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In: Statistics and Probability

You are considering two investments. Let X represent the proportional rate of return on the first...

You are considering two investments. Let X represent the proportional rate of return on the first investment, and let Y represent the proportional rate of return on the second investment. These are annual rates of return.

X is approximately normally distributed with mean 0.25 and standard deviation 0.2. Y is approximately normally distributed with mean 0.30 and standard deviation 0.4.

The first six questions are about the rates of return, X and Y.

What is the probability of a negative rate of return on the first investment?  

What is the probability of a negative rate of return on the second investment?  

If the rates of return on these investments are independent, what is the probability that the rates of return on both investments will be negative?

What is the expected amount by which Y exceeds X? HINT: The amount by which the rate of return on the second investment is higher than the rate of return on the first investment is Y - X.   

If the rates of return on these investments are independent, what is the probability that the second investment will have a higher rate of return than the first? HINT: Restate the question in terms of the rate of return on the second investment minus the rate of return on the first investment.  

If instead X and Y have a correlation of – 0.4 (a negative correlation), what is the probability that the second investment will have a higher rate of return than the first? HINT: Be careful! You’re given the correlation, not the covariance!  

The remaining questions are about the value of your portfolio after one year (the dollar amount you end up with), taking into account both the amount you invest and the yield, which depends on the rate of return. For example, if you invest $100, and the proportional rate of return is X, the yield is $100X, and you end up with $100 + $100X.

You have $100 to invest. Suppose you invest $100 in the first investment. What is the expected dollar amount you will end up with? (What is the expected value of your portfolio?)

What is the standard deviation of the dollar amount you will end up with?

What is the probability you will end up with less than $80?    

Alternatively suppose you invest $100 in the second investment. What is the probability you will end up with less than $80?  

Suppose you invest $30 in the first investment and $70 in the second investment. What is the expected dollar amount you will end up with?  

If X and Y are independent, what is the standard deviation of the dollar amount you will end up with?  

What is the probability you will end up with less than $80?  

Suppose instead that X and Y have a correlation of – 0.4 (a negative correlation). You invest $30 in the first investment and $70 in the second investment. What is the standard deviation of the dollar amount you will end up with?  

What is the probability you will end up with less than $80?  

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orchestra answered 2 years ago
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