3  Case Studies and Calculations

3. Case studies and examples

This section presents a series of case studies that are used throughout the remainder of the Monograph. It starts with an overview of the aggregate Python package used to generate most exhibits and graphs.

3.1 The aggregate Python Package

My AAS paper S. Mildenhall (2024).

3.2 Python Environment Setup

How to setup env. Prefob code. Etc.

Basic autotime/reload set up with aimports done
to load auto time ip.magic('load_ext autotime')
doesn't work with qmd
{'autoreload', 'storemagic'}

3.3 Case Study 1: Cat/Non-Cat

This Study uses the Cat/NonCat Case Study portfolio from S. J. Mildenhall and Major (2022), described in Chs 2 and 7. The two units proxy a tame non-cat exposed unit and a severely cat exposed one.

port = build('''
port CNC
    agg Cat    1 claim sev lognorm 20 cv 1.00 fixed
    agg NonCat 1 claim sev gamma   80 cv 0.15 fixed
''')
hGT(port.describe)

Table 3.1: Validation statistics for Cat/NonCat case study portfolio.

unit	X	E[X]	Est E[X]	Err E[X]	CV(X)	Est CV(X)	Err CV(X)	Skew(X)	Est Skew(X)
Cat	Freq	1.00
	Sev	20.00	20.00	-0.00	1.00	1.00	-0.00	4.00	4.00
	Agg	20.00	20.00	-0.00	1.00	1.00	-0.00	4.00	4.00
NonCat	Freq	1.00
	Sev	80.00	80.00		0.15	0.15	0.00	0.30	0.30
	Agg	80.00	80.00		0.15	0.15	0.00	0.30	0.30
total	Freq	2.00
	Sev	50.00	50.00	-0.00	0.68			0.12
	Agg	100.00	100.00	-0.00	0.23	0.23	-0.00	2.56	2.56

Table 3.2 shows quantiles for the portfolio.

quantile_ps = np.array([.5, .8, .9, .95, .98, .99, .995, .999, .9999, .99999, .999999, 1-1e-7, 1-1e-8, 1-1e-9, 1-1e-10, 1])
impact = pd.DataFrame({'p': quantile_ps, 'Independent': port.q(quantile_ps)}).set_index('p')
f = lambda x: f'{x:,.1f}' if x > 1 else ('1' if x== 1 else (f'{x:.4g}' if x < 1-1e-5 else f'1-{1-x:.0e}'))
hGT(impact, table_float_format=f)

Table 3.2: Quantiles for the Cat/NonCat case study portfolio.

p	Independent
0.5	96.2
0.8	113.5
0.9	126.0
0.95	139.9
0.98	161.7
0.99	181.1
0.995	203.3
0.999	267.2
0.9999	394.1
1-1e-05	573.6
1-1e-06	820.5
1-1e-07	1,151.2
1-1e-08	1,569.4
1-1e-09	1,959.5
1-1e-10	2,048.0
1	2,048.0

The regulatory capital standard is set at 99.9% and the total cost of capital at 10% mirroring S. J. Mildenhall and Major (2022). These quantities are used to calibrate a consistent set of distortions, with parameters shown in Table 3.3

reg_p = 0.999
coc = 0.10
port.calibrate_distortions(COCs=[coc], Ps=[reg_p])
hGT(port.distortion_df[['P', 'param', 'error']].droplevel([0, 1]), 5)

Table 3.3: Standard distortion parameters at 99.9% capital standard and a 10% overall cost of capital.

method	P	param	error
ccoc	115.14968	0.10000
ph	115.14968	0.59651	0.00000
wang	115.14968	0.61082	0.00000
dual	115.14968	2.46322	-0.00000
tvar	115.14968	0.48158	0.00001

Table 3.4 shows the lifted natural allocation (S. J. Mildenhall and Major (2022) Chapter 14, and compare exhibits in Chapter 15.4) premium, margin, loss ratio, allocated assets and capital, leverage (PQ) and cost of capital by unit and by distortion.

ans = port.analyze_distortions(p=reg_p, efficient=False, add_comps=False)
# rename
ans.comp_df = ans.comp_df.rename(index={'ROE': 'COC'})
ans.comp_df = ans.comp_df.rename(index=lambda x: x.replace('Dist ', ''), level=0)

# extract loss ratios and coc
bit1 = ans.comp_df.xs('LR', 0, 1)
bit1 = bit1.iloc[[0, 2, 4, 1, 3]]

bit2 = ans.comp_df.xs('COC', 0, 1)
bit2 = bit2.iloc[[0, 2, 4, 1, 3]]
bit = pd.concat((bit1, bit2), keys=['LR', 'COC'])
hGT(bit.T, ratio_cols='all', vrule_widths=(1.5, 0, 0))

Table 3.4: Loss ratio and cost of capital by unit by distortion, lifted allocation.

	LR					COC
line	ccoc	ph	wang	dual	tvar	ccoc	ph	wang	dual	tvar
Cat	48.1%	60.7%	63.8%	67.3%	69.6%	10.7%	12.5%	12.3%	10.9%	9.8%
NonCat	108.6%	97.2%	95.4%	93.6%	92.5%	12.9%	4.7%	6.5%	8.7%	10.2%
total	86.8%	86.8%	86.8%	86.8%	86.8%	10.0%	10.0%	10.0%	10.0%	10.0%

Table 3.5 and subsequent exhibits focus on the Wang distortion, a “middle of the road” choice. Two units have COCs to NA Q which are hard to interpret. The loss ratio for Cat is lower than NonCat as expected, but the COCs are the opposite of a basic intuition. The reason is the transfers between units caused by limited liability.

dist = 'wang'
basic_order = ['L', 'P', 'M', 'LR', 'a', 'Q', 'PQ', 'COC']
hGT(ans.comp_df.loc[dist].T[basic_order])

Table 3.5: Pricing statistics for all-equity capital structure. Wang distortion; all equity capital with lifted natural allocation.

line	L	P	M	LR	a	Q	PQ	COC
Cat	19.96	31.26	11.31	0.64	123.43	92.17	0.34	0.12
NonCat	79.99	83.89	3.90	0.95	143.75	59.87	1.40	0.07
total	99.95	115.15	15.20	0.87	267.19	152.04	0.76	0.10

Table 3.6 shows the stand-alone pricing. Here we see the expected ordering of cost of capital for the two units. In this table, assets are determined as the 99.9% quantile of each unit’s losses and the pricing is determined using the Wang distortion calibrated on the total portfolio. Notice that the loss payments are different: the Cat line recovery is lower and NonCat higher, reflecting the inter-unit payments brought about by the limited liability pool. The sum of the stand-alone premiums is obviuosly higher than the pool’s total premium, resulting in a higher cost of capital. In this exhibit we see the expected lower cost of capital for the cat line.

capital_structure = [reg_p]
capital_structure_names = ['Equity', 'Debt', 'Senior']
bit, bit2 = optimal.standalone(port, dist, capital_structure, capital_structure_names)
hGT(bit)

Table 3.6: Cumulative stand-alone pricing statistics for all-equity capital structure, Wang distortion.

Unit	Tranche	L	P	M	LR	a	Q	PQ	COC
Cat	Equity	19.95	32.82	12.87	0.61	185.28	152.46	0.22	0.08
NonCat		80.00	87.50	7.51	0.91	122.25	34.75	2.52	0.22
Sum of parts		99.94	120.32	20.38	0.83	307.53	187.21	0.64	0.11

Figure 3.1 shows how the natural allocation premium compensates for inter-unit transfers caused by limited liability. The effect is most pronounced at low pool asset levels and can result in a negative margin for the NonCat line.

def rner(x):
    x = x.split('_')
    if x[0] == 'T.P':
        return x[1] + ' premium'
    elif x[0] == 'T.L':
        return x[1] + ' loss'
    else:
        return x[1] + ' margin'

fig, axs = plt.subplots(1, 2, figsize=(7, 2.75), constrained_layout=True)
ax0, ax1 = axs.flat

t = ans.augmented_dfs[dist].loc[port.q(.00001):port.q(reg_p)].filter(regex='T\.P_[A-Zt]')
t = t.rename(columns=rner)
t.plot(ax=ax0)

t = ans.augmented_dfs[dist].loc[port.q(.00001):port.q(reg_p)].filter(regex='T\.L_[A-Zt]')
t = t.rename(columns=rner)
t.plot(ax=ax0, lw=.5, color=['C0', 'C1', 'C2'])

ax0.legend(ncols=2, loc='lower right')
ax0.set(title='Cumulative premium and loss', xlabel='Pool assets')

tg = ans.augmented_dfs[dist].loc[port.q(.00001):port.q(reg_p)].filter(regex='T\.M_[A-Zt]')
tg = tg.rename(columns=rner)
tg.plot(ax=ax1)
ax1.axhline(0, lw=.5, c='k')
ax1.legend(loc='lower right')
ax1.set(title='Cumulative margin', xlabel='Pool assets');

Figure 3.1: Cumulative premium, loss, and margin by pool assets, up to 99.9% level.

Table 3.7 compares the natural allocation of capital and the constant allocation for an all-equity capital structure. The two allocations are markedly different. The margin and loss ratios are unchanged.

capital_structure = np.array([0, port.q(reg_p) ])
capital_names = ['All Equity' ]

bit = optimal.make_tranching(ans, dist, capital_structure, capital_names)
hGT(bit, ratio_cols=['LR', 'COC', 'ΔQ%'])

Table 3.7: Natural allocation vs. average return allocation of capital. Wang distortion and linear natural allocation.

Tranche	Unit	L	P	M	LR	a	COC	Q	Q_const	ΔQ	ΔQ%
All Equity	Cat	19.96	31.26	11.31	63.8%	123.43	12.3%	92.17	113.06	20.89	22.7%
	NonCat	79.99	83.89	3.90	95.4%	143.75	6.5%	59.87	38.98	-20.89	-34.9%
	total	99.95	115.15	15.20	86.8%	267.19	10.0%	152.04	152.04		0.0%

Table 3.8 uses a debt and equity capital structure. Consistent with rating agency limits on the ratio of debt to total capital, the debt attachment probability is computed to a 30% limit. The table shows that almost all of the difference in allocation is due to the equity layer. This is to be expected from Figure 3.1, which shows all the “weirdness” happens at low asset levels. Table 3.9 shows the same thing for a more complicated capital structure not constrained by the 30% limit.

debt_to_capital_limit = 0.3
# from last table, pull out premium
premium = bit.iloc[-1, 0]
# figure total capital
q = port.q(reg_p) - premium
# debt attachment level
da = premium + q * (1 - debt_to_capital_limit)
debt_attach_prob = port.cdf(da)

capital_structure = np.array([0, port.q(debt_attach_prob), port.q(reg_p), 0, port.q(reg_p)])
capital_names = ['Equity', 'Debt', 'drop', 'Total Capital']

bit = optimal.make_tranching(ans, dist, capital_structure, capital_names)
# x = bit.T.swaplevel(1)
hGT(bit, ratio_cols=['LR', 'COC', 'ΔQ%'])

Table 3.8: Natural allocation vs. average return allocation of capital with a debt and equity capital structure.

Tranche	Unit	L	P	M	LR	a	COC	Q	Q_const	ΔQ	ΔQ%
Equity	Cat	19.89	30.87	10.98	64.4%	87.64	19.3%	56.78	76.27	19.49	34.3%
	NonCat	79.96	83.72	3.76	95.5%	129.36	8.2%	45.64	26.14	-19.49	-42.7%
	total	99.85	114.59	14.74	87.1%	217.00	14.4%	102.41	102.41		-0.0%
Debt	Cat	0.07	0.40	0.33	16.9%	35.79	0.9%	35.39	35.15	-0.25	-0.7%
	NonCat	0.03	0.17	0.14	17.7%	14.40	1.0%	14.23	14.48	0.25	1.7%
	total	0.10	0.56	0.47	17.2%	50.19	0.9%	49.63	49.63		0.0%
Total Capital	Cat	19.96	31.26	11.31	63.8%	123.43	12.3%	92.17	113.06	20.89	22.7%
	NonCat	79.99	83.89	3.90	95.4%	143.75	6.5%	59.87	38.98	-20.89	-34.9%
	total	99.95	115.15	15.20	86.8%	267.19	10.0%	152.04	152.04		0.0%

capital_structure = np.array([0, port.q(.95), port.q(.98), port.q(.99), port.q(reg_p), 0, port.q(reg_p)])
capital_names = ['Equity', 'Junk', 'Debt', 'Senior', 'drop', 'Total Capital']

bit = optimal.make_tranching(ans, dist, capital_structure, capital_names)
hGT(bit, ratio_cols=['LR', 'COC', 'ΔQ%'])

Table 3.9: Natural allocation vs. average return allocation of capital with a multi-tranche debt and equity capital structure.

Tranche	Unit	L	P	M	LR	a	COC	Q	Q_const	ΔQ	ΔQ%
Equity	Cat	19.25	28.36	9.11	67.9%	41.33	70.2%	12.97	23.66	10.69	82.4%
	NonCat	79.41	81.77	2.37	97.1%	98.61	14.1%	16.84	6.15	-10.69	-63.5%
	total	98.65	110.13	11.48	89.6%	139.94	38.5%	29.81	29.81		0.0%
Junk	Cat	0.35	1.21	0.86	28.9%	11.52	8.3%	10.31	10.23	-0.08	-0.8%
	NonCat	0.35	1.12	0.77	31.4%	10.20	8.5%	9.08	9.16	0.08	0.9%
	total	0.70	2.33	1.63	30.1%	21.72	8.4%	19.39	19.39		0.0%
Debt	Cat	0.16	0.65	0.49	24.4%	11.48	4.5%	10.83	10.79	-0.04	-0.4%
	NonCat	0.12	0.46	0.34	26.0%	7.96	4.6%	7.49	7.53	0.04	0.5%
	total	0.28	1.11	0.83	25.1%	19.44	4.5%	18.33	18.33		0.0%
Senior	Cat	0.20	1.05	0.85	19.3%	59.11	1.5%	58.06	56.47	-1.58	-2.7%
	NonCat	0.11	0.53	0.42	20.6%	26.99	1.6%	26.46	28.04	1.58	6.0%
	total	0.31	1.58	1.27	19.7%	86.09	1.5%	84.51	84.51		0.0%
Total Capital	Cat	19.96	31.26	11.31	63.8%	123.43	12.3%	92.17	113.06	20.89	22.7%
	NonCat	79.99	83.89	3.90	95.4%	143.75	6.5%	59.87	38.98	-20.89	-34.9%
	total	99.95	115.15	15.20	86.8%	267.19	10.0%	152.04	152.04		0.0%

Figure 3.2 shows that outside the equity layer the return (discount rate) changes quite slowly, explaining why the average allocation is very similar to the natural allocation for the non-equity layers. All of the allocations “add-up” and so the total capital row in Table 3.7, Table 3.8 and Table 3.9 are all identical.

fig, axs = plt.subplots(1, 2, figsize=(2 * 3.5, 2.75), constrained_layout=True)
ax0, ax1 = axs.flat

g = port.dists[dist]
S = port.density_df.loc[:port.q(reg_p), 'S']
gS = g.g(S)
d = (gS - S) / (1 - S)

ax0.plot(S.index, S, label='S')
ax0.plot(S.index, d, label='Discount return')
for x in capital_structure:
    if x > 0:
        ax0.axvline(x, lw=.5, c='C4')
ax0.legend(loc='upper right')
ax0.set(title='S and rate of discount by layer', xlabel='Asset level')

S = S.loc[port.q(0.01):]
d = d.loc[port.q(0.01):]
ax1.plot(S.index, np.gradient(d, S.index))
for x in capital_structure:
    if x > 0:
        ax1.axvline(x, lw=.5, c='C4')
ax1.set(title='Gradient of layer rate of discount', xlabel='Asset level');

Figure 3.2: Discount rate by layer (left) and derivative of discount rate (right). The vertical lines show the capital structure tranche boundaries.

3.4 Case Study 2:

3.5 Case Study 3:

Mildenhall, S. (2024). Aggregate: fast, accurate, and flexible approximation of compound probability distributions. Annals of Actuarial Science, 1–40. https://doi.org/10.1017/S1748499524000216

Mildenhall, S. J., & Major, J. A. (2022). Pricing insurance risk: Theory and practice. John Wiley & Sons, Inc.