Question and Problem Answers Chapter 4- Diversification 4-1: 4-2: page 1

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1 Question and Problem Answers Chapter 4- Diversification page 1 4-1: The arguments brought forth by Buffet, Loeb, and Keynes present arguments in favor of intelligent, informed, investing rather than against diversification. In each case they emphasize the need to research companies and invest according to good sound fundamentals. Buffet states the case most forcefully, stating that the investor should choose five to ten sensibly-priced companies that possess important long-term competitive advantages. The probabilities are that these five to ten companies come from a variety of industries and so diversification is a natural byproduct of intelligent investing. The conventional diversification that makes no sense for you recognizes that many investors, possessed of an introductory level of investment education, often choose companies because they provide diversification rather than because they provide a good investment. 4 - : Consider the efficient portfolio frontier and the portfolio allocation line in figure 4-3 (below). If the risk free rate increases then the portfolio allocation line becomes more horizontal as the y-intercept increases. This moves the tangency up along the curve towards the 100% Hypothetical Resources point. Essentially as the risk free return increases we can afford to take on more risk to gain a higher return in our portfolio of risky assets. To explore the limits to this construction we can see that if we increase the risk free rate far enough then the portfolio allocation line goes beyond the horizontal and begins to slope downwards. Now the entire portfolio of risky assets is dominated by the risk free rate: everyone invests all their money in Treasury Bills and the stock market collapses. Conversely, if the risk free rate decreases then the portfolio allocation line becomes more vertical as the y-intercept decreases. This moves the tangency down along the curve towards the 100% Tardis Intertemporal point. FIGURE 4-3: EFFICIENT PORTFOLIO FRONTIER AND PORTFOLIO ALLOCATION LINE (PAGE 78)

2 FINANCIAL MARKETS... AND THE INSTRUMENTS THAT TRADE IN THEM 4-3: Consider the case where everyone invests his portfolio in the risk free asset and an index portfolio. A new company, Google, seeks funding and initiates a public offering. Since it is not in the index, no one invests. So at the first level of analysis we point out that any company not already in the index has no investors. We devise a mechanism whereby every company seeking to become publically owned is added to the index automatically as soon as they are offered. This is easy enough to do. The NYSE index, for example, consists of all companies traded on the New York Stock Exchange. But at what price? The index is constructed based on the price revealed in the market from moment to moment. If the stocks themselves are never traded because everyone buys and sells only the index portfolio then there is no pricing information coming into the index construction. True, buyers and sellers must trade the index, but the relative prices of the companies in the index cannot be fixed or, once fixed, cannot change. This is a classic example of the micro/macro dichotomy. Many theories of economics and finance work this way. The classic example is saving. If you double your saving rate you begin to accumulate wealth. If everyone in the economy doubles their savings rate then consumption decreases and the economy goes into a recession. The dominance of the risk free rate and index portfolio relies on a market with enough informed investors to efficiently reflect all relevant information. In that environment, other investors use that efficiency to form their portfolios. These investors, often called liquidity traders because they bring liquidity rather than information to the market, can rely on the index. But a market comprised entirely of liquidity traders reflects liquidity, but no information. 4-4: A. In July and November all three stocks had positive returns; in March, June, and September all three stocks had negative returns. In the remaining seven months one of the three stocks moved in a direction different from the other two. B. In each month the portfolio return is the weighted average of the three component stocks. C. There is no easy way to calculate the standard deviation of a portfolio; you can't just take the average of the standard deviations of the three stocks because sometimes the prices of these stocks move in the same direction and sometimes they don't. We do not know the degree of correlation between the rates of return on the three stocks. We can calculation the pairs of covariances but with a portfolio of three stocks this becomes extremely cumbersome. The best way to calculate the standard deviation on the portfolio is to calculating the rate of return on the portfolio for each month and then calculate the standard deviation just as we do for the individual stocks. Portfolio January % February % March % April 6.08% May 1.648% June % July 6.738% August % September -1.39% October 7.135% November % December -0.38% Total Time Weighted % Average Time Weighted % Standard Deviation Total % Average -.417% E[X ] = E[X] = ó = ó = %

3 CHAPTER 4 - DIVERSIFICATION 3 D. The standard deviation of % is lower than the standard deviation of Sun and Boeing but higher than the standard deviation of P&G. It is also higher than the standard deviation of the S&P 500. E. The key is to draw the normal distribution curve from the mean minus three standard deviations to the mean plus three standard deviations for each investment. In excel you can set the rates of return and have excel calculate the probability using the function =NORMDIST() Time Weighted Rate of Return S&P 500 PG SUNW BA Total % 0.881% % -41.4% Average % 0.073% % % Statistical Rate of Return S&P 500 PG SUNW BA Total % 3.19% % % Average % 0.61% % % Standard Deviation E[X ] = E[X] = ó = ó = % % % % 4-5: A. A 50/50 portfolio of TI and HR has an expected return of 18.00% and a standard deviation of 4.90%

4 4 FINANCIAL MARKETS... AND THE INSTRUMENTS THAT TRADE IN THEM B. If TI and HR are perfectly correlated then the standard deviation of the portfolio is the weighted average of the standard deviation of the two stocks. C. If TI and HR are perfectly correlated then the portfolio standard deviation is 30%. If TI and HR are anything less than perfectly correlated then the diversification effect gives us a portfolio standard deviation of less than 30%. Perfect negative correlation gives us a portfolio standard deviation of 10%. 4-6: A portfolio consisting of Galifrey Educational is not efficient because a portfolio 67% TI & 33% HR also gives a 17% return but at a lower risk (ó=1.4%). If you have $1,000 of Galifrey Educational shares in the portfolio you could replace them with $670 of Tardis Intertemporal and $330 of Hypothetical Resources and reduce the overall risk of the portfolio without affecting the expected rate of return. 4-7: This is possible because Hypothetical Resources (HR) and Tardis Intertemporal (TI) have perfect negative correlation. Since Hypothetical Resources has twice the variance as Tardis this means that whenever TI generates a return 1% above its expected value HR generates a return % below its expected value. We can offset these risks exactly by weighting the portfolio so that TI has twice the weight of HR. To see how this works use the variance equation with two thirds in Tardis, one third in Hypothetical Resources and ñ = -1:

5 CHAPTER 4 - DIVERSIFICATION 5 4-8: A. The correlation coefficient is calculated as ñ = 0.36 B. The expected return is 10.80% and the standard deviation is 19.78% C. The expected return would still have been 10.80% but the standard deviation would have been 1.6% ELISABETH OLTHETEN AND KEVIN G. WASPI 006Not to be transmitted, copied, or distributed without express written permission from the authors.