Sept 2014 Exam - Q 6

Discussion in 'SP9' started by Viki2010, Feb 21, 2015.

  1. Viki2010

    Viki2010 Member

    Question 1.

    The ASET solution uses the following formula for calculating VaR:

    VaR at 95% = number of standard deviations from the mean for 99.5% distribution * the size of the standard deviation for that distribution * times total claims costs


    How is it derived mathematically?


    I am trying to relate the ASET approach for VaR calculation to the mathematical formula given for VaR in the Act Ed notes e.g.

    VaR = F ^ (-1) (L) and for normal distribution
    VaR = average + std. dev * Invese Distribution

    Question 2

    Where in the Act Ed notes or the books can I find reference to the formula for Total Capital that the ASET is referring to? - the square root of the sum of squares?

    Question 3

    If two risks are 100% correlated, we can simply add (sum) their VaR. Where in the course do we learn that?
     
    Last edited by a moderator: Feb 21, 2015
    Viki2010, Feb 21, 2015
    #1
  2. Edwin

    Edwin Member

    Q1) I don’t have ASET since I bought one last year and they don’t sell the SEP14 one in isolation, so sorry can’t help here.

    Q2) Here, take a pen and a piece of paper or a keyboard and an excel spreadsheet. Since the risks are independent it means you have a correlation matrix with a diagonal of 1’s and zero’s everywhere;-
    Suppose you computed capital for two risks A and B to be 3 and 4 and your represent this in a vector;- [3 4], and that these are independent so you have;-
    [1 0]
    [0 1]

    to calculate capital you have;-
    [3 4]*[1 0]*[3]
    .........[0 1] [4]

    = sqrt(25), this is the sqrt of the sum of squares;- sqrt(3^2+4^2).

    Q3) Now suppose the risks are 100% correlated and replace the matrix above with one with 1’s everywhere, you get sqrt(49) = 7, this is just the sum of 3+4 or the sum of your initial for A and B.
     
    Last edited by a moderator: Feb 22, 2015
    Edwin, Feb 22, 2015
    #2
  3. Viki2010

    Viki2010 Member

    Great, thanks!

    For Q1, I guess there are different approaches to calculations of VaR. ASET shows a different method than the actual Exam Report, but both methods result in similar answers.
     
    Last edited by a moderator: Feb 22, 2015
    Viki2010, Feb 22, 2015
    #3
  4. Simon James

    Simon James ActEd Tutor Staff Member

    Q1

    The basic formula for VaR is \( \sigma \Phi ^{-1}\left ( \alpha \right ) \)

    We are given the standard deviation \( \sigma \) as a % of the mean

    We are also told that the 99.5th percentile (ie the equivalent of the \( \Phi ^{-1}\left ( \alpha \right ) \) for whatever our distribution is) is a given multiple of the mean.

    The approach we used in ASET was to then combine these with our estimate of the mean to give the VaR.

    Q2 & 3

    These results come from the covariance formula:
    \(\sigma _{x+y}^{2} = \sigma _{x}^{2} + \sigma _{y}^{2} + 2\rho \sigma _{x}\sigma _{y}\)

    If x and y are independent then \( \rho=0\) and the joint s.d. is the square root of sum of the squares of the individual s.d.

    Note that this is only an approximation as it fails to weight the portfolios (ie it assumes x and y are of equal size)

    If x and y are 100% correlated \( \rho=1\) and \(\sigma _{x+y}^{2} = (\sigma _{x} + \sigma _{y})^{2} \) and \(\sigma _{x+y} = \sigma _{x} + \sigma _{y} \) so the VaRs are simply added
     
    Simon James, Feb 23, 2015
    #4
  5. Viki2010

    Viki2010 Member

    Hello Simon, thank you for your explanation.


    VaR can be calculated as

    miu + std dev * inverse distr

    or

    std dev * invesrse distr.



    these approaches are fully interchangable?
     
    Viki2010, Feb 27, 2015
    #5
  6. Simon James

    Simon James ActEd Tutor Staff Member

    A more formal statement of VaR for losses that are \( N(\mu ,\sigma ^{2}) \) is \( \mu + \sigma \Phi ^{-1}\left ( \alpha \right ) \)

    For short durations it is common to ignore the \( \mu \) (ie a zero mean)

    When communicating or interpreting VaR it is necessary to be careful to know if the VaR is being expressed as a loss relative to the current or some expected position.
     
    Simon James, Feb 28, 2015
    #6

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