Loss distributions and reinsurance: questions

Pareto claims under excess of loss

Paper A style · Losses and reinsurance · Straightforward · 8 marks

Individual claim amounts on a household contents portfolio follow a Pareto distribution with parameters $\alpha=4$ and $\lambda=2{,}400$ , so that $F(x)=1-\left(\frac{2400}{2400+x}\right)^{4}$ for $x$ $>0$ . The insurer has an individual excess of loss reinsurance treaty with retention £2,000.

(i)
Calculate the probability that a claim involves the reinsurer, to 4 significant figures.
1 mark
Your answer
(ii)
Calculate the insurer's expected payment per claim, to the nearest penny.
3 marks
Your answer
(iii)
Calculate the reinsurer's expected payment on a claim that involves the reinsurer, to the nearest penny.
2 marks
Your answer
(iv)
Hence calculate the reinsurer's expected payment per claim (including claims that do not involve it), to the nearest penny, and check your answers against the mean claim amount.
2 marks
Your answer

Inflation and a fixed retention

Paper A style · Losses and reinsurance · Straightforward · 7 marks

Claim amounts on a commercial vehicle account are exponentially distributed with mean £4,000. The insurer holds individual excess of loss reinsurance with retention £10,000.

(i)
Calculate the reinsurer's expected payment per claim, to the nearest penny.
2 marks
Your answer
(ii)
Next year all claim amounts are expected to be 8% higher, and the retention remains £10,000. Calculate the percentage increase in the reinsurer's expected payment per claim, to 2 decimal places.
3 marks
Your answer
(iii)
Explain why the increase in (ii) is much greater than 8%, and suggest how the treaty could be changed to avoid this.
2 marks

Weibull claims under quota share

Paper A style · Losses and reinsurance · Straightforward · 6 marks

Claim amounts $X$ (in £000) on a marine cargo account follow a Weibull distribution with $F(x)$ $=1-\exp(-c x^{\gamma})$ for $x$ $>0$ , where $c$ $=0.2$ and $\gamma=0.5$ . The insurer cedes 30% of every claim under a quota share treaty, so it retains $Y$ $=0.7X$ .

(i)
Show that $Y$ has a Weibull distribution with the same $\gamma$ , and find its parameter $c$ .
2 marks
(ii)
Calculate the probability that the insurer's retained amount on a claim exceeds £20,000, to 4 significant figures.
2 marks
Your answer
(iii)
Given that $E[X]$ $=\Gamma(1+1/\gamma)\,c^{-1/\gamma}$ , calculate the insurer's expected retained amount per claim, in £000.
2 marks
Your answer

Pareto shape parameter by maximum likelihood

Paper A style · Losses and reinsurance · Straightforward · 9 marks

An insurer models claims from a travel portfolio by a Pareto distribution with density $f(x)$ $=\alpha\lambda^{\alpha}/(\lambda+x)^{\alpha+1}$ for $x$ $>0$ , with $\lambda=500$ known. A random sample of 25 claims $x_1,\dots,x_{25}$ gives $\sum_{i=1}^{25}\ln\left(1+x_i/500\right)=9.6213$ .

(i)
Derive the maximum likelihood estimate of $\alpha$ , to 4 significant figures.
4 marks
Your answer
(ii)
Using the asymptotic distribution of the maximum likelihood estimator, calculate an approximate 95% confidence interval for $\alpha$ . Give the upper limit, to 3 significant figures. You are given that the upper 2.5% point of the standard normal distribution is 1.96.
3 marks
Your answer
(iii)
The variance of a Pareto distribution is finite only if $\alpha>2$ . Comment on what your interval suggests about the variance of the claim amount distribution.
2 marks

Lognormal losses with a policy excess

Paper A style · Losses and reinsurance · Standard · 8 marks

Losses on a pet insurance portfolio follow a lognormal distribution with parameters $\mu=6.2$ and $\sigma=1.1$ . Each policy has an ordinary deductible (excess) of £300: the policyholder bears the first £300 of each loss and the insurer pays the rest. You may use $\Phi(-0.4511)$ $=0.3260$ and $\Phi(-1.5511)$ $=0.0604$ , where $\Phi$ is the standard normal distribution function, and $\ln 300$ $=5.70378$ .

(i)
Calculate the probability that a loss gives rise to a payment by the insurer, to 4 decimal places.
1 mark
Your answer
(ii)
Calculate the insurer's expected payment per loss, to the nearest £.
4 marks
Your answer
(iii)
Calculate the insurer's expected payment per claim, that is per loss that leads to a payment, to the nearest £.
1 mark
Your answer
(iv)
Next year all losses are expected to increase by 5% while the deductible stays at £300. Without further calculation, explain whether the insurer's expected payment per loss will rise by more or less than 5%.
2 marks

Exponential fit with capped claim payments

Paper A style · Losses and reinsurance · Standard · 9 marks

A liability insurer's policies have a limit of £15,000: the insurer pays the loss up to £15,000 and nothing above it. Losses are assumed to be independent and exponentially distributed with parameter $\lambda$ . Of the last 40 losses, 34 were below the limit, with total amount £182,400, and 6 reached the limit, so only the payment of £15,000 is known for each of them.

(i)
Write down the likelihood and derive the maximum likelihood estimate of $\lambda$ . Give $1/\hat\lambda$ to the nearest £.
4 marks
Your answer
(ii)
Explain the effect on the estimate of $\lambda$ of wrongly treating the six capped payments as complete observations of £15,000.
2 marks
(iii)
Using your fitted model, estimate the expected amount per loss that falls on policyholders because of the limit, to the nearest £.
3 marks
Your answer

Weibull fit with known shape

Paper A style · Losses and reinsurance · Standard · 10 marks

An insurer assumes that claim amounts $X$ (in £000) on a small-business property account follow a Weibull distribution with $\gamma=2$ known, so that $F(x)$ $=1-\exp(-c x^2)$ for $x$ $>0$ . A random sample of 30 claims gives $\sum x_i$ $=186.3$ and $\sum x_i^2$ $=1352.4$ . You are given that for this distribution $E[X]$ $=\Gamma(1.5)\,c^{-1/2}$ and $E[X^2]$ $=1/c$ , with $\Gamma(1.5)$ $=\sqrt\pi/2$ $=0.886227$ .

(i)
Derive the maximum likelihood estimate of $c$ , to 4 significant figures.
4 marks
Your answer
(ii)
Calculate the method of moments estimate of $c$ based on the sample mean, to 4 significant figures.
3 marks
Your answer
(iii)
Using the maximum likelihood estimate, estimate the probability that a claim exceeds £10,000, to 3 significant figures.
1 mark
Your answer
(iv)
Comment on the difference between the two estimates in (i) and (ii), and on what it suggests about the model.
2 marks

Pareto fit by the method of moments

Paper A style · Losses and reinsurance · Standard · 8 marks

A sample of 50 claims from a liability account has mean £1,850 and standard deviation £3,100 (calculated with divisor $n$ ). The insurer fits a Pareto distribution with $F(x)=1-\left(\frac{\lambda}{\lambda+x}\right)^{\alpha}$ , for which $E[X]$ $=\frac{\lambda}{\alpha-1}$ and $\mathrm{Var}(X)$ $=\frac{\alpha\lambda^2}{(\alpha-1)^2(\alpha-2)}$ when $\alpha>2$ .

(i)
Calculate the method of moments estimate of $\alpha$ , to 4 significant figures.
3 marks
Your answer
(ii)
Calculate the method of moments estimate of $\lambda$ , to the nearest whole number.
1 mark
Your answer
(iii)
Estimate the probability that a claim exceeds £5,000, to 3 significant figures.
2 marks
Your answer
(iv)
Comment on the suitability of the method of moments for fitting this distribution.
2 marks

Chi-square test of an exponential fit

Paper A style · Losses and reinsurance · Standard · 11 marks

An insurer fitted an exponential distribution to 200 claims on a home emergency account by maximum likelihood, using the individual amounts; the fitted mean is £2,500. The claims are grouped as follows.

Claim amount (£)	0-1,000	1,000-2,000	2,000-4,000	4,000-8,000	over 8,000
Number of claims	58	41	52	34	15

You are given that the upper 5% point of the $\chi^2_3$ distribution is 7.815.

(i)
Calculate the expected number of claims in each group under the fitted model, and give the expected number in the group over £8,000, to 2 decimal places.
3 marks
Your answer
(ii)
Calculate the chi-square goodness-of-fit statistic, to 3 decimal places.
3 marks
Your answer
(iii)
State the degrees of freedom and the conclusion of the test at the 5% significance level.
2 marks
(iv)
Comment on the fit, with particular reference to the largest claims.
3 marks

Lognormal fit and the insurer's retained cost

Paper A style · Losses and reinsurance · Hard · 11 marks

A sample of 60 claims (in £) from a commercial property account gives $\sum\ln x_i$ $=494.7$ and $\sum(\ln x_i)^2$ $=4116.2$ . The claims are assumed to be independent and lognormal with parameters $\mu$ and $\sigma^2$ . The insurer holds individual excess of loss reinsurance with retention £15,000. You may use $\ln 15000$ $=9.61581$ , $\Phi(1.7363)$ $=0.9587$ and $\Phi(0.9468)$ $=0.8281$ .

(i)
Calculate the maximum likelihood estimate of $\mu$ .
2 marks
Your answer
(ii)
Calculate the maximum likelihood estimate of $\sigma$ , to 4 significant figures.
2 marks
Your answer
(iii)
Using the fitted model, calculate the insurer's expected payment per claim net of reinsurance, to the nearest £.
5 marks
Your answer
(iv)
Explain why an error in the estimate of $\sigma$ matters much more for the reinsurer's expected payment per claim than for the insurer's.
2 marks

The reinsurer's claims under a Pareto model

Paper A style · Losses and reinsurance · Hard · 10 marks

Claim amounts $X$ on an engineering account follow a Pareto distribution with $\alpha=3$ and $\lambda=6{,}000$ , so $P(X>x)=\left(\frac{6000}{6000+x}\right)^{3}$ . The insurer buys individual excess of loss reinsurance with retention $M$ . For a Pareto distribution with parameters $\alpha$ and $\lambda$ , $E[X]$ $=\frac{\lambda}{\alpha-1}$ and $E[X^2]$ $=\frac{2\lambda^2}{(\alpha-1)(\alpha-2)}$ .

(i)
Show that, for any retention $M$ , the reinsurer's payment on a claim that exceeds $M$ has a Pareto distribution with parameters $\alpha$ and $\lambda+M$ .
3 marks
(ii)
With $M$ $=4{,}000$ , calculate the standard deviation of the reinsurer's payment per claim, including claims on which it pays nothing, to the nearest £.
4 marks
Your answer
(iii)
The insurer wants the reinsurer's expected payment per claim to be 10% of the expected claim amount. Find the retention $M$ needed, to the nearest £.
3 marks
Your answer

Lognormal or gamma for claim amounts

Paper B style · Losses and reinsurance · Standard · 18 marks

The data set lo-p12.csv gives 120 claim amounts (column amount, in £) from a household buildings portfolio; load it with df <- read.csv("lo-p12.csv"). Assume the amounts are independent and identically distributed. In R, dlnorm(x, meanlog, sdlog) is the lognormal density with parameters $\mu$ and $\sigma$ , and dgamma(x, shape, rate) is the gamma density $\lambda^\alpha x^{\alpha-1}e^{-\lambda x}/\Gamma(\alpha)$ .

lo-p12.csv: 120 rows, 1 column

Show the data

amount
1282
2076
1986
2168
1487
1944
4198
2972
476
2863
1370
802
2206
3315
2408
498
915
376
1895
1605
2573
2794
453
1164
1417
1231
1447
960
2940
1603
1434
1557
1756
2365
1545
808
363
1546
2636
1030
1631
1714
2031
371
1113
2963
179
786
1110
1438
3817
2042
1545
1227
2603
2456
701
3026
820
1881
729
158
1062
664
3660
959
944
2580
1585
472
1469
865
2303
1248
3710
1923
2560
1851
1936
1005
1389
2353
477
1116
4368
1970
634
1921
574
638
2311
633
304
1278
3583
1646
998
3638
924
474
363
1844
1808
404
1614
723
3060
154
657
923
6961
1155
5824
1578
582
1364
3545
181
1255
5031

(i)
Fit a lognormal distribution by maximum likelihood. Give $\hat\sigma$ to 4 significant figures.
3 marks
Your answer
(ii)
Use optim to fit a gamma distribution by maximum likelihood, starting from the method of moments estimates. Give $\hat\alpha$ to 4 significant figures.
4 marks
Your answer
(iii)
Calculate the AIC of each fitted model and state which model is preferred. Give the AIC of the gamma model to 1 decimal place.
3 marks
Your answer
(iv)
Group the claims into the bands $(0,750]$ , $(750,1500]$ , $(1500,2250]$ , $(2250,3250]$ and above 3,250, and carry out a chi-square goodness-of-fit test of the fitted gamma model. Give the p-value to 3 significant figures.
5 marks
Your answer
(v)
Comment on your results and on whether the gamma model is suitable for pricing excess of loss reinsurance on this portfolio.
3 marks

Fitting gross claims from net reinsured data

Paper B style · Losses and reinsurance · Hard · 18 marks

An insurer holds individual excess of loss reinsurance with retention £50,000 on a commercial property account. The data set lo-p13.csv gives 150 recent claims; load it with df <- read.csv("lo-p13.csv"). Column paid is the insurer's net payment in £000, $\min(X,50)$ , and column reins is 1 if the claim exceeded the retention (so the gross amount is known only to exceed 50) and 0 otherwise. Assume the gross claims $X$ (in £000) are independent and lognormally distributed with parameters $\mu$ and $\sigma$ .

lo-p13.csv: 150 rows, 2 columns

Show the data

paid	reins
41.97	0
16.55	0
3.23	0
24.63	0
9.53	0
8.54	0
21.93	0
50	1
12.77	0
4.7	0
12.68	0
50	1
11.69	0
7.85	0
9.97	0
19.44	0
2.39	0
17.94	0
7.84	0
13.23	0
27.17	0
18.04	0
16.82	0
14.62	0
21.89	0
42.47	0
50	1
1.07	0
50	1
10.78	0
7.04	0
49	0
6.89	0
13.79	0
28.2	0
50	1
7.74	0
50	1
50	1
21.79	0
15.09	0
50	1
4.55	0
25.36	0
45.04	0
10.53	0
18.98	0
8.31	0
18.26	0
19.01	0
21.95	0
12.33	0
12.54	0
50	1
3.07	0
41.84	0
29.41	0
14.34	0
28.69	0
30.77	0
50	1
6.92	0
45.27	0
14.15	0
18.02	0
31.32	0
21.54	0
50	1
50	1
5.54	0
36.71	0
9.7	0
26.35	0
22.82	0
10.44	0
17.92	0
7.84	0
50	1
9.99	0
13.8	0
21.93	0
2.96	0
50	1
15.56	0
7.21	0
50	1
50	1
9.26	0
13.58	0
10.88	0
23.71	0
35.15	0
25.24	0
45.24	0
50	1
9.31	0
16.07	0
38.95	0
4.3	0
6.08	0
49.09	0
50	1
27.91	0
20.42	0
19.35	0
25.62	0
13.11	0
50	1
50	1
50	1
50	1
50	1
12.71	0
21.38	0
10.93	0
13.87	0
16.83	0
9.04	0
17.79	0
19.93	0
13.85	0
6.5	0
13.94	0
14.16	0
33.54	0
20.75	0
31.31	0
18.24	0
36.06	0
48.81	0
49.64	0
32.04	0
50	1
50	1
45.99	0
44.25	0
14.97	0
30.4	0
33.28	0
10.39	0
8.12	0
12.05	0
9.29	0
21.52	0
50	1
30.29	0
45.97	0
9.61	0
14.07	0
46.28	0

(i)
Calculate the proportion of claims that involved the reinsurer and the insurer's average net payment per claim. Give the average net payment in £000 to 4 significant figures.
2 marks
Your answer
(ii)
Write the log-likelihood for $\mu$ and $\sigma$ allowing for the censoring, and maximise it with optim. Give $\hat\mu$ to 4 significant figures.
6 marks
Your answer
(iii)
State $\hat\sigma$ , to 4 significant figures.
1 mark
Your answer
(iv)
Using the fitted model, estimate the reinsurer's expected payment per claim (including claims that do not reach it), in £000 to 4 significant figures.
4 marks
Your answer
(v)
The reinsurer offers to raise the retention to £75,000. Estimate the reinsurer's expected payment per claim under the new retention, in £000 to 4 significant figures.
2 marks
Your answer
(vi)
Explain what would go wrong if the net payments were treated as complete observations of the gross claims, and comment on the reliability of the estimate in (v).
3 marks