Allen: Modifications to Your Asset Allocation Model - Premises
I read with great interest your mini-dissertation on asset allocation models and your approach in determining what proportion of one's portfolio should be invested in a stock such as WIND. I agree with you whole-heartedly on a number of things but feel that your model is too simplistic and could yield results that are not practical and go against natural common sense (I think you recognize this yourself in one of your examples where you "adjust" the results for this). After giving it a lot of thought, I have created another model which is largely based on your approach but extends and refines it (in my not-so-humble opinion). The results I get are quite different to yours. I would very much appreciate your taking the time to criticize it as appropriate.
I apologize at the outset for the detailed nature of this note, which some of the others may find difficult to follow.
A. Where I agree with you
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1. Your disdain for conventional asset allocation models. In my profession, I have come across many asset allocation models for pension fund portfolios. They are all based on forward simulations of asset returns for different classes of investments. The drivers behind these simulations are past expected returns and variances on these returns, for each asset class. I have always questioned the validity of using past experience in making forward projections in an economy and market place that is changing constantly. Even if there is any validity in all this for a large pension fund, the approach would certainly not be valid for an individual investor with a very limited number of stocks and other classes of investments.
2. Using utility functions is more appropriate than pure portfolio size.
3. Your "Gambler's Ruin" approach is intuitive, easy to use and rather compelling.
B. What I see as the shortcomings to your model.
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1. Selecting the most appropriate utility function for an individual investor is of course a very crucial part of the exercise and has the greatest impact on the final results. I have no disagreements over your utility function for the "logarithmic" end of the richness scale. Most of us will never reach there and I have not given it much thought. However, I would not feel comfortable with using a linear 1-to-1 function (where delta(U(Z)) = delta(Z)) where Z= portfolio size) even at the lower end of the wealth spectrum. If other investors are like me, the utility function for portfolio sizes less than I, where I = initial portfolio size, will be different in shape to the utility function that governs when Z>I, reflecting an aversion to asset loss that would be greater than the desire for asset gains.
2. In your straight application of the Gambler's Ruin approach, there are only two results: win (in which case the value of WIND holdings doubles over a period of time, or loose (in which case WIND tanks completely). You have assigned a probability of 90% for winning and 10% for loosing. The advantage of this win/loose approach is of course that the algebra is simple and you are able derive results with a paper and pencil. However, the following criticisms can be leveled on it:
a. No sensible investor is going to hold on to an investment that is tanking until the value of the investment is zero.
b. With a stock like WIND, there is a definite possibility that the stock will more than double in value over the period of time. The simple win/loose approach cannot take this into account in a useful manner.
c. Whether he/she realizes it or not, an investor is intuitively making decisions based on certain criteria. (e.g. "At no event do I want to end up with a portfolio that is less than 50% of what I currently have AND I want there to be only a very small chance of my portfolio reducing in value over the period"). The simple win/loose model is incapable of testing such criteria.
d. Most investors are likely to have more than one favorite stock (each with its own return/probability function), which they would want to factor into the equation. The maths of course becomes a lot more tedious, if not impossible.
3. If the model is changed to overcome some of the shortcomings mentioned above, a paper and pencil method of arriving at results will no longer be possible. Simulations are required.
C. Modifications to your model
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I have made the following changes to your model in an effort to overcome the shortcomings mentioned above.
1. New utility function.
Z = size of portfolio
I = Initial portfolio size
U(Z) is a quadratic of the form AZ^2 + BZ + C, where:
when Z=<I, A= -2/I, B=5, C=-2I - a quadratic whose value is I when Z=I, whose value is 0 when Z=I/2, and whose slope is 1 when Z= I.
when Z>I, I have basically used your quadratic formula: A=-0.00000004, B=1, C=A x I^2
2. A new return/probability function for stock W (WIND).
Let R be the return over the period of time in question, say 3 to 4 years. That is, an initial investment of IW will be worth IW x (1+R). Let Y be a random variable between 0 and 1, as generated by any common random number generator. Then let R= 5Y-1.
What this function is saying is that there is an equal 10% chance that any of the following may happen:
R is between -1 and -.5
R is between -.5 and 0
R is between 0 and .5
----------and so on-----
R is between 3.5 and 4
I will leave it you and others more knowledgeable on the subject to debate whether the above represents a realistic return/probability function, but suffice it to say that it represents a much more bullish assessment of WIND than your win/loose function. As others have said, garbage in = garbage out, and we must be careful in selecting the return/probability function.
3. Bring in another favorite stock.- Alternate A
Let A have a different return probability function, totally independent of the return on W, of R = 1.2Y-.2. This is a stock which can at most double during the period, but under the worst circumstances, will retain 80% of its value.
4. Let B be the balance of the portfolio, which is expected to return 25% with 100% certainty.
5. Make the following allocation criteria:
Maximize total portfolio return subject to:
a. 0% chance that my utility is 0 (equivalent to saying that in no event will my portfolio be less than 50% of I).
b. The chance that my portfolio ends up less than I is less than 10%.
6. Run simulations using the above on different initial proportions of W, A, and B and select the proportion mix that best meets the above criteria. Each proportion tested should have at least 500 simulation trials, so that the various confidence limits can be established. All of this is quite simple to do in an Excel spreadsheet. Please note that independent random numbers must be generated for W and A because we are assuming that W and A are companies in totally different markets with the return on one not affecting the return on the other. In practice of course, both will be affected by the general direction of the overall market but we won't worry about that here.
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In my next message, I will display the results of the above modified model.
Erwin |
