| Home | My photos | Forex | My trading | Contacts |
|
|
You Can't Become Rich In Your Pocket Until You Become Rich In Your Mind | ||||
|
In this refinement of portfolio selection, all investors choose the same optimal stock portfolio and then adjust how much risk theyre willing to take by increasing or decreasing the percentageBeta-Is It Better? Returning to more quantitative matters, we choose stocks so that when some are down, others are up (or at least not as down), giving us a healthy rate of return with as little risk as possible. More precisely, given any portfolio of stocks, we grind the numbers describing their past performances and come up with estimates for their expected returns, volatilities, and covariances, and then use these to determine the expected returns and volatilities of the portfolio as a whole. We could, if we had the time, the price data, and fast computers, do this for a variety of different portfolios. The Nobel prize-winning economist Harry Markowitz, one of the originators of this approach, developed mathematical techniques for carrying out these calculations in the early 1950s, graphed his results for a few portfolios (computers werent fast enough to do much more then), and defined what he called the efficient frontier of portfolios. If we were to use these techniques and construct comparable graphs for a wide variety of contemporary portfolios, what would we find? Arraying the (degree of) volatility of these portfolios along the graphs horizontal axis and their expected rates of return along its vertical axis, we would see a swarm of points. Each point would represent a portfolio whose coordinates would be its volatility and expected return, respectively. Wed also notice that among all the portfolios having a given level of risk (that is, volatility, standard deviation), there would be one with the highest expected rate of return. If we single out the portfolio with the highest expected rate of return for each level of risk, we would obtain a curve, Markowitzs efficient frontier of optimal portfolios. The more risky a portfolio on the efficient frontier curve is, the higher is its expected return. In part, this is because most investors are risk-averse, making risky stocks cheaper. The idea is that investors decide upon a risk level with which theyre comfortable and then choose the portfolio with this risk level that has the highest possible return. Call this Variation One of the theory of portfolio selection. Dont let this mathematical formulation blind you to the generality of the psychological phenomenon. Automobile engineers have noted, for example, that safety advances in automobile design (say anti-lock brakes) often result in people driving faster and turning more sharply. Their driving performance is enhanced rather than their safety. Apparently, people choose a risk level with which theyre comfortable and then seek the highest possible return (performance) for it. Inspired by this trade-off between risk and return, William Sharpe proposed in the 1960s what is now a common measure of the performance of a portfolio. It is defined as the ratio of the excess return of a portfolio (the difference between its expected return and the return on a risk-free treasury bill) to the portfolios volatility (standard deviation). A portfolio might have a hefty rate of return, but if the volatility the investor must endure to achieve this return is roller coasterish, the portfolios Sharpe measure wont be very high. By contrast, a portfolio with a moderate rate of return but a less anxiety-inducing volatility will have a higher Sharpe measure. There are many complications to portfolio selection theory. As the Sharpe measure suggests, an important one is the existence of risk-free investments, such as U.S. treasury bills. These pay a fixed rate of return and have essentially zero volatility. Investors can always invest in such risk-free assets and can borrow at the risk-free rates as well. Moreover, they can combine risk-free investment in treasury bills with a risky stock portfolio. Variation Two of portfolio theory claims that there is one and only one optimal stock portfolio on the efficient frontier with the property that some combination of it and a risk-free investment (ignoring inflation) constitute a set of investments having the highest rates of return for any given level of risk. If you wish to incur no risk, you put all your money into treasury bills. If youre comfortable with risk, you put all your money into this optimal stock portfolio. Alternatively, if you want to divide your money between the two, you put p% into the risk-free treasury bills and (100 - p)% into the optimal risky stock portfolio for an expected rate of return of [p x (risk-free return) + (1 - p) x (stock portfolio)]. An investor can also invest more money than he has by borrowing at the risk-free rate and putting this borrowed money into the risky portfolio. In this refinement of portfolio selection, all investors choose the same optimal stock portfolio and then adjust how much risk theyre willing to take by increasing or decreasing the percentage, p, of their holdings that they put into risk-free treasury bills. This is easier said than done. In both variations the required mathematical procedures put enormous pressure on ones computing facilities, since countless calculations must be performed regularly on new data. The expected returns, variances, and covariances are, after all, derived from their values in the recent past. If there are twenty stocks in a portfolio, we would need to compute the covariance of every possible pair of stocks, and there are (20 x 19)/2, or 190, such covariances. If there were fifty stocks, wed need to compute (50 x 49)/2, or 1,225 covariances. Doing this for each of a wide class of portfolios is not possible without massive computational power. As a way to avoid much of the computational burden of updating and computing all these covariances, efficient frontiers, and optimal risky portfolios, Sharpe, yet another Nobel Prize winner in economics, developed (with others) whats called the single index model. This Variation Three relates a portfolios rate of return not to that of all possible pairs of stocks in the portfolio, but simply to the change in some index representing the market as a whole. If your portfolio or stock is statistically determined to be relatively more volatile than the market as a whole, then changes in the market will bring about exaggerated changes in the stock or portfolio. If it is relatively less volatile than the market as a whole, then changes in the market will bring about attenuated changes in the stock or portfolio. This brings us to the so-called Capital Asset Pricing Model, which maintains that the expected excess return on ones stock or portfolio (the difference between the expected return on the portfolio, R P, and the return on risk-free treasury bills, Rf) is equal to the notorious beta, symbolized by (3, multiplied by the expected excess return of the general market (the difference between the markets expected return, Rm, and the return on risk-free treasury bills, Rf). In algebraic terms: (R P - Rf) = P (R.,, - Rf). Thus, if you can get a sure 4 percent on treasury bills and if the expected return on a broad market index fund 10 percent and if the relative volatility, beta, of your portfo1.5, then the portfolios expected return is obtained by solving (RP - 4%) = 1.5(10% - 4%), which yields 13 percent for RP. A beta of 1.5 means that your stock or portfolio gains (or loses) an average of 1.5 percent for every 1 percent gain (or loss) in the market as a whole. Betas for the stocks of high-tech companies like WorldCom are often considerably more than 1, meaning that changes in the market, both up and down, are magnified. These stocks are more volatile and thus riskier. Betas for utility company stocks, by contrast, are often less than 1, which means that changes in the market are muted. If a company has a beta of .5, then its expected return is obtained by solving (RP - 4%) = .5(10% - 4%), which yields 7% for Rp, the expected return on the portfolio. Note that for short-term treasury bills, whose returns dont vary at all, beta is 0. To reiterate: Beta quantifies the degree to which a stock or a portfolio fluctuates in relation to market fluctuations. It is not the same as volatility. This all sounds neat and clean, but you beta watch your step with all of these portfolio selection models. Specifically with regard to Variation Three, we might wonder where the number beta comes from. Who says your stock or portfolio will be 40 percent more volatile or 25 percent less volatile than the market as a whole? Heres the rough technique for finding beta. You check the change in the broad market for the last three months-say its 3 percent-and check the change in the price of your stock or portfolio for the same period-say its 4.1 percent. You do the same thing for the three months before that-say the numbers this time are 2 percent and 2.5 percent, respectively-and for the three months before that-say -1.2 percent and -3 percent, respectively. You continue doing this for a number of such periods and then on a graph you plot the points (3%, 4.1%), (2%, 2.5%), (-1.2%, -3%), and so on. Most of the time if you squint hard enough, youll see a sort of linear relationship between changes in the market and changes in your stock or portfolio, and you then use standard mathematical methods for determining the line of closest fit through these points. The slope or steepness of this line is beta. One problem with beta is that companies change over time, sometimes rather quickly. AT&T, for example, or IBM is not the same company it was twenty years ago or even two years ago. Why should we expect a companys relative volatility, beta, to remain the same? In the opposite direction is a related difficulty. Beta is often of very limited value in the short term and varies with the index chosen for comparison and the time period used in its definition. Still another problem is that beta depends on market returns, and market returns depend on a narrow definition of the market, namely just the stock market rather than stocks, bonds, real estate, and so forth. For all its limitations, however, beta can be a useful notion if its not turned into a fetish. You might compare beta to different peoples emotional reactivity and expressiveness. Some respond to the slightest good news with outbursts of joy and to the tiniest hardship with wails of despair. At the other end of the emotional spectrum are those who say ouch when they accidentally touch a scalding iron and allow themselves an oh, good when they win the lottery. The former have high emotional beta, the latter low emotional beta. A zero beta person would have to be unconscious, perhaps from ingesting too many betablockers. Unfortunate for the prospect of predicting the behavior of people, however, is the commonplace that peoples emotional betas vary depending on the type of stimulus a person faces. Ill leave out the examples, but this may be betas biggest limitation as a measure of the relative volatility of a portfolio or stock. Betas may vary with the type of stimulus a company faces. Whatever refinements of portfolio theory are developed, one salient point remains: Portfolios, although often less risky than individual stocks, are still risky (as millions of 401(k) returns attest). Some mathematical manipulation of the notions of variance and covariance and a few reasonable assumptions are sufficient to show that this risk can be partitioned into two parts. There is a systematic part that is related to general movements in the market, and there is a non-systematic part that is idiosyncratic to the stocks in the portfolio. The latter, nonsystematic risk, specific to the individual stocks in the portfolio, can be eliminated or diversified away by an appropriate choice of thirty or so stocks. An irreducible core, however, remains inherent in the market and cannot be avoided. This systematic risk depends on the beta of ones portfolio. Or so the story goes. To the criticisms of beta above should be added the problems associated with forcing a non-linear world into a linear mold. |
|
|||||||||||||||
Previous Issues
|
| ©2007 Olesia | Home • My photos • Forex • News • My trading • Contacts |