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STANDARD DEVIATION AND
PROBABILITY
PROBABILITY
- Probability refers to the likelihood that something will happen.
- A probability of 0.95 (or 95%) means it is expected to happen 19 times out of 20 on average.
- 0.05 (or 5%) means it is expected to happen 1 time out of 20 on average.
- (^) Note: If we sum up all the various probabilities that an event will occur, they must add up to 100% or to 1.00.
STANDARD DEVIATION TERMINOLOGY
Mean and average: symbol is ‘μ’
- These terms are synonymous:
- Total, divided by the number of observations.
STANDARD DEVIATION: symbol ‘SD’ or ‘σ’
- Risk implies a degree of uncertainty about an outcome.
- (^) The degree of uncertainty can be quantified by the standard deviation, or volatility, of possible outcomes.
Variance
- Square of the standard deviation. Var = SD 2 or σ^2.
- Standard deviation is the square root of variance. σ = √Var.
- Both are measures of risk.
The higher the standard deviation or variance, the greater the risk.
EXAMPLE
An investment has a set of possible returns of 4%, 5%, 6%, and 9%, which average 6%. So expected value is 6%.
This is a more volatile (risky) return than an investment with a guaranteed return of 6% which no variability.
The variability which is relevant in finance and risk management is the future variability (around the average expected value).
The standard deviation of the investment with a guaranteed 6% return with no variation will be zero. There will always be zero difference between the 6% reported for each period and the 6% mean.
- The larger the standard deviation, the larger the variance (variance is 2 ), the larger the volatility of the dispersion around the mean and the greater the risk.
- When comparing STDEVs for risk purposes, a coefficient of variation should be used or the data should be reported in percentages.
- Higher the standard deviation, higher the risk.
NORMAL DISTRIBUTION CURVE
For the above to apply, the data points must be normally distributed: 50% of the results must be below the mean and 50% above the mean.
Z SCORES
- To find the modelled probabilities using the normal distribution, we first convert the x value for the outcome into its corresponding Z value, as follows:
- Where:
- X = value of the outcome
- (^) μ = mean
- σ = standard deviation of outcomes
- Z = number of SDs represented by the difference between X and mean
HOW TO USE THE NORMAL DISTRIBUTION TABLE TO CALCULATE
PROBABILITIES
Over a period of time, an investment’s mean value (μ) has been USD 16 and its standard deviation (σ) was USD 2
Calculate the probability it being worth between USD 14 and USD 18.
Method: Calculate the ‘Z scores’ for values of USD 14 and USD 18 and then look up these Z probabilities in the normal distribution table.
HOW TO USE THE NORMAL DISTRIBUTION TABLE TO CALCULATE
PROBABILITIES
So, a range of USD 14 - USD 18 is +/- 1 standard deviation from the mean of USD 16.
Answer:
z score for 13.30 is (13.30 - 10.00 ) / 2 = 1.
1.65 Z score = 0.4505 from normal distribution tables. (ie 45%)
- 50% probability < 10m
- 45% probability between 10 & 13.30 (this is from the 1.65 Z score)
- 95% between 0 and 13.3m
- 5% above 13.3m by a process of elimination (95% <13.30, so 5%> 13.30)
- 100%
Answer: 95% chance that weekly sales will be below £13.30m.
QUANTIFYING PROBABILITY AND IMPACT
Expected value, based on probability and impact
Multiply the probability of a loss, on a scale from 0 percent (impossible) to 100 percent (certain) by the amount of the potential loss to produce an ‘expected value’.
For example, an event with 5 percent probability and an adverse impact of EUR 10m has an expected value (adverse) of:
EUR 10m x 0.05 = EUR 0.5m
