Asset Allocation - Part I
Another Way
Traditional approaches to Asset Allocation insist that each stock can only be appreciated as it contributes to the whole portfolio, not by itself. Thus, a stock which may have unappealing aspects in its own right, may be quite useful in a larger context. For this reason, foreign investments often are heralded as essential to the complete portfolio.
But let's take another route. Consider one generic stock which we label ABC (think WIND) in isolation from your remaining portfolio. Assume that you have done your homework and think ABC is a great company, and you think you should invest heavily in the company. However, being a realist, you realize that ABC could tank for reasons that seem remote, but certainly possible. Because of this possibility, you realize that you must limit your financial exposure to ABC, but you don't know by how much.
Now, (here we go with the math) let M = initial dollar value of your portfolio. Let B = the amount you decide to invest in ABC. Time to get drastic. Assume that over a designated period of time your investment will either double or effectively go to zero, the worse possible outcome. Let the probability of ABC doubling in value during the designated period of time = P. Finally, during this same period, assume your remaining portfolio enjoys a rate of return = R. Finally, let U(x) denote your utility for your portfolio, as a function of its dollar value. (U defines how much more you value a portfolio of $10,000,000 versus a portfolio of $1,000,000, or one of size $100,000, for example.)
What is the optimum value for B? How does it relate to the size M of the initial portfolio? I hope you can see that answering these questions gives a nice approximation to the question you first asked me about how much of your portfolio to concentrate in WIND. Especially, since the only thing that would keep you from putting everything you have in WIND must be a deep-felt fear that something catastrophic might occur. After all, even the co-founders found it sensible to consider this possibility when they sold 30% of their WIND holdings in the secondary.
To answer the question, we need to rely on Utility Theory. It can be shown that a rational investor (including Warren Buffett, all traditional financial analysts, and even me) accept that it is always optimal to maximize the expected utility associated with any investment decision. There are no exceptions to this maximum; although, the foundation axioms that lead to this conclusion can be challenged if you want to get extremely esoteric.
We assume two possible outcomes: Either ABC doubles, with probability P, or it tanks, with probability 1-P. Consequently, the expected utility of the investment is given by the formula:
O(B) = P*U{(M-B)*(1+R)+2B} + (1-P)*U{(M-B)*(1+R)}
Accordingly, the optimum value for B depends, among other things, on the shape of the utility function that represents your value of your portfolio, at least within the range of your starting capital and all possible outcomes over the period in question. For our purposes, we will limit B to be strictly greater than zero and less than M; although, in general there is no requirement to be so restrictive. B greater than M implies that leverage is used to enhance the investment in ABC. B less than zero implies that the investor is short ABC, placing the borrowed proceeds in the remaining portfolio to obtain an improved return. Let's take some examples.
1. Linear Utility Function
Probably no one alive values money strictly in accordance with a linear utility function. Almost no one would value $10 billion as being worth 10 times $1 billion, for instance. But most people value $2 as being worth exactly twice the value of $1, meaning that most people's utility function for money is linear for small amounts of money. This means that a high income person that has yet to establish a sizable portfolio may well have a linear utility for money, at least for a few years. As a rough rule of thumb, any portfolio not exceeding about 50% of your annual cost of lifestyle probably is valued accurately with a linear utility function for money.
In such a case, the optimum value for B is either M or zero, depending on whether or not 2P exceeds 1+R. Simply stated, if the probably of doubling your money with ABC is sufficiently high, then mortgage the house and invest everything in ABC; else invest nothing. This aggravating bang-bang optimal control is characteristic of linear objective functions, and is probably one of the primary motivations for complicating decision making with Utility Theory. Nevertheless, it is true, and is why I suggested previously that someone with small capital relative to annual lifestyle pick a couple of good stocks and go for broke. That is the theoretical optimal choice.
2. Logarithm Utility Function
Logarithms are monotonically increasing for positive numbers at a decelerating rate, making them interesting candidates for utility functions. The problem is they decelerate too rapid to suit most people. If $1,000 is valued at 3, and $10,000 is valued at 4, then valuing $100,000 at 5 and $1,000,000 at 6 is inadequate. $1 million is worth more than twice $1,000.
However, logarithms work well for super wealthy people, in the range of their investment decisions. These are people with investment portfolios around 50 to 100 times their annual lifestyle expenses. For example, it may be entirely reasonable to scale $100 million, $1 billion and $10 billion as 8, 9 and 10 in relative utility. For this reason, it is useful to solve the Logarithm case.
To find the solution, you must substitute Log for U in the above equation, determine the first derivative, and set it to zero. (Check to make sure the inflection point is maximum by checking that the 2nd derivative is negative.) When done, the equation for B is:
B = M*[2P - (1+R)]/(1-R), where 2P > 1+R
In other words, as with the linear case, invest in ABC if 2P exceeds 1+R. The proportion of M to be invested is independent of M, and always equals [2P - (1+R)]/(1-R). Let's take a simple example to illustrate. Suppose your remaining portfolio is expected to return 7% annually after taxes. Assume you believe an investment in ABC will double in three years with 90% probability (or tank). Then R = 1.07^3 - 1, or .225, so exactly 74.2% of your portfolio that should be invested in ABC, irrespective of how rich you are, as long as you are very rich. This is why I indicated before that a rich person may well be heavily weighted in WIND.
3. Quadratic Utility Function
If your investment portfolio is significant, but not gigantic, then neither the linear nor the logarithmic Utility Function will meet your needs. Since all smooth functions can be reasonably approximated by piece-wise quadratic equations, let's consider the quadratic Utility Function.
Taking the derivatives of a quadratic function in the form X - A*X^2 (with A > 0) substituted for U( ) above, is laborious but straightforward. Solving for zero, gives the following solution for B:
B = [2P - 2P*A*M*(1+R)*(1-R) - (1-R) + 2A*M*(1-P)*(1+R)^2] /
[-2P*A*(1+R)*(1-R) + 4P*A*(1-R) + 2A*(1-P)*(1+R)^2]
Unlike either the linear or the logarithmic Utility Functions, the optimal investment value for the quadratic Utility Function depends on M as well all the usual factors, including the deceleration parameter, A. The result is ugly and extremely sensitive to the combination of all the parameters. In addition, you must be careful to make sure that the quadratic function meets your needs appropriately within the entire range of possible dollar amounts to be considered. For practical work, you should consider reverting to numerical techniques at this point, and not worry about a closed-form solution, which is a bit old fashioned anyway. A trivial goal-seeking optimization program on a PC could handle any Utility Function calibrated closely to meet anyone's peculiarities, and would probably exhibit much more stability than can be expected from stretching pieces of quadratic functions.
Nevertheless, the quadratic Utility Function works, and give answers like "invest 37.4% in ABC", when P = .86, A = 0.00000004, R = 15% per annum for 3 years, and M = $5,000,000. (The value of A assumed is calibrated to work well in the $5,000,000 to $12,000,000 million range.) This is why I indicated in my earlier post that the optimum percentage of medium-size portfolios that should be invested in WIND might vary considerably from the optimum percentage of either very small portfolios or very large portfolios.
Allen |
