Randolph and William Leverage run their own private investment pool which has grown out of the proceeds of a trust established by their grandfather Horace Leverage. Horace Leverage was known as a high roller during the wild years of the 1920's and 1930's. He regaled his grandchildren with tales of borrowing money from a friend and investing in 1932 in Anaconda at $ 3, General Motors at $2, General Foods at $4 and so on. The result of these shrewd investments is that the current value of the grandfather Leverage's legacy is $ 50 million dollars. Now that grandfather is gone, William, 27 and Randolph, 32 are the sole, equal partners in Leverage Brothers or (L Bro). Both are married, both have one child (Randolph Jr. and William Jr. respectively). Both figure that their annual living expenses, what with private schools and vacations and so on, amount to $750 thousand each. Of course, this is after taxes. The difference between the two is that Randolph went to business school to learn the state-of-the-art investment theory, while William went to the school of hard (sic) knocks.

Randolph is particularly interested in the CAPM model, that recommends holding the market portfolio and leveraging up or down to do so. He figures that Leverage Brothers is less risk averse than the average investor, and so they should borrow money and invest it in the market. Of course, L Bro can't borrow at the risk-free rate, but they can borrow at prime, which currently stands at 9.5 %.

Randolph and William have one additional problem: their brother Charles. Charles is the black sheep of the family. Against grandfather's wishes, Charles shunned the investment business, went to medical school, then volunteered for the Peace Corps. Although Horace cut Charles out of his will, he has instigated a suit, claiming $10 million. Charles claims that he needs the money to finance an AIDs research facility in Zaire. L Bro's lawyers admit that the suit may have some merit, although it will take 5 years before any judgement. Just in case, however, they advise L Bro to make provisions to be prepared to pay the $10 million in 5 years' time.

Randolph finds that William rarely listens to his theoretical arguments about the wisdom of buying the market portfolio. Consequently, he decides to illustrate an investment policy by simulation. He wants to show the following: suppose L Bro borrows 50 million at 9.5 % (assuming a 5 year note at 9.5 %) and invests in the S&P 500. After taxes and expenses and debt service, what will be the chance that, after 5 years they will have less than $40, $30, 20$ or $10 million? He also wishes to see what the effect of a $10 million loss 5 years' hence will be upon their wealth 10 years out. Assume the 5 year note may be rolled over at 9.5 % in five years.

Prepare Randolph's study, under the assumption that the stock market follows a random walk. Thus, you may create a pseudo-history by randomly drawing (with replacement) from the returns over the 1926-1989 period. Assume dividends are re-invested, and that the current tax rates remain unchanged (ignore city, state & property taxes). Assume interest rates are fixed. Assume living expenses grow at 5% per year. Perform 100 simulations.

Write a report that uses simulations of stock performance and draws inferences from the distribution of returns. Include the following analysis:

1) Show the distribution of L Bro's wealth 5 years hence, without the effect of Charles' lawsuit. Is the distribution skewed? What is the median return? Is the standard deviation a useful measure of risk to the Leverage Brothers? Plot a 90% confidence band that shows the bounds within which the L Bro's wealth is expected to lie in each of the next 5 years. Show the distribution of wealth 10 years hence, assuming that Charles' lawsuit succeeds.

2) Discuss the risk of loss. Is there a chance that L Bro will lose the family fortune? What is the probability that they will not make enough to meet their living expenses and debt service in any of the first five years?

Assume that L Bro does not leverage their portfolio, and that the families agree to consume $500,000 each or half of each year's return, whichever is a higher amount. Answer 1 and 2 under these assumptions.

3) Simulate some policies of your own invention and compare them to the preceding strategies using stochastic dominance methods. Can you find a policy that dominates?

Conclude with some recommendations for L Bro, or some observations on the results of your study.