PRM I – probability

[Rassim]

Hello!
Please tell me how can I solve the following problem:
Well suppose we have portfolio A and B. port.A has an expected return 10% and stdev=20%, whereas port. B has an expected return 12% and stdev=15%. Correlation is -0.33. What is the probability that porfolio weighted of 50%A+50%B will outperform 60%A+40%B portfolio?

Please if you know how it should be solved explain it to me. If I need to compute z-value for getting the probability, how can I do it with Exam calculator or without calculators by hand?

[Anonymous]

So we have the following
A B
mean 0.1 0.12
std dev 0.2 0.15
Correlation AB = -0.33

What we want is P(0.5A + 0.5B > 0.6A +0.4B) which reduces to
P(0.5A +0.5B – 0.6A – 0.4B)>0 which futher reduces to
P(0.1B – 0.1A)>0

Because the difference 0.1B – 0.1A is a linear combination of a random variable,it is itself normally distributed.

Its mean is therefore E(0.1B -0.1A) = 0.1*0.12 – 0.1*0.1 = -0.002

Its standard dev is SQRT((0.1^2*0.15^2 + 0.1^2*0.2^2 -2(0.1)(0.1)(0.2)(0.15)(-0.33))=0.028687976

The probability we are calculating is therefore P(Z> 0-(-0.002)/0.028687976) which reduces to

P(Z>0.0697156)= 47.21%

I AM NOT SURE WHAT HAPPENS WITH THE Z VALUE IN THE EXAM. MAYBE SOMEONE WHO HAS SAT FOR THE EXAM WILL KNOW IF THEY DO GIVE A RANGE OF Z VALUES TO CHOOSE FROM.

[Author]

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