Question

Stock ABC has monthly returns over the previous year given by the following: Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec 0.09 0.04 -0.02 -0.04 0.05 -0.03 -0.05 0.02 0.01 0.08 0.11 -0.03

1)Estimate the first two moment of the stock’s monthly return distribution

2)Suppose you want to estimate the probability of the stock having monthly returns below -0.06. In the data above, we don’t see any such months so would the probability of such an event be zero? Explain your answer for full credit

3) If we assume the stock’s monthly returns are normally distributed, we could use the method of moments to estimate parameter values and then plug those parameter values a computer program like excel in order to estimate the probability of returns being below -0.06. In Excel this can be done by typing in: NORM.DIST(x,µ,σ,TRUE) to get the value of the cumulative distribution function F(x) at any value “x.” Give the values of x, µ and σ that you would plug in.

4) Is your answer to part c) the population or sample probability? Briefly explain for full credit

Answer 1

1) For the given data, we find sample mean m= 0.01916667 (the first raw moment)

Variance=0.002757639( Second Central moment of data)

2) Although, there is no data value below -.06, but we can not say that the event probability is zero. Because getting monthly returns below -.06 is not impossible. However,  only for the given data we do not find any data value below -.06, which is possibly due to sampling fluctuation and for other data we might get data value less than -.06.

3) Here x=-.06, and the method of moments estimates are mu=0.01916667 (sample mean m) and sigma=0.05251323 (square root of variance)

Thus we compute the probability as  0.06583402 (using EXCEL)

4) The probabilty above is an estimate of the population probability. The sample estimate of the above probability is the proportion of data values less than -.06, which for this data is 0.

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