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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.

/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.

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

/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.

/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."

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?

/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 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.

/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.

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.

/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..

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

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.

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

In probem #2. Can we assume anything about the independence of the ages of the husband and wife?

/m21 answer: An algebraic expression is ideal, but failing that, an expression to be maximized numerically is almost as useful in practice.

In both cases, one or two figures to show qualitatively how m depends on c, d, s^2, and n would be informative.

/m20 answer: Yes, if we are risk-neutral and we don't have the oracle, we will always invest in the stock market. However, the answer from the oracle may cause us to invest in cash for the first m years. We will always invest in the stock market for the last n-m years.

The oracle tells us the outcome of the random variable SUM xi for i=1 to i=m. Knowing the actual outcome is better than knowing its expectation!

/19 answer: I just mean that every y value is a member of the set {0,1}. y can be dependent on x. This means that p(y|x1) =/= p(y|x2) in general.

In problem 4, are we expected to generate an algebraic expression for m where the gain is maximized? That is, something of the form, gain is maximized at m = F(c,d,s), or is it sufficient to give an function, F(c,d,s,m), which should be maximized (presumably numerically) with respect to m?

I'm sorry, but I guess I'm still missing something after your clarification of Problem 4. If we only look at expected values, don't we always expect the stock market outperform the cash investment, since d>c. If we don't care about risk, why wouldn't we always invest in the stock market?

I guess I'm just having trouble formulating the problem so that we can find a optima for m that will correlate to maximium expected wealth or something to that effect.

For problem 1c, do you mean that we assume p(y|x) = k for y=1 and (1-k) for y=0? or is it something more complicated so that y can be 0 or 1, but dependent on x also?

For problem 4, just pay attention to the expected value of your wealth under alternative policies. That is, do not pay attention to risk. From this point of view, a 10% chance of $1000 is worth the same as a 50% chance of $200: you are neither risk-loving (like a gambler) nor risk-averse (like most real investors).

The value of the information is the expected extra profit that the information allows.

On problem 4, I can see how we can get to a point where the value of investing in the stock market will be above the value of a cash investment with a certain probability.

Selecting m will change that probability. Namely, we can find an m corresponding to the cash investment being k standard deviations from the mean of the stock investment. The question is how many standard deviations, or equivalently, what probability of being better than the cash investment is desired?

This seems like it is dependent on how much you are willing to "gamble". Is it OK to leave the selection of m as a function of k standard deviations or probability?

Also, for part b, I'm a bit confused as to what makes a good price for the oracle. I initially thought that paying something less than the difference between the cash and expected stock market returns would be good, but that doesn't seem ok, since a large percentage of the time, you may make more than the cash but less than the expected value of stock market, resulting in a loss.

Can you say something about how to go about the selection of the appropriate cost, given the method for selecting m I described above. Thanks.

Someone asked me whether instead of Matlab you can use Octave, an open-source alternative. The answer is yes, but I recommend Matlab.

Matlab is expensive, but a full student version is available at the UCSD bookstore for only $100.

Compared to Octave, Matlab is more comprehensive, and more widely used by other researchers. Matlab can increase research your productivity by more than it costs.

I am generally favorable to the open-source philosophy, but I am not a fan of open-source projects that mostly just aim to copy the functionality of existing software products. Doing that seems like a waste of human talent. Innovation should be rewarded and extended rather than copied.

Assume that the age difference can be negative, i.e. do not take the absolute value. As you say, this would make the problem much harder. Also, it would be scientifically less interesting, if this was a real-world scenario.

/m12 Thanks for your answer. I have another question on problem 2. Did you mean the age difference can be negative? Is it x-y or |x-y|? By definition, it seems to be latter one, but it makes the problem much complicated.

I've received a lot of feedback saying it would be helpful to extend the deadline, so we'll extend it by one week, to Tuesday January 18. Please use the extra time to review basic statistics and probability!

/m11 answer: You are right that when you generate simulated data, the std. devs will not be exactly 4.0 and 3.0. That's ok. Every simulation only captures some aspects of the real-world phenomenon being studied. It's a judgment call whether the important aspects are being captured. The same is true for every mathematical model: it never captures all aspects of the real world.

Use real-valued ages for mathematical simplicity. Forcing ages to be integers would be an unnecessary layer of complexity.

Edited by author 01-06-2005 07:59 PM
For the simulation in problem 2, you told me that assuming Gaussian would be sufficient. I have a few more questions on this.

1. The problem says that the standard deviation of the husbands' and wives' ages are four and three, respectively. Even if I assume a Gaussian distribution with the stdev, the stdev of n sampled data might not be exactly 4.0 and 3.0. Would it be okay? I guess so, but it looks like not okay by the statement itself in the problem. "In a sample of n married couples, the standard deviation of the husbands' ages is four years, and the standard deviation of the wives' ages is three years."

2. The ages are integers usually. Do you want us to use integer value of ages, or just a real value?

Thanks,

Hyun

I have made a couple of small corrections to the notes about Question 4--nothing major.

/m8 Thanks for the response. I confused the question because I misread Table 1 as Figure 1 in the homework sheet, so I thought we needed to implement Anticor in order to replicate the Figure. Just implementing UBAH and CBAL is easy.

Edited by author 01-06-2005 12:00 AM
/m5 answer: The obvious implementation of each algorithm has two nested "for" loops: one to iterate over days, and one to iterate over stocks. It is possible to eliminate both these loops, and to get a one-line implementation of each algorithm.

You don't need standard deviations (the Matlb "std" function) here. The algorithms are very simple!

The point of the paper is algorithms that use window sizes, but UBAH and CBAL do not. 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.

/m4 answer: No, there is no typo. It is simpler to assume that x_i has a Gaussian distribution, rather than r_i. The reason is that the sum of Gaussian random variables is itself Gaussian. This fact makes the problem solvable.

/m3 answer: P is a probability distribution; it is a bivariate probability density function to be precise. x and y are real-valued random variables.

Regarding Problem 1.3, what sort of times should we be shooting for? The way I see it, you need to have one loop to vary the window size, and one to step through t. Is this correct? I've been thinking about how to remove stepping through t, but I can't see how. The timing I'm getting doesn't seem to be too fast, which leads me to believe I'm missing something.

These are the only loops I'm using and all other calculations are done through matrix multiplications and array multiplication as well as matlab commands like mean and std.

In question 4, bullet 2, is there a typo? I think it's not the case that x_i ~ N(d, s^2), but instead r_i ~ N(d, s^2).

In problem 1, what does "P" mean in "(x,y) ~ P." ? Is "P" a distribution or does the notation simply mean that x and y are probabilities in [0,1]?

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Please ask questions here about Assignment 1 in CSE 291, "Statistical Learning."