Data analysis and correlation: questions

Aims and stages of an analysis

Paper A style · Data analysis · Straightforward · 9 marks

A general insurer's analytics team has been asked to carry out three pieces of work using its motor portfolio data:

(a) summarising the average claim cost in each region last year for the annual report;

(b) using a random sample of 2,000 policies to assess whether the claim frequency of drivers aged under 25 differs from that of older drivers across the whole portfolio, with a measure of the uncertainty;

(c) forecasting the number of claims the insurer will receive next quarter.

(i)
State, with a reason, whether each of (a), (b) and (c) has a descriptive, an inferential or a predictive aim.
3 marks
(ii)
Describe the main stages the team should follow to carry out piece of work (c), from defining the problem to reporting the results.
4 marks
(iii)
Suggest two tools or techniques the team could use in the exploratory stage of piece of work (a), and say what each would show.
2 marks

Summary statistics for ten claims

Paper A style · Data analysis · Straightforward · 8 marks

The amounts (£) of ten claims settled last month on a small commercial property account were:

420, 380, 1250, 510, 460, 390, 2900, 610, 450, 530

You may use $\sum x$ $= 7{,}900$ and $\sum x^2$ $= 11{,}772{,}600$ .

(i)
Calculate the sample mean claim amount.
1 mark
Your answer
(ii)
Calculate the sample median.
1 mark
Your answer
(iii)
Calculate the sample standard deviation, using divisor $n-1$ . Give your answer to 1 decimal place.
2 marks
Your answer
(iv)
Comment on the shape of the distribution of these claim amounts, and state which measures of location and spread you would prefer to report.
2 marks
(v)
It is found that the claim recorded as 2,900 should have been 290. Recalculate the sample mean, and state the effect of the correction on the median and on the standard deviation.
2 marks
Your answer

Telematics data as a big data source

Paper A style · Data analysis · Standard · 10 marks

A motor insurer plans to offer a policy that uses a 'black box' to record each insured vehicle's position, speed and acceleration once every second while the vehicle is moving. It expects 50,000 policyholders to take up the policy in the first year.

(i)
Assuming each vehicle is driven for an average of 1.5 hours a day on every day of a 365-day year, calculate the number of records collected in the first year. Give your answer in standard form to 4 significant figures.
1 mark
Your answer
(ii)
Describe four characteristics of this data set that make it an example of 'big data', and the difficulty each creates for the analysis.
4 marks
(iii)
State whether the telematics data are primary or secondary data for the insurer, and give one example of secondary data the insurer might combine with them.
2 marks
(iv)
Explain three issues, other than the size of the data set, that the insurer should consider before using the data to set premiums.
3 marks

Making an analysis reproducible

Paper A style · Data analysis · Standard · 9 marks

Last year a pricing actuary produced a home insurer's rating factors using an R script, a spreadsheet in which several adjustments were typed by hand, and an extract of claims data taken from the live claims system on one particular day. This year a colleague tries to reproduce last year's results and cannot.

(i)
Explain what is meant by reproducible research, and how it differs from replication of a study.
2 marks
(ii)
Describe four elements that would have made last year's analysis reproducible.
4 marks
(iii)
Explain two benefits to the insurer of reproducible analysis.
2 marks
(iv)
Part of the script draws bootstrap samples with R's sample function, and gives slightly different results every time it is run. Explain how this should be dealt with.
1 mark

Premium level and claim frequency

Paper A style · Data analysis · Standard · 10 marks

For 12 sales regions, an insurer records the average annual premium $x$ (£00) and the claim frequency $y$ (claims per 100 policies). The summary statistics are:

$n$ $= 12$ , $\sum x$ $= 99.5$ , $\sum y$ $= 137.2$ , $\sum x^2$ $= 890.49$ , $\sum y^2$ $= 1604.06$ , $\sum xy$ $= 1164.40$ .

(i)
Calculate the sample (Pearson) correlation coefficient between $x$ and $y$ . Give your answer to 4 decimal places.
3 marks
Your answer
(ii)
Test, at the 5% significance level, the null hypothesis that the population correlation coefficient $\rho$ is zero against the alternative that $\rho > 0$ . You are given that the upper 5% point of the $t_{10}$ distribution is 1.812. Give the value of your test statistic to 3 decimal places.
4 marks
Your answer
(iii)
State the assumptions needed for the test in (ii) to be valid.
2 marks
(iv)
A manager concludes that higher premiums cause higher claim frequencies. Comment on this conclusion.
1 mark

Fund manager returns in consecutive years

Paper A style · Data analysis · Standard · 9 marks

An investment consultant compares the returns (%) achieved by eight fund managers in two consecutive years:

Manager	A	B	C	D	E	F	G	H
Year 1	4.2	7.9	3.1	6.5	9.4	5.0	2.6	8.3
Year 2	5.1	6.0	2.2	7.4	8.8	3.9	4.4	6.9

Under the hypothesis of no association between the years, the exact distribution of Spearman's rank correlation coefficient $r_s$ for $n$ $= 8$ gives $P(r_s$ $\geq 0.643)$ $= 0.048$ and $P(r_s$ $\geq 0.619)$ $= 0.057$ .

(i)
Calculate Spearman's rank correlation coefficient between the two years' returns. Give your answer to 4 decimal places.
4 marks
Your answer
(ii)
Test, at the 5% significance level, whether there is positive association between managers' returns in the two years. State your hypotheses and conclusion.
3 marks
(iii)
Give two reasons why the consultant might prefer Spearman's coefficient to Pearson's coefficient for these data.
2 marks

Marketing spend and new policies

Paper A style · Data analysis · Standard · 10 marks

A life insurer records its monthly marketing spend $x$ (£000) and the number $y$ of new protection policies sold in the same month, for ten months:

Month	1	2	3	4	5	6	7	8	9	10
$x$	12	15	9	20	17	11	24	14	19	22
$y$	310	335	290	360	320	305	395	340	330	380

Under the hypothesis of no association, Kendall's $\tau$ for a sample of size $n$ may be taken as approximately normal with mean 0 and variance $\dfrac{2(2n+5)}{9n(n-1)}$ . You are given that the upper 2.5% point of the standard normal distribution is 1.9600.

(i)
Calculate Kendall's rank correlation coefficient $\tau$ for these data. Give your answer to 4 decimal places.
4 marks
Your answer
(ii)
Using the normal approximation, test at the 5% significance level whether there is any association between marketing spend and new policies sold, against a two-sided alternative. Give your test statistic to 3 decimal places.
3 marks
Your answer
(iii)
Kendall's $\tau$ can be read as $P(\text{concordant})$ ${}- P(\text{discordant})$ for two months chosen at random. Use this to estimate the probability that, of two months chosen at random, the month with the higher marketing spend also sold more policies.
1 mark
Your answer
(iv)
Give two advantages of Kendall's $\tau$ over Pearson's correlation coefficient for data of this kind.
2 marks

Fisher's transformation for pension scheme data

Paper A style · Data analysis · Hard · 13 marks

For a random sample of 40 occupational pension schemes, the sample correlation coefficient between the average age of active members and the scheme's annual ill-health retirement rate is $r$ $= 0.72$ . Assume that the pairs are independent observations from a bivariate normal distribution.

You are given that the upper 2.5% and 5% points of the standard normal distribution are 1.9600 and 1.6449 respectively.

(i)
State the approximate distribution of $z$ $= \tanh^{-1} r$ for a sample of size $n$ from a bivariate normal population with correlation $\rho$ .
1 mark
(ii)
Calculate an approximate 95% confidence interval for $\rho$ . Give the lower limit to 4 decimal places as your answer.
3 marks
Your answer
(iii)
State the upper limit of your interval in (ii) to 4 decimal places, and comment on the shape of the interval.
1 mark
Your answer
(iv)
Test, at the 5% level, $H_0: \rho$ $= 0.5$ against $H_1: \rho$ $> 0.5$ . Give your test statistic to 3 decimal places.
3 marks
Your answer
(v)
An independent random sample of 35 schemes in another country gives a sample correlation of 0.55 between the same two variables. Test, at the 5% level, whether the two population correlation coefficients are equal, against a two-sided alternative. Give your test statistic to 3 decimal places.
3 marks
Your answer
(vi)
Explain why the inference above is based on $\tanh^{-1} r$ rather than on $r$ itself.
2 marks

Two risk indicators reduced to one

Paper A style · Data analysis · Hard · 13 marks

For each of 50 postcode sectors an insurer records $X_1$ , the number of burglaries per 1,000 households, and $X_2$ , the number of theft claims per 1,000 home policies. The sample means are $\bar x_1$ $= 12$ and $\bar x_2$ $= 5$ , and the sample covariance matrix is

$S = \begin{pmatrix} 9 & 4 \\ 4 & 3 \end{pmatrix}.$

(i)
Show that the eigenvalues of $S$ are 11 and 1.
3 marks
(ii)
Find the unit-length vector of loadings for the first principal component, taking the loading on $X_1$ to be positive. Give the loading on $X_2$ to 4 decimal places.
2 marks
Your answer
(iii)
Calculate the proportion of the total variance explained by the first principal component.
1 mark
Your answer
(iv)
A sector has $x_1$ $= 15$ and $x_2$ $= 6.5$ . Calculate its score on the first principal component, using mean-centred data. Give your answer to 4 decimal places.
2 marks
Your answer
(v)
An analyst repeats the analysis using the sample correlation matrix instead of $S$ . Calculate the proportion of total variance explained by the first principal component in that case. Give your answer to 4 decimal places.
3 marks
Your answer
(vi)
Explain why the two analyses give different answers, and state which you would recommend here.
2 marks

Broker complaints and satisfaction scores

Paper A style · Data analysis · Hard · 12 marks

A general insurer distributes its policies through twelve brokers. For each broker it records $x$ , the number of complaints about the broker received last year, and $y$ , the broker's average customer satisfaction score (out of 10) from a survey this year:

Broker	A	B	C	D	E	F	G	H	I	J	K	L
$x$	3	7	2	5	7	1	4	9	3	6	2	8
$y$	7.2	6.8	8.1	6.0	7.4	7.9	6.5	5.9	7.0	7.4	6.6	6.2

Tied values are given the average of the ranks they occupy. Under the hypothesis of no association, the statistic $T$ $= r_s\sqrt{n-2}\big/\sqrt{1-r_s^2}$ may be taken to have approximately a $t_{n-2}$ distribution, where $r_s$ is Spearman's rank correlation coefficient. You are given that the upper 5% and 1% points of $t_{10}$ are 1.812 and 2.764.

(i)
Calculate Spearman's rank correlation coefficient $r_s$ between $x$ and $y$ , as the Pearson correlation coefficient of the ranks. Give your answer to 4 decimal places.
5 marks
Your answer
(ii)
Calculate the value given by the formula $1$ ${}- \dfrac{6\sum d^2}{n(n^2-1)}$ , where $d$ is the difference between a broker's two ranks, to 4 decimal places. Explain why it differs from your answer to (i).
2 marks
Your answer
(iii)
Test the hypothesis of no association between complaints and satisfaction against the alternative of negative association, at the 5% level and at the 1% level. State your hypotheses, give the test statistic to 3 decimal places and state your conclusions.
3 marks
Your answer
(iv)
An analyst suggests repeating the calculation with the logarithm of each satisfaction score. State, with a reason, the effect this would have on $r_s$ , and give one reason why a rank-based coefficient is more suitable than Pearson's coefficient for these data.
2 marks

Exploring motor claims by region

Paper B style · Data analysis · Straightforward · 14 marks

The data set da-p10.csv gives, for a random sample of 60 motor claims settled last year, the policy number (policy), the policyholder's region (region: North, Midlands or South, to be treated as a factor), the age of the main driver in years (age) and the claim amount in pounds (claim). Load it with df <- read.csv("da-p10.csv").

da-p10.csv: 60 rows, 4 columns

Show the data

policy	region	age	claim
1	North	32	1064
2	Midlands	43	1016
3	North	61	1110
4	South	28	2207
5	Midlands	56	2362
6	North	18	1864
7	South	44	740
8	Midlands	26	933
9	South	64	2096
10	North	57	616
11	South	69	901
12	South	51	2631
13	Midlands	23	4659
14	North	76	647
15	Midlands	48	1273
16	South	20	10731
17	North	59	1356
18	North	73	1169
19	North	59	942
20	South	21	2342
21	North	46	1711
22	Midlands	48	874
23	South	41	1391
24	Midlands	76	860
25	Midlands	79	404
26	Midlands	74	566
27	Midlands	65	233
28	North	49	1491
29	South	30	1604
30	South	74	1313
31	North	48	611
32	Midlands	56	807
33	North	73	1588
34	North	74	828
35	North	29	624
36	Midlands	62	873
37	South	54	869
38	North	73	530
39	North	51	1895
40	South	65	2349
41	Midlands	40	2863
42	Midlands	29	2280
43	South	44	2707
44	North	31	1177
45	South	23	2042
46	South	43	1623
47	North	25	774
48	Midlands	59	1358
49	South	71	1135
50	Midlands	48	634
51	Midlands	40	3663
52	Midlands	53	1488
53	South	57	786
54	Midlands	33	1645
55	North	76	2139
56	Midlands	24	2234
57	North	31	618
58	South	56	3308
59	South	28	2213
60	South	32	2282

(i)
Use R to produce summary statistics of the claim amounts, including the standard deviation. Give the standard deviation to 1 decimal place.
3 marks
Your answer
(ii)
Comment on what the output in (i) suggests about the shape of the distribution of claim amounts.
2 marks
(iii)
Calculate the mean and standard deviation of the claim amounts in each region, and draw side-by-side boxplots of claim amount by region. Give the mean for the South region to 2 decimal places.
3 marks
Your answer
(iv)
The boxplot rule flags as outliers any values more than 1.5 times the interquartile range beyond the hinges. Use boxplot.stats to find how many claim amounts (all regions together) are flagged, and state their values.
2 marks
Your answer
(v)
Calculate Pearson's and Spearman's correlation coefficients between driver age and claim amount. Give Spearman's coefficient to 4 decimal places.
2 marks
Your answer
(vi)
Comment on the results of (iii) to (v), including whether you would prefer Pearson's or Spearman's coefficient here and what you would do next.
2 marks

Correlation between two fund returns

Paper B style · Data analysis · Standard · 17 marks

The data set da-p11.csv gives the monthly returns (%) over 30 consecutive months of a UK equity fund (equity) and a commercial property fund (property), with the month number (month). Month 18 was a month of severe market stress. Load the data with df <- read.csv("da-p11.csv"). Use a 5% significance level and two-sided alternatives throughout, and assume the months are independent.

da-p11.csv: 30 rows, 3 columns

Show the data

month	equity	property
1	-1.26	1.9
2	-6.65	0.53
3	4.24	4.88
4	3.17	0.86
5	4.61	-1.22
6	2.74	-0.47
7	-0.59	-1.91
8	-2.11	-3.78
9	1.55	0.71
10	0.64	2.43
11	-0.7	-0.89
12	-2.06	0.81
13	1.41	-0.46
14	-3.87	0.88
15	-1.24	5.05
16	-1.14	0.42
17	-2.67	2.08
18	-11.8	-9.6
19	3.35	4.09
20	1.16	1.58
21	-2.73	-1.92
22	0.02	-0.61
23	-0.52	0.55
24	2.59	5.89
25	-1.55	-1.47
26	2.52	5.51
27	0.45	-2.09
28	-0.91	-5.06
29	-0.83	0.59
30	-1.6	0.74

(i)
Use cor.test to test whether the Pearson correlation between the two funds' returns is zero. Give the sample correlation to 4 decimal places.
3 marks
Your answer
(ii)
State the conclusion of the test in (i), and the assumption on which the test and the confidence interval rely.
2 marks
(iii)
Repeat the test using Spearman's rank correlation, with exact = FALSE. Give the sample value of Spearman's coefficient to 4 decimal places.
2 marks
Your answer
(iv)
Repeat the test using Kendall's $\tau$ , with exact = FALSE. Give the sample value of $\tau$ to 4 decimal places.
2 marks
Your answer
(v)
Remove month 18 and repeat the Pearson test. Give the new sample correlation to 4 decimal places.
3 marks
Your answer
(vi)
Comment on the results of (i) to (v). What should an investor seeking diversification between the two funds conclude?
3 marks
(vii)
Show, using R, that the 95% confidence interval in (i) is obtained from Fisher's transformation $z$ $= \tanh^{-1} r$ .
2 marks

Principal components of branch performance

Paper B style · Data analysis · Hard · 18 marks

The data set da-p12.csv gives five performance measures for each of 40 branches of a general insurer: claim frequency per 100 policies (freq), average claim severity in £00 (severity), annual lapse rate in % (lapse), expense ratio in % (expense) and growth in the number of policies in % (growth). Load it with df <- read.csv("da-p12.csv").

da-p12.csv: 40 rows, 5 columns

Show the data

freq	severity	lapse	expense	growth
10.7	34	7.7	26.9	5.5
3.5	23.6	13	27.1	5.2
6.8	30.7	12.3	29.7	4.4
6.8	34.1	17.8	29.8	1.4
5.5	27.2	9.7	28.8	4.4
5.9	25.5	15.1	29.9	0
13.4	45.8	14.9	25	3.9
10.8	31.5	6.3	23.2	6.1
6.4	29.1	14.5	29.6	2.8
3.9	23.3	9.8	28.3	4.6
4.9	25	13.6	30.5	3.5
5.5	28	13.4	29	3.2
5.6	30.9	12.8	30.3	3.6
7.5	33.1	17.4	31.3	-0.1
8.7	37.1	12.9	28	2
7.5	33.2	13.4	29.5	4.7
6.7	31.3	9.7	26.7	4.9
6.3	30.6	12.9	28.8	3.3
6.4	27	10.5	26	5.2
8.5	35.3	12.5	26.6	4.2
10.1	31.9	8.7	26.3	4.4
7.1	21.4	11	29.2	3.1
8.2	30.1	7.7	26.5	3.3
9	27.9	12.6	30	3.7
7.9	35.7	15	30.9	0.7
9.9	39.4	11.6	29.7	2.7
4.8	29.7	10.8	30.3	5.1
6	29.8	10.9	27.7	5.2
9.2	35.5	11.2	28.5	0.3
8.4	31.7	12.2	28.1	3.2
6.6	25.3	8.2	26.4	6.8
9	33.2	11.9	26.5	4.7
7.4	27.2	14.5	28.5	3
9.1	35.7	12.3	25.2	3.5
10.8	36.4	12.8	28.1	1.5
8.6	34.6	12.3	27.8	4.7
5.8	30.2	11.6	30.1	3.2
7.2	30.9	13.5	28	2.3
7.6	28.5	9.5	27.7	6.6
5.7	25.3	7.7	25.4	7.4

(i)
Calculate the sample variance of each of the five variables. Give the variance of severity to 3 decimal places.
2 marks
Your answer
(ii)
Explain why the principal component analysis should be carried out on the standardised variables (the correlation matrix) here.
2 marks
(iii)
Carry out a principal component analysis on the standardised data using prcomp, and print its summary. Give the cumulative proportion of variance explained by the first two components to 4 decimal places.
3 marks
Your answer
(iv)
Calculate the eigenvalues of the correlation matrix from the prcomp output, and state how many components the Kaiser criterion (retain components whose eigenvalue exceeds 1) would keep.
2 marks
Your answer
(v)
Print the loadings of the first two components, rounded to 3 decimal places, and interpret each component.
4 marks
(vi)
Give the size of branch 1's score on the first principal component, to 4 decimal places. (The sign of a component is arbitrary: software may reverse every loading and score.)
2 marks
Your answer
(vii)
Comment on how useful the principal component analysis is for summarising branch performance, and give one limitation.
3 marks