**Reading Source:** https://www.bis.org/publ/joint25.pdf

**Topics Covered in this Reading:**

- VarCovar Approach
- Distribution-based aggregation
- Scenario-based aggregation
- Coherent risk measures
- Correlations vs dependencies
- Positive semi-definiteness

**Reading Source:** https://www.bis.org/publ/joint25.pdf

**Topics Covered in this Reading:**

- VarCovar Approach
- Distribution-based aggregation
- Scenario-based aggregation
- Coherent risk measures
- Correlations vs dependencies
- Positive semi-definiteness

Is someone able to explain the reasoning behind the Frechet Upper and Lower Bounds? They are just listed in the readings I have without much detail.

Thanks!

IHi @smoore29

Letâ€™s consider it this way. C(u1,u2) is a way of saying what is the probability of U1 < u1 and U2 < u2. We donâ€™t know the dependence structure so we are placing a minimum and maximum bound on this probability.

Letâ€™s apply some numbers here to make it easier. Scenario 1: u1=0.2 and u2=0.4. Scenario 2: u1=0.6 and u2=0.7

For scenario one, we know the probability of U1 being less than 0.2 is 0.2, and the probability of U2 being less than 0.4 is 0.4. However what is the probability of U1 being less than 0.2 AND U2 being less than 0.4? It depends on the dependence structure.

Lower bound:

this will be when the variables are perfectly negatively correlated. This would indicate that if one variable is low, the other would be high. So the probability of both variables being below certain values will be at its minimum (because typically the variables both wonâ€™t be low).

Scenario 1: the 0.2 probability is placed on the number line from 0 to 0.2. However since U2 is perfectly negatively correlated, the 0.4 probability will be placed on the number line from 0.6 to 1. If a low value occurs for U1, a high value will occur for U2. There is no overlap between the U1 and U2 placements on the number line therefore it is not possible for U1 to be less than u1 and U2 to be less than u2 at the same time. Therefore C(u1,u2)=0

Scenario 2: the 0.6 probability is placed from 0 to 0.6 on the number line. The 0.7 probability is placed from 0.3 to 1 on the number line. This is still as spread out as possible indicating the perfect negative correlation. The overlap here is from 0.3 to 0.6 indicating a 0.3 chance of U1 being below 0.6 AND U2 being below 0.7 at the same time. For example, U1 could be 0.35 and therefore U2 would be 0.65. The event U1<0.6 and U2< 0.7 would have occurred.

Notice that the probability of this happening, 0.3, is equal to 0.6+0.7-1. Therefore, if we assume perfect negative independence, the probability of C(u1,u2) is either 0 (if there is no overlap along the number line), or u1+u2-1 (if there is an overlap).

So the lower bound for C(u1,u2) is max(0,u1+u2-1)

Upper bound: here we assume perfect positive correlation. A low event for U1 will create the exact same low event for U2.

Scenario 1: the probability 0.2 is placed from 0 to 0.2. The U2 probability is placed from 0 to 0.4 since low events will happen together. So the overlap here is 0.2. In other words, in order for U1 to be less than 0.2 AND U2 to be less than 0.4, U1 will have to be less than 0.2. Because if U2 is 0.3, this means U1 will also be 0.3 and the condition is not met.

Scenario 2: similarly, the 0.6 is placed from 0 to 0.6 and the 0.7 is placed from 0 to 0.6. The overlap is 0.6.

In both these cases, the probability of the event being satisfied is simply the size of the smallest individual event, or min(u1,u2).

So to summarize, if we assume a perfectly negative correlation, C(u1,u2) will be max(0, u1+u2-1). And if we assume a perfectly positive correlation, C(u1,u2)=min(u1,u2).

Of course the probability C(u1,u2) could be somewhere in between these bounds as these bounds are simply the lowest and highest possible values.

This is a bit of a long winded post and a little tricky to explain over text. Let me know if things are unclear and we can talk about it further.