Valuation of real-estate losses via Monte Carlo simulation

Figures & data

Figure 1. SCR aggregation diagram.

Source: Granito and De Angelis (2015).

Figure 2. Annual percentage change in the Housing Price Index-HPI-.

Source: National Statistics Institute (INE, 2018).

Figure 3. Theoretical frequencies.

Source: Own work.

Figure 4. Theoretical frequencies.

Source: Own work.

Figure 5. Theoretical and practical frequencies Variation of the HPI.

Source: Own work.

Figure 6. Cumulative theoretical and practical frequencies variation of the HPI.

Source: Own work.

Table 1. Ranking of distributions.

Kolmogorov–Smirnov		Chi-2		Anderson–Darling
0,0939	Normal	5	Normal	0,4714	Normal
0,1225	Gamma	6	Gamma	0,4777	Log-normal
0,1518	Log-normal	7	Log-normal	0,5064	Gamma
0,1494	Beta	10	Beta	1,5405	Beta
0,4080	Exponential	77	Exponential	2,2582	Exponential
0,3162	Poisson	77	Geometric	2,2780	Geometric
0,3345	Binomial	48.836.071	Poisson	7,9921	Poisson
0,4097	Geometric	62.593.385.224	Binomial	12,8741	Binomial

Source: Own work.

Figure 7. Frequency distribution and VaR of the property portfolio.

Source: Own work.

Table A1. Annual percentage change in the Housing Price Index.

Year / Quarter.	Q1	Q2	Q3	Q4
2017	5,3	5,6	6,7	7,2
2016	6,3	3,9	4	4,5
2015	1,5	4	4,5	4,2
2014	−1,6	0,8	0,3	1,8
2013	−14,3	−12	−7,9	−7,8
2012	−12,6	−14,4	−15,2	−12,8
2011	−4,1	−6,8	−7,4	−11,2
2010	−2,9	−0,9	−2,2	−1,9
2009	−7,6	−7,7	−7	−4,3
2008	2,8	−0,3	−3	−5,4
2007	13,1	11,6	9,2	5,7

Source: National Statistics Institute (INE, 2018).

Table B1. HPI statistics.

Average	Typical Deviation	Variance	α-Gamma
−1.552273	7.395111	54.687669	0.044060
β- Gamma	Log-normal Media	Typical Log-normal Dev.	λ-Exponential
−35.230709	1.515018	0.944659	−0.644217

Source: Own work.

Table B2. Common distributions in the insurance sector.

Discrete	Continuous
Binomial Poisson Binomial Negative	Gamma Beta Pearson Log-gamma Normal Log-normal

Source: Own work.

Table B3. HPI variation statistics.

Average	Typical deviation	α-Gamma	β-Gamma	Media log-normal
−1.552273	7.395111	47.215223	73.951.111644	1.515018
Min	−15.200000	α-Beta	α-Gamma	Typical deviation Log-normal
Max	13.100000	1.309944	0.000033	0.944659
Difference between legs	5	Average-Beta	β-Gamma	λ-Exponential
No. of sections	6	0.416272	−47.640.540444	−0.644217
Interannual variation	Frequency accumulated	Frequency Individual	Frequency acumulated (%)	Frequency
−10.48	7	7	0.225806	0.225806
−5.77	14	7	0.451613	0.225806
−1.05	22	8	0.709677	0.258065
3.67	28	6	0.903226	0.193548
8.38	29	1	0.935484	0.032258
13.10	31	2	1.000000	0.064516

Source: Own work.

Table B4. Theoretical frequencies of the HPI variation.

Individual frequencies (%)
Observed	Normal	Gamma	Log-normal	Beta	Exponential	Poisson	Binomial	Geometric
0.0313	0.0449	0.0266	0.0344	0.0000	0.4392	0.0000	0.0000	0.4409
0.1563	0.1120	0.1269	0.1499	0.0770	0.0882	0.0000	0.0000	0.0881
0.0938	0.2182	0.2503	0.2487	0.1818	0.0744	0.0511	0.0234	0.0742
0.3750	0.2695	0.2671	0.2358	0.2480	0.0627	0.9213	0.9674	0.0625
0.1875	0.2108	0.1846	0.1610	0.2746	0.0528	0.0276	0.0092	0.0527
0.1563	0.1045	0.0925	0.0901	0.2185	0.0445	0.0000	0.0000	0.0444
Cumulative Frequencies (%)
Observed	Normal	Gamma	Log-normal	Beta	Exponential	Poisson	Binomial	Geometric
0.0313	0.0449	0.0266	0.0344	0.000000	0.4392	0.0000	0.0000	0.4409
0.1875	0.1569	0.1535	0.1843	0.077017	0.5275	0.0000	0.0000	0.5290
0.2813	0.3751	0.4038	0.4331	0.258805	0.6018	0.0511	0.0234	0.6032
0.6563	0.6446	0.6708	0.6689	0.506852	0.6645	0.9724	0.9908	0.6657
0.8438	0.8554	0.8555	0.8299	0.781453	0.7173	1.0000	1.0000	0.7184
1.0000	0.9598	0.9480	0.9200	1.000000	0.7618	1.0000	1.0000	0.7627

Source: Own work.

Table B5. Goodness-of-fit of the HPI variation. Test Kolmogorov–Smirnov.

Kolmogorov–Smirnov
		Normal	Gamma	Log-normal	Beta	Exponential	Poisson	Binomial	Geometric
		0.094	0.123	0.152	0.149	0.408	0.316	0.335	0.410
Level 0.1%.	0.220	Ok	Ok	Ok	Ok	x	x	x	x
Level 0.05%.	0.240	Ok	Ok	Ok	Ok	x	x	x	x
Level 0.01%.	0.288	Ok	Ok	Ok	Ok	x	x	x	x

Source: Own work.

Table B6. Goodness-of-fit of the HPI variation. Test Chi-2.

Chi-2
		Normal	Gamma	Log-normal	Beta	Exponential	Poisson	Binomial	Geometric
		4.690	6.176	7.414	9.968	76.981	48.836.070	62.593.385.223	77.281
Level 0.1%.	7.78	Ok	Ok	Ok	x	x	x	x	x
Level 0.05%.	9.49	Ok	Ok	Ok	x	x	x	x	x
Level 0.01%.	13.28	Ok	Ok	Ok	Ok	x	x	x	x

Source: Own work.

Table B7. Goodness-of-fit of the HPI variation. Anderson test–Darling.

Anderson–Darling
		Normal	Gamma	Log-normal	Beta	Exponential	Poisson	Binomial	Geometric
		0.471	0.506	0.478	1.541	2.258	7.992	12.874	2.278
Level 0.1%.	1.929	Ok	Ok	Ok	Ok	x	x	x	x
Level 0.05%.	2.502	Ok	Ok	Ok	Ok	Ok	x	x	x
Level 0.01%.	3.907	Ok	Ok	Ok	Ok	Ok	x	x	x

Source: Own work.

Granito, I., & De Angelis, P. (2015). Capital allocation and risk appetite under Solvency II framework. Cornell University Library. doi:10.13140/RG.2.1.1136.8404

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Instituto Nacional de Estadística (INE). (2018). Índice de Precio de la Vivienda. https://www.ine.es/consul/serie.do?s=HPI948

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