1 Return 13 11 2 6 5 8 7 9 7 6 2 Difference from the mean 5.6 3.6 –5.4 –1.4 –2.4 0.6 -0.4 1.6 –0.4 –1.4 3 Difference squared 31.36 12.96 29.16 1.96 5.76 0.36 0.16 2.56 0.16 1.96 86.4 4 Average variance (86.4/10) 8.64
The standard deviation of a set of data is simply the square root of the variance and is the most commonly used measure of dispersion.
Although the variances and standard deviations of both ordered and raw frequency distribution data can be calculated quickly and easily by using a scientific calculator, it is useful to understand how to work through their calculations manually. In summary, the steps you should take in making these
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To ensure precision in the calculation of the variance and standard deviation, statistical rules require a slight change to the formula if measuring a sample rather than the whole population. A small data set might not provide a representative picture of the population as a whole and so sampling error may arise. As a result, a slight adjustment to the standard deviation formula is made by reducing the number of observations by one.
Population standard deviation =
∑ (x – x)2
n
√
Sample standard deviation =
∑ (x – x)2
n – 1
√
In effect, by taking the square root of the variance, the standard deviation represents the average amount by which the values in the distribution deviate from the mean.
2.4
Relevance and Application of Dispersion and Variance
Measures
Learning Objective
5.2.4 Understand the relevance and application of measures of dispersion and variance within risk analysis
The variance of a set of stock returns provides a measure of the returns’ dispersion and is used to calculate the beta of the stock – a measure of how closely its movement mirrors that of the general market (see below). However, it results in a value with different units from the original values, whereas it is easier to conceptualise the dispersion of a set of values when its value is expressed in the same units as the returns themselves. To do this, we use the standard deviation.
The variability of returns generated by an asset or portfolio – its volatility – is a measure of its risk. The volatility of an investment’s returns is expressed mathematically by the standard deviation of their values – because, as we have seen, standard deviation measures how widely the values are dispersed, or fluctuate, around their mean position.
The more volatile an investment’s return is, the greater is the standard deviation. Low standard deviation implies low risk; high standard deviation implies high risk. Knowing the standard deviation can help us to know the range of different values of return we might expect from an investment. Experience shows that for about two-thirds of the time, we can expect the return to be within one standard deviation above or below the average return.
2.5
Understanding the Terminology
Learning Objective
5.2.5 Understand the terms distribution analysis, confidence intervals, normal distribution and fat- tailed distribution, and how they are used within risk analysis
2.5.1 Distribution Analysis
Distribution analysis is a statistical means of using historical data to predict future events and relies on an understanding of probability distributions – of which one such distribution is the ‘normal distribution’, which we discussed in Section 2.3. The distribution of UK incomes was another example; it was positively skewed – unlike the normal, which is symmetrical about its mean.
A normal distribution curve has the following attributes:
• it is symmetrical about its mean
• it is defined by its mean and its standard deviation (represented by the Greek letter sigma,
σ
). Statistical analysis shows that for a normal distribution:• two thirds or 68.3% of the data will be within one standard deviation either side of the mean • 95.5% of the data will be within two standard deviations either side of the mean
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So, using these twin distribution concepts of central tendency value, and confidence interval, we can start to include a measure of confidence in our reported measure of returns.
Instead of using simply the median or mean value, we can describe it by saying ‘there is a 95.5% chance
that the return will be within two standard deviations of the mean’ – where we will have calculated the
value of ‘two standard deviations’.
2.5.3 Fat-Tailed Distribution
It can be seen from the graph above, that after three, four, or perhaps five standard deviations, the edges of the graph, or its ‘tails’ will have touched the horizontal x axis.
However, this is one of the aspects of using normal distribution as a simplifying assumption that is often not true in ‘real life’. Many probability distributions have what are known as fat tails, as illustrated by the weekly income graph in Section 2.3. This means that using a normal distribution for simplicity when in fact the distribution is fat-tailed will result in periodically very large deviations from our expected returns.
2.6
Risk Measurement and Control Concepts
Learning Objective
5.2.6 Understand the following concepts used in risk measurement and control: probability, volatility, regression, correlation coefficients alpha and beta, optimisation
2.6.1 Probability
Probability is the measure of how likely an event is. In order to measure probabilities, mathematicians have devised the following formula for finding the probability of an event:
The number of ways the event can occur The total number of possible outcomes
So as a simple example, if a single six-sided die is rolled, the probability of rolling a six would be:
The number of ways to roll a six
= 1/6
The total number of sides
The probability of rolling an even number would be:
The number of ways to roll an even number
2.6.2 Volatility
When discussing market prices, the volatility is a description of how variable they are. So, typically, bond prices would be less volatile than equities.
Technically, volatility is a measure of the standard deviation of the returns of a financial instrument within a specific time horizon. It is often used to quantify the risk of the instrument over that time expressed in annualised terms, and it may either be an absolute number (£5) or a fraction of the mean return (5%).
2.6.3 Regression
Regression analysis is a statistical tool for the investigation of relationships between variables. It is used to ascertain the effect of one variable upon another – the effect of a price increase upon demand, for example, or the effect of changes in the money supply upon the inflation rate.
To explore such issues, data is gathered and regression analysis is employed to estimate the quantitative effect of one variable upon another. The statistical significance of the estimated relationships is also assessed, that is, the degree of confidence that the true relationship is close to the estimated relationship.
The simplest way of determining potential relationships between two variables, is to measure their differing values over time and then plot a scatter-gram. The example below shows that there is a clear relationship between the variables plotted on the x and y axes. What it will not show is which variable is causing the change, and that will need to be determined using other methods.