# Hedging and Cash Flows in the Presence of Taxes and Expenses in Life and Pension Insurance

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## Abstract

**:**

## 1. Introduction

## 2. The Complete Financial Market: A Review

#### 2.1. Traded Assets

#### 2.2. Payment Processes

#### 2.3. Trading and the Value Process with Payment Processes

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**1.**

**Theorem**

**1.**

#### 2.4. A Short Overview of the Review on Complete Financial Markets

## 3. Hedging with Taxes and Expenses

**Definition**

**3.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

#### 3.1. Decomposition of Prices and Replicating Strategies

**Proposition**

**1.**

**Proof.**

#### 3.2. Valuation with Forward Rates

#### 3.3. A Short Overview of the Main Results on Taxes and Expenses

## 4. An Affine Term Structure Model

**Lemma**

**3.**

#### 4.1. The Payment Process

#### 4.2. The Price and Decomposition

#### 4.3. Hedging of the Payments

**Lemma**

**4.**

**Remark**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

#### 4.3.1. Decomposition of the Hedging Strategies

#### 4.3.2. Hedging in the Vasicek and CIR Model

#### 4.4. A Short Overview of the Main Results for the Affine Term Structure Model

## 5. Numerical Example

#### 5.1. Valuation with Forward Rates

#### 5.2. A Short Overview of the Main Results in the Numerical Example

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A simulated path for the Vasicek interest rate model (left axis) and relevant associated value processes (right axis). The total value of the hedging portfolio is illustrated as the black line “Bnf + Tax + Exp”. The value of the benefits alone is illustrated as the blue line “Bnf”, and the value of the benefit and tax payments is illustrated as the red line “Bnf + Tax”. The value of the investment in the bond (the dashed purple line “Bond-inv”) is, here in the Vasicek case, identical to the value of the whole hedging portfolio and therefore coincides with the black line “Bnf + Tax + Exp”.

**Figure 2.**A simulated path for the CIR interest rate model (left axis) and relevant associated value processes (right axis). The illustrated lines are similar to Figure 1. Here, in the CIR case, the value of the investment in the bond does not coincide with the total value of the hedging portfolio, and in particular, we see that the value of the bond investment (the purple line “Bond-inv”) is larger than the value of the portfolio, because the investment in the bond is leveraged.

**Figure 3.**A simulated path for the Vasicek interest rate model (left axis) and the hedging strategy ${\tilde{h}}_{0}^{{A}^{\mathrm{b}}}$ and ${\tilde{h}}_{1}^{{A}^{\mathrm{b}}}$ in the presence of taxes and expenses (right axis). As we argued above, here in the Vasicek case, the investment in the bank is zero corresponding to ${\tilde{h}}_{0}^{{A}^{\mathrm{b}}}\left(t\right)=0$. We see that we buy a bit more than one bond, and the difference in value is the value of the tax and expense payments combined.

**Figure 4.**A simulated path for the CIR interest rate model (left axis) and the hedging strategy ${\tilde{h}}_{0}^{{A}^{\mathrm{b}}}$ and ${\tilde{h}}_{1}^{{A}^{\mathrm{b}}}$ in the presence of taxes and expenses (right axis). In this (the CIR) case, there is a small investment in the bank account.

**Figure 5.**Difference between the approximation based on the forward interest rate curve and the correct theoretical prices in the Vasicek and CIR models. The Vasicek and CIR models with parameters from Table 1 are shown, as well as two variations, where the volatilities are scaled up respectively down by 50%. The approximated value overestimates the correct value by 0.01–0.4% in our numerical examples.

**Table 1.**Parameters for the two interest rate models: the Vasicek model ($\alpha =0$) and the CIR model ($a=0$).

Model | $\mathit{r}\left(0\right)$ | b | $\mathit{\beta}$ | $\sqrt{\mathit{a}}$ | $\sqrt{\mathit{\alpha}}$ |
---|---|---|---|---|---|

Vasicek | 0.01 | 0.007006001 | −0.162953 | 0.015384 | - |

CIR | 0.01 | 0.003801358 | −0.092540 | - | 0.06467 |

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## Share and Cite

**MDPI and ACS Style**

Buchardt, K.; Møller, T.
Hedging and Cash Flows in the Presence of Taxes and Expenses in Life and Pension Insurance. *Risks* **2018**, *6*, 68.
https://doi.org/10.3390/risks6030068

**AMA Style**

Buchardt K, Møller T.
Hedging and Cash Flows in the Presence of Taxes and Expenses in Life and Pension Insurance. *Risks*. 2018; 6(3):68.
https://doi.org/10.3390/risks6030068

**Chicago/Turabian Style**

Buchardt, Kristian, and Thomas Møller.
2018. "Hedging and Cash Flows in the Presence of Taxes and Expenses in Life and Pension Insurance" *Risks* 6, no. 3: 68.
https://doi.org/10.3390/risks6030068