Question

Suppose you are the risk manager for UPS and you self-insure the property – damage liability risk you face from the fleet of trucks that operate in the Southeast region. Assume each exposure’s loss distribution is independent and identically distributed and, based upon historical data, you know the mean of the loss distribution for one truck is $500 and the variance of loss is 421,000. Assume an actuarial consultant has concluded that you can be 99 percent certain there will be enough funds available to pay all loss obligations if you set aside the expected loss plus a 10 percent safety loading on each truck. Determine the number of trucks you have in the Southeast region.

Answer 1

Let N be the number of trucks in the fleet.

µ = mean loss for each truck = $ 500

Mean loss for the fleet = µN = $ 500 x N = 500N

σ = standard deviation of loss on each truck = 421,000

Standard deviation of loss on the fleet = σN^1/2 = 421,000N^1/2

P(z) = 0.99

z = 2.326

(X - µN) /  σN^1/2 = 2.326

X = the expected loss plus a 10 percent safety loading = µN x (1 + 10%) = 1.1µN

Hence, (1.1µN - µN) /  σN^1/2 = 2.326

Or, 0.1µN^1/2 / σ = 2.326

Or, 0.1 x 500 x N^1/2 / 421,000 = 2.326

Hence, N = 2.326 x 421,000 / (500 x 0.1)² =  383,569,091