 12.49% | 210.23% | 0.43% | 14.69 |

3-year | 10.99% | 12.42% | 199.34% | 0.33% | 11.42 |

4-year | 10.96% | 12.47% | 190.87% | 0.26% | 8.38 |

5-year | 20.36% | 21.32% | 260.24% | 0.50% | 13.93 |

Naive—Sub-portfolio horizon | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | −0.10% | 0.77% | 20.68% | 1.32% | 23.93 |

2-year | 0.26% | 1.77% | 29.77% | 1.08% | 22.19 |

3-year | −0.53% | 1.52% | 24.34% | 0.42% | 13.69 |

4-year | −0.39% | 1.92% | 29.46% | 0.44% | 20.68 |

5-year | −0.72% | 2.15% | 61.18% | 0.47% | 15.24 |

Panel A—3-Year Immunization Horizon | |||||

Sub-Portfolio Horizon | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | −0.36% | 0.56% | 23.95% | 0.89% | 25.47 |

2-year | −1.11% | 0.62% | 23.52% | 0.54% | 17.12 |

3-year | −2.03% | 0.44% | 15.39% | 0.40% | 13.37 |

Panel B—5-year immunization horizon | |||||

Sub-portfolio horizon | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | −0.27% | 0.61% | 26.32% | 0.89% | 26.57 |

2-year | −1.04% | 0.51% | 19.09% | 0.51% | 16.29 |

3-year | −1.81% | 0.28% | 9.58% | 0.38% | 12.85 |

4-year | −2.63% | −0.15% | −4.72% | 0.38% | 12.18 |

5-year | −3.36% | −0.38% | −11.16% | 0.43% | 12.92 |

Panel A—3-Year Immunization Horizon | |||||

Pre-Crisis | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 1.74% | 0.88% | 54.32% | 1.57% | 29.65 |

2-year | 4.26% | 1.78% | 82.93% | 0.92% | 19.05 |

3-year | 4.95% | 0.73% | 30.61% | 0.75% | 17.54 |

ZIRP | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | −1.15% | 0.33% | 15.26% | 1.77% | 28.07 |

2-year | −1.98% | 0.29% | 56.04% | 1.57% | 25.37 |

3-year | −1.92% | 0.41% | 42.49% | 0.98% | 21.78 |

Panel B—5-year immunization horizon | |||||

Pre-crisis | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 1.74% | 0.88% | 54.32% | 1.57% | 29.67 |

2-year | 4.26% | 1.78% | 82.64% | 0.92% | 19.05 |

3-year | 4.95% | 0.73% | 30.61% | 0.75% | 18.44 |

4-year | 9.19% | 2.19% | 77.52% | 0.46% | 14.99 |

5-year | 11.09% | 1.55% | 51.08% | 0.41% | 15.15 |

ZIRP | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | −1.55% | 0.29% | 22.50% | 2.12% | 28.69 |

2-year | −2.34% | 0.34% | 67.21% | 3.31% | 22.00 |

3-year | −1.86% | 0.62% | 58.55% | 0.71% | 20.96 |

4-year | −1.79% | −0.14% | −10.14% | 1.26% | 16.68 |

5-year | −1.21% | −0.74% | −45.56% | 1.00% | 18.23 |

Panel A—3-Year Immunization Horizon | |||||

Pre-Crisis | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 7.28% | 10.00% | 234.60% | 0.59% | 21.74 |

2-year | 14.52% | 19.76% | 343.34% | 0.44% | 16.80 |

3-year | 15.62% | 23.79% | 395.66% | 0.33% | 13.48 |

ZIRP | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 4.30% | 7.79% | 168.46% | 0.64% | 19.21 |

2-year | 3.42% | 10.50% | 167.90% | 0.39% | 10.36 |

3-year | 0.43% | 10.66% | 164.61% | 0.32% | 7.29 |

Panel B—5-year immunization horizon | |||||

Pre-crisis | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 7.28% | 7.40% | 173.59% | 0.58% | 21.74 |

2-year | 14.52% | 14.24% | 247.48% | 0.44% | 16.80 |

3-year | 15.62% | 15.19% | 252.59% | 0.33% | 13.48 |

4-year | 17.72% | 16.89% | 259.45% | 0.29% | 10.85 |

5-year | 26.49% | 24.50% | 288.68% | 0.39% | 13.24 |

ZIRP | $\overline{LC}$ | $\overline{ER}$ | R/R_{h} | $\overline{TC}$ | $\overline{T}$ |

1-year | 6.61% | 8.55% | 190.82% | 0.69% | 20.72 |

2-year | 5.73% | 9.34% | 148.98% | 0.43% | 11.57 |

3-year | 2.66% | 7.45% | 112.98% | 0.33% | 7.76 |

4-year | 1.20% | 4.51% | 69.42% | 0.20% | 3.68 |

5-year | 9.32% | 15.59% | 205.91% | 0.75% | 15.58 |

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## Share and Cite

**MDPI and ACS Style**

Simões, C.; Oliveira, L.; Bravo, J.M. Immunization Strategies for Funding Multiple Inflation-Linked Retirement Income Benefits. *Risks* **2021**, *9*, 60.
https://doi.org/10.3390/risks9040060

**AMA Style**

Simões C, Oliveira L, Bravo JM. Immunization Strategies for Funding Multiple Inflation-Linked Retirement Income Benefits. *Risks*. 2021; 9(4):60.
https://doi.org/10.3390/risks9040060

**Chicago/Turabian Style**

Simões, Cláudia, Luís Oliveira, and Jorge M. Bravo. 2021. "Immunization Strategies for Funding Multiple Inflation-Linked Retirement Income Benefits" *Risks* 9, no. 4: 60.
https://doi.org/10.3390/risks9040060