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Stochastic Loss Reserving Using Bayesian MCMC Models

Section 5

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Stochastic Loss Reserving Using Bayesian MCMC Models

Bayesian Markov Chain Monte Carlo (MCMC)

ODP-Over Dispersed Poisson

Probably the two most popular nonproprietary stochastic loss reserve models are the Mack (1993, 1994) chain-ladder model and the England and Verrall (2002) bootstrap

ODP model.

Mack Model -- Stochastic version of Chain Ladder Method (link ratio method)

f: age to age factor. (link ratio)

Let’s begin with the classic chain-ladder model. Let Cw, d denote the accumulated loss amount, either incurred or paid, for accident year, w, and development lag, d, for 1 ≤ w ≤ K and 1 ≤ d ≤ K. Cw, d is known for the “triangle” of data specified by w + d ≤ K + 1. The goal of this model is to estimate the loss amounts in the last column of data, Cw, K for w = 2, . . . , K. To use the chain-ladder model, one first calculates the age to age factors given by

then

Bootstrap ODP is designed for work with Incremental Losses. Where Incremental Losses are described by ODP

the sample size is very large, this will rarely be the case because of random fluctuations.

A visual test of uniformity that allows one to test for statistical significance is the

p–p plot combined with the Kolmogorov–Smirnov (K–S) test.

over dispersion : Overdispersion means that the variance of the response is greater than what's assumed by the model

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Section 5

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<!DOCTYPE html>

<head>

<title>Collecting Data</title>

<script src=

"https://ajax.googleapis.com/ajax/libs/jquery/3.5.1/jquery.min.js">

</script>

"https://cdn.jsdelivr.net/npm/bootstrap@4.5.3/dist/css/bootstrap.min.css"

integrity=

"sha384-TX8t27EcRE3e/ihU7zmQxVncDAy5uIKz4rEkgIXeMed4M0jlfIDPvg6uqKI2xXr2"

crossorigin="anonymous">

</head>

<body class="container" style="margin-top: 50px;

width: 50%; height:auto;">

<h2 class="text-primary" style=

"margin-left: 15px; margin-bottom: 10px">

Hey There,Help Us In Collecting Data

</h2>

Enter Your Name

</label>

<input type="text" class="form-control"

id="name" placeholder="Enter your name">

</div>

Email address

</label>

<input type="email" class="form-control"

id="email" placeholder="name@example.com">

</div>

<button type="submit" class="btn btn-primary"

style="margin-left: 15px; margin-top: 10px">

Submit

</button>

</div>

</form>

<script src=

"https://www.gstatic.com/firebasejs/3.7.4/firebase.js">

</script>

var firebaseConfig = {

apiKey: "Use Your Api Key Here",

authDomain: "Use Your authDomain Here",

databaseURL: "Use Your databaseURL Here",

projectId: "Use Your projectId Here",

storageBucket: "Use Your storageBucket Here",

messagingSenderId: "Use Your messagingSenderId Here",

appId: "Use Your appId Here"

};

firebase.initializeApp(firebaseConfig);

var messagesRef = firebase.database()

.ref('Collected Data');

document.getElementById('contactForm')

.addEventListener('submit', submitForm);

function submitForm(e) {

e.preventDefault();

// Get values

var name = getInputVal('name');

var email = getInputVal('email');

saveMessage(name, email);

document.getElementById('contactForm').reset();

}

// Function to get form values

function getInputVal(id) {

return document.getElementById(id).value;

}

// Save message to firebase

function saveMessage(name, email) {

var newMessageRef = messagesRef.push();

newMessageRef.set({

name: name,

email: email,

});

}

</script>

</body>

</html>

</pre>

Chain Ladder Method. Cape Cod, Bornhutter-Ferguson Method