Question 11.1: Suppose that a car owner has an 80% probability of zero acci......
Suppose that a car owner has an 80% probability of zero accidents in the year, a 20% probability of one accident and a zero probability of being in more than one accident in a year. If an accident does occur, then there is a 50% probability that the repairs will cost €500, a 40% probability that the repairs will cost €5000 and a 10% probability that the car will need to be replaced at a cost of €15,000.
Combine the frequency and severity distributions to determine the car owner’s expected loss in the year, and determine the variability of the loss by calculating the standard deviation.
The overall loss distribution L(x) is determined from the frequency distribution F(x) of the frequency of loss random variable, and the severity distribution S(x) of the amount of loss random variable. The car owner’s loss is then calculated as the expected value (i.e., mean) of this distribution. The car owner’s loss is zero where there are no accidents, and when an
accident has occurred the loss may be €500, €5000 or €15,000 (depending on the severity of the accident). The calculation of the overall loss function L(x) is given in Table 11.2.
Therefore, the expected value of the overall loss function is given by
L\left(x\right)={\left\{\begin{array}{l}{0.8,\;x=0}\\ {0.1,\;x=500}\\ {0.08,\;x=5000}\\ {0.02,\;x=15,000}\end{array}\right.}ΣxL(x) = 0.8 * 0 + 0.1 * 500 + 0.08 * 5000 + 0.02 * 15000
= 0 + 50 + 400 + 300
= €750
That is, the average car owner spends €750 on repairs after accidents, and we determine the variability from the standard deviation:
\sigma^{2}\ =\ \Sigma(X-\mu)^{2}{L}(x)=\Sigma(X-{\mathrm{E}}(X))^{2}{ L}(x)= (0 – 750)² * 0.8 + (500 – 750)² * 0.1 + (5000 – 750)² * 0.08
+ (15,000 – 750)² * 0.02
= 0.8 * 750² + 0.1 * (-250)² + 0.08 * 4250² + 0.02 * 14250²
= 450,000 + 6250 + 1,445,000 + 4,061,250
= 5,962,500
Therefore, σ = √5,962,500 = €2442, and so there is quite a large variation in outcomes.
| TABLE 11.2 | |||
| Loss Function | |||
| F(x) | S(x) | L(x) | |
| 0.8 | – | 0.8, 0 | |
| 0.2 | 0.5 | 0.1, 500 | |
| 0.4 | 0.08, 5000 | ||
| 0.1 | 0.02, 15000 | ||