quantitative methods for finance - lecture 1
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Quantitative methods for finance
Lecture 1
Serafeim Tsoukas
Chapter 1: Descriptive statistics
• Descriptive statistics summarises a mass of
information.
• We may use graphical and/or numerical methods.
• Examples of the former are the bar chart and XY
chart, examples of the latter are averages and
standard deviations.
Graphical techniques
• Education and employment data
Higher A levels Other No Total
education qualification qualification
In work 9,713 5,479 10,173 1,965 23,852
Unemployed 394 432 1,166 382 2,374
Inactive 1,256 1,440 3,277 2,112 8,084
Total 11,362 7,352 14,615 4,458 37,788
Table 1.1 Economic status and educational qualifications, 2009 (numbers in
000s) Source: Adapted from Department for Children, Schools and Familits, Education and Training Statistics for
the UK 2009, http://www.education.gov.uk/rsgateway/DB/VOL/v000891/, contains public sector information
licensed under the Open Government Licence (OGL) v1.0. http://www.nationalarchives.gov.uk/doc/opengovernment-
licence/open-government
0
2,000
4,000
6,000
8,000
10,000
12,000
Higher education Advanced level Other qualifications No qualifications
Num
ber
of
people
(000s)
The bar chart
Figure 1.1 Educational qualifications of people in work in the UK, 2009
Note: The height of each bar is determined by the associated frequency. The first bar is 9,713 units high,
the second is 5,479 and so on. The ordering of the bars could be reversed (‘no qualifications’ becoming
the first category) without altering the message.
9,713
A multiple bar chart
Figure 1.2 Educational qualifications by employment category
0
2,000
4,000
6,000
8,000
10,000
12,000
Higher education
Advanced level Other qualifications
No qualifications
Nu
mb
er
of p
eo
ple
(0
00
s)
In work
Unemployed
Inactive
The stacked bar chart
Figure 1.3 Stacked bar chart of educational qualifications and employment status
0
2,000
4,000
6,000
8,000
10,000
12,000
14,000
16,000
Higher education
Advanced level Other qualifications
No qualifications
Num
ber
of people
(000s)
Inactive
Unemployed
In work
A stacked bar chart (percentages)
Figure 1.4 Percentages in each employment category, by educational qualification
0%
20%
40%
60%
80%
100%
Higher education Advanced level Other qualifications
No qualifications
Inactive
Unemployed
In work
The pie chart
Figure 1.5 Educational qualifications of those in work
Data on wealth in the UK
Table 1.3 The distribution of wealth, UK, 2005 Source: Adapted from HM Revenue and Customs Statistics, 2005, contains public sector information licensed
under the Open Government Licence (OGL) v1.0.http://www.nationalarchives.gov.uk/doc/open-governmentlicence/
open-government
Class interval (£) Numbers (thousands)
0–9,999 1,668
10,000–24,999 1,318
25,000–39,999 1,174
40,000–49,999 662
50,000–59,999 627
60,000–79,999 1,095
80,000–99,999 1,195
100,000–149,999 3,267
150,000–199,000 2,392
200,000–299,000 2,885
300,000–499,999 1,480
500,000–999,999 628
1,000,000–1,999,999 198
2,000,000 or more 88
Total 18,667
A (misleading!) bar chart
Figure 1.7 Bar chart of the distribution of wealth in the UK, 2005
0
500
1,000
1,500
2,000
2,500
3,000
3,500
0
10,0
00
25,0
00
40,0
00
50,0
00
60,0
00
80,0
00
100,0
00
150,0
00
200,0
00
300,0
00
500,0
00
1,0
00,0
00
2,0
00,0
00
Num
ber
of
indiv
iduals
Income class (lower boundary)
The histogram – the correct picture
Figure 1.9 Histogram of the distribution of wealth in the UK, 2005
Histogram versus bar chart
• The bar chart gives the wrong picture because of varying class widths.
• These two classes have similar frequencies (similar heights for bar chart) but the second is over 13 times wider. Adjusting for width, its frequency should be 111 (1,480 15/200).
Class interval £) Numbers (thousands)
10,000–24,999 1,318
: :
300,000–499,000 1,480
The frequency density
• Applying this principle leads to calculation of the
frequency densities:
Range Number, or
Frequency
Class width Frequency density
0– 1,668 10,000 0.1668
10,000– 1,318 15,000 0.0879
25,000– 1,174 15,000 0.0783
40,000– 662 10,000 0.0662
50,000– 627 10,000 0.0627
60,000– 1,095 20,000 0.0548
80,000– 1,195 20,000 0.0598
Numerical techniques
• We examine the measures of
– Location
– Dispersion
– Skewness.
Measures of location
• Mean – strictly the arithmetic mean, the well known
‘average’
• Median – the wealth of the person in the middle of
the distribution
• Mode – the level of wealth that occurs most often
• These different measures can give different answer.
The mean of the wealth distribution
875.186677,18
260,490,3
f
fx
Mean wealth is £186,875
Range x f fx
0 - 5.0 1,668 8,340
10,000 - 17.5 1,318 23,065
25,000 - 32.5 1,174 38,155
40,000 - 45.0 662 29,790
50,000 - 55.0 627 34,485
60,000 - 70.0 1,095 76,650
80,000 - 90.0 1,195 107,550
100,000 - 12 5.0 3,267 408,375
150,000 - 175.0 2,392 418,600
200,000 - 250.0 2,885 721,250
300,000 - 400.0 1,480 592,000
500,000 - 750.0 628 471,000
1,000,000 - 1500.0 198 297,000
2,000,000 - 3000.0 88 264,000
Total 18,677 3,490,260
Locating the mean
The median
• The wealth of the ‘middle person’ – i.e. the one
located halfway through the distribution.
• The median is little affected by outliers, unlike the
mean.
This person’s wealth
Poorest Richest
Simple example
• Values 45, 12, 33, 80, 77
• What is the median value?
• Put the values in order: 12, 33, 45, 77, 80
Median of the five values
Range Frequency
Cumulative
frequency
0– 1,668 1,668
10,000– 1,318 2,986
25,000– 1,174 4,160
40,000– 662 4,822
50,000– 627 5,449
60,000– 1,095 6,544
80,000– 1,195 7,739
100,000– 3,267 11,006
Calculating the median – wealth
• 18,677 (thousand) observations, hence person 9,338.5 in rank order
has the median wealth
• This person is somewhere in the £100–150k interval
Number with wealth
less than £100k
Number with wealth
less than £150k
• To find the precise median value, use
• Median wealth is £124,480.
f
FN
xxx 2LUL
Calculating the median (Continued)
4801243,267,000
000100000150000100
7,739,0002
18,677,000
,,,,
The mode
• The mode is the observation with the highest
frequency
Size Sales
8 7 10 25 12 36 14 11 16 3 18 1
Modal dress size = 12
Range Number, or
Frequency
Class
width
Frequency
density
0– 1,668 10,000 0.1668
10,000– 1,318 15,000 0.0879
25,000– 1,174 15,000 0.0783
40,000– 662 10,000 0.0662
50,000– 627 10,000 0.0627
: : : :
• For grouped data, the mode corresponds to the
interval with greatest frequency density.
Modal
class
Mode = £0–10,000
The mode – wealth data
Differences between mean,
median and mode
Figure 1.12 The histogram with the mean, median and mode marked
0 50 10 100 80 60 40 200 150 25
Class widths squeezed
Wealth (£000)
Mode Median Mean
Measures of dispersion
• The range – the difference between smallest and
largest observation. Not very informative for
wealth.
• Inter-quartile range – contains the middle half of
the observations.
• Variance – based on all observations in the
sample.
Inter-quartile range
• First quartile – one-quarter of the way through the
distribution, person ranked 4,669.25
• Third quartile – three quarters of the way through the
distribution, person ranked 14,007.75 hence
Q3 = 221,135.1
• IQR = Q3 – Q1 = 221,135 – 47,693 = 173,442.
6.692,47662
160,425.669,4000,40000,50000,401
Q
100
200
300
400
x
Wealth
(£000)
Box and whiskers plot
Median
First quartile
Third quartile
Outlier
IQR
The variance
• The variance is the average of all squared
deviations from the mean:
• The larger this value, the greater the dispersion of
the observations.
f
xf2
2
Small
variance Large
variance
The variance (Continued)
Range
Mid-
point x
(£000)
Frequency,
f
Deviation
(x – )
(x – )2
f(x – )
2
0– 5.0 1,668 –181.9 33,078.4 55,174,821.9
10,000– 17.5 1,318 –169.4 28,687.8 37,810,535.3
25,000– 32.5 1,174 –154.4 23,831.6 27,978,261.2
40,000– 45.0 662 –141.9 20,128.4 13,325,033.3
50,000– 55.0 627 –131.9 17,391.0 10,904,128.1
: : : : : :
1,000,000– 1500.0 198 1,313.1 1,724,297.9 341,410,980.4
2,000,000– 3000.0 88 2,813.1 7,913,673.6 696,403,275.4
Totals 18,677 1,499,890,455.1
Calculation of the variance
830680
22
67718
14558904991.,
f
xf
,
.,,,
The standard deviation
• The variance is measured ‘squared £s’ (because
we used squared deviations).
• Hence take the square root to get back to £s.
This gives the standard deviation:
or £283,385.
385283830680 ..,
Sample measures
• For sample data, use to calculate the sample variance.
• This gives an unbiased estimate of the population variance.
• Take the square root of this for the sample standard deviation.
1
2
2
n
xxfs
Measuring skewness
Right skewed
CS > 0
Left skewed
CS < 0
3
3
N
xfskewnessoftCoefficien
Skew of the wealth distribution
8985
7147572267718
02355188250623
3
.,,,
,,,,
N
xf
Range x f x- (x- 3 f (x- 3
0 - 5.0 1,668 -181.9 -6,016,132 -10,034,907,815
10,000 - 17.5 1,318 -169.4 -4,858,991 -6,404,150,553
: : : : : :
500,000 - 750.0 628 563.1 178,572,660 112,143,630,236
1,000,000 - 1500.0 198 1313.1 2,264,219,059 448,315,373,613
2,000,000 - 3000.0 88 2813.1 22,262,154,853 1,959,069,627,104
Total 18,677 3,898.8 24,692,431,323 2,506,882,551,023
Summary
• We can use graphical and numerical measures to
summarise data.
• The aim is to simplify without distorting the
message.
• Measures of location, dispersion and skewness
provide a good description of the data.
Descriptive statistics: Time series data
• Slightly different techniques are used for time
series data – data on one or more variable over
time.
• We look at investment data in the UK by way of
example.
Investment data: 1977–2009
Not very informative – we need a graph
Year Investment Year Investment Year Investment
1977 28,351 1988 97,956 1999 161,722
1978 32,387 1989 113,478 2000 167,172
1979 38,548 1990 117,027 2001 171,782
1980 43,612 1991 107,838 2002 180,551
1981 43,746 1992 103,913 2003 186,700
1982 47,935 1993 103,997 2004 200,415
1983 52,099 1994 111,623 2005 209,758
1984 59,278 1995 121,364 2006 227,234
1985 65,181 1996 130,346 2007 249,517
1986 69,581 1997 138,307 2008 240,361
1987 80,344 1998 155,997 2009 204,270
Source: Data adapted from the Office for National Statistics licenced under the Open Government Licence v.1.0.
http://www.nationalarchives.gov.uk/doc/open-government-licence/open-government
Time series chart of investment
Figure 1.16 Time–series graph of investment in the UK, 1977–2009
0
50,000
100,000
150,000
200,000
250,000
300,000
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
2007
2009
Investm
ent
Chart of the change in investment
Figure 1.17 Time−series graph of the change in investment
-40,000
-30,000
-20,000
-10,000
0
10,000
20,000
30,000
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
2007
2009
Change in investm
ent
The logarithm of investment
Figure 1.18 Time–series graph of the logarithm of investment expenditures
8.0
8.5
9.0
9.5
10.0
10.5
11.0
11.5
12.0
12.5
13.0
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
2007
2009
Log investm
ent
Graphing several series
Figure 1.20 A multiple time–series graph of investment
0.00
10,000.00
20,000.00
30,000.00
40,000.00
50,000.00
60,000.00
70,000.00
80,000.00
90,000.00
100,000.00
1977
1979
1981
1983
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
2007
2009
Investm
ent expenditure
s
Dwellings
Transport
Machinery
Intangible fixed assets
Other buildings
An area graph of the same data
Figure 1.22 Area graph of investment categories, 1977–2009
These are Excel’s
default colours!
Using separate axes – investment and the interest rate
Figure 1.21 Time–series graph using two vertical scales: investment (LH scale)
and the interest rate (RH scale), 1985–2005
0
2
4
6
8
10
12
14
16
0
50,000
100,000
150,000
200,000
250,000
1985
1987
1989
1991
1993
1995
1997
1999
2001
2003
2005
Inte
rest ra
te (
%)
Investm
ent (£
m)
Interest rates
Investment
Numerical summary measures
• It makes more sense to calculate the average
growth rate of investment, rather than the level.
• The growth rate is similar each year, the level
continuously increases.
• Calculate the growth factor over the whole time
period:
• Take the T−1 root:
• Subtract 1: 1.0637 − 1 = 0.0637
• The average growth rate is 6.4% p.a.
The growth rate of investment
2050.7351,28
270,204
1
x
xT
06371205732 ..
An approximate alternative
• The average growth rate can also be calculated
as the arithmetic mean of the annual growth rates:
i.e. 6.6%.
• This gives approximately the right answer, as long
as the growth rate is not too big.
0664.132
850.0963.0190.1142.1
Variance of the growth rate
• The stability of growth can be measured by calculating the
variance of the growth rate.
0766.0
0059.0
31
066.032323.0
12
22
2
s
n
xnxs
Year Investment
Growth
rate, x x2
1978 32,387 0.142 0.020
1979 38,548 0.190 0.036
1980 43,612 0.131 0.017
: : : :
2006 227,234 0.083 0.007
2007 249,517 0.098 0.010
2008 240,361 -0.037 0.001
2009 204,270 -0.150 0.023
Totals 2.1253 0.3230
Bivariate data
• We examine the relationship between investment and
GDP.
Figure 1.24 Scatter diagram of investment (vertical axis) against GDP (horizontal
axis) (nominal values)
0
50,000
100,000
150,000
200,000
250,000
300,000
0 200,000 400,000 600,000 800,000 1,000,000 1,200,000
Inve
stm
en
t (£
m)
GDP (£m)
• High values of investment seem associated with
high values of GDP, there is a close relationship.
• As both variables are growing over time, later
observations are at the top right of the graph, but
this does not have to be so.
• Both variables are influenced by inflation, so it
might be better to graph the real series, after
adjusting for inflation.
Bivariate data (Continued)
Figure 1.25 The relationship between real investment and real output
Real investment versus real GDP
50 000
100 000
150 000
200 000
250 000
300 000
500 000 700 000 900 000 1100 000 1300 000 1500 000
Real In
vestm
ent
Real GDP
Summary
• Slightly different graphical and numerical techniques are
used for time–series data
• A variety of time–series charts are available, both for single
and multiple series.
• The mean and variance are both useful descriptive
devices, but it makes more sense to apply them to the
growth rate, rather than the level, of a trended variable.
• Data transformations can be useful, e.g. taking logs or
differences and deflating to real terms.