Quantitative Methods
Time Value of Money
Interest rates
3 ways to consider IR:
1. Required rate of return- minimum an investor will accept for an investment
2. Discount rates- what the future value is worth today
3. Opportunity cost- Amount forgone by choosing a particular idea
Taking the perspective of investors, can view IR rate as being composed of a real risk-free IR plus
four premiums that are required returns or compensation: R equals:
1. Risk-free IR rate- single-period interest rate for completely risk-free security if no
inflation expected over the maturity of debt. Reflects time preferences.
2. Inflation premium- compensates investors for expected inflation and reflects average
inflation rate expected over the life of debt. Inflation reduces purchasing power of a
currency. Sum of Rf and inflation premium known as nominal Rf IR.
3. Default risk premium- Compensate investors for possible defaults
4. Liquidity premium- Compensates investors for the risk of loss relative to an investments
fair value if it needs to be converted into cash quickly. Also, the costs of selling can be
considered.
5. Maturity premium- Compensate investors for the increased sensitivity of the market
value of debt to a change in IR as maturity is extended. PV and FV important points
PV and FV are separated has the important consequences:
o Can add money only if indexed at same point in time
o For given IR, FV increases with N periods
o For given number N, FV increases with IR
Non-annual compounding (FV)
Usually quote the annual interest rate known as the stated annual IR/quoted IR. Instead of the
monthly IR rate.
As the number of compounding periods increases EAR increases at a decreasing rate
Stated and effective rates
Stated IR does not give the fv directly
FV of a series of CF, FV annuities
Annuity- a finite set of level sequential cfs
Ordinary annuity- has a first CF that occurs one period from now; indexed at t=1
Annuity due- has a first cf that occurs immediately; indexed at t=0
Perpetuity- in a perpetual annuity, or a set of level never-ending sequential cfs with the first cfs
occurring one period from now. Annuity that goes on forever.
PV evaluation
For a given discount rate, the farther in the future the amount to be received, the smaller the
amount’s PV
Holding time constant, the larger the DR the smaller the PV of a future amount
A lump sum can fund an annuity by taking the equivalence between PV and FV
, Organising, visualising, and describing data
Numerical Vs Categorical data
Numerical- continuous or discrete data:
o Continuous- data that can be measured and can take on any numerical value in a
specified range of values.
o Discrete- These data are limited to a finite number of values.
o Ordinal data- Ordinal data are categorical values that can be logically ordered or ranked.
But cannot be subject to arithmetic operations
Categoric- Qualitative data. Values that describe a quality of a group E.g bankrupt vs not
bankrupt. Mutually exclusive usually.
o Nominal data-Can’t be organised in logical order. E.g classifying stocks into 11 sectors.
Ordinal data can be ordered/ranked.
o Cross sectional data- based on specific time for different variables E.g January inflation
rates (variable) for each Euro-area countries (observation) in the EU for a given year
o Time-series data- Different times but a single variable e.g closing price of the stock for a
month.
o Panel data- mix of time-series and cross-sectional. Observations through time on one or
more variables for multiple observational. Observations usually in matrix called data
table. Might be quarterly EPS (variable) for 3 companies (observed unit) in given year
per quarter. Each column would show time-series and each row shows cross-sectional
data
Structured versus Unstructured data
Structured data: are highly organised in a pre-defined manner, usually with repeated patterns.
Structured data are relatively easy to enter, store, query and analyse without much manual
processing.
Unstructured data: Data that doesn’t follow any conventional organised forms.
Organising data for quantitative analysis (QA)
Depending on no.variables can be organised into two forms of QA:
1. One-dimensional array: simplest format for representing a collection of the same data
type.
2. Two-dimensional array: also known as a data table is a popular way for organising data
for processing by computers. Summarising data using Frequency distributions
A frequency distribution (one-way table) is a display of data constructed either by counting the
observations of a variable by distinct values or by tallying the values of a numerical variable into
a set of numerically ordered in bins.
Absolute frequency- Would be the raw frequency the number of observations counted for each
unique value of a variable. E.g 5
Express frequencies in terms of percentages to show the relative frequency. E.g 15.2% are tech
Contingency table
Use to summarise date for two variables simultaneously-two-way
table
Listing all the levels e.g categories of one variable as rows and all
the levels of other variables as columns in the table
Can be ordered (ordinal data) or unordered (nominal data)
, Data displayed in cells can be frequency- count or relative frequency- %
The marginal frequency is the total of the row or column
If test statistic is greater than chi-squared distribution value then evidence of association
Data visualisation
Histograms: Distribution by the numerical data using the height of a bar or column to represent
the absolute frequency of each bin or interval in the distribution. Height of bar shows the
absolute frequency and width shows the bin
Frequency polygon: Plot the midpoint on the X axis (bottom) and the absolute frequency for the
bon on the Y axis. Distribution of numerical data.
Cumulative frequency: Cumulative relative/absolute frequency on the Y-axis against the upper
limit of the interval. View the number of percentage of observations that lie below a certain
value
Bar chart: Each bar represents a different category. Bars height is frequency. If cumulative then
will have a line known as a pareto chart. If have two categorical variables can use grouped bar
chart or clustered bar charts or stacked bar charts
Tree map: Is like a heat map on Bloomberg. Can have more than one categorical value for
example spit each sector it mid, small and large cap. Value differences of categorical groups
Word cloud: Use structured and textual data. Can adjust the size
Line chart: Change of data series over time. Plot against X and Y e.g 10-daily closing prices
against time. Can use a bubble line chart if need a third dimension of data. A bubble line chart is
a version of a line chart where data points are replaced with varying-sized bubbles to represent
a third dimension of the data. A line chart is very effective at visualizing trends in three or more
variances.
Scatter plot: Type of graph for visualising the joint variation in two numerical variables. patterns
and associations between variables. Scatter plot matrix combines heat map and scatter
Heat map: Colour based on high/low values
Measures of central tendency
The use of quantitative measures that explains characteristics of data
Measures of central tendency specifies where data is centred
Measures of location include central tendency and illustrate the location or distribution of data
A statistic is a summary measure of a set of observations and descriptive statists summarise the
central tendency and spread variation in distribution of data
If summarise All possible observations of a population make statistic a parameter
If summarise a subset refer to as sample statistic
Outliers
1. Do nothing use data with no adjustments- if the values are legitimate and takes away
judgement of the analyst to decide if values are extreme- arithmetic mean
2. Delete all the outliers- trimmed mean: remove e.g lowest and highest 2.5%.
3. Replace all the outliers with another value- winsorized mean. Sub values for extreme
values. Calculated by assigning a stated percentage of the lowest values equal to one
specified low value and a stated percentage of the highest values equal to one specified
high value, and then it computes a mean from the restated data
Both the trimmed and winsorized lead to means being higher than arithmetic mean
Time Value of Money
Interest rates
3 ways to consider IR:
1. Required rate of return- minimum an investor will accept for an investment
2. Discount rates- what the future value is worth today
3. Opportunity cost- Amount forgone by choosing a particular idea
Taking the perspective of investors, can view IR rate as being composed of a real risk-free IR plus
four premiums that are required returns or compensation: R equals:
1. Risk-free IR rate- single-period interest rate for completely risk-free security if no
inflation expected over the maturity of debt. Reflects time preferences.
2. Inflation premium- compensates investors for expected inflation and reflects average
inflation rate expected over the life of debt. Inflation reduces purchasing power of a
currency. Sum of Rf and inflation premium known as nominal Rf IR.
3. Default risk premium- Compensate investors for possible defaults
4. Liquidity premium- Compensates investors for the risk of loss relative to an investments
fair value if it needs to be converted into cash quickly. Also, the costs of selling can be
considered.
5. Maturity premium- Compensate investors for the increased sensitivity of the market
value of debt to a change in IR as maturity is extended. PV and FV important points
PV and FV are separated has the important consequences:
o Can add money only if indexed at same point in time
o For given IR, FV increases with N periods
o For given number N, FV increases with IR
Non-annual compounding (FV)
Usually quote the annual interest rate known as the stated annual IR/quoted IR. Instead of the
monthly IR rate.
As the number of compounding periods increases EAR increases at a decreasing rate
Stated and effective rates
Stated IR does not give the fv directly
FV of a series of CF, FV annuities
Annuity- a finite set of level sequential cfs
Ordinary annuity- has a first CF that occurs one period from now; indexed at t=1
Annuity due- has a first cf that occurs immediately; indexed at t=0
Perpetuity- in a perpetual annuity, or a set of level never-ending sequential cfs with the first cfs
occurring one period from now. Annuity that goes on forever.
PV evaluation
For a given discount rate, the farther in the future the amount to be received, the smaller the
amount’s PV
Holding time constant, the larger the DR the smaller the PV of a future amount
A lump sum can fund an annuity by taking the equivalence between PV and FV
, Organising, visualising, and describing data
Numerical Vs Categorical data
Numerical- continuous or discrete data:
o Continuous- data that can be measured and can take on any numerical value in a
specified range of values.
o Discrete- These data are limited to a finite number of values.
o Ordinal data- Ordinal data are categorical values that can be logically ordered or ranked.
But cannot be subject to arithmetic operations
Categoric- Qualitative data. Values that describe a quality of a group E.g bankrupt vs not
bankrupt. Mutually exclusive usually.
o Nominal data-Can’t be organised in logical order. E.g classifying stocks into 11 sectors.
Ordinal data can be ordered/ranked.
o Cross sectional data- based on specific time for different variables E.g January inflation
rates (variable) for each Euro-area countries (observation) in the EU for a given year
o Time-series data- Different times but a single variable e.g closing price of the stock for a
month.
o Panel data- mix of time-series and cross-sectional. Observations through time on one or
more variables for multiple observational. Observations usually in matrix called data
table. Might be quarterly EPS (variable) for 3 companies (observed unit) in given year
per quarter. Each column would show time-series and each row shows cross-sectional
data
Structured versus Unstructured data
Structured data: are highly organised in a pre-defined manner, usually with repeated patterns.
Structured data are relatively easy to enter, store, query and analyse without much manual
processing.
Unstructured data: Data that doesn’t follow any conventional organised forms.
Organising data for quantitative analysis (QA)
Depending on no.variables can be organised into two forms of QA:
1. One-dimensional array: simplest format for representing a collection of the same data
type.
2. Two-dimensional array: also known as a data table is a popular way for organising data
for processing by computers. Summarising data using Frequency distributions
A frequency distribution (one-way table) is a display of data constructed either by counting the
observations of a variable by distinct values or by tallying the values of a numerical variable into
a set of numerically ordered in bins.
Absolute frequency- Would be the raw frequency the number of observations counted for each
unique value of a variable. E.g 5
Express frequencies in terms of percentages to show the relative frequency. E.g 15.2% are tech
Contingency table
Use to summarise date for two variables simultaneously-two-way
table
Listing all the levels e.g categories of one variable as rows and all
the levels of other variables as columns in the table
Can be ordered (ordinal data) or unordered (nominal data)
, Data displayed in cells can be frequency- count or relative frequency- %
The marginal frequency is the total of the row or column
If test statistic is greater than chi-squared distribution value then evidence of association
Data visualisation
Histograms: Distribution by the numerical data using the height of a bar or column to represent
the absolute frequency of each bin or interval in the distribution. Height of bar shows the
absolute frequency and width shows the bin
Frequency polygon: Plot the midpoint on the X axis (bottom) and the absolute frequency for the
bon on the Y axis. Distribution of numerical data.
Cumulative frequency: Cumulative relative/absolute frequency on the Y-axis against the upper
limit of the interval. View the number of percentage of observations that lie below a certain
value
Bar chart: Each bar represents a different category. Bars height is frequency. If cumulative then
will have a line known as a pareto chart. If have two categorical variables can use grouped bar
chart or clustered bar charts or stacked bar charts
Tree map: Is like a heat map on Bloomberg. Can have more than one categorical value for
example spit each sector it mid, small and large cap. Value differences of categorical groups
Word cloud: Use structured and textual data. Can adjust the size
Line chart: Change of data series over time. Plot against X and Y e.g 10-daily closing prices
against time. Can use a bubble line chart if need a third dimension of data. A bubble line chart is
a version of a line chart where data points are replaced with varying-sized bubbles to represent
a third dimension of the data. A line chart is very effective at visualizing trends in three or more
variances.
Scatter plot: Type of graph for visualising the joint variation in two numerical variables. patterns
and associations between variables. Scatter plot matrix combines heat map and scatter
Heat map: Colour based on high/low values
Measures of central tendency
The use of quantitative measures that explains characteristics of data
Measures of central tendency specifies where data is centred
Measures of location include central tendency and illustrate the location or distribution of data
A statistic is a summary measure of a set of observations and descriptive statists summarise the
central tendency and spread variation in distribution of data
If summarise All possible observations of a population make statistic a parameter
If summarise a subset refer to as sample statistic
Outliers
1. Do nothing use data with no adjustments- if the values are legitimate and takes away
judgement of the analyst to decide if values are extreme- arithmetic mean
2. Delete all the outliers- trimmed mean: remove e.g lowest and highest 2.5%.
3. Replace all the outliers with another value- winsorized mean. Sub values for extreme
values. Calculated by assigning a stated percentage of the lowest values equal to one
specified low value and a stated percentage of the highest values equal to one specified
high value, and then it computes a mean from the restated data
Both the trimmed and winsorized lead to means being higher than arithmetic mean
