Asset Allocation - Part II
Case Studies
The following case studies are fictional except for the last one, which you will recognize as one that was much discussed one year ago on this thread. Each case has been constructed to illustrate optimal investment strategies for various situations.
Case 1 - The High Roller
After numerous divorces and failed business ventures, Sue has never been able to accumulate a sizable portfolio of financial investments. Even though Sue is a heavy-hitting lawyer, making $500K, her portfolio is a paltry (for her) $50K. Her stockbroker suggests that, for a change, she diversify her $50K portfolio over a basket of large-Cap stocks that have low betas and pay reasonable dividends. Meanwhile, Sue discovered WIND, and believes that the stock will double in three years with a 90% probability. When pressed, Sue can't defend against the possibility that WIND could tank for reasons like Microsoft waking up, or INTS pRISM+ actually working and cutting into sales, or I2O failing to become a I/O standard. Sue hasn't bothered, nor does she have the time, to conjure up another stock that he likes as much a WIND.
How much of Sue's portfolio should be allocated to WIND? Answer: Sue's utility for money in the range of her small portfolio is linear. Therefore, the most rational thing Sue can do is to invest 100% of her portfolio in the best company she can find: WIND.
Case 2 - The Senior Citizen
Sam is 85 years old and has adapted to living on $20,000 per year, while still saving money from a combination of defined benefits retirement programs amounting to almost $3,000 per month. The surprise is that Sam has a portfolio worth $1,000,000. While Sam still prefers the predictable flow of income from CD's and the Money Market, all of which left an indelible impression on him throughout most of the 80's, he has long since recognized the importance of investing in the highest growth component of the economy -- high tech. The question is, since he only knows about WIND, how much of his portfolio should he allow WIND to comprise?
Since Sam's portfolio is 50 times greater than his annual lifestyle spending, which is protected anyway by a defined flow of retirement funding, he qualifies nicely for a logarithmic Utility Function, as long as we make sure the range of practical outcomes remain substantial relative to his lifestyle. Consequently, if CD's are returning an after tax 4%, for example, and Sam believes WIND should double every two years, and therefore will double every three years with a probability of 90%, then the portion of Sam's million dollar portfolio that should be invested in WIND is equal to:
[2*.90 - 1.125]/(1 - .125) = (1.8 - 1.125)/.875 = 77%
Wizened with age, however, Sam is too smart to take the 77% literally. He knows that in a worse case scenario, in which WIND tanks, his resulting portfolio would be reduced to:
$1,000,000 * (1-.77) * 1.125 = $258,750, which would be pushing the assumption of logarithmic utility for money. Accordingly, Sam mentally adjusts the 77% downward to adjust for this technicality, using a ink pen to scratch in 60% instead. Sam didn't become a millionaire by taking foolish risks.
Case 3 - Average Investors
Karen and Harry live adequately on $50,000 a year, and have accumulated $250,000 in their tax deferred pension portfolio consisting of a balance between bond funds, mutual funds and money funds. Lately, they have been more than pleased with annual returns approximating 22%, but they fear a market correction, and want to make sure they will be decent shape when they retire in 15 years. They figure that the best return they can expect their stock mutual funds to yield on average is the historical return for stocks at less than 12%, less expenses of the funds of at least 2%, for a net return of 10% per annum. The bonds and money market can be assumed to return 5%, for an overall return of about 8%. Karen has convinced Harry that WIND has a 60% chance of doubling every two years, but there is always a risk with small companies that something terrible could happen.
Harry set about to establish how much of their portfolio they should invest in WIND, realizing immediately that he could reasonably approximate his and Karen's current joint utility for money by the quadratic function x - A*x^2, where A = 0.0000005. After calculating the most rational amount for an investment in WIND, he was slightly disappointed when he realized it would consists of a mere 9.9% of his growing portfolio. He made an management decision, after discussing it with Karen, and rounded it up to an even 10%.
Case 4 - Jerry Fiddler's Secondary Stock Offering
Remember when Jerry Fiddler sold 30% of his stock to diversify his portfolio. At the time, I said his action was rational when viewed from the point of view of Utility Theory. Turn the clock back and let's look at his decision again with the help of the equations from Part I.
The stock was sold at a split-adjusted price of $18. According to the recent Forbes article that was amply discussed and clarified by Jerry on this thread, Jerry's portfolio value probably greatly exceeded his annual lifestyle costs, meaning that the logarithmic Utility Function adequately describes his value of money in the range of his likely portfolio outcomes.
Since Jerry diversifies expressly to avoid a catastrophic WIND meltdown that would render his assets worthless, the model presented in Part I absolutely is appropriate to his situation as it was in the summer of 1996.
Based on the formula presented in Part I, Jerry should have exactly [2P - (1+R)]/(1-R) of his assets in WIND. Assuming that Jerry invested the proceeds of his sale in "safe" investments like bonds and other (boring) investments, to balance his high-return investment in WIND, then he might have assumed an after tax return of 7% per annum. This return would compound to 22.5% over three years. Since we know he targeted to have 70% of his holding in WIND after the sale, this means that, if he were completely rational, he must have believed that the chances that WIND would double within 3 years from last summer was 88.3%. Had he assumed that he could return a net 15% in other investments, he must have assumed that the chances that WIND would double were near certainty, at 94.7%.
Say this again. As a perfectly rational person, Jerry Fiddler last summer sold stock in WIND believing with near certainty that the stock would double in price within three years. He did it to optimally diversify his portfolio, not because he thought the stock was over priced.
People on this thread that suggested that WIND's stock is worth $18/share because Jerry sold a bundle at that price do not appreciate implications of optimal diversification. The fact that the stock more than doubled within the year following his sale merely confirms his good judgment about probabilities, not that he thought the stock was only worth what he sold it for then. Anyone believing with near certainty that a stock will double within three years does not value the stock at the current price. Nevertherless, rational behavior may dictate that that same person sell some stock at the current price - like it or not.
Allen |
