This document discusses various numerical analysis techniques used in insurance underwriting. It covers measures of central tendency like mean, median, and mode to summarize claim data. It also discusses principles of probability, the law of large numbers, frequency and severity of claims, correlation and regression analysis to determine relationships between variables, and expected value calculations. Mathematical techniques like regression analysis are used to predict future claims. Homogeneous risk pools in personal and motor insurance allow for more precise underwriting based on past data trends. However, some claims like catastrophes are difficult to predict accurately.

1. Chapter 5
NUMERICAL ANALYSIS OF RISKS AND
RELATIONSHIPS AND THEIR SIGNIFICANCE IN
UNDERWRITING PROCESS

2. NUMERICAL ANALYSIS
Organizations senior members, shareholders, or external parties
such as auditors and regulators are interested in statistical and
numerical information.
In statistical analysis, first stage is data collection and insurance
departments generate large volume of data from which information
is generated for decision making.

3. MEASURE OF CENTRAL TENDENCY
Representative value provide information about the data. Popular measure of averages are:
Arithmetic mean (most common average) - Adding all values and divide by the
number of values. X = Σ x / Σ ƒ
Frequency distribution – The total number of times (frequency) each
value occurred. E.g. 25 motor insurance claims: 100,80,150,65, 70,65,100,
125,85,80,65,125,100,150,85,100,65,180,,215,400, 180,125,70 and 140.
Claim size (X) : 65 70 80 85 100 125 140 150 180 215 400 Σ ƒ
Frequency : 4 2 3 2 4 3 1 2 2 1 1 = 25
It is used to summarise all type of numerical data including premium,
driver’s age etc. Mean is influenced by extreme values.
Arithmetic mean Median Mode

4. MEASURE OF CENTRAL TENDENCY
Distribution of data: e.g. 30 claims take 20 days to settle claims
X = Σ ƒ . X / Σ ƒ = 25
frequency distribution
can be used to compare
different subsets of data
e.g. no. of size of theft
claims against no. of size
of bodily injury claims
arising under motor.
This comparison provide
insight to underwriters.
Time taken to
settle claims in
days (X)
Frequency
(f)
f . X
10 20 200
20 30 600
30 40 1200
40 10 400
50 4 200
Total 104 = Σ ƒ 2600=Σ ƒX

5. MEASURE OF CENTRAL TENDENCY
Median – Mid value of the ascending ordered data e.g.:
In case of odd no. Values: 1, 3, 8, 12, 20, 23, 26, 33, 40 . Median = 20
In case of even no. Values: 45, 56, 90, 100, 120, 150, 175, 200
Median= (100 + 120) / 2 =110
Data in frequency distribution median will be calculated as:
X : 10 20 30 40 50 Σ ƒ
f : 20 30 40 10 4 104
C.f: 20 50 90 100 104
No. of values are even so the (n+1)/2 term will be our median.
Therefore, (104+1)/2 = 52.5th term is median i.e. 52.5th value lie in C.f of 90.
hence 30 days is the median.
Not influenced by extreme values.

6. MEASURE OF CENTRAL TENDENCY
Mode – Number which occur most often in data set and mode can be more
than 1 value.
Comparison with mean & median:
Mode can be suitable than mean if extreme value exist in data set
Median ignore the other important values due to selection of mid value(s).
Mode is quoted where audience is interested in measuring e.g.:
• Preferences;
• Average number of bedrooms in a house;
• Average time it takes a student to pass all their exams.

7. PRINCIPLES OF PROBABILITY
Probability quantify the likelihood of an event happening.
Probability is measured on a scale of 0 to 1, whereby 0 implies impossibilities of
an event occurrence and 1 represent certainties and may be expressed in
decimal, fraction or percentage.
Notion (P) and (E) is used to represent the probability of an occurrence and
event e.g. P(E) means Probability of an Event occurrence.
Methods of deriving probabilities:
Method 1 – Easiest and used where:
Probability of selecting a black jack from a deck of cards – P( black jack) = 2/52
All possible outcomes of an event are known All of outcomes are equally likely

8. PRINCIPLES OF PROBABILITY
Method 2 – The relative frequency of a particular loss event is the observed
frequency of that loss event expressed as a %age of the total number of observations.
Consider that a fire insurance co. incurred 85 incidents last year distributed by claims
costs:
It assumes that likelihood of an event occur in
future is same as it was in the past.
Cost Freq. Relative freq.
0< 50 10 10/85
50< 100 20 20/85
100< 150 35 35/85
150< 200 12 12/85
> = 200 8 8/85
Total 85 1

9. PRINCIPLES OF PROBABILITY
Method 3 – Subjective probabilities
Where the underwriters use their own skill and judgement to
determine the associated probability of an event. It is not the
quantitative measure of probability as values are assigned on the
basis of personal judgement.

10. THE LAW OF LARGE NUMBER & COMMON POOL
Law of large numbers is beneficial for underwriters to seek the
benefit for predicting the future claims costs and numbers.
The common pool – Insurer brings together a large number of
individual risks into a common pool i.e. risks of similar types are
brought together and insurer charge equitable premium from each
insured according to their degree of hazards.

11. FREQUENCY AND SEVERITY
Low frequency and High severity
e.g. Aircraft accidents Underwriter take decisions on the
basis of frequency and severity for
pricing the risks.
F
S
Insurer decision is based on frequency and severity.
Many classes of business exhibit the standard relationship between frequency and severity.

12. CORRELATION AND REGRESSION
Dependent variable value depends upon the independent
variables.
Scatter plot – depicts the relationship between variables:
• Positive relationship – Two variables moves in the same direction.
• Negative relationship – Value of one variable increases other variable
decreases.
• No relationship - No relationship exist between variables.
Underwriter observe the strength of relationship for decision making.
Co-efficient of correlation and Co-efficient of determination is used to
measure the strength of relationship.

13. APPLICATION TO INSURANCE (MATHEMATICAL
TECHNIQUES)
Mathematical techniques mostly used in insurance:
Expected value
Underwriter use it to calculate the expected value of future claims by using frequency &
severity. Table below shows probabilities that No. of claims and Costs occurring next years.
So the number of claims expected to arise next year is 2.41.
Expected value of claims
Regression analysis to predict the future
Correlation to determine the strength of relationship
No. of claims (X) 1 2 3 4 5 Total
Prob. P(X) 0.20 0.30 0.40 0.09 0.01 1
Expected value =
X . P(X)
0.20 0.60 1.20 0.36 0.05 2.41 =
Σ X.P(X)

14. APPLICATION TO INSURANCE (MATHEMATICAL
TECHNIQUES)
Expected cost of claims next year is ₤ 401.
Therefore, The Expected Value = 2.41 (₤401) = ₤966.41
Cost (₤) of
claims X
200 500 750 1300 1500 Total
P(X) 0.5 0.4 0.06 0.02 0.02 1
X . P (X) 100 200 45 26 30 401

15. APPLICATION TO INSURANCE (MATHEMATICAL
TECHNIQUES)
Regression analysis is a statistical technique by which underwriter can predict in
advance the value of dependant variables.
Dependent variable = a + b (Independent variables)
a is where the straight line will cross the Y-axis.
b is the gradient, or slope of the straight line.
b = (ΣXY – n XY) / (Σ X*2 – n X*2 ) And a = (Σ Y - b Σ X) / n
Claims
Y
1 2 3 4 5 6 7 9 Total
37
Age X 52 43 50 37 35 29 22 19 287
XY 52 86 150 148 175 174 154 171 1,110
X*2 2,704 1,849 2,500 1,369 1,225 841 484 361 11,333
Y*2 1 4 9 16 25 36 49 81 221

16. APPLICATION TO INSURANCE (MATHEMATICAL
TECHNIQUES)
Y = a + b X = 12.144 + (- 0.2096) X - Line of best fit
This line can be used to predict future values for any given values of X.
- 1 < r < + 1
r = n Σ XY - Σ X ΣY / [(n Σ X2 – (ΣX)2 ) (nΣY2 - (ΣY)2]
r = - 0.9559
Value of r shows the strength of relationship.
Co-efficient of determination = r * 2 = (- 0.9559)*2 = 0.9137 . Value is between 0 and 1.
It determines how much the variation in one variable can be explained by the other variable
i.e. variability in the number of motor claims can be explained by the age of an insured
drivers, or rather the relationship between the number of claims and the driver’s age.
Any other variations may due to some unexplained variables like sex or value of car etc.

17. APPLICATION TO INSURANCE (MATHEMATICAL
TECHNIQUES)
Importance of homogeneous risks to personal & motor risk
portfolios
Personal and motor business tends to generate a large group of homogeneous
claims. The degree of certainty arising from large homogeneous data sets is
particularly important for underwriters in these markets as the underwriting approach
is based on pricing sophistication and system-driven acceptance criteria.
Result of conclusion is dependant on data quality & its relevance.
It is difficult to predict the timing, frequency or size of catastrophe perils e.g. Earth
quake.
Some claims like EL difficult to predict & are latent claims which can exhibit a long
delay between incidence & manifestation.
So, some allowance should be made for these claims in pricing structure.