# Rational Savings Account Models for Backward-Looking Interest Rate Benchmarks

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Derivatives on Backward-Looking Rates

#### 2.1. Swaplet

#### 2.2. Forward Rate Agreement

#### 2.3. Caplet

#### 2.4. Swaps and Swaptions

#### 2.5. Considerations

## 3. Rational Savings Account Models

**Assumption**

**1.**

**Lemma**

**1.**

**Proof.**

**Remark**

**1.**

**Assumption**

**2.**

**Remark**

**2.**

#### 3.1. Linear Derivatives

**Proposition**

**1.**

**Proof.**

#### 3.2. Caplets and Swaptions

**Proposition**

**2.**

**Proof.**

**Proposition**

**3.**

**Proof.**

#### 3.3. Futures

**Proposition**

**4.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Proposition**

**5.**

**Proof.**

#### Example: Multidimensional CIR Process

## 4. Conclusions and Outlook

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. General Results

**Lemma**

**A1.**

**Lemma**

**A2.**

**Proof.**

**Lemma**

**A3.**

**Proof.**

## Appendix B. Affine Processes

**Lemma**

**A4.**

**Proof.**

**Lemma**

**A5.**

**Proof.**

**Lemma**

**A6.**

**Proof.**

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1. | One may expect that the Euro Overnight Index Average (EONIA), essentially a one-day EURIBOR, may in time become outdated as it is made superfluous by an established STR. |

2. | After consecutive applications of the tower property one sees that, for $t\le T\le U$, ${B}_{t}{\mathbb{E}}_{t}\left[{B}_{T}^{-1}\overline{R}(t,T)\right]={B}_{t}{\mathbb{E}}_{t}\left[{B}_{T}^{-1}R(t,T)\right]$, and it thus follows that pricing linear derivatives, i.e., any derivative that is linear in the rate, is unaffected by the continuous compounding approximation in the case of $t\le T\le U$. |

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**MDPI and ACS Style**

Macrina, A.; Skovmand, D.
Rational Savings Account Models for Backward-Looking Interest Rate Benchmarks. *Risks* **2020**, *8*, 23.
https://doi.org/10.3390/risks8010023

**AMA Style**

Macrina A, Skovmand D.
Rational Savings Account Models for Backward-Looking Interest Rate Benchmarks. *Risks*. 2020; 8(1):23.
https://doi.org/10.3390/risks8010023

**Chicago/Turabian Style**

Macrina, Andrea, and David Skovmand.
2020. "Rational Savings Account Models for Backward-Looking Interest Rate Benchmarks" *Risks* 8, no. 1: 23.
https://doi.org/10.3390/risks8010023