Chapter 13 FM

8/7/2019 Chapter 13 FM

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Chapter 13: Time Series Models

Outcomes of learning

Calculate

o Moving averages

o Exponentially Smoothed Values

Plot the graphs of

o Moving Averages

o Exponentially Smoothed Values

Forecast future values

Identify buying and selling signals in stock prices

Choose the best forecasting model

Most of the financial data are recorded at different points of time. These data are known as time

series data. Table 12.1 shows an example of time series data. It is recorded annually, and therefore isan annual time series data. Other frequencies: quarterly, monthly, weekly, daily.

Table 12.1: Malaysian Money Demand Data

Year

M2

Money Demand

(M2) in Millions

Ringgit Year

M2

Money Demand (M2)

in Millions Ringgit

1969 3718.500 1988 64072.100

1970 4122.300 1989 74392.800

1971 4668.200 1990 83902.900

1972 7551.900 1991 96092.500

1973 7551.900 1992 114481.000

1974 8713.900 1993 139800.000

1975 9981.500 1994 160366.000

1976 12748.200 1995 198873.000

1977 14819.000 1996 238209.000

1978 17466.500 1997 292217.000

1979 21706.400 1998 296472.000

1980 27991.800 1999 337138.000

1981 32772.700 2000 354702.000

1982 37899.900 2001 362512.000

1983 42264.100 2002 383542.000

1984 47733.200 2003 426061.000

1985 50412.200 2004 534163.000

1986 56096.800 2005 616178.000

1987 59771.700 2006 718216.000


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Figures 12.1 and 12.2 show the plots of the Malaysian Money Demand for different sample periods.

Figure 12.1: Malaysian Money Demand (1965 2006)

Over the long run, there is a linear increasing trend.

Movement of the series is predictable.

0.000

100000.000

200000.000

300000.000

400000.000

500000.000

600000.000

700000.000

800000.000

1975 1980 1985 1990 1995 2000 2005 2010

MalaysiaM

oneyDemand(MillionsRinggit)

Year

Malaysian Money Demand (1980-2006)

-200000.000

-100000.000

0.000

100000.000

200000.000

300000.000

400000.000

500000.000

600000.000

700000.000

800000.000

1975 1980 1985 1990 1995 2000 2005 2010MalaysiaMoneyDeman

d(MillionsRinggit)

Year



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Some data are lying outside the straight line due to random variations/disturbances irregular

changes that are not following the long-term trend.

over the long run, there is a linear increasing curvilinear trend.

Movement of the series is predictable.

random variations/disturbances are again observed.

0.000

100000.000

200000.000

300000.000

400000.000

500000.000

600000.000

700000.000

800000.000

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

MalaysiaM

oneyDemand(MillionsRinggit)

Year


0.000

100000.000

200000.000

300000.000

400000.000

500000.000

600000.000

700000.000

800000.000

900000.000

1965 1970 1975 1980 1985 1990 1995 2000 2005 2010

MalaysiaMoneyDeman

d(MillionsRinggit)

Year



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Long-term trend is less clear for higher frequency data.

Movement/Behavior of these series are less obvious/predictable.

Need the help of time series models to trace/capture the movement.

Time Series Models:

o Moving Average

o Exponential Smoothingo Autoregressive Process

12.2 Moving Average

A financial analysis tool that shows the average value of a securitys price over a period of time.

Use to smooth out the random variation so as to uncover the direction of a trend in the time series.

To compute the three-period moving average for any time period:

3

VVV 1tt1t

where

Vt = Value of the time series at that time

Vt-1 = Value in the previous time period

Vt+1 = Value in the following time period

Time Period Stock Price 3-Period Moving Average

1965 27 -

1966 17.3 (27+17.3+32.5)/3=25.6

1967 32.5 (17.3+32.5+42.8)/3=30.9

1968 42.8 39.1

1969 42.0 44.1

1970 47.4 32.5

1971 8.2 27.8

1972 27.7 25.6

1973 41 29.3

1974 19.2 27.8

1975 23.1 -


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The MA is placed at the center of the group of values being averaged.

To compute the 5-period MA,

o Obtain the average value of 5 consecutive values and place the MA in the center of

the group of values.

Time Period Stock Price 5-Period Moving Average

1965 271966 17.3

1967 32.5 34.8

1968 42.8 31.7

1969 42 33.4

1970 47.4 34.9

1971 8.2 30.9

1972 27.7 27.8

1973 41 23.8

1974 19.2

1975 23.1

4-period MA

Time Period Stock Price 4-Period

Moving

Average

4-Period Centered Moving

Average

1965 27

1966 17.3

29.9

1967 32.5 (29.9+33.7)/2=31.8

33.7

1968 42.8 37.4

41.2

1969 42 38.1

35.1

1970 47.4 33.2

31.3

1971 8.2 31.2

31.1

1972 27.7 27.6

24.01973 41 25.9

27.8

1974 19.2

1975 23.1

The moving averages fall in between the time periods.

Creating various problems graphing difficulty.

Centering the moving averages corrects the problem.

Compute the 2-period MA of the Moving averages.


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Commonly use moving average in analysis the movement of daily stock prices

30-day MA, 50-day MA and 100-day MA.

Different time span tell different story.

The shorter the time span, the more sensitive the MA will be to prices changes.

The longer the time span, the less sensitive or the more smoothed the MA will be. The choice of time frame is subjective.

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30 per. Mov. Avg. (Series1)


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Usefulness of Moving Average

Forecast future values

o The last MA value is the predicted value for the future.

Determine the direction of the long-term trend

Identify turning point/reversal.

Benchmark for trading strategy.

o Crossovers:

When the price moves below the moving average, it is expected to fall-selling signal,

sell the stock, if any, before you lose your investment.

When the price protrudes the moving average, it is expected to rise- buying signal

Crossovers may not always give correct signals.

Filters: Filtering is used to increase our confidence about a signal/an indicator.

One may wait until the price crosses above the MA and is at least 10% above the MA to

make sure that it is a true crossover.

Setting the percentile to high could result in missing the boat and buying the stock at the

peak.

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Double crossovers

When shorter MA crosses the longer MA from above, it is a strong sign of selling signal (price

is going to fall).

When shorter MA crossovers the longer MA from below, it is a strong sign of buying signal

(price is going to rise).

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Tripple Crossovers

For extra insurance.

The shortest moving average must pass through the two higher ones.

Even stronger signal.

Using Excel to compute MA

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Press Enter

Copy


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Use Excel to Plot MA

Plot the data

Point to any part of the graph, right click the mouse


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Excel Built-in function for MA


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Note: the first value is placed at the end of the group of 3.


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12.3 Exponential Smoothing

2 drawbacks of moving average method:

Do not have MAs for the first and last sets of data loss of important information.

MA forgets most of the previous time-series values.

The 3-year moving average for year 2000 depends on the values in year 1999, 2000and 2001. Values in year 1998 and before have no influence on this MA.

Exponential Smoothing

Addresses the above problems.

Takes into account of all previous observations.

Assigns more weight to the latest observations.

Also known as Exponential Smoothing Moving Average or Exponential Moving

Average (EMA).

(the MA we previously study is also known as Simple Moving Average).

EMA formula

1ttt S)w1(wyS , )2tfor(

Where

St = Exponential smoothed time series at time t

Yt = Time series at time t

St-1 = Exponentially smoothed time series at time t-1

W= Smoothing constant, where 1w0

Note: Set tt yS

t22 S)w1(wyS

t

2

233

t233

y)w1(y)w1(wwyS

y)w1(wy)w1(wyS

t

3

2

2

343

t

2

2344

y)w1(y)w1(wy)w1(wwyS

y)w1(y)w1(wwy)w1(wyS

Smoothing constant w is chosen on the basis of how much smoothing is required.

The smaller the w, the smoother is the resulted series.


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Time Period Stock Price w=0.3 w=0.8

1965 27 27 27

1966 17.3 0.3(17.3)+0.7(27)=24.1 19.2

1967 32.5 0.3(32.5)+0.7(24.1)=26.6 29.8

1968 42.8 0.3(42.8)+0.7(26.6)=31.5 40.21969 42 34.6 41.6

1970 47.4 38.5 46.2

1971 8.2 29.4 15.8

1972 27.7 28.9 25.3

1973 41 32.5 37.9

1974 19.2 28.5 22.9

1975 23.1 26.9 23.1

(1-w) = damping factor.

The larger the damping factor, the smoother will be the resulted series.

For w=0.3, damping factor =0.7.

For w=0.8, damping factor = 0.2.


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Usefulness of EMA

Similar to MA. Forecasting the last EMA is the predicted future values

Trace the direction of trend

Identify the reversal

Trading strategies.

EMA with w=0.8 represents a shorter term trend

EMA with w=0.2 represents a longer term trend

The longer term trend is smoother.

Buying signal

shorter term trend cuts the longer term trend from below.

Selling signal shorter term trend cuts the longer term trend from above.


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Forecasting

One of the main objectives of modelling a financial series is to forecast/predict the future values of

the series.

Normally, a financial analyst will estimate a few models for the same set of data, and then choosethe bet model for the purpose of forecasting, based on certain objective criteria.

Few commonly used forecast accuracy criteria include:

Mean Absolute Error

n

yy

MAE

n

1t

tt

Mean Absolute Percentage Error

%100n

y

yy

MAPE

n

1t t

tt

Root Mean Square Error

%100

n

yy

RMSE

n

1t

2

tt

Among a group of models, the model with the smallest values of MAE, RMSE or MAPE is regarded as

the best (most accurate) model.

Time

Period

actual Forecasted (3-period

MA)

1966 17.3 25.6 8.3 68.89 0.479769

1967 32.5 30.9 1.6 2.56 0.049231

1968 42.8 39.1 3.7 13.69 0.086449

1969 42 44.1 2.1 4.41 0.05

1970 47.4 32.5 14.9 222.01 0.314346

1971 8.2 27.8 19.6 384.16 2.3902441972 27.7 25.6 2.1 4.41 0.075812

1973 41 29.3 11.7 136.89 0.285366

1974 19.2 27.8 8.6 73.96 0.447917

average 8.066666667 101.22 0.464348

Forecast Accuracy

Criterion MAE RMSE MAPE

tt yy 2tt yy t

tt

y

yy


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Time

Period

actual Forecasted (3-period

MA)

1966 17.3

1967 32.5 34.8 2.3 5.29 0.070769

1968 42.8 31.7 11.1 123.21 0.259346

1969 42 33.4 8.6 73.96 0.204762

1970 47.4 34.9 12.5 156.25 0.263713

1971 8.2 30.9 22.7 515.29 2.768293

1972 27.7 27.8 0.1 0.01 0.00361

1973 41 23.8 17.2 295.84 0.419512

1974 19.2

average 10.64285714 167.1214286 0.570001

Forecast Accuracy

Criterion MAE RMSE MAPE

The 3-period MA has a smaller MAE value as compared to the 5-period MA. Thus, by the MAE

forecast accuracy criteria, 3-period MA model is more accurate than the 5-period moving average

model. This conclusion is supported by the RMSE and MAPE criterion. S, 3-period MA model should

be chosen to forecast the future value of stock price.

tt yy 2tt yy t

tt

y

yy

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Chapter 13 FM