## Question 1 from UoL exam 2016, Zone B, Post 2

For the problem statement and first part of the solution see Question 1 from UoL exam 2016, Zone B, Post 1.

Let denote the return on From Table 1 we can derive the probabilities table for this return:

**Table 2**. Joint table of returns on separate portfolios

From Table 2 we conclude that the return on the combined portfolio looks as follows:

**Table 3**. Total return

Table 3 shows that

for

for

for and

for

Try to follow the procedure used in Post 1 and you will see that

for

for

for

for and

for

This implies In statistics, we always have to watch if the numbers we get make sense. The last number doesn't and in fact leads to a contradiction: This is because the quasi-inverse notion doesn't have anything to do with probabilities. With a more realistic return, the VaR should be negative for small values of

**(b)** The subadditivity definition requires amounts opposite in sign to ours. That is, we define from and then say that VaR thus defined is **sub-additive** if We have been using the definition It's easy to see that Thus, in our case we have which is not smaller than Sub-additivity does not hold in this example. Absence of sub-additivity means that riskiness of the whole portfolio, as measured by VaR, may exceed riskiness of the sum of the portfolio parts.

**(c)** The problem uses the definition of the expected shortfall that yields positive values. I use everywhere the definition that gives negative values: Since the setup is static, this is the same as By definition, so .

In Post 1 we found that for each of The condition places no restriction on so from Table 1

As a result,

Since from Table 3

Therefore Converting everything to positive values, we have so that sub-additivity holds.

The returns in percentages can be easily converted to those in dollars.

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