The daily returns on a portfolio are normally distributed wi

The daily returns on a portfolio are normally distributed with a mean of 0.001 and a standard deviation of 0.004. a. What is the probability that the daily returns on that portfolio are negative for at least 50 out of the next 110 days? (10 points) b. What is the probability that the average return for the portfolio over the next 110 days is less than 0.001? (5 points) c. What is the probability that the average return for the portfolio over the next 50 days is greater than 0.001? (5 points) d. What is the probability that the average return for the portfolio over the next 50 days is negative? (10 points) e. What is the probability that a daily return would be positive? (8 points)

Solution

a. What is the probability that the daily returns on that portfolio are negative for at least 50 out of the next 110 days? (10 points)

The probability that the daily returns on that portfolio are negative is

P(X<0) = P((X-mean)/s <(0-0.001)/0.004)

=P(Z<-0.25) = 0.4013 (from standard normal table)

Then X follows Binomial distribution with n=110 and p=0.4013

P(X=x)=110Cx*(0.4013^x)*((1-0.4013)^(110-x))

So the probability at least 50 out of the next 110 days is

P(X>=50) = P(X=50)+P(X=51)+...+P(X=110)

=110C50*(0.4013^50)*((1-0.4013)^(110-50))+

110C51*(0.4013^51)*((1-0.4013)^(110-51))+...+

110C110*(0.4013^110)*((1-0.4013)^(110-110))

=0.1487909

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b. What is the probability that the average return for the portfolio over the next 110 days is less than 0.001? (5 points)

P(xbar<0.001) = P((xbar-mean)/(s/vn) <(0.001-0.001)/(0.004/sqrt(110)))

=P(Z<0) = 0.5 (from standard normal table)

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c. What is the probability that the average return for the portfolio over the next 50 days is greater than 0.001? (5 points)

P(xbar>0.001) = P(Z>(0.001-0.001)/(0.004/sqrt(50)))

=P(Z>0) = 0.5 (from standard normal table)

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d. What is the probability that the average return for the portfolio over the next 50 days is negative? (10 points)

P(xbar<0) = P(Z<(0-0.001)/(0.004/sqrt(50)))

=P(Z<-1.77) = 0.0384 (from standard normal table)

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e. What is the probability that a daily return would be positive? (8 points)

P(X>0) = P(Z>(0-0.001)/0.004)

=P(Z>-0.25) = 0.5987 (from standard normal table)