The Loss Distribution

Cumulative Probability

Cumulative probabilities turn a loss distribution into quantiles.

highlighted = computed this step

Cumulative probability

Cumulative probability adds the exact probabilities in ascending loss order. At loss $200.00, the cumulative probability reaches 19/20.

P(L≤$200.00)=19/20P(L\le \$200.00)=19/20

The quantile idea

A quantile asks for the smallest loss whose cumulative probability reaches a chosen alpha. The table shows the cumulative spine used by VaR.

VaRα=min⁡{L:P(Loss≤L)≥α}\text{VaR}_\alpha=\min\{L:P(\text{Loss}\le L)\ge\alpha\}
Loss distributionVaR and expected-shortfall rows are recomputed from the distribution.Loss distributionScenarioLossProbabilityCumulative1$-100.003/53/52$0.001/54/53$200.003/2019/204$500.001/5097/1005$1,500.003/1001

Distribution dependent

A cumulative distribution is only as good as the stated loss model behind it. In reality that model is estimated from data with uncertainty. This is descriptive, not investment advice.

quantiles depend on the stated distribution\text{quantiles depend on the stated distribution}