Chapter 14 Axioms of (In)coherence and VaR risk measure

[Bill SD]

Hi, (sorry me again; encourage other SP9 students to also post their Qs as this isn't my private forum!)

Appreciate any answers to six Questions below on Chapter 14 Intro to Risk Measurement.

Page 5 of the Acted notes defines subadditivity as "A merger of risk situations does not increase the overall level of risk. Indeed, it may decrease the overall level of risk, as a consequence of diversification"

Page 6 explains the 'Positive homogeneity principle' with "if you consider two identical organisations that carry out exactly the same transactions, you would indeed expect the total capital required to protect both of them to be just double the amount required for each"

Q1: Surely a merger of risk could also concentrate risk and so a firm with 100% (£1mn) of assets invested in Bond A should hold more capital than the sum of 2 firms invested 50% (£.5mn) each in Bond A? So please explain the rationale for the subadditivity and Positive homogeneity principles.

Page 5 continues: "subadditivity makes decentralisation of risk-management systems possible, since constraints can be placed on business units and if they stay within these constraints then the overall risk level cannot exceed the sum of the parts."

Q2: How is this derived from the 'sub-additivity' principle?

Q3: For the purpose of SP9, do I need to understand why the Value-at-risk (VaR) measure is only sub-additive for elliptical distributions (page 37) while tail Var is always sub-additive? Or enough to just know this fact. [Guess it's because VaR & any percentile-measure requires ordering losses by size and this order will change with the addition of a new risk, resulting in an adjusted percentile value]

Section 1.2 of the Acted notes (page 6) gives a one-line formula for 'convexity' and adds the fact: "Convexity follows from the axioms of subadditivity and positive homogeneity." The solution to Q14.10 (iii) (bottom of page 37) defines convexity "that by diversifying across different projects the amount of risk is reduced and the corresponding amount of risk capital is reduced."

Q4: This description (page 37) of convexity appears identical to the axiom of subadditivity rather than positive homogeneity. Please provide a clearer explanation in words of what a convex risk measure actually is. Or is it an unintuitive measure that simply ticks the subadditivity and positive homogeneity boxes?

Page 16 of the Acted notes mentions a disadvantage of the empirical approach to calculate VaR: "the practical difficulties and limitations of [linear] interpolation"

Q5: What are these difficulties and limitations and surprised they still exist even with modern computing capabilities?

Page 17 of the Acted notes discussing the derivation of a stochastic VaR includes: "bootstrapped – the losses used to calculate the VaR are derived from random sampling of past observed returns."

Q6: For the purpose of SP9, do I need to understand how bootstrapping works? If yes, appreciate if someone can provide a quick explanation of a bootstrapping process or helpful source to see a relevant example .

Many thanks

 

[Alvin Kissoon]

Hi Bill,

Thanks for the questions:

Q1. These principles are talking about whether a risk measure of losses behaves sensibly when risks are combined ie is coherent. For example, we could expect that if we combine losses this could lead to a risk measure total equal to or less than if we just summed the risk measures individually (subadditivity) and that if we double the portfolio, we double the risk (positive homogeneity). It is provided in the context of discussing properties of the risk measures in the Chapter.

Q2. This is because if we have a risk measure which is subadditive, then by definition summing the risk measure over the losses of the business units separately will result in a risk measure which will be equal to or greater than if we took the risk measure of the combined losses. Therefore, if the total risk capacity has been split into risk limits at a Business Unit level, and each Business Unit sticks within its risk limit, then we know that we will remain within the risk capacity of the organisation.

Q3. The syllabus objective is to describe the properties and limitations of the risk measures so knowing the property should be enough.

Q4. It is essentially meeting the subadditive and positive homogeneity principles. The way I think of it is saying that if you have two portfolios of losses, you scale them individually and then combine them, the result under the risk measure will be equal to or less than the result of the risk measure of the sum of the scaled portfolios. As before, it is then behaving as we would expect as a risk measure.

Q5. Difficulties can be, for example, that there may be insufficient data to accurately model the percentiles. As for limitations, for example, VaR may not act linearly between the two points so interpolation is only an estimate.

Q6. Not in any great detail, however bootstrapping involves sampling with replacement. Sweeting discusses this in p.295-296 (Section 13.3.1).

I hope this helps, let me know if you have any followup questions.

Alvin.

 

[Bill SD]

Thanks very much Alvin for all your helpful responses to my many posts.

Two follow-up questions (A & B) on your above post:

Question A: Seems like I need to clarify what a risk measure is. I interpret it is as a formula F(x) which converts a firm's exposure (eg. bond portfolio or no. of staff) into an amount ('measure') of a specific risk (eg. risk of bond defaults or fraud by staff) . This amount can then be compared to the amounts of risk from different exposures, firms or even the same exposure to a different risk. [eg. compare the amount of default risk on bond portfolio to (i) amount of default risk on alternative available investments, (ii) default risk on another firm or last year's bond portfolio, or (ii) interest rate risk (rather than default risk) on same portfolio, etc.]

The formula could be simple (eg. the risk= double the exposure) or more complex like a stochastic VaR (simulating potential losses).

The Acted notes and your response to my Qs 1 and 4 imply that a risk measure converts a firm's losses into an amount of risk. Is this actual, known losses or potential future losses (and if so how are these future losses calculated before input into the risk measure)? Or is my understanding of a risk measure incorrect?

Question B: A theme running through this chapter is that "diversification can reduce risk and the amount of risk capital needed." But still would expect that a concentration of similar assets/liabilities (or a combination of risks with strong positive correlations between them) would exacerbate riskiness and so unsure how to understand your responses to Q1 and Q4.

Thanks again.

 

[Alvin Kissoon]

Hi Bill,

A: Yes, the Course Notes considers these risk measures as a function of possible losses on a portfolio (which will depend on its exposure) over a time period. It will include both known and future losses, and there will be an assumption on what future losses could occur based on a set of potential outcomes (eg it could come from a distribution, there could be a probability of certain events occurring with a certain loss etc). Then, the function is applied to these possible losses to provide the risk measure.

B: Q1 and Q4 are talking about convex risk measures (linked to subadditivity and positive homogeneity). It is the idea of a convex risk measure that under the risk measure, diversification can be beneficial, and your scenario does not occur. In other words, by diversifying across different investments, you would reduce the overall proportion of each in the total portfolio and therefore you would potentially gain from a diversification benefit. Later on the course management of risk is covered, and diversification is presented as a fundamental concept of portfolio management; you would select your investments with the aim to diversify (and not to concentrate risk!) and therefore reduce your specific risk.

Let me know if this doesn't quite answer the question, particularly B!

Alvin.

 

[CapitalActuary]

It would actually be pretty unintuitive if positive homogeneity did not hold - that's why it's a good property for a risk measure, and why it's a property of 'coherent' risk measures. If you hold N shares in a company, then the risk should intuitively be N times as big as having just one share. i.e. the risk scales linearly with position size.

You have mentioned the idea of 'concentration of risk' a couple of times above, and that you think it should or could lead to higher risk than just summing things up. Can you come up with a clear example though? (I posit that in trying to come up with an example you will realise there is not a simple one.*)

Surely the most 'concentrated' your risks can be is 100% correlated with each other, for example holding all your positions in the same company. In this case, per above, the risk would be (N * risk per share) = (sum of the risk on each share). So in the most concentrated case your total risk would still be the sum of the individual risks, i.e. would satisfy sub-additivity. In less correlated cases the total risk would be lower due to diversification between risks, so would also still satisfy sub-additivity.


*One might come up with reasons why risk would scale non-linearly with holding a larger position in a company. For example, it may become more and more difficult to liquidate your position. If you have a large number of shares compared to the number that are typically traded then you will move the market if you try to sell lots of your shares. In a scenario where you have to liquidate your portfolio you may incur more trading costs. There are other effects too. But I would argue these things change the underlying return distribution of each share you hold, so this example doesn't violate positive homogeneity or sub-additivity. That is, holding M >> N shares in a company, the return distribution of each share you hold changes from X (when holding N shares) to Y (when holding M). It probably is the case that risk(M * Y) > M * risk(X), but this isn't the same as saying risk(M * X) > M * risk(X). The point of positive homogeneity is only that risk scales linearly when the return distribution doesn't change.

This other post may or may not help: https://www.acted.co.uk/forums/index.php?threads/ch-8-return-on-capital-page-9.18955/