point of limitation; therefore, prices move much fur-ther up during periods of extreme shortage relative to demand.
The measures of central tendency discussed in the previous section 2 are used to qual-ify the shape and extremes of price movement shown in the frequency distribution. Thegeneral relationship between the results when using the three principal means is
arithmetic mean > geometric mean > harmonic mean
Median and Mode
‘Two other measurements, the median and the mode, are often used to define distribution.The median, or middle item, is helpful for establishing the center of the data; it halves thenumber of data items. The median has the advantage of discounting extreme values, whichmight distort the arithmetic mean, The mode is the most commonly occurring value. InFigure 2-3 the mode is the highest point.
In a normally distributed price series, the mean, median, and mode will all occur at thesame value; however, as the data become skewed, these values will move farther apart. Thegeneral relationship is: :ieeeFIGURE 2-2 Hypothetical price-frequency distribution.
, Frequency
 
Lower - HigherPricesDISTRIBUTION - 15
 
aFIGURE 2-3 Hypothetical price distribution skewed to the right, showing the relationship ofthe mode, median, and mean.
 
Frequency
Mean
Lower Mode Median HigherPrices
 
mean > median > mode
The mean, median, and mode help to tell whether data is normally distributed or skewed.A normal distribution is commonly called a bell curve, and values fall equally on both sidesof the mean. For much of the work done with price and performance data, the distributionstend to extend out toward the right (positive values) and be more cut off on the left (nega-tive values). If you were to chart a distribution of trading profits and losses based on a trendsystem with a fixed stop-loss, you would get profits that could range from zero to very largevalues, while the losses would be theoretically limited to the size of the stop. Skewed dis-tributions will be important when we try to measure the probabilities later in this chapter.
Characteristics of the Principal Averages
Each averaging method has its unique meaning and usefulness. The following summarypoints out their principal characteristics:
The arithmetic mean is affected by each data element equally, but it has a tendency toemphasize extreme values more than other methods. It is easily calculated and is sub-ject to algebraic manipulation. - :The geometric mean gives less weight to extreme variations than the arithmetic meanand_is most important when using data representing ratios or rates of change. It can-not always be used for a combination of Positive and negative numbers and is also sub-ject to algebraic manipulation. ~ _
The harmonic mean is most applicable to time chariges and, along with the geometricmean, has been used in economics for price analysis. The added complications of com-putation have caused this to be less popular than either of the other averages, althoughit is also capable of algebraic manipulation.
The mode is not affected by the size of the variations from the average, only the distri-bution. It is the location of greatest cqncentration and indicates a typical value for areasonably large sample. With an unordered set of data, the mode is time consumingto locate and is not capable of algebraic manipulation.16 BASIC CONCEPTS
 
The median is most useful when the center of an incomplete set is needed. It is notaffected by extreme variations and is simple to find if the number of data points areknown. Although it has some arithmetic properties, it is not readily adaptable to com-
putational methods.
DISPERSION AND SKEWNESS
The center or central tendency of a data series is not a sufficient description for price analy-sis. The manner in which it is scattered about a given point, its dispersion and skewness,are necessary to describe the data. The mean deviation is a basic method for measuringdistribution and may be calculated about any measure of central location, for example, thearithmetic mean. It is found by computing
price; |
> |pricemp2
  
 
# or @Avg(@AbsValue(price — @Avg(price,7)),n)nwhere MD is the mean deviation, thé average of the differences between each price and thearithmetic mean of the prices, or other measure of central location, with signs ignored.The standard deviation is a special form of measuring average deviation from themean, which uses the root-mean-square
Standard deviation = or
 
= @Sqrt(@Avg((price - @Avg(price,n))’,n))
where the differences between the individual prices and the mean are squared to empha-size the significance of extreme values, and then total final value is scaled back using thesquare root function. This popular measure, found throughout this book, is available in allspreadsheets and software programs as @Std or @Stdev. For n prices, the standard devia-tion is simply @Std(price,7).
The standard deviation is the most popular way of measuring the degree of dispersionof the data. The value of one standard deviation about the mean represents a clustering ofabout 68% of the data, two standard deviations from the mean include 95.5% of all data,and three standard deviations encompass 99.7%, nearly all the data. These values repre-sent the groupings of a perfectly normal set of data, shown in Figure 2-4.
Probability of Achieving a Return
If we look at Figure 2-4 as the annual returns for the stock market over the past 50 years,then the mean is about 8% and one standard deviation is 16%. In any one year we canexpect the compounded rate of return to be 8%; however, there is a 32% chance that it willbe either greater than 24% (mean plus one standard deviation) or less than —8% (the meanminus one standard deviation). If you would like to know the probability of a return of
20% or greater, you must first rescale the values,
objective — meanPi ili: if hit bjective = -———_—___——_robability of reaching objec: Standard deviation
If your objective is 20%, we calculate20% - 8% _
Probability = 16%
75DISPERSION AND SKEWNESS, V7
 
 
FIGURE 2-4 Normal distribution showing the percentage area included within one standarddeviation about the arithmetic mean,
 
68.26%
Mean
 
We look in Appendix Al under the probability for normal curves, and find that a stan-dard deviation of .75 gives 27.34%, a grouping of 54.68% of the data. That leaves one-halfof the remaining data, or 22.66%, above the target of 20%.
Skewness .
Most price data, however, are not normally distributed. For physical commodities, such asgold, grains, and interest rates (yield), prices tend to spend more time at low levels andmuch less time at extreme highs; while gold peaked at $800 per ounce for one day, it hasremained between $375 and $400 per ounce for most of the past 10 years. The possibilityof falling below $400 by the same amount as its rise to $800 is impossible, unless youbelieve that gold can go to zero. This relationship of price versus time, in which marketsspend more time at lower levels, can be measured as skewness—the amount of distortionfrom a symmetric shape that makes the curve appear to be short on one side and extendedon the other. In a perfectly normal distribution, the median and mode coincide. As pricesbecome extremely high, which often happens for short intervals of time, the mean willshow the greatest change and the mode will show the least. The difference between themean and the mode, adjusted for dispersion using the standard deviation of the distribu-tion, gives a good measure of skewness
mean — mode
ke =(Skewness) Sx standard deviation
Because the distance between the mean and the mode, in a moderately skewed distribu-tion, is three times the distance between the mean and the median, the relationship canalso be written as:
3 x (mean - median)
kc =(Skewness) Sx =" C ndard deviation
This last formula may be more practical for computer applications, because the moderequires dividing the data into groups and counting the number of occurrences in eachbar. When interpreting the value of S,, the distribution leans to the right when Sx is posi-tive (the mean is greater than the median), and it is skewed left when Sx is negative.18 BASIC CONCEPTS
 
Kurtosis .
One last measurement, that of kurtosis, should be familiar to analysts. Kurtosis is the“peakedness” of a distribution, the analysis of “central tendency.” For most cases a smallerstandard deviation means that prices are clustered closer together; however, this does notalways describe the distribution clearly. Because so much of identifying a trend comesdown to deciding whether a price change is normal or likely to be a leading indicator of anew direction, deciding whether prices are closely grouped or broadly distributed may beuseful. Kurtosis measures the height of the distribution.
Transformations
The skewness of a data series can sometimes be corrected using a transformation on thedata. Price data may be skewed in a specific pattern. For example, if there are % of theoccurrences at twice the price and % of the occurrences at three times the price, the origi-nal data can be transformed into a normal distribution by taking the square root of eachdata item. The characteristics of price data often show a logarithmic, power, or square-rootrelationship. .
Skewness in Price Distributions
Because the lower price levels of most commodities are determined by production costs,price distributions show a clear boundary of resistance in that direction. At the high levels,prices can have a very long tail of low frequency. Figure 2-5 shows the change in the distri-bution of prices as the mean price (over shorter intervals) changes. This pattern indicatesthat a normal distribution is not appropriate for commodity. prices, and that a log distrib-ution would only apply to overall long-term distributions. =|
Choosing between Frequency Distribution and Standard Deviation _
You should note that it is more likely that unreliable probabilities will result from using toolittle data than from the choice of method. For example, we might choose to look at thedistribution of one month of daily data, about 23 days; however, it is not much of a sam-ple. The price or equity changes being measured might be completely different during thenext month. Even the most recent five years of S&P data will not show a drop as large asOctober 1987.
 
FIGURE 2-5 Changing distribution at different price levels. A,B, and C are increasing meanvalues of three shorter-term distributions. —
 
a ~\
- /- / \
/ AQ Long:term distribution
 DISPERSION AND SKEWNESS 19
 
Although we can identify and measure skewness, it is difficult to get meaningful prob-abilities using a standard deviation taken on yery distorted distributions. It is simpler touse a frequency distribution for data with long tails on one side and truncated results onthe other. To find the likelihood of returns using a trend system with a stop-loss, you cansimply sort the data in ascending order using a spreadsheet, then count from each end tofind the extremes. You will notice that the largest 10% of the profits cover a wide range,while the largest 10% of the losses is clustered together.
A standard deviation is very helpful for giving some indication that a price move, largerthan any we have seen in the data, is possible. Because it assumes a normally shaped curve,a large clustering of data toward one end will force the curve to extend further. Although theusefulness of the exact probabilities is questionable, there is no doubt that, given enoughtime, we will see price moves, profits, and losses that are larger than we have seen in the past.
Student t-test
Throughout the development and testing of a trading system, we will want to know if theresults we are seeing are as expected. The answer will keep referring back-to the size of thesample and the amount of variance that is typical of the data during this period. Readersare encouraged to refer to other sections in the book on sample error and chi-square test.Another popular method for measuring whether the average price of the data is signifi-cantly different from zero, that is, if there is an underlying trend bias or if the patternexhibits random qualities, is the student t-test,
= average of price changes VN. of data hems
standard deviation of price changes
and where degrees of freedom = number of data items — 1. The more trades in the sample,the more reliable the results, The values of t needed to be significant can be found inAppendix 1, Table A1.2, “7-Distribution.” The column headed “.10” gives the 90% confi-dence level, “.05” is 95%, and “.005” is 99.5% confidence.If we separate data into two periods and compare the average of the two periods forconsistency, we can decide whether the data has changed significantly. This is done with a2-sample t-test:
 
where x, and X, are the averages of data periods 1 and 2, v, and v, are_the variances of peri-ods 1 and 2, and 7; and n; are the number of data items in periods 1 and 2. The degrees offreedom, df needed to find the confidence levels in Table A1-2, can be calculated as:
ss s¥ -_ GeysiY syny nz
(—n) * Gam)
The-student ¢-test can also be used to compare the profits and losses generated by a trad-ing system to show that the underlying system process is sound. Simply replace the dataitems by the average profit or loss of the system, the number of data items by the number20 BASIC CONCEPTS
 
of trades, and calculate all other values using the profit/loss to get the student ¢-test valuefor the trading performance. 1
STANDARDIZING RETURNS AND RISK
To compare one trading method with another, it is necessary to standardize both the testsand the measurements used for evaluation. If one system has total returns of 50% and theother of 250%, we cannot decide which is best unless we know the duration of the test. Ifthe 50% return was over 1 year and the 250% return over 10 years, then the first one is best.Similarly, the return relative to the risk is crucial to performance as will be discussed inChapter 21 (“Testing”). For now it is only important that returns and risk be annualized orstandardized to make comparisons valid.
Calculating Returns
The calculation of rate of return is essential for assessing performance as well as for manyarbitrage situations. In its simplest form, the one-period rate of return R, or the holdingperiod rate of return is
_ Ending value — Starting value _ Ending value
 
=>,where Py is the initial investment or starting value, and P, is the value of the investmentafter one period. In most cases, it is desirable to standardize the returns by annualizing.This is particularly helpful when comparing two sets of test results, in which each covers adifferent time period. Although calculations on government instruments use a 360-day rate(based on 90-day quarters), a 365-day rate is common for most other purposes. The fol-lowing formulas show 365 days; however, 360 may be substituted.
The annualized rate of return on a simple-interest basis for an investment over n days isPie Pi-Po y 365 -
Po n \The annualized compounded rate of return isP, —Po B65/n)
Pi oO )The geometric mean is the basis for the compounded growth associated with interest rates.If the initial investment is $1,000 (Py) and the ending value is $1,600 (P,) after 12 years(y = 12), there has been an increase of 60%. The simple rate of return is 5%, but the com-pounded growth shows :
Ending value= Starting value x (1 + Compounded return)”
P,=PoX (1+R)” “
Reimpte =
Reompounded = ( +
or
P,\ayR= (2) -1P o
(ee can2y 1~ i500) ~R= .0398 or about 4.0%STANDARDIZING RETURNS AND RISK 2
 
The use of the standard deviation and compounded rate of return are combined to find theprobability of a return objective. In the following calculation,’ the arithmetic mean of con-tinuous returns is In(1 +.R,), and it is assumed that the returns are normally distributed.
In (Z) —In(1+R,)neS Vsxn
where z= standardized variable (can be looked up in Appendix Al)T = target value or rate-of return objectiveB = beginning investment valueR, = geometric average of periodic returnsn = number of periodss = standard deviation of the logarithms of the quantities 1 plus the periodicreturns
Indexing Returns
The Federal Government has defined standards for calculating returns in the FuturesIndustry Commodity Trading Advisors (CTAs). This is simply an indexing of returns basedon the current period percentage change in equity. It is the same process as creating anyindex, and it allows trading returns to be compared with, for example, the S&P Index orthe Lehman Brothers Treasury Index, on equal footing. Readers should refer to the sectionJater in this chapter, “Constructing an Index.”
Calculating Risk
Although we would always like to think about returns, it is even more important to be ableto assess risk. With that in mind, there are two types of risk that are important for very dif-ferent reasons. The first is catastrophic risk, which will cause fatal losses or ruin. This is acomplicated type of risk, because it may be the result of a single price shock or a steadydeterioration of equity by being overleveraged. This form of risk will be discussed in detaillater in the book.
Standard risk measurements are useful for comparing the performance of two systemsand for understanding how someone else might evaluate your own equity profile. The sim-plest estimate of risk is the variance of equity over a time interval commonly used by mostinvestment managers. To calculate the variance, it is first necessary to find the mean return,or the expected return, on an investment:
E(R) = pi + Pet + PaRnThe most common measure of risk is variance, calculated-by squaring the deviation ofeach return from the mean, then multiplying each value by its associated probability.
variance = > DR; mefal 7The sum of these values is called the variance, and the square root of the variance is calledthe standard deviation. This was given in another form in the early section “Dispersion andSkewness.”
standard deviation = Vp,[Ri — E(R) |? + Palo — E(R)F + 9 + PalRn — ECR)?
=> {2 -E@)P
1 This and other very clear explanations of returns can be found in Peter L. Bernstein's The Portable MBA inInvestment (John Wiley & Sons, New York, 1995).
 n BASIC CONCEPTS
 
The greater the standard ‘deviation of returns, the greater the risk. In the securities indus-try, annual returns are most common, but monthly returns may be used if there are notenough years of data. There is no clear way to infer annual returns from monthly returns.
Downside Risk
Downside equity movements are often more important than profit patterns. It seems sen-sible that, if you want to know the probability of a loss, then you should study the historyof equity drawdowns. The use of only the equity losses is called lower partial moments, inwhich lower refers to the downside risk and partial means that only one side of the returndistribution is used. A set of relative lower partial moments (RLPMs) is the expected valueof the tracking error (equity drawdowns, the difference between the actual equity and theannualized returns) raised to the power of n:
RLPM, = £[(R - B)"], over the range where R < B=0, over the range where R >B
where R= return on investmentB= benchmark or corresponding regression return at that point.in time£ = expected or mean return (described at the beginning of this-section)
Therefore, the elements of the probability have only losses or zeros. The value n representsthe order or ranking of the RLPMs. When 7 = 0, RLPM is the probability of a shortfall, Prob-ability (R < B); when 7 = 1, RLPM is equal to the expected shortfall, E[R ~ B]; and whenn= 2, RLPM is equal to the relative lower partial variance.
One concern about using only the drawdowns to predict other drawdowns is that itlimits the number of cases and discards the likelihood that higher than normal profits canbe related to higher overall risk. In situations where there are limited amounts of test data,both the gains and losses will offer needed information.
THE INDEX
The purpose of an average is to transform individuality into classification. When doneproperly, there is useful information to be gained. Indices have gained popularity in thefutures markets recently; the stock market indices are now second to the financial marketsin trading volume. These contracts allow both individual and institutional participants toinvest in the overall market movement rather than take the higher risk of selecting indi-vidual securities. Furthermore, investors can hedge their current market position by takinga short position in the futures market against a long position in the stock market.
A less general index, the Dow Jones Industrials, or a grain or livestock index can helpthe trader take advantage of a more specific price without having to decide which productsare more likely to do best. An index simplifies the decision-making process for trading. Ifan index does not exist, it can be constructed to satisfy most purposes.
Constructing an Index
An index is traditionally used to determine relative value and normally expresses change asa percentage. Most indices have a starting value of 100 or 1,000 on a specific date. Theindex itself is a ratio of the current or composite values to those values during the baseyear. The selection of the base year is often chosen for convenience, but usually is farenough back to show a representative, stable price period. The base year for U.S. produc-tivity and for unemployment is 1982, consumer confidence is 1985, and the composite ofleading indicators is 1987. For example, for one market, the index for a specific year is
current price (year t)
x 100starting price (base year)
Index (year f) =PROBABILITY - 23
 
If the value of the index is less than 100, the current value (year f) is lower than during thebase year. The actual index value represents the percentage change.
For each year after the base year, the index value is the sum of the previous index valueand the percentage change in price over the same period,
current price
current index value = previous index value + - ,previous price
It is very convenient to create an index for two markets that trade in different unitsbecause they cannot be otherwise compared. For example, if you wanted to show thespread between gold and the U.S. Dollar Index, you could index them both beginning atthe same date. The new indices would both be in the same units, percent, and would beeasy to compare.
Most often, an index combines a number of related markets: into a single number. ASimple aggregate index is the ratio of unweighted sums of market prices in a specific yearto the same markets in the base year. Most of the popular indices, such as the New YorkStock Exchange Composite Index, fall into this class. A weighted aggregate index biasescertain markets by weighting them to increase or decrease their effect on the compositevalue. The index is then calculated as in the simple aggregate index. When combining mar-kets 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.
U.S. Dollar Index
A practical example of a weighted index is the U.S. Dollar Index, traded on the New YorkFutures Exchange. In order of greatest weighting, the 10 currency components are theDeutschemark 20.8%, Japanese yen 13.6%, French franc 13.1%, British pound 11.9%,Canadian dollar 9.1%, Italian lira 9.0%, Netherlands guilder 8.3%, Belgian franc 6.4%,Swedish kroner 4.2%, and the Swiss franc 3.6%. This puts a total weight of 75.5% in Euro-pean currencies with only the Japanese yen representing Asia, not a practical mix for aworld 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 nota-tion, where there are 1.25 Swiss francs per U.S. dollar, instead of .80 dollars per franc asquoted on the Chicago Mercantile Exchange’s IMM. For example, when the Swiss francmoves from 1.25 to 1.30 per dollar, there are more Swiss francs per dollar; therefore, eachSwiss franc is worth less.
In the daily calculation of the Dollar Index, each price change is represented as a per-cent. If, in our previous example, the Swiss franc rises .05 points, the change is 5/125 = .04;this is multiplied by its weighting factor .208 and contributes +.00832 to the Index.
PROBABILITY a
Calculation must measure the incalculable.Dixon G. Watts
Change is a term that causes great anxiety. However, the effects and likelihood of a chanceoccurrence can be measured, although not predicted. The area of study that deals withuncertainty is probability. Everyone uses probability in daily thinking and actions. Whenyou 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.24 BASIC CONCEPTS
 
You will drive at a predictable speed.You will have the normal number of green lights. :
All these circumstances are extremely probabilistic, and yet everyone makes the sameassumptions. Actually, the 30-minute arrival is intended only as an estimate of the averagetime it should take for the trip. If the arrival time were critical, you would extend your esti-mate to 40 or 45 minutes, to account for unexpected events. In statistics, this is calledincreasing the confidence interval. You would not raise the time to 2 hours, because thelikelihood of such a delay would be too remote. Estimates imply an allowable variation, allof which is considered normal.
Probability is the measuring of the uncertainty surrounding an average value. Proba-bilities are measured in percent of likelihood. For example, if M numbers from a total of Nare expected to fall within a specific range, the probability P of any one number satisfyingthe criteria is
M
P= N’ 0<P<1
When making a trade, or forecasting prices, we can only talk in terms of probabilitiesor ranges. We expect prices to rise 30 to 40 points, or we have a 65% chance of a $400profit from a trade. Nothing is certain, but a high probability of success is very attractive.
Laws of Probability
Two basic principles in probability are easily explained by using examples with playingcards. In a deck of 52 cards, there are 4 suits of 13 cards each. The probability of drawinga specific card on any one turn is %. Similarly, the chances of drawing a particular suit orcard number are % and %s, respectively. The probability of any one of these three possibil-ities occurring is the sum of their individual probabilities. This is known as the law ofaddition. The probability of success in choosing a numbered card, suit, or specific card is1 1 1 18
Peas tg * 32732 79%Another basic principle, the Jaw of multiplications, states that the probability of twooccurrences happening simultaneously or in succession is equal to the product of theirseparate probabilities. The likelihood of drawing a three and a club from the same deck intwo consecutive turns (replacing the card after each draw) or of drawing the same cardsfrom two decks simultaneously is
Joint and Marginal Probability
Price movement is not as clearly defined as a deck of cards. There is often a relationshipbetween successive events. For example, over two consecutive days, prices must have oneof the following sequences or joint events: (up, up), (down, down), (up, down), (down,up), with the joint probabilities of 40, .10, .35, and’715, respectively. In this example,there is the greatest expectation that prices will rise. The marginal probability of a pricerise on the first day is shown in Table 2-2. Thus there is a 75% chance of higher prices onthe first day and a 55% chance of higher prices on the second day.
Contingent Probability
What is the probability of an outcome contingent on the result of a prior event? In theexample of joint probability, this might be the chance of a price increase on the second dayPROBABILITY - 25SS
Se aeTABLE 2-2 Marginal Probability
 
Day 2Up = DownUp 40 35 75 Up MarginalDay | ProbabilityDown AS 10 25 Down on Day |55 5Up DownuoMarginal Probabilityon Day 2
 
when prices declined on the first day. The notation for this situation (the probability of Aconditioned on B) is
P(|B) = P(Aand B) _ joint probability of A and B- P®) - marginal probability of B
then
 
joint probability of (down, up)
Pi Day 2|d Day 1) = ————_(up Day 2|down Day 1) marginal probability of (down Day 1)
 
=.60
.25The probability of either a price increase on Day 1 or a price increase on Day 2 isP(either) = P(up Day 1) + P(up Day 2) — P(up Day 1 and up Day 2)=.75+.55—.40=.90 _ ~
Markov Chains -
If we believe that today’s price movement is based in some part on what happened yester-day, we have a situation called conditional probability. This can be expressed as a Markovprocess, ot Markov chain. The results, or outcomes, of a Markov chain express the proba-bility of a state or condition occurring. For example, the possibility of a clear, cloudy, orrainy day tomorrow might be related to today’s weather.
The different combinations of dependent possibilities are given by a transitionmatrix. In our weather prediction example, a clear day has a 70% chance of being fol-lowed by another clear day, a 25% chance of a cloudy day, and only a 5% chance of rain.In Table 2-3, each possibility today is shown on the left, and its probability of changingtomorrow is indicated across the top. Each row totals 100%, accounting for all weathercombinations. The relationship between these events can be shown as a continuous net-work (see Figure 2-6).
: The Markov process can reduce intricate relationships to a simpler form. First, con-Sider a two-state process. Using the markets as an example, what is the probability of an upor down day following an up day, or following a down day? If there is a 70% chance of ahigher day following a higher day and a 55% chance of a higher day following a lower day,what is the probability of any day within an uptrend being up?"26 BASIC CONCEPTS
 
TABLE 2-3 Transition Matrix
TomorrowClear Cloudy RainyClear 70 25 05Today Cloudy 20 60 20Rainy 20 40 40
 
Start with either an up or down day, and then calculate the probability of the next daybeing up or down. This is done easily by simply counting the number of cases, given inTable 2-4a, then dividing to get the percentages, as shown in Table 2-4b.
Because the first day may be designated arbitrarily as up or down, it is an exception tothe general rule and, therefore, is given the weight of 50%. The probability of the secondday being up or down is the sum of the joint probabilities
P(UP); = (.50 x .70) + (.50 x .55)= 625
The probability of the second day being up is 62.5%. Continuing in the same manner, usethe probability of an up day as .625, the down as .375, and calculate the third day,
P(UP); = (.625 x .70) + (.375 X .55)= 64375and the fourth day,P(UP)4 = (.64375 X .70) + (35625 x .55)= 64656
which can