Credit-linked Notes and Rating Sensitive Notes

Overview

Basic Concepts

Credit-linked Notes

A credit-linked note (CLN), also called a credit default note, is a fixed or floating rate note where the coupon and principal payments are referenced to a reference credit, which can be a single name or multiple names. If there is no credit event of the reference credit, all the coupons and the redemption value will be paid in full. However, if there is a credit event, the payments of the note will be altered. For the note’s principal only the recovery rate will be paid. For the coupons payment can continue or discontinue, depending on the contract.

CLNs are normally created by a Special Purpose Vehicle (SPV), or trust which typically holds highly rated securities and credit default swap agreements between itself and highly rated counterparties. Either the SPV or its counterparties are bankruptcy-remote. The dominant credit risk that the buyer of a CLN assumes is that of the reference credit.

The buyer of a CLN is selling the credit protection in exchange for higher yield on the note. The other side of the deal, the seller, is buying credit protection. Hence there is close relationship between a CLN and a credit default swap. Roughly speaking a CLN is a regular note with an embedded credit default swap.

Rating Sensitive Notes

A rating sensitive note, also known as a credit-sensitive note, is a fixed or floating rate note with a coupon rate reset in the event of a change in the issuer's credit rating. Take a fixed rate rating sensitive note for example. Instead of having a single coupon rate, which is the case of a common fixed rate note, a rating sensitive note has a list of coupon rates with each rating category having a different coupon rate. If the issuer does not default, the coupon rate will be determined at the reset date. The value will be the coupon rate in the list corresponding to the issuer’s current rating. At maturity redemption will be paid in full. If the issuer does default, only the recovery value of the redemption will be paid.

Valuation

Credit-linked Notes

Under the assumption that interest rates, default rates and recovery rates are independent, the valuation of a CLN is similar to the valuation of the payoff leg of a credit default swap. See the Credit Default Swaps FINCAD Math Reference document for details.

Rating Sensitive Notes

Under the assumption that interest rates, the rating transition matrix and recovery rates are independent, the valuation of a rating sensitive note is straightforward. The value of a rating sensitive note is the present values of the expected coupons and redemption. Using the default probabilities derived from the issuer’s rating transition matrix one can calculate the value with the same method for the valuation of CLNs. The expected coupon value of a coupon payment is the weighted average of the coupon values of all rating categories.

FINCAD Functions

Credit-linked Notes

aaCredit_L_Fix(d_v, d_t, d_e, d_f_cpn, d_l_cpn, cpn, princ, redem_value, freq, acc_rate, acc_accrued, hl, d_rul, position, cpn_pmt, intrp, df_crv, prob_crv, intrp_prob, rate_recover, calc_type, stat):

Calculates fair value and risk statistics of a credit-linked fixed rate note.

aaCredit_L_FRN(d_v, d_e, d_t, freq_pay_reset, position, npa, redem_value, acc_frn, d_rul_frn, reset_mktdays, reset_time, mgn_pay, mgn_reset, fixed_reset_table, cpn_pmt , intrp, df_crv_acc, df_crv_disc, hl, prob_crv, intrp_prob, rate_recover, calc_type, stat):

Calculates fair value and risk statistics of a credit-linked floating rate note.

Rating Sensitive Notes

aaCredit_L_Fix_RS_CF(d_v, d_t, d_e, d_f_cpn, d_l_cpn, cpn_tbl, princ, redem_value, freq, acc_rate, acc_accrued, hl, d_rul, reset_mktdays, position, intrp, df_crv, rating, trans_matrix, intrp_prob, rate_recover, calc_type, table_type):

Calculates expected cash flows and their present values of a rating sensitive fixed rate note.

aaCredit_L_Fix_RS(d_v, d_t, d_e, d_f_cpn, d_l_cpn, cpn_tbl, princ, redem_value, freq, acc_rate, acc_accrued, hl, d_rul, reset_mktdays, position, intrp, df_crv, rating, trans_matrix, intrp_prob, rate_recover, calc_type, stat):

Calculates fair value and risk statistics of a rating sensitive fixed rate note.

Description of Inputs

Input Argument	Description
d_v	valuation date
d_e	effective date
d_t	maturity date
d_f_cpn	date of first coupon after dated date
d_l_cpn	date of last coupon prior to maturity date
cpn	coupon
princ	principal
redem_value	redemption value per 100 par
freq	cash flow frequency
acc_rate	accrual method for coupons
acc_accrued	accrual method for accrued interest
d_rul	business day convention (see Glossary)
freq_pay_reset	pay and reset frequencies, a switch, see also aaFRN
position	trade position, a switch
cpn_pmt	Type of coupon payment - a switch. 1 = coupon payment ends after a credit event of the reference credit. 2 = coupon payment continues after a credit event of the reference credit.
npa	notional principal amount
acc_frn	The coupon accrual method table. It can be a one-entry or a two-entry table. If it has only one entry, the value is the accrual method for both future coupon payments and accrued interests. Otherwise the two entries are the accrual methods for coupon and accrued interests respectively.
d_rul_frn	The business day convention table. It can be a one-entry or a two-entry table. If it has only one entry, the value is the business convention for both future coupon payment days and reset dates. Otherwise the two entries are the business day convections for coupon payment days and reset days respectively.
reset_mktdays	number of business days prior to the reset date that the rate is fixed
reset_time	the hour of the day when the rate is reset
mgn_pay	flat margin above or below compounded rate
mgn_reset	margin above or below a reset rate
fixed_reset_table	reset rates (already fixed)
intrp	interpolation method
df_crv	discount factor curve or a yield table. If it is a yield table, it can be a one entry table of a yield value (annual, actual/365), or a two entry table of yield and compounding frequency or a three entry of yield, a compounding frequency and an accrual method.
df_crv_acc	discount factor curve or yield table for accruing rates. If it is a yield table, it can be a one entry table of a yield value (annual, actual/365), or a two entry table of yield and compounding frequency or a three entry of yield, a compounding frequency and an accrual method.
df_crv_disc	discount factor curve or yield table for discounting rates. If it is a yield table, it can be a one entry table of a yield value (annual, actual/365), or a two entry table of yield and compounding frequency or a three entry of yield, a compounding frequency and an accrual method.
hl	holiday list
prob_crv	Default probability curve. It can be a two or three column table. The first column can be time in years (>=0 and <1000 ), or dates (>=1000) and the second default probability values. Time and date must be increasing and probability must be non-decreasing and between 0 and 1. If the first column is time in years, the time accrual method (default = 30/360) can be put in the first row of the third column. For a date based curve, the first value of the second column must be 0 and the first date cannot be bigger than a valuation date. Excel users may pass term strings to represent time intervals. Term strings must be in the format kB, kD, kW, kM, or kY where k is an integer and D, W, M and Y stands for calendar day, week, month and year respectively, e.g., 3M = 3 months.
intrp_prob	interpolation method of default probability curve
rate_recover	The recovery rate table. It can be a two-, or three- or four-column table. The first column has IDs of the entities. If it is a two column table, both the principal and the coupon have the same recovery rate (>=0 and <1) in the second column. Otherwise, the second column has the principal recovery rate and the third coupon recovery rate. The fourth column can hold the recovery delay time (in years) after default.
calc_type	calculation type, a switch.
stat	statistics, a switch
cpn_tbl	one-dimensional coupon array of all ratings arranging from highest to lowest
rating	current rating of issuer
trans_matrix	one year rating transition matrix of issuer

Description of Outputs

Except for the output survival probability at maturity and par coupon rates, the outputs of the functions aaCredit_L_fix and aaCredit_L_FRN have the same outputs as those of the functions aaFixlg_p and aaFRN, respectively. For aaFixlg_p and aaFRN see the Swaps and Floating Rate Notes and Floating Legs of Swaps FINCAD Math Reference documents respectively.

Output Statistic	Description
Survival probability	the opposite of a default probability. For example, suppose an entity has a probability of 5% to default in one year. Then we say that the entity has a one-year survival probability of 95% = 100%-5%.
Par coupon rate of aaCredit_L_fix	is the coupon rate that makes the fixed coupon credit linked note have a fare value equal to the principal of the note.
Par spread over a compounded rate:	is the spread over the compounded rate of the floating rate note, assuming that the spread over the reset rate is 0, that makes the credit linked note have the same value as the floating rate note that is not linked to a credit reference and has no spread over either compounded rates or reset rates.
Par spread over a reset rate	is the spread over the reset rates of the floating rate note, assuming that the spread over the compounded rates is 0, that makes the credit linked note have the same value as the floating rate note that is not linked to a credit reference and has no spread over either compounded rates or reset rates.

The function aaCredit_L_fix_RS has the same outputs as those of the function aaCredit_L_fix, but they are based on the expected cash flows.

table_type = 1

Output	Type	Description
Column 1	Date	Date
Column 2	Number	Expected interest
Column 3	Number	Principal
Column 4	Number	Total expected cashflow
Column 5	Number	Expected accrued interest
Column 6	Number	Present value of expected interest
Column 7	Number	Present value of principal
Column 8	Number	Present value of total expected cashflow
Column 9	Number	Survival probability
Column 10	Number	Risk-free discount factor

table_type = 2

Output	Type	Description
Column 1	Date	Date
Column 2	Number	Total expected cashflow
Column 3	Number	Present value of total expected cashflow
Column 4	Number	Expected interest

Note that expected value means expected future values and present value is the discounted expected value.

Examples

Examples are given for valuing a credit-linked fixed rate note, credit-linked floating rate note and a rating sensitive note.

Example 1: Credit-linked fixed rate note

An investor bought a fixed rate note, effective Oct. 1, 2003, from a special purpose company (SPC) that is linked to the credit of country B. The note has a notional of $1,000,000 and a coupon of 10% paid quarterly on an actual/360 basis. It will mature Oct. 1, 2005. If there is no credit event of country B during the life of the note, all the coupons will be paid in full and the notional principal will be paid at maturity. If there is a credit event before maturity; however, only a portion of the notional principal and coupon, i.e., the recovery value, will be paid. Suppose the default probability curve of country B is given as follows:

Default Probability Curve

Years	Probability Value
1	0.05
2	0.1
3	0.15
4	0.2

and the expected recovery rate of principal is 25% and that of coupon is 0. Suppose the SPC and counterparties are bankruptcy-remote and so their default probabilities can be ignored. Today’s date is Dec. 3, 2003 and spot risk-free discount factor is given as follows:

Discount Factor Curve

Grid Date	Discount Factor
3-Dec-2003	1
11-Dec-2003	0.999708
1-Jan-2004	0.998922
2-Mar-2004	0.996662
2-Jun-2004	0.993272
2-Sep-2004	0.989830
2-Nov-2004	0.987323
3-Jun-2005	0.974558
5-Dec-2005	0.963094
5-Jun-2006	0.945682

Further suppose that the business day convention is next good business day and holidays are ignored. Using the function aaCredit_L_fix we value the note as follows:

aaCredit_L_fix

Argument	Description	Example Data	Switch
d_v	value (settlement) date	3-Dec-2003
contra_d	CDS contract dates	see above
cpn	coupon	0.06
princ	principal	1,000,000
redem_value	redemption value per 100 par	100
freq	frequency	3	quarterly
acc_rate	accrual method for coupons	2	actual/360
acc_accrued	accrual method for accrued interest	2	actual/360
hl	holiday list	0
d_rul	business day convention	2	next good business day
position	trade position	1	long
cpn_pmnt	type of coupon payment	1	coupon payment ends after credit event
intrp	interpolation method	1	linear
df_crv	discount factor curve	see above
prob_crv	default probability curve	see above
intrp_prob	interpolation method of probability curve	1	linear
rate_recover	recovery rate table	0.25
calc_type	calculation type	1	simplified calculation method
stat	stat list	1…16

Results

Statistics	Description	Value
1	fair value	1005674.807
2	accrued interest	10500
3	fair value plus accrued interest	1016174.807
4	yield: annually compounded	0.057963166
5	yield: semi-annually compounded	0.057146729
6	quarterly compounded	0.05674424
7	yield: monthly compounded	0.056478009
8	yield: money market (actual/ 365)	0.058791427
9	yield: money market (actual/ 360)	0.057963166
10	duration	1.757482459
11	modified duration	1.732899444
12	convexity	3.549005028
13	basis point value	-176.0748437
14	yield value change per 1bp increase in price	-5.67882E-09
15	probability of survival at maturity	0.908219178
16	par coupon rate	0.056749664

If the recovery rate is only the recovery rate of principal and the coupon recovery rate is 0, the recovery rate table of the following should be used:

Recovery Rate Table

principal recovery rate	coupon recovery rate
0.25	0

Example 2: Credit-linked floating rate note

Suppose in example 1 the note is a floating rate note. Suppose its reset frequency is also quarterly and rates are reset 2 days prior to a next coupon payment date. The current reset rate 0.015. Call the function aaFRN_tables to build reset date table and then call the function aaCredit_L_FRN to value the note. Here are the results:

Active Reset Table

ID	reset date	effective date	terminating date	rate
1	29-Sep-2003	1-Oct-2003	1-Jan-2004	0.015

aaCredit_L_FRN

Argument	Description	Example Data	Switch
d_v	value (settlement) date	3-Dec-2003
contra_d	CDS contract dates	see example 1
freq_pay_reset	pay and reset frequencies	3	pay freq quarterly / reset freq quarterly
position	trade position	1	long
npa	notional principal amount	1,000,000
redem_value	redemption value per 100 par	100
acc_frn	accrual method(s)	2
d_rul_frn	business date convention table	2
reset_mktdays	number of business days prior to the reset date that the rate is fixed	2
reset_time	the hour of the day when the rate is reset	0
mgn_pay	margin above or below compounded rate	0
mgn_reset	margin above or below a reset rate	0
fixed_reset_table	reset rates (already fixed)	see above
cpn_pmnt	type of coupon payment	1	coupon payment ends after credit event
intrp	interpolation method	1	linear
df_crv_acc	discount factor curve for accruing rates	see example 1
df_crv_disc	discount factor curve for discounting rates	see example 1
hl	holiday list	0
prob_crv	default probability curve	see example 1
intrp_prob	interpolation method of probability curve	1	linear
rate_recover	recovery rate table	0.25
calc_type	calculation type	1	simplified calculation method
stat	stat list	1…12

Results

Statistics	Description	Value
1	fair value	1072182.54
2	accrued interest	2625
3	fair value plus accrued interest	1074807.54
4	yield: annually compounded	0.05389616
5	yield: semi-annually compounded	0.05318889
6	yield: quarterly compounded	0.05283989
7	yield: monthly compounded	0.05260891
8	yield: money market (actual/ 365)	0.05466481
9	yield: money market (actual/ 360)	0.05389616
10	probability of survival at maturity	0.90821918
11	par spread over a compounded rate	0.03681978
12	par spread over a reset rate	0.03681978

Example 3: Rating sensitive note

An investor bought a rating-sensitive note with a notional of 1,000,000 issued by country T. The note matures Oct. 1, 2010 and pays the coupon semi-annually on an actual/360 basis with each coupon rate reset 2 days prior to the effective date of the payment period according to the following coupon table:

Coupon Table

coupon rate

0.05

0.07

0.09

0.1

For example, if at a reset date the rating of the country is at level 1, the coupon rate for the next period will be 5%; and if its rating is level 4, the coupon will be 10%. The note also pays the full amount of notional principal at maturity if the reference country does not default. If the country defaults during the life of the note, coupon payment will terminate and the note will pay the recovery value of the principal. The current rating of the country is level 2 and so the coupon for the current payment period is 7%. Suppose the average rating transition matrix of country T is given as follows:

one year rating transition matrix

percent at level 1	percent at level 2	percent at level 3	percent at level 4	percent at default
85	5	4	4	2
4	80	6	6	4
4	4	78	7	7
2	3	5	75	15
0	0	0	0	100

and its recovery rate is 50% in the case that it defaults. Call the function aaCredit_L_fix_RS with the recovery rate table:

Recovery Rate Table

principal recovery rate	coupon recovery rate
0.5	0

and contract dates:

Contract Date Table

maturity date	effective date
1-Oct-2005	1-Oct-2003

aaCredit_L_fix_RS

Argument	Description	Example Data	Switch
d_v	value (settlement) date	3-Dec-2003
contra_d	CDS contract dates	1-Oct-2005
cpn_tbl	coupon table	see above
princ	principal	10,000,000	long
redem_value	redemption value per 100 par	100
freq	frequency	2	semi-annual
acc_rate	accrual method for coupons	2	actual/360
acc_accrued	accrual method for accrued interest	2	actual/360
hl	holiday list	0
d_rul	business day convention	2	next good business day
reset_mktdays	number of business days prior to coupon payment date that the rate is fixed	2
position	trade position	1	long
intrp	interpolation method	1	linear
df_crv	discount factor curve – risk free	see example 1
rating	rating	2
trans_matrix	one year rating transition matrix	see above
intrp_prob	interpolation method of probability curve	1	linear
rate_recover	recovery rate table	see above
calc_type	calculation type	1	simplified calculation method
stat	stat list	1…15

to get the following results:

Results

Statistics	Description	Value
1	fair value	9604225.183
2	accrued interest	122500
3	fair value plus accrued interest	9726725.183
4	annually compounded	0.097204123
5	semi-annually compounded	0.094950236
6	quarterly compounded	0.093849275
7	monthly compounded	0.093124721
8	money market (actual/ 365)	0.098618678
9	money market (actual/ 360)	0.097204123
10	duration	1.755058086
11	modified duration	1.675512913
12	convexity	3.712699638
13	basis point value	-1629.544802
14	yield value change per 1bp increase in price	-6.136E-10
15	probability of survival at maturity	0.920388889

References

[1] Arvanitis, A and Gregory, J., (2001), Credit: the Complete Guide to Pricing, Hedging, and Risk Management, RiskBooks.

[2] Francis, J., Frost, J. and Whittaker, J., (1999), The Handbook of Credit Derivatives, New York, McGraw-Hill.

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