# Sharp Probability Tail Estimates for Portfolio Credit Risk

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## Abstract

**:**

## 1. Introduction

## 2. Sharp Tail Asymptotics

#### 2.1. Tail Approximation for the Large Loss Regime

**Theorem 1.**

**Lemma 1.**

**Proof**

**of Lemma 1.**

**Proof**

**of Theorem 1.**

#### 2.2. Some Consequences of Theorem 1

**Proposition 1.**

#### 2.3. Tail Approximation in the Small-Default Regime

**Theorem 2.**

## 3. Some Extensions and Refinements

#### 3.1. Multiple Types

#### 3.2. Tail Asymptotics for Divergent Number of Factors

**Theorem 3.**

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Collamore, J.F.; de Silva, H.; Vidyashankar, A.N.
Sharp Probability Tail Estimates for Portfolio Credit Risk. *Risks* **2022**, *10*, 239.
https://doi.org/10.3390/risks10120239

**AMA Style**

Collamore JF, de Silva H, Vidyashankar AN.
Sharp Probability Tail Estimates for Portfolio Credit Risk. *Risks*. 2022; 10(12):239.
https://doi.org/10.3390/risks10120239

**Chicago/Turabian Style**

Collamore, Jeffrey F., Hasitha de Silva, and Anand N. Vidyashankar.
2022. "Sharp Probability Tail Estimates for Portfolio Credit Risk" *Risks* 10, no. 12: 239.
https://doi.org/10.3390/risks10120239