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Statistics question :/

  1. Sep 23, 2009 #1
    Really stuck on this question.

    Stock A has an expected return mean of 0.03 and standard deviation of 0.02
    Stock B has an expected return mean of 0.02 and standard deviation of 0.01
    Investor invests in 20 lots of stock A and 15 lots of Stock B (as in 4/7 in A and 3/7 in B)
    What is the probability that the portfolio will have an expected return of > 0?

    Im guessing you need to find the pooled mean and sd then use z score = X - mu / sd but I'm really not sure, hope somebody is willing to help :0
     
  2. jcsd
  3. Sep 23, 2009 #2

    CompuChip

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    So his total portfolio has a return distributed as Z = 20X + 15Y, where X and Y are the returns of stock A and B respectively. Since X and Y are normally distributed, so is Z. Therefore, as you say, start by finding the mean E(Z) and standard deviation σ(Z) of Z and calculate P(Z > 0).
     
  4. Sep 23, 2009 #3
    Yeah im not sure how to find the mean and standard deviation.
     
  5. Sep 23, 2009 #4

    CompuChip

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    These are standard formulas, that you are probably supposed to know :)

    For two normally distributed variables X and Y,
    E(X + Y) = E(X) + E(Y)
    Var(X + Y) = Var(X) + Var(Y)

    There are straightforward generalisations to n variables. A particular version is that for a normally distributed variable X and integer n,
    E(nX) = a E(X)
    Var(nX) = n Var(X)
     
  6. Sep 23, 2009 #5
    So for E(X + Y) = E(X) + E(Y)
    E (X + Y) = 4/7 (0.03) + 3/7 (0.02) = 9/ 350 = 0.025714

    and for Var(X + Y) = Var(X) + Var(Y)

    Var (X + Y) = 0.02^2 + 0.01^2 = 1/2000
    Standard deviation = 0.0223606

    We are finding P (X > 0)

    then for z = X - Mu/ sd
    = 0 - 0.025714 / 0.0223606
    = -1.149969

    0.0668 + 0.5 = 56.68% chance that return > 0?
    Does this look okay Compuchip?
     
    Last edited: Sep 24, 2009
  7. Sep 24, 2009 #6
    I was searching on the internet and just found that Var(X + Y) = Var(X) + Var(Y) + 2COV(X,Y) therefore the above is most likely wrong.

    How would i find the covariance of stocks A and B? Is there a quick way?
     
  8. Sep 24, 2009 #7

    CompuChip

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    Yes, noticing that both variables are statistically independent, for example :P

    Also, shouldn't you include the 4/7 and 3/7 in the variance? You don't want Var(X + Y), but Var(4/7 X + 3/7 Y), don't you?
     
  9. Sep 24, 2009 #8
    found out we can find the sd using

    root (sd1/number of stocks + sd2/number of stocks)
     
  10. Sep 24, 2009 #9

    CompuChip

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    Except that the sd1 and sd2 in that formula should be squared.
    And that, too, is exactly what I told you ;)
     
  11. Sep 24, 2009 #10
    Well i have my stats exam tommorow thanks for the help compuchip, really appreciated ciao.
     
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