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The greater the standard deviation of retums, the greater the risk. In the securities industry, annual retums are mosl common, but monthly retums may be used if there are not enough years of data. There is no clear way to infer annual retums fran monthly retums. Downside Risk Downside equity movements are often more important than profit pattems. It seems sensible that, if you want to know the probability of a loss, then you should study the history of equity drawdowns. The use of only the equity losses is called lower partial moments in which lower refers to the downside risk and partial means that only one side of the retum distribution is used. A set of relative lower partial moments (RLPMs) is the expected value of the tracking error (equity drawdowns, the difference between the actual equity and the annualized retums) raised to the power of n: , =£(R  ), over die rarge wliere < =0, over tlie range v.taJ(>  benchmarlc or co¹(nadii$ regression return a ihai point in time f=expected or return (descnbed die begnnmg of diis section) Therefore, the elements of the probability have only losses or zeros. The value n represents the order or ranking of the RLPMs. When n = 0, RLPM is the probability of a shortfall. Probability (R =: B); when n = 1, RLPM is equal to the expected shortfall. \  ] ~ and when n = 2, RLPM is equal to the relative lower partial variance. One concem about using only the drawdowns to predict other drawdowns is that it limits the number of cases and discards the likelihood that higher than normal profits can be related to higher overall rid;. In situations where there are limited amounts of test data, both the gains and losses will offer needed information. 1 ) The pu ose of an average is to transform individuality into classification. When done properly, there is useful information to be gained. Indices have gained popularity in the futures markets recently; the stock market indices are now second to the financial markets in trading volume. These contracts allow both individual and institutional participants to invest in the overall market movement rather than take the higher rid; of selecting individual securities. Furthermore, investors can hedge their current market position by taking a short position in the futures market against a long position in the stock market. A less general index, the Dow Jones Indudrials, or a grain or livestock index can help the trader take advantage of a more specific price without having to decide which products are more likely to do best. An index simplifies the decisionmaking process for trading, if an index does not exist, it can be conshncted to satisl most pu oses. Conshiicting an Index An index is traditionally used to determine relative value and normally expresses change as a percentage. Mosl indices have a starting value of 100 or 1,000 on a specific date. The index itself is a ratio of the current or composite values to those values during the base year. The selection of the base year is often chosen for convenience, but usually is far enough bad; to show a representative, stable price period. The base year for US. produdivity and for unemployment is 1982, consumer confidence is 1985, and the composite of leadiig mdicators is 1987. For exanple. for one market, the index for a specific year is infage.,,) cu„en,p„ce(ye.r,) staning price (base year) If the value of the index is less than 100, the current value (year t) is lower than during the base yeai. The adual index value represents the percentage change. For each year after the base year, the index value is the sum of the previous index value and the percentage change in price over the same period.
11 price It is very convenient to create an index for two markets that trade in different units because they caimot be otherwise compared. For example, if you wanted to show the spread between gold and the US. Dollar Index, you could index them both beginning at the same date. The new indices would both be in the same units, percent, and would be easj to compare. Most often, an mdex combines a number of related markets mto a smgle number. A simple aggregate mdex is the ratio of unweighted sums of market prices in a specific year to the same markets in the base year. Most of the popular indices, such as the New York Stock Exchange Composite Index, fall into this class. A weighted aggregate index biases certain markets by weighting them to increase or decrease their effect on the composite value. The index is then calculated as in the simple aggregate index. When combining markets into a single index value, the total of all the weighting normally totals to the value one, although you may also divide the composite value by the total of all the individual weights. US. Dollar Index A praaical exanple of a weighted index is the U.S. Dollar Index, traded on the New York Futures Exchange In order of greatest weighting, the 10 currency components are the Deutschemark 20.8° o, Japanese yen 13.6° o, French franc 13.Po, British pound 11.9°o, Canadian dollar 9.Po, Italian lira 9.0°o, Netheriands guilder 83"o, Belgian franc 6.4"o, Swedish kroner 4.2"o, and the Swiss franc 3.6S0. This puts a total weight of 75.5"o in European currencies with only the Japanese yen representing Asia, not a practical mix for a world economy that has become dependent on Far Eastern trade. Within Europe, however, allocations seem to be proportional to the relative size of the economies The Dollar Index rises when the U.S. dollar rises. Quotes are in foreign exchange notation, where there are 1.25 Swiss francs per U.S. dollar, instead of .80 dollars per franc as quoted on the Chicago Mercantile Exchanges IMM. For example, when the Swiss franc moves from 1.25 to 130 per dollar, there are more Swiss francs per dollar: therefore, each Swiss franc is worth less. In the daily calculation of the Dollar Index, each price change is represented as a percent. If, in our previous , the Swiss franc rises .05 points, the change is 5/12 5 04; this is multiplied by its weighting factor .208 and confributes +. 00832 to the Index. PROBABILITY Calculation must measure the incalculable. Dixon G. Watts Change is a term that causes great anxiety. However, the effects and likelihood of a chance occurrence can be measured, although not predirted. The area of study that deals with uncertainty is probability. Everyone uses probability in daily thinking and actions. When you tell someone that you will be there in 30 minutes, you are assuming: Your car will start. You will not have a breakdown. You will have no unnecessary delays. You will drive at a predictable speed. You will have the normal number of green lights. All these circumstances are exffemely probabilistic, and yei everyone makes die same assumptions. Actually, the 30minute arrival is intended only as an estimate of the average time it should take for the frip. If the arrival time were critical, you would extend your estimate to 40 or 45 minutes, to account for unexpected events. In statistics. This
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