[Paper Reading] Instance-based Credit Risk Assessment for Investment Decisions in P2P Lending

Title: Instance-based Credit Risk Assessment for Investment Decisions in P2P Lending
Author(s): Yanhong Guo, Wenjun Zhou, Chunyu Luo, Chuanren Liu, Hui Xiong
Journal: European Journal of Operation Research
Year: 2015

Abstract

Objective: Effective allocation of personal investors' money across different loans by accurately assessing the credit risk of each loan.

Key contributions: 
  • Guo et al. proposed a data-driven investment decision-making framework for the P2P market.
  • They designed an instance-based credit risk assessment model, which has the ability to evaluate the return and risk of each individual loan.
  • Given the estimate of return and risk, they formulated the investment decision in P2P lending as a portfolio optimization problem with boundary constraints.

Data Description

  • 2016 loan samples from Lending Club
  • 4128 loan samples from Prosper
  • Features include the borrower's credit score from FICO, the amount of the loan, the number of inquiries in the past six months, debt-to-income ratio, number of current delinquencies, and home ownership status

Instance-based Credit Risk Assessment

Instance-based Models

The so-called 'instance-based' method is to use other instances to make predictions about a particular instance. In this paper, since there are very few historical observations for each individual borrower to make predictions on his/her new loan request, the authors used past loans with similar attributes to predict the return and risk of each loan. The model is as follows: 

For a given loan i, based on n past loans, each with an observed return rate $R_j$ (j = 1,2,...,n), they directly predict loan i's return, $\mu_i$, using a weighted average of the performance of past loans
$$
\mu_i = \sum_{j=1}^n w_{ij} R_j,
$$
where $w_{ij}$ represents the weight of loan j for predicting the return of loan i. 
They quantify the risk values of a new loan i, $\sigma_i^2$, as the weighted variance among the past loans
$$
\sigma_i^2 = \sum_{j = 1}^ n w_{ij} (R_j - \mu_i)^2.
$$
The likelihood between two loans is measured by the default probability as defined below
$$
d_{ij} = |p_i - p_j|.
$$
To determine the function that maps the raw distances into optimal weight, they employ kernel regression.

Kernel Weights for Instance-based Modeling

Kernel regression is a statistical technique to find non-linear relation between a pair of random variables. With observations of n instances, $\{(x_j,y_j)|j=1,2,...,n\}$, the estimation of an outcome y given its predictive observation x, will be 
$$
y = \hat{f}(x) = \frac{\sum_{j=1}^n K(\frac{x-x_j}{h})y_j}{\sum_{j=1}^n K(\frac{x-x_j}{h})}.
$$
Here, $h (h > 0)$ is called the 'bandwidth,' which determines the proportion of local versus remote information used in the summation. $K(\cdot)$ is a Gaussian kernel function specified as below
$$
K(u) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2} u^2}.
$$
In the context of the instance-based risk modeling, the default probability $p_j$ serves as the predictive variable, and the return rate $R_j$ is the response variable. Thus, we obtain
$$
\mu_i = \frac{\sum_{j=1}^n K(\frac{x_i-x_j}{h})y_j}{\sum_{j=1}^n K(\frac{x_i-x_j}{h})} = \frac{\sum_{j=1}^n K(\frac{p_i-p_j}{h})R_j}{\sum_{j=1}^n K(\frac{p_i-p_j}{h})}.
$$
It is straightforward to extract the optimal weight as 
$$
w_{ij} =\frac{ K(\frac{p_i-p_j}{h})}{\sum_{j=1}^n K(\frac{p_i-p_j}{h})} = \frac{K(\frac{d_{ij}}{h})}{\sum_{j=1}^n K(\frac{d_{ij}}{h})}.
$$
The above formula indicates that loans with a smaller default likelihood distance $d_{ij}$ will carry more weights.

Optimization of Bandwidth - h

The authors proposed to use the leave-one-out least-squares cross-validation method. Specifically, the bandwidth is chosen to minimize the following cross-validation error
$$
CV(h) = \frac{1}{n} \sum_{i = 1}^n (\hat{f}_h(x_{-i})- y_i)^2,
$$
where $\hat{f}(x_{-i})$ is the leave-one-out estimation of $y_i$, using kernel regression
$$
\hat{f}_h(x_{-i}) = \frac{\sum_{j=1,j \neq i}^{n}K(\frac{x_i-x_j}{h})y_j}{\sum_{j=1,j \neq i}^{n}K(\frac{x_i-x_j}{h})}.
$$
In the context of instance-based risk modeling,
$$
CV(h) = \frac{1}{n} \sum_{j=1}^n [\frac{\sum_{k=1,k \neq j}^{n}K(\frac{p_j-p_k}{h})R_k}{\sum_{j=1,j \neq i}^{n}K(\frac{p_j-p_k}{h})} - R_j]^2.
$$
To find the optimal bandwidth efficiency, they adopt a mixed bandwidth selection strategy, which searches for the optimal bandwidth between $0.25h_0$ and $1.5h_0$, where $h_0$ is given by 
$$
h_0 = (4/3n)^{1/5}\sigma .
$$

Portfolio Optimization

Given the estimated return and risk, they formulate the investment decision problem as a portfolio optimization problem as follows
Minimize: $\sum_{i=1}^l \lambda_i^2 \sigma_i^2$;
Subject to:$$
\sum_{i=1}^l \lambda_i \mu_i = R^* \\
\sum_{i=1}^l \lambda_i = 1 \\
\left\{
\begin{aligned}
m \leq \lambda_i M & \leq e_i, & \text{if loan i is invested} \\
\lambda_i & = 0, & \text{otherwise}
\end{aligned}
\right.
$$
where $M$ is the total amount of money invested, m is the minimum amount to invest in a single loan (In Lending Club, the minimum investment is 25 dollars), and $e_i$ is the maximum available funding for loan i.

Integrated Model and Results


The stated $R^2$ for Prosper is 0.2603, and that for Lending Club is 0.2070.




Comments

Post a Comment

Popular posts from this blog

Online Optimization Specialization (1/4): Review of 'Online Convex Programming and Generalized Infinitesimal Gradient Ascent' by Martin Zinkevich

Specialization Introduction This specialization covers five selected grounding papers in online optimization. In each blog, I will discuss one paper, where I aim to include Brief introduction and summary to the paper Key takeaways of the paper Notice that all the discussion and summary in this specialization are based on the reviewed papers. None of the algorithms or theorems is proposed by myself.  Summary Paper Detail Title: Online Convex Programming and Generalized Infinitesimal Gradient Ascent Author(s): Martin Zinkevich URL:  https://www.aaai.org/Library/ICML/2003/icml03-120.php Abstract Convex programming involves a convex set $F \subseteq R^n$ and a convex function $c : F \to \mathbb{R}$. The goal of convex programming is to find a point in F which minimizes c. In this paper, we introduce online convex programming.  In online convex programming, the convex set is known in advance, but in each step of some repeated optimization problem, one must select a point in F ...
Read more

[Paper Reading] In-sample and Out-of-sample Sharpe Ratios of Multi-factor Asset Pricing Models

Image
Title:  In-sample and Out-of-sample Sharpe Ratios of Multi-factor Asset Pricing Models Author(s):  Raymond Kan*, Xiaolu Wang, Xinghua Zheng Year:  2019, visited the version revised on 22 Feb 2021 URL:  https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3454628 Definitions Optimal portfolio weight: For a mean-variance investor who wants to hold a portfolio with a target standard deviation of $\sigma$, then their optimal portfolio has weights of  $$ w^* = \frac{\sigma}{\theta} \Sigma^{-1} \mu, $$              where $\mu = \mathbb{E}[r_t]$, $\Sigma = Var[r_t]$, and $r_t$ is the return of risky assets in excess of risk-free rate. $\theta$ is the Sharpe ratio. Sharpe Ratio: $\theta = \sqrt{\mu' \Sigma^{-1}\mu}$. In practice, the investor does not know the mean and covariance matrix of the factors; he has to estimate $\theta$ using historical data. Thus, we have the in-sample and out-of-sample Sharpe ratio as follows. In-sample...
Read more

Online Optimization Specialization (4/4): Review of 'Projection-free Online Learning'

Image
Specialization Introduction This specialization covers five selected grounding papers in online optimization. In each blog, I will discuss one paper, where I aim to include Brief introduction and summary to the paper Key takeaways of the paper Notice that all the discussion and summary in this specialization are based on the reviewed papers. None of the algorithms or theorems is proposed by myself.  Summary Paper Detail Title:  Projection-free Online Learning Author(s):  Elad Hazan, Satyen Kale URL:  https://icml.cc/2012/papers/292.pdf Abstract The computational bottleneck in applying online learning to massive data sets is usually the projection step. We present efficient online learning algorithms that eschew projections in favor of much more efficient linear optimization steps using the Frank-Wolfe technique. We obtain a range of regret bounds for online convex optimization, with better bounds for specific cases such as stochastic online smooth convex optimization...
Read more