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# LTV (Life Time Valuation), Regression

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Question 1) Consider a direct marketing company where customers become members, pay periodical fees, and cancel membership at some time, e.g., a CD club. Assume the annual discount rate is 8%.
1. Suppose that the expected monthly cash flow from customers is \$15. Compute the expected number of months until canceling and LTV for retention rates of 80%, 85%, 90%, 95%, 97%, and 99%. Plot LTV against retention rate. Hint: This part is easy. See the attached spreadsheet (example1.xls) for a hint and starting point. Simply change the cash flow and discount rate cells, copy the cells down, and read off the LTV numbers for the discount rates.
2. Now suppose that the business is seasonal. Suppose that the expected monthly cash flow during Oct, Nov, and Dec is \$20 and \$10 for the other months. Suppose that the retention rate is constant across all months. Consider customers acquired as of June 1, so period 1 is June, 2 July, ..., 5 October, 6 November, 7 December, ..., 17 October, 18, November, .... Compute LTV for retention rates of 80%, 90%, and 95%.

Question 2. Consider the topic of business failures. See the attached data in BUSFAIL.xls and import into your favorite statistic software program (SPSS, MiniTab, etc.) The table gives data for each state on the number of failed businesses and the population in thousands from the Statistical Abstract of the United States.
1. Construct box plots or histograms of the two variables separately. Describe the distribution of each (e.g., are they normal, right skewed, left skewed, etc.?)
2. Construct a scatter plot (Graph / Plot in Minitab) of business failures (Y) against population (X). Describe the relationship.
3. Does the linear regression model appear to hold? Why or why not?
4. Compute the logarithm of each variable (logpop and logfail). In Minitab go into Calc / Calculator. Type logpop into the "Store result in variable box." Type LOGE(pop) in the "Expression" box. Do the same for failures. In SPSS use Transform / Compute, and the ln function.
5. Construct boxplots or histograms of the logged versions of the two variables separately. Describe the distribution of each (e.g., are they normal, right skewed, left skewed, etc.?)
6. Construct a scatterplot (Graph / Plot in Minitab) of log(business failures) (logfail) against log(population) (logpop). Describe the relationship.
7. Regress logfail (dependent variable) on logpop (predictor variable). Report the value of the regression equation.
8. Find the two-sided 95% confidence interval for the slope. Hint: take the "Coef" (slope estimate) plus or minus 1.96 times the standard error (SE Coef in the next column).
9. Test at the 5% level to see whether there is a significant relationship between the logs of failure and of population. Explain why this is reasonable. Hint: Use the P-value in the 5th column of the coefficient estimates. Be sure to state the null and alternative hypotheses.
10. Extra credit. If the slope for the logs were exactly 1, then business failures would be proportional to population. A value greater than 1 would say that large states have proportionately more failures, and a slope less than 1 would suggest that the smaller states have proportionately more failures. Explain why this interpretation is correct. Test at the 5% level to see whether the population slope for the logs is significantly different from 1 or not, and briefly discuss your conclusion. Hint: you have to take the number from the Minitab or SPSS output and compute the P-value "by hand."

See the attached Word document (ass4.doc) for a detailed decription of the problems. Attached also a spreadsheet for the first problem to help speed your work. (example1.xls) The dataset for problem 2 is in (BUSFAIL.xls).