Calculating Probability of Expected Return for Stock Portfolio

[anjunabeats]

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

 

[CompuChip]

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).

 

[anjunabeats]

Yeah I am not sure how to find the mean and standard deviation.

 

[CompuChip]

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)

 

[anjunabeats]

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)

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?

 

[anjunabeats]

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?

 

[CompuChip]

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?

 

[anjunabeats]

found out we can find the sd using

root (sd1/number of stocks + sd2/number of stocks)

 

[CompuChip]

Except that the sd1 and sd2 in that formula should be squared.
And that, too, is exactly what I told you ;)

 

[anjunabeats]

Well i have my stats exam tommorow thanks for the help compuchip, really appreciated ciao.