## Solution to Question 1 from UoL exam 2016, Zone B

This problem is a good preparation for Question 2 from UoL exam 2015, Zone A (FN3142), which is more difficult.

## Problem statement

Two corporations each have a 4% chance of going bankrupt and the event that one of the two companies will go bankrupt is independent of the event that the other company will go bankrupt. Each company has outstanding bonds. A bond from any of the two companies will return % if the corporation does not go bankrupt, and if it does, a bondholder will lose the face value of the investment, i.e., %. Suppose an investor buys $1000 worth of bonds of the first corporation, which is then called portfolio , and similarly, an investor buys $1000 worth of bonds of the second corporation, which is then called portfolio .

(a) [40 marks] Calculate the VaR at % critical level for each portfolio and for the joint portfolio .

(b) [30 marks] Is VaR sub-additive in this example? Explain why the absence of sub-additivity may be a concern for risk managers.

(c) [30 marks] The expected shortfall at the % critical level can be defined as . Calculate the expected shortfall for the portfolios and . Is this risk measure sub-additive?

## Solution

There are a couple of general ideas to understand before embarking on calculations. The return on the bond of one company is a binary variable taking values 0% and -100%. All calculations involving it are similar to the ones for the coin. After doing calculations the return figures can be translated to dollar amounts by multiplying by $1000.

While the use of the notions of the distribution function and generalized inverse can be avoided, I prefer to use them to show the general approach.

(a) The return on one bond is described by the table

**Table 1**. Probability table for return on one bond

Return values | Probability |

0 | 0.96 |

-100 | 0.04 |

Therefore its distribution function can be found in the same way as for the coin:

The distribution function is shown in red. It is zero for **generalized inverse** defined by

see the definition of the infimum here. In our case

What we do next is very similar to the derivation of the sampling distribution for two coins.

**Table 2**. Joint probability table for returns on two portfolios

First portfolio | |||

0 | -100 | ||

Second portfolio | 0 | ||

-100 |

The main body of the table contains probabilities of pairs

**Table 3**. Probability table for return on the total portfolio

Total return | Probabilities |

0 | 0.9216 |

-50 | |

-100 | 0.0016 |

This table results in the following distribution function:

Since 0.0784>0.05, the Value at Risk is -50% (use the generalized inverse).

(b) Translating the percentages to dollars, at 5% the risk for each of the bonds is $0 and for the total portfolio it is $1000 (50% of $2000; I am passing from negative percentages to positive loss figures).

We say that Value at Risk is **sub-additive** if

(c) Here we have to apply the definition of the conditional expectation:

For the total portfolio we get

In monetary terms, this translates (again passing to positive values) to $40 for each bond and to $1020.40 for the total portfolio. The conclusion is that expected shortfall is not sub-additive.

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