Financial Model Overview:

      Aim of this project is to design an insurance system for airlines which compensate and assist passengers for delayed flights, cancelled flights, denied boarding (i.e. oversold flights) for which an. To calculate the premium imposed on airlines which yields at least a 5% profit more than 99% of the time and to ensure that the insurance company does not go bankrupt, we need to have an efficient model which is amenable to analysis. Therefore, we carry out Burning Cost Model and Ruin Model to assess the premium and expected loss for possible compensation events respectively. The difference of the two nomenclature is the amount which an insurance company needs to maintain in their account to avert losses.

Burning Cost Model:

      Burning cost is the estimated cost of claims in the forthcoming insurance period, calculated from previous years’ experience adjusted for changes in the numbers insured. 

      The fee for the insurance protection, F, must take into account the cost of processing the claims. In addition to disruption event service providers also include additional costs to account for variability in the underlying distribution, cost of capital, profit, and a risk management adjustment. The fee is equal to the sum of the Expected Payout (also known as the Burning Cost), the 99.5 th percentile risk, cost of capital and claim processing fees.

F = E [CP] + Risk and Cost of Capital + Cost of Claim Processing + Profit Margin + Management Risk Adjustment

Where:

• E[CP] = Burning Cost = Probability of the Event P(e) * Claim Payout (CP)

• Risk and Cost of Capital = 99.5 th Percentile of Variance in P(e) * Claim Payout (CP) * Cost of Capital

• Cost of Capital ~ 6%

• 99.5 th Percentile = inverse of the normal cumulative distribution for a specified mean and standard deviation.

• Cost of Claim Processing C CP = fixed cost per claim ~ $0.30

• Profit Margin = % of fee

• Management Adjustment = % of fee

Premium Cost for an Event = Burning Cost + Risk/Capital Cost + Management Adjustment

Where:

• Burning Cost = Probability of the Event * Payout

• Risk/Capital Cost = 99.5 th Percentile * Payout * Cost of Capital

• Cost of Capital ~ 6%

• 99.5 th Percentile = inverse of the normal cumulative distribution for a specified mean and standard deviation.

• Management Adjustment (e.g. CR) = (Burning Cost / CR ) – Burning Cost CR ~ 100% – 80%

      Firstly, we determine the quarterly mean occurrence rate and standard deviation of each compensation event in Microsoft Excel by analyzing historical flight data for all the airlines to calculate the 99.5th percentile of the occurrence rates as required in the formula above. Secondly, using the formula for Insurance Protection Fee, we calculate the premium cost for each compensation event and then add them up to get total premium for each airline for all the four quarters. Following terminologies were assumed to be constant for each compensation event calculation.

      Our Burning Cost Model is automatized in such a way that Premium can be assessed for each airline operating in DCA for every quarter. Using the total premium value for each airline, we compute the average and target profit margin. These values are later used in Ruin Model Calculation to estimate the amount required in escrow account for the insurance company.

Ruin Model

      Risk theory refers to a body of techniques to model and measure the risk associated with a portfolio of insurance contracts. A first approach consists in modeling the distribution of total claims over a fixed period of time using the classical collective model of risk theory. A second input of interest to the actuary is the evolution of the surplus of the insurance company over many periods of time. In ruin theory, the main quantity of interest is the probability that the surplus becomes negative, in which case technical ruin of the insurance company occurs.

      In the classical risk model, the insurer's surplus at time t, S (t) being the aggregate claims between 0 and t, c being the rate at which premiums are received, given an initial surplus u, is U (t) where

U (t) = u + ct- S (t)

      The aggregate claims process {S (0}t≥0 is a compound Poisson process, with Poisson parameter λ. We denote by P the distribution function of individual claim amounts, and assume that P (0) = 0. Let Pk denote the kth moment of this distribution. We assume that the insurer's premium income is received continuously at rate ‘c’ per unit time, where c = (1+θ)λp1 and θ is the premium loading factor. Without loss of generality we can set both λ and p1 to be 1 and these values will be assumed in all numerical illustrations in this paper.

The probability of ultimate ruin from initial surplus u is denoted ψ (u). We denote by L the maximum of the aggregate loss process so that, Ψ (u) = Pr(L > u)

Probability of survival,

1-Ψ (u) = Pr(L <= u),  u>=0

The aggregate loss process {L (t)}t>0 is defined by ,

L (t) = S (t) - ct

The probability of ruin by time t from initial surplus u is denoted ψ (u, t) and given by,

Ψ (u, t) = Pr (T< t)

so that,   

Pr (Tc ≤ t) = Pr(T ≤ t|T< ∞) = ψ(u, t) /ψ(u) 

is the distribution function of the time to ruin given that ruin occurs.

      Aggregate expected loss for a single quarter, S(1) is estimated first by calculating the 99.5th percentile of operational cost for each compensation event and then added up. Total Premium from Burning Cost Model calculation acts as levied premium in this model.  Difference between aggregate expected loss and levied premium gives the minimum amount required in escrow to cover the "worst case" claims-costs. Using the automatized Premium Assessment workbook in Microsoft Excel, escrow amount for each airline and for each quarter can be determined easily.