Liquidity risk in derivatives valuation: an improved credit proxy method

Figures & data

Table 1. Distribution of 1182 entities across different regions, sectors and ratings based on 30 December 2015 Markit data after merging with equity data from Bloomberg.

Region	N	Original Sector	N	Rating	N
Asia	141	Basic Materials	85	AAA	6
Europe	301	Consumer Goods	141	AA	64
N. America	663	Consumer Services	123	A	338
Other	77	Energy	93	BBB	526
		Financials	318	BB	180
		Health care	59	B	68
		Industrials	152
		Technology	49
		Telecommunications	65
		Utilities	97
		Renamed Sector
		Cyclical	567
		Financials	318
		Non Cyclical	297

Table 2. Standard deviation within different buckets in North American Financials based on 30 December 2015.

Region	Sector	Rating	Mean proxy	Bucket
			estimate (bps)	RSD (%)
North American	Financials	AA	56.73	31.79
North American	Financials	A	83.69	85.60
North American	Financials	BBB	112.98	79.77
North American	Financials	BB	263.14	49.38
North American	Financials	B	308.66	153.21

Figure 1. Time series of CDS spreads for 39 (North American, Financials, A) rated entities from January 2014 to December 2015.

Figure 2. Box plot of CDS spreads for North American, Financials across all rating levels for 30 December 2015.

Figure 3. Distribution of number of entities within each bucket for Markit data on 30 December 2015.

Table 3. Correlation coefficient between normalized CDS spreads and average daily return, and normalized CDS spreads and volatility based on 1 July 2015 to 30 December 2015 data.

		Corr. coeff. return	Corr. coef. volatility
Region	Asia	0.436	0.287
	Europe	0.307	0.598
	N. America	0.459	0.537
	Other	0.395	0.701
Sector	Cyclical	0.419	0.633
	Non Cyclical	0.404	0.707
	Financials	0.407	0.501
Rating	AAA	0.203	0.655
	AA	0.279	0.272
	A	0.152	0.285
	BBB	0.546	0.522
	BB	0.326	0.481
	B	0.545	0.514

Table 4. Regression coefficients for proxy model using region, sector, rating, equity returns and volatility based on 30 December 2015 Markit data.

Coefficient	Value	Coefficient	Value	Coefficient	Value
Intercept	5.09	Intercept ret	135.72	Intercept vol	8.81
AA	0.39	AA ret	65.61	AA vol	6.6
BBB	0.26	BBB ret	38.9	BBB vol	14.04
BB	1.32	BB ret	55.44	BB vol	7.95
B	1.99	B ret	73.16	B vol	23.59
AAA	0.72	AAA ret	539.76	AAA vol	8.12
Asia	0.55	Asia ret	24.03	Asia vol	26.99
N. Amer	0.3	N. Amer ret	32.73	N. Amer vol	19.43
Other	0.55	Other ret	153.54	Other vol	6.94
Consumer Goods	0.9	Consumer Goods ret	70.17	Consumer Goods vol	32.34
Basic Materials	0.74	Basic Materials ret	76.22	Basic Materials vol	25.29
Utilities	0.54	Utilities ret	0.09	Utilities vol	13.77
Telecommunications Services	-0.54	Telecommunications Services ret	58.04	Telecommunications Services vol	16.13
Industrials	1.15	Industrials ret	63.3	Industrials vol	44.89
Consumer Services	0.39	Consumer Services ret	187.68	Consumer Services vol	-0.85
Energy	0.3	Energy ret	16.55	Energy vol	5.55
Technology	0.59	Technology ret	48.04	Technology vol	12.24
Health care	0.65	Health care ret	81.84	Health care vol	0.72

Table 5. Adjusted values for models based on different explanatory variables.

Explanatory variables	Adjusted
Region, Sector and Rating	0.276
Equity return	0.173
Equity volatility	0.396
Equity return and volatility	0.427
Region, Sector, Rating and equity return	0.370
Region, Sector, Rating and equity volatility	0.463
Region, Sector, Rating, equity return and volatility	0.487

Table 6. Root-mean-square error (see equation 11(11) ) in basis points for intersection (first column), cross-section (second column) and equity proxy (last three columns) methodologies using real equity data for entities with redefined sectors for 30 December 2015.

	Intersection	Cross-section	Return	Volatility	Ret. and Vol.
Region
Asia	43.72	40.88	43.78	36.98	41.41
Europe	64.33	60.70	59.46	58.56	57.87
N. Amer	116.77	124.93	111.94	108.72	103.32
Other	189.89	181.04	171.08	157.32	161.41
Sector
Cyclical	130.29	134.76	122.94	118.71	115.25
Financials	70.97	71.89	67.90	67.43	68.20
Non Cyclical	81.89	94.36	80.80	77.34	66.12
Rating
AAA	2.68	3.11	10.06	3.27	2.71
AA	21.99	22.47	20.77	25.07	20.11
A	35.03	34.68	33.30	31.82	30.89
BBB	103.14	109.77	94.95	92.12	87.84
BB	172.09	176.05	166.24	159.47	156.43
B	177.55	191.93	178.39	175.06	162.28
Total	97.18	101.07	93.19	89.60	87.09

Table 7. Root-mean-square error (see equation 11(11) ) in basis points for cross-section (first column) and equity proxy (last three columns) methodologies using real equity data for entities with original sectors for 30 December 2015.

	Cross-section	Return	Volatility	Ret. and Vol.
Region
Asia	48.86	42.90	35.49	37.35
Europe	63.29	61.36	59.46	59.27
N. Amer	122.00	114.55	107.18	107.61
Other	159.41	158.28	158.36	160.64
Sector
Basic Materials	107.06	108.69	105.03	105.93
Consumer Goods	87.68	73.66	74.95	65.14
Consumer Services	106.97	107.23	123.49	122.58
Energy	193.34	174.39	167.30	165.05
Financials	73.58	68.52	67.46	67.81
Health care	110.67	88.06	79.25	59.41
Industrials	123.19	121.49	104.40	108.49
Technology	71.39	71.70	71.15	70.54
Telecommunications	144.63	143.60	175.00	164.19
Utilities	86.95	86.43	61.56	82.33
Rating
AAA	13.38	11.60	13.06	12.40
AA	20.54	19.11	24.32	20.82
A	31.25	30.85	29.14	29.15
BBB	99.47	91.83	89.84	88.85
BB	174.08	170.84	173.41	179.08
B	214.21	196.00	186.26	167.24
Total	99.92	94.41	88.84	89.37

Table 8. Root-mean-square error (see equation 11(11) ) in basis points for cross-section (first column) and equity proxy methodologies (last three columns) using proxy equity data for entities with original sectors for 30 December 2015.

	Cross-section	Return	Volatility	Ret. and Vol.
Region
Asia	48.86	50.46	49.12	51.35
Europe	63.29	63.83	64.78	64.17
N. Amer	122.00	116.47	114.37	112.22
Other	159.41	189.35	166.65	237.02
Sector
Basic Materials	107.06	104.10	115.53	116.70
Consumer Goods	87.68	83.18	87.85	90.85
Consumer Services	106.97	109.51	109.88	108.47
Energy	193.34	177.83	174.47	204.41
Financials	73.58	72.19	71.88	70.28
Health care	110.67	73.82	70.12	49.37
Industrials	123.19	127.43	119.89	116.58
Technology	71.39	80.96	87.05	81.55
Telecommunications Services	144.63	144.34	142.29	138.99
Utilities	86.95	100.52	64.40	65.06
Rating
AAA	13.38	17.05	10.57	16.44
AA	20.54	22.38	21.98	23.98
A	31.25	31.69	31.28	31.81
BBB	99.47	97.49	95.12	104.73
BB	174.08	173.79	170.70	169.02
B	214.21	199.95	188.40	179.70
Total	99.92	98.65	96.37	98.92

Table 9. Relative errors for buckets showing the strongest improvements in relative errors in proxying error using different methdologies on 30 December 2015.

Region	Sector	Rating	N	SD (bps)	Int (%)	CS (%)	Eq (%)	Eq (bps)
Eur.	Fin	AA	8	48.5	114.2	116.3	49.4	24.0
Asia	Fin	A	15	45.2	107.1	103.1	74.5	34.1
N. Amer	NCyc	BB	22	62.2	104.7	109.5	76.4	47.6
Eur.	Fin	A	33	34.9	103.0	115.1	80.2	28.0

Table 10. Errors in basis points for buckets showing the highest improvements in errors in proxying error using different methdologies on 30 December 2015.

Reg	Sec	Rat	N	SD	Int	CS	Eq	Eq ()
N. Amer	NCyc	BB	22	152.5	159.7	167.0	116.5	76.5
N. Amer	NCyc	B	12	175.5	191.5	257.3	142.9	81.4
N. Amer	Cyc	BBB	144	144.3	145.2	158.2	118.7	82.3

Table 11. Relative errors for buckets showing the highest improvements in relative errors in proxying error using proxy equity using cross-section and equity methodologies on 30 December 2015.

Region	Sector	Rating	N	SD (bps)	CS(%)	Eq (%)	Eq (bps)
Europe	Utilities	A	5	20.5	178.1	138.9	28.5
N. Amer	Health care	BB	8	37.1	146.9	113.1	41.9
N. Amer	Energy	BBB	36	51.7	123.1	107.7	55.7

Table 12. Errors in bps for buckets showing the highest improvements in errors in proxying error using proxy equity using cross-section and equity methodologies on 30 December 2015.

Region	Sector	Rating	N	SD (bps)	CS	Eq	Eq (%)
N. Amer	Utilities	BB	3	140.1	356.1	241.3	172.2
N. Amer	Health care	BB	8	37.1	137.9	106.1	113.1
N. Amer	Energy	BBB	36	51.7	188.5	165.1	107.7

Table 13. Standard deviation, and quantile for the pdf of daily CDS proxy changes for (Asia, Financials, BB) bucket on 30 December 2015.

	Std. dev.	quantile	quantile
Intersection	21.14	82.92	78.55
Cross-section	3.21	10.02	11.81
Equity	2.88	9.19	10.83

Figure 4. Intersection, cross-section and equity proxy for (Asia, Financials, BB) bucket based on data from 1 January 2015 to 30 December 2015.

Figure 5. Intersection, cross-section and equity proxy daily changes for (Asia, Financials, B) bucket based on data from 1 January 2015 to 30 December 2015. The solid lines correspond to the probability density functions of the daily changes in proxy calculated by different methodologies, and the dashed vertical lines correspond to the 1 and quantiles of the corresponding density functions.

Figure 6. Intersection and equity proxy for (Asia, Financials, B) bucket based on data from 1 January 2015 to 30 December 2015.