/m25 answer: You should investigate what follows *if* you assume independence. This is the central point of the problem.

It seems more reasonable to invest in cash for the last n-m years. Say you start investing in stock at 20 to withdraw at 65. At 60 you transfer all assets to cash. For the last five years c works on the higher rate, but shields against drops in the stock market due to high s.

If the opposite strategy is taken, and you start investing in cash at 20 then transfer to stock at 25, c works on a much smaller amount, and large s could make final w< initial w.

For problem 3, did everybody got the same results to the paper? I think my algorithm is correct, but my results are different.

/m28 answer: Yes, you will get a numerical answer that is slightly different from what the paper reports.

Unfortunately, this is very common when you try to replicate results in a research paper. It is impossible to know why, but one reasonable guess is that the authors made corrections to the data, and then forgot to rerun their algorithms..

Hyun Min Kang: I too had this problem at first. However it was solved by noticing that my result for UBAH was the true result of UCBAL. Make sure you understand what the two are doing and then implement accordingly.

/m27 answer: Shifting into cash towards the end of the period is the "conventional wisdom" in investing.

However, with the assumptions suggested in the exercise, the correct strategy is to invest in the stock marker always, unless the oracle tells you in advance that the outcome of the stock market will be bad.

Edited by author 01-17-2005 12:20 AM
/m30 answer: What I said in /m29 about getting slightly different results applies to the window-based algorithms later in the paper. I don't recall now whether or not my results for UBAH and UCBAL were exactly the same as in the paper.

/m30 My results are same to neither UBAH nor UCBAL. I'm afraid that I may misunderstand the data. DJIA dataset is 507 x 30 matrix. I see that sum(DJIA(1,:))=30.08 and sum(DJIA(507,:)) = 30.20, which means the change of market is very little. But the results in the paper says that the monetary returns of U-BAH is 0.76, which is very low. How can this happen? Am I wrong at some point?

Edited by author 01-17-2005 04:14 AM
For problem 4, I used the same assumption stated in the problem. In computing optimal value of m, I came up with pretty complicated differential equations where exp(m) and erf(\sqrt{m}) is involved. I think it is almost impossible to be solved by hand. Do you want me to solve this equation, or is there any other way to obatin optimal value of m?

/m33 answer: The notes in the assignment say "These are matrices of relative stock prices, called X in the paper." Read the paper carefully (top of the second page) to find out exactly what this means.

From /m8: "Part of the point of this exercise is to develop the ability to get what you need rapidly from a paper, while temporarily ignoring what you don't need--this is an important ability for a researcher."

/m34 answer: See /m24.

Most real-world problems do not have short algebraic solutions. The skill one develops with practice is how to make simplifications that enable one to obtain qualitative insights (and numerical solutions), while preserving the essence of the problem.

Here, you can simplify the scenario even further by assuming that the return from cash is zero. This does not change the problem qualitatively.

/31 in that case, why not simply ask for m=n, and pay less than the difference between which ever wins: stock or cash

/m37 answer: For any long period, it is almost certain that the stock market will go up over the entire length of the period. So, the answer from the oracle for m=n, where n is large, e.g. n=30, is very likely not to change the default policy (which is to invest in the stock market for all n years).

Over any short period (e.g. five years, m=5) there is a non-trivial probability that the market will go down. The answer from the oracle can let you avoid one of these negative short periods.

Followup to /m31: It is completely beyond the scope of this course, but if anyone wants a summary of what modern academic research implies about investment advice, see

"Portfolio Advice for a Multifactor World" by John H. Cochrane,
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=217489

Start reading with the "Conclusion" section on page 26.