Basic Examples (3) 

Compute the distribution of the amount in excess of 2/3 for a uniform distribution on the interval from 0 to 1:

In[1]:=

Out[1]=

Compute the mean of the excess:

In[2]:=

Out[2]=

Compute the distribution of the amount in excess of a point for a uniform distribution in four dimensions:

In[3]:=

Out[3]=

Compute the mean of the excess:

In[4]:=

Out[4]=

Compute the distribution of the amount in excess of 10 for a log-normal distribution with a mean of 9 and a median of 3:

In[5]:=

Out[5]=

Compute the mean and standard deviation:

In[6]:=

Out[6]=

Scope (2) 

The function works with discrete distributions:

In[7]:=

Out[7]=

In[8]:=

Out[8]=

One can compute a symbolic non-negative excess distribution from an underlying symbolic univariate distribution:

In[9]:=

Out[9]=

Compute various statistics of the resulting distribution:

In[10]:=

Out[11]=

Calculate for a numeric value:

In[12]:=

Out[12]=

Properties and Relations (2) 

One can nest non-negative excess distributions. Compute the mean amount owed by an insured who has a primary policy that covers up to 2/3 and a secondary policy that will pay any amount remaining (but not exceeding 1/10) after the primary has paid its 2/3:

In[13]:=

Out[13]=

In[14]:=

Out[14]=

The result is no different than if there was a primary policy with a limit of 2/3 and 1/10:

In[15]:=

Out[15]=

Applications (2) 

An excess insurer agrees to pay for damages in excess of $250,000 suffered by an insured where the distribution of losses is logarithmic, the mean loss is $10,000, and the median loss is $2,800. What premium would cause the excess insurer to break even:

In[16]:=

Out[16]=

A reinsurer agrees to pay 80% of the excess insurer's loss that is in turn over $100,000. What premium would cause the reinsurer to break even:

In[17]:=

Out[17]=

Possible Issues (2) 

A mismatch between the dimension of value and the dimension of the distribution dist will gives an invalid distribution:

In[18]:=

Out[18]=

When one attempts to work with the resulting distribution errors can occur:

In[19]:=

Out[19]=

In[20]:=

Out[20]=

In[21]:=

Out[21]=

Neat Examples (6) 

Consider a distribution of judgments in a lawsuit that may be modeled as a log-normal distribution:

In[22]:=

Imagine an insurance policy with a limit of 20 and consider the expected positions of the insurer and insured:

In[23]:=

Out[23]=

Now, consider a second distribution with the same mean but a higher median:

In[24]:=

Examine the positions of the insurer and the insured:

In[25]:=

Out[25]=

Note the conflict in perspectives. From the perspective of the insurer, the second distribution leaves them in a worse expected position. From the perspective of the insured, the second distribution leaves them in a better position. Plot the difference in case values as the median goes from 1 to 9.9:

In[26]:=

Out[26]=

Consider how risk aversion alters the analysis. Suppose that when the insured considers two outputs from a distribution, they focus on the worst case:

In[27]:=

Out[27]=

In[28]:=

Out[28]=