# Value at Risk - Ch.17: VAR and Risk Budgeting in Investment Management

#1

Reading Source: Textbook - Value at Risk

• VaR Applications to Investment Management
• Sell Side Vs Buy Side
• Investment Process
• Hedge Funds
• What are the Risks?
• Absolute and Relative Risks
• Policy Mix and Active Management Risk
• Funding Risk
• Risk Budgeting
• Budgeting across Asset Classes
• Budgeting across Active Managers
• Conclusions

#2

VaR can be used to set consistent guidelines. I am quite confused what the definition of consistent guidelines is? I guess VaR can provide a consistent manner to measure different types of risks, is that correct?

In addition, I am also confused about the calculation of potential surplus deficit in page 433 Jorion.

• Why it can be calculated as VaR less the ending surplus?
• VaR is defined as : VaR = Asset value * 2.33 * Surplus volatility, what does this VaR refer to?

Any help will be appreciated!

#3

Yes, using VaR as the risk measure for all risks just ensures you are using the same methodology for all risks. You can then compare the impacts of different risks to budget, rank the risks, etc. In other words, for budgeting purposes, you wouldnâ€™t want a bunch of different metrics being used for different risks because youâ€™d have a hard time determining which are the most severe risks.

For the calculation: I think the key formula to look at is the Rs = Î”S/A

The volatility of the Surplus return is 9% (given in article). Therefore, at the 99% level, VaR = 0.09 x 2.33 = 0.20 (as in paper -> 2.33 is not the exact z value). This is the VaR for the surplus return, or Rs. Therefore, the VaR for Î”S is VaR(Rs) x A = 0.20 x 1000M = 220M. So this 220M VaR amount now refers to the value of the Î”S at the worst 1% of event.

Now, the expected surplus is 150M, we know that at the 99% level, the change in surplus could change by 220M. Therefore, the worst 1% value of the surplus in a year would be 150M-220M=-70M

#4

BTW, does it mean there is no more 1% probability that Î”S will greater than 220M? if so, the ending surplus will be 100 - 220 = -120M, and the surplus deficit will be 150 - (-120) = 270M. I think less surplus is not favorable.

#5

BTW, does it mean there is no more 1% probability that Î”S will greater than 220M? if so, the ending surplus will be 100 - 220 = -120M, and the surplus deficit will be 150 - (-120) = 270M. I think less surplus is not favorable.

Iâ€™m not positive at what youâ€™re asking.

First step, do you agree the expected value of surplus is \$150M?
Second step, do you agree that the 99% VaR of the Î”S is \$220M?

Because we are dealing with the symmetric normal distribution, this means there is a 1% chance that the new value of the surplus is \$150M+\$220M=\$370M or greater AND there is also a 1% chance that the new value of the surplus is \$150M-\$220M=-\$70M or less.

Like you said, having less surplus is bad so the -\$70M is the tail we are concerning ourselves with when talking about risk (the â€śbadâ€ť tailâ€¦ not the tail where things go right)

#6

Thanks for you patience,

For the first step and second step, I completely agree with your statements.
I also understand the Î”S either will be +220M or -220M.

My confusion is the starting point that are used to calculate ending surplus.

• Since beginning surplus is 100 and change in surplus is 50, then we get the ending surplus is 150. We can say that expected ending surplus is calculated based the staring point from 100.
• The problem is the starting point about 220. In my opinion, this number is the maximum variation from beginning surplus 100 (the worst reduction on surplus), so we get the conclusion that there is no more than 1% likelihood the actual surplus will be lower
than -120(=100 - 220). Since our expected surplus is 150, the deficit in surplus is 150 - (-120) = 270. (I am not sure whether this is correct or not)
• I think the key question is the definition about the VaR number 220. l agree that this number is 99% VaR of the Î”S, but what starting point are we used to calculate the 220? Beginning surplus 100 or Ending surplus 150?

#7

Beginning surplus 100 or Ending surplus 150?

I think at the end of the day the calculation is intended to specify the surplus VaR at the end of one year. The sensitivity of surplus return is also specified to indicate the year-long sensitivity.
This would indicate to me that they are looking for the Surplus VaR position. From Chapter 5 of the textbook, there are two definitions of VaR:

1. Relative VaR = E(W)-W*
2. Absolute VaR = 0-W*

Where W* is the tail position of the surplus in one year.

What do you think the answers for the two respective definitions are?

#8