DEBENTURE PRICING MANUAL
31/03/2020 PUBLIC INFORMATION
INFORMAÇÃO PÚBLICA – PUBLIC INFORMATION Debenture Pricing Manual
CONTENT
INTRODUCTION ...... 3
1 GLOSSARY OF VARIABLES ...... 4
2 PRICING MODELS FOR FIXED INCOME CONTRACTS ...... 7
3 PRICED DEBENTURES ...... 10
4 PROCEDURES ADOPTED DUE TO CHANGE IN DEBENTURE CHARACTERISTICS ...... 12
ANNEX – METHODOLOGIES ...... 14
1 DEBENTURES SPREAD CURVE ...... 14
2 SYSTEMATIC BIAS CORRECTION MODEL FOR TRADED DEBENTURES ...... 22
CHANGE LOG ...... 23
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INTRODUCTION
This manual presents the calculation methodologies used to generate debentures reference prices and rates disclosed by B3 in the Private Fixed Income segment. Also listed are the characteristics that must be observed in a given bond for it to be eligible for pricing by one of the methods set forth herein.
Section 1 defines the variables used in pricing models. Section 2 presents the pricing model for the categories in which the debentures are grouped. Section 3 provides the economic and financial indices covered by the pricing models and the exceptions. Lastly, section 4 describes the procedures adopted due to changes in debenture characteristics. The Annex shows the methodologies for calculating the parameters used for pricing.
The calculations presented throughout this document are performed without applying any truncation or rounding. Only the end product of the calculation, the one sent to users, is rounded to the sixth decimal place.
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1 GLOSSARY OF VARIABLES
Debentures contracts are divided into four groups: (i) Fixed Rate, (ii) Floating Rate with Percentage Spread, (iii) Floating Rate with Multiplicative Spread and (iv) Indexed. The methodology used to calculate the reference price and rate is specific to each group. However, the variables and concepts are common to all. This section presents a glossary of the variables used in fixed income contract pricing methodologies and their descriptions.
풕 Mark-to-market date.
풕−ퟏ Business day prior to mark-to-market date.
풕ퟎ Bond yield start date.
푨푴푻%(풆풊) Par Value percentage to be paid on payment flow dates (풆풊).
%푰풏풄(풆풊) Percentage of the future value of the i-th interest rate flow to be incorporated into the issuance par value (VNE).
풆풊 Payment flow dates, 0 ≤ 𝑖 ≤ 푛; dates on which the issuer must pay the investor. In the notation used in this document,
푒푖 is the i-th payment flow date; 푒0 is assumed to be the bond
yield start date (푡0) and 푒푛 is the last payment date. The set of all these dates will be named 퐸 and the subset will be
named 퐸푡, where 푒푖 ≥ 푡.
푵풕 Value in points on the 풕 calculation date of the index correcting the bond issuance par value.
푵풕ퟎ Value in points on the 풕ퟎ bond calculation start date of the index correcting the bond issuance par value.
풓풆풊 Interest rate expressed in % per annum (p.a.) that pays the
푒푖 interest rate event. All rates contained in the models presented are expressed in % p.a., on the basis of 252
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business days (d.u.) capitalized under the compound interest regime. If the interest rates of any bond differ from those mentioned above, they shall be converted to an equivalent rate so that the calculations made are coincident with the characteristics of the bond’s issuance.
푽푵푬 Bond issuance par value.
푽푵푹 Remaining par value, i.e., the amount of the issuance par value that the issuer still owes the investor, minus amortizations already made.
푉푁푅(푡) = (푉푁퐸 + ∑ 푭(풆풌) ∗ (1 − %퐼푛푐(푒푘)) − ∑ 퐴푀푇(푒푘) 푒푘:푒푘∈피푡 푒푘:푒푘≤푡
푻푼 Single contract discount rate obtained through the bisection method.
The variables below are defined only in 퐸푡, that is, in payment flows that occur on
dates where 푒푖 ≥ 푡.
푨푴푻(풆풊) Amortization Cash Value.
In the event of amortization on the issuance par value:
퐴푀푇(푒푖) = 푉푁퐸 + ∑ 푭(풆풊) ∗ (1 − %퐼푛푐(푒푖))) ⋅ 퐴푀푇%(푒푖) 푒푘:푒푘∈피푡
In the event of amortization on the remaining par value:
퐴푀푇(푒푖) = 푉푁푅 + ∑ 푭(풆풊) ∗ (1 − %퐼푛푐(푒푖))) ⋅ 퐴푀푇%(푒푖) 푒푘:푒푘∈피푡
푫푼(풆풊) Business days count between 푡 and 푒푖.
푫푼푪풖풑풐풎(풆풊) Business days count between 푒푖−1 and 푒푖 Payment Flows.
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푫풖풓(풕) Fixed income contract duration.
∑푒 :푒 ∈피 퐷푈(푒푘) ⋅ 푃(푒푘) 1 퐷푢푟(푡) = 푘 푘 푡 ⋅ 푃푈 252
푭(풆풊) Cash value of the interest rate event to be paid on the 푒푖 date.
푰(풕) Premiums released and not paid by the issuer.
풑 Percentage premium, which may be a premium or a discount, applied to the debenture credit spread. The premium is differentiated by debenture and allows incorporating other factors into the price other than the credit profile.
푷(풆풊) Present value related to the total to be paid at the 퐹(푒푖) flow.
푷푼(풕) Present value of contract, reference price (PU = Unit Price).
푃푈(푡) = 퐼(푡) + ∑ 푃(푒푘) 푒푘:푒푘∈피푡
푹(풆풊) Discount interest rate for the 푒푖 maturity calculated through exponential interpolation. This rate depends on the contract index and will be detailed in the following sections.
푺(풆풊) Credit spread for the 푒푖 maturity calculated through exponential interpolation. The spread is obtained from the credit spread curves for the debenture credit profile. Each fixed income contract is assigned a credit profile and used to obtain the corresponding credit spread curve.
The characteristics and payment flows of the debentures used in pricing are those defined in the public issuance deeds of debentures and are reflected in B3's securities registration system.
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Below, we present the calculation specificities of different debenture groups, using the notation presented herein.
2 PRICING MODELS FOR FIXED INCOME CONTRACTS
The reference price for liquid debentures, i.e., those that have a minimum trading frequency over a time window, is the average day's trading prices that meet the maximum dispersion criterion weighted by quantity. For other debentures, the price is determined via the models described below.
The types of fixed income contracts differ depending on the financial index used for indexation, which is reflected in the methodology used for pricing the bond. Fixed income contracts are classified as fixed rate, floating rate (percentage or multiplicative spread) and indexed. The following are formulas for pricing bonds.
2.1 Fixed rate contracts
Are contracts characterized for having their interest rate (푟) and yield known upon issuance.
퐷푈퐶푢푝표푚(푒 )⁄252 ( ) 𝑖 푎푐푐퐹 푒푖 = (1 + 푟푒𝑖) − 1
퐹(푒푖) = 푉푁푅(푒푖−1) ∗ 푎푐푐퐹(푒푖)
퐹(푒푖) ∗ (1 − %퐼푛푐(푒푖)) + 퐴푀푇(푒푖) 푃(푒푖) = 퐷푈(푒𝑖)⁄252 [(1 + 푅(푒푖)) ∗ (1 + 푆(푒푖)) ∗ (1 + 푝)]
퐷푈(푒𝑖)⁄252 퐷푈(푒𝑖)⁄252 [(1 + 푇푈)] = [(1 + 푅(푒푖)) ∗ (1 + 푆(푒푖)) ∗ (1 + 푝)]
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2.2 Floating rate contracts
Their indexation is based on an index (퐼푛푑𝑖푐(푡)) with an annualized rate on a compound interest basis of 252 business days, such as the CDI (Interbank Certificate of Deposits), which is used for most contracts.
Floating rate contracts may be of two types: percentage spread and multiplicative spread.
2.2.1 Floating rate contracts with percentage spread
Percentage spread contracts are traded as a percentage (휙푒𝑖) of the index interest rate defined at the issuance of the contract.
If 푒푖 is the first event immediately following the 푡 mark-to-market date, then:
퐷푈(푡,푒 ) 1⁄ 𝑖 ( ) ( ) 252 푎푐푐퐹 푒푖 = 푓푎푡표푟퐼푛푑𝑖푐 ∗ {[(1 + 푅 푒푖 ) − 1] ∗ 휙푒𝑖 + 1}
푡 1 ( ) ⁄252 푓푎푡표푟퐼푛푑𝑖푐 = ∏ [ 1 + 퐼푛푑𝑖푐푘 − 1] ∗ 휙푒𝑖 + 1 푘=푒𝑖−1
Otherwise:
퐷푈(푒 ) 1 𝑖 ( ) ( ) 252 푎푐푐퐹 푒푖 = [1 + (((1 + 푅 푒푖 ) ) − 1) ∗ 휙푒𝑖]
푎푐푐퐹(푒푖) 퐹(푒푖) = 푉푁푅(푒푖−1) ∗ [ − 1] 푎푐푐퐹(푒푖−1)
퐹(푒푖) ∗ (1 − %퐼푛푐(푒푖)) + 퐴푀푇(푒푖) 푃(푒푖) = 퐷푈(푒𝑖)⁄252 [(1 + 푅(푒푖)) ∗ (1 + 푆(푒푖)) ∗ (1 + 푝)]
퐷푈 1 𝑖 252 [[(1 + 푅(푒푖)) − 1] ∗ (1 + 푇푈)]
퐷푈(푒𝑖)⁄252 = [(1 + 푅(푒푖)) ∗ (1 + 푆(푒푖)) ∗ (1 + 푝)]
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2.2.2 Floating rate contracts with multiplicative spread
Flow payments of floating rate contracts with multiplicative spread consider the composition of the contract coupon rate (r) with the index interest rate for the calculation of the contract’s yield. Cash flow discount considers the composition
of the market credit spread for the contract and the discount rate (푅(푒푖)) for the index (퐼푛푑𝑖푐).
If 푒푖 is the first event immediately following the 푡 mark-to-market date, then:
퐷푈(푡,푒𝑖) 퐷푈(푒𝑖) ( ) ( ) 252 252 푎푐푐퐹 푒푖 = 푓푎푡표푟퐼푛푑𝑖푐 ∗ (1 + 푅 푒푖 ) ∗ (1 + 푟푒𝑖)
푡 1⁄ 푓푎푡표푟퐼푛푑𝑖푐 = ∏ (1 + 퐼푛푑𝑖푐푘) 252
푘=푒𝑖−1
Otherwise:
퐷푈(푒𝑖) ( ) ( ) 252 푎푐푐퐹 푒푖 = [(1 + 푅 푒푖 )(1 + 푟푒𝑖)]
푎푐푐퐹(푒푖) 퐹(푒푖) = 푉푁푅(푒푖−1) ∗ [ − 1] 푎푐푐퐹(푒푖−1)
퐹(푒푖) ∗ (1 − %%퐼푛푐(푒푖)) + 퐴푀푇(푒푖) 푃(푒푖) = 퐷푈(푒𝑖)⁄252 [(1 + 푅(푒푖)) ∗ (1 + 푆(푒푖)) ∗ (1 + 푝)]
퐷푈 𝑖 퐷푈(푒 )⁄252 252 𝑖 [(1 + 푅(푒푖)) ∗ (1 + 푇푈)] = [(1 + 푅(푒푖)) ∗ (1 + 푆(푒푖)) ∗ (1 + 푝)]
2.3 Indexed contracts
Indexed bonds update their issuance par value (VNE) according to the variation of a given 푵 index, which is defined at the contract’s issuance, such as the IGP- M (General Market Price Index) or the IPCA (Extended Consumer Price Index).
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The contract coupon interest rate (푟푒𝑖) is then applied over the updated par value to determine the event’s cash value to be paid by the issuer.
퐷푈퐶푢푝표푚(푒 )⁄252 ( ) 𝑖 푎푐푐퐹 푒푖 = (1 + 푟푒𝑖) − 1
푁푡 퐹(푒푖) = 푉푁푅(푒푖−1) ∗∗ ∗ 푎푐푐퐹(푒푖) 푁푡0
푁푡 퐹(푒푖) ∗ (1 − %%퐼푛푐(푒푖) + 퐴푀푇(푒푖) ∗ 푁푡0 푃(푒푖) = 퐷푈(푒𝑖)⁄252 [(1 + 푅(푒푖)) ∗ (1 + 푆(푒푖)) ∗ (1 + 푝)]
퐷푈(푒𝑖)⁄252 퐷푈(푒𝑖)⁄252 [(1 + 푇푈)] = [(1 + 푅(푒푖)) ∗ (1 + 푆(푒푖)) ∗ (1 + 푝)]
3 PRICED DEBENTURES
Public issuances performed through the Brazilian Securities & Exchange Commission (CVM) Normative Instructions No. 400 and No. 476.
3.1 Economic and financial indices covered by the models
▪ CDI with [d-1] business day displacement.
▪ IPCA with anniversary date on the 15th day and pro-rata capitalization of 252 business days under the compound interest regime.
▪ IGP-M with anniversary date on the 1st day and pro-rata capitalization of 252 business days under the compound interest regime.
3.2 Economic and financial indices not covered by the models
▪ Indices other than IPCA, DI and IGP-M. If the index update features are equivalent to the IPCA, DI or IGP-M features, the model with the closest features will be used.
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▪ IPCA with anniversary date other than the 15th day. Pricing will be performed through the IPCA with anniversary date on the 15th day.
▪ IGPM with anniversary date other than the 1st day. Pricing will be performed through the IGP-M with anniversary date on the 1st day.
▪ IPCA with a lag of more than 1 month.
▪ Assets with amortization events prior to interest payment events with incorporation.
▪ Perpetual maturity.
▪ Interest-free amortization: the payment flow comprises amortization only.
▪ Convertibility into shares or any other asset offered by the issuer as a conversion object.
▪ Formalization of premium payment by performance (participation event).
▪ Issuers with defaulted interest flows and amortizations and/or with restricted confirmed status in B3's securities system.
▪ Liquidity-free issuances that are concentrated on few holders
▪ Issuances made by securitization companies with credit rights-backed payment flow
▪ Issuers that do not have a credit profile assigned.
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4 PROCEDURES ADOPTED DUE TO CHANGE IN DEBENTURE CHARACTERISTICS
The changes deliberated by the General Meeting of Debenture Holders (AGDs) will be reflected in the pricing following the formalization of the change with B3’s Fixed Income Securities. The changes deliberated at the AGDs and treatments applied are listed below.
▪ Change of event dates (interest rate, amortization, premium, etc): the new dates are updated in the debentures register used for pricing.
▪ Payment of premium at a future date: the premium’s cash value is updated in field I(t) (premiums due) and incorporated into the reference price according to the unit price (PU) calculation formula available in the glossary of this document.
▪ Change in the debentures return rate between payment events: the return rate used in the debenture pricing is replaced with the equivalent rate for the period, calculated from the original rate and the changed rate within their respective timeframes.
▪ Extraordinary amortization with redefinition of future amortization flows: the new amortization flows are updated in the debentures register used for pricing.
▪ Total early redemption: the debenture redemption date is updated at its maturity date.
▪ Generic event: the calculated or estimated event value (interest rate, amortization, premium, etc.) is incorporated into the debenture price.
▪ Change in amortization type (from remaining base amount to issuance base value, or vice versa): debt amortization percentages are updated to reflect the new amortization type.
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▪ Non-payment of event (interest rate, amortization, premium, etc): in this case, the debenture becomes ineligible for pricing.
▪ Unscheduled amortization without redefining future flows: when the issuer does not deliberate on the redistribution of flows, the unscheduled payment will be debited uniformly in the remaining installments expiring.
▪ Interest incorporation: in this case, interest amounts are incorporated into the debenture par value as of the date of incorporation.
Changes made through AGDs that are not treated by the available pricing models are not to be incorporated into the debenture reference price.
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ANNEX – METHODOLOGIES
This annex presents the methodologies for estimating the inputs needed for pricing.
1 DEBENTURES SPREAD CURVE
1.1 Model
The debentures spread curve model uses the Nelson-Siegel approach to adjust the level and slope of the corporate credit curve. These features are observed in stylized interest rate curve facts:
1 − 푒−휆휏 푟 (휏) = 훽 + 훽 ( ) + 휖(휏) (1) 푥 푥 퐶푟é푑푖푡표 휆휏
Where:
• x is one of the possible credit profiles. For notation purposes, in this manual credit profiles A, B, C and C- are used. Details of the methodology used to define the credit profiles are described in the document “Methodology for Assigning Debentures Credit Risk Profiles”;
• 푟푥(휏) is the temporal structure of the credit spread for the 푥 credit profile;
• 휏 is the annualized curve term;
• λ is the 푟푥 speed of decay;
• 훽푥 is the long-term rate level of the credit spread curve for the 푥 credit profile;
• 훽퐶푟é푑푖푡표 is the slope common to all interest rate curves;
• 휖(휏) is the model’s error term.
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In order to find a homogeneous term structure among debentures, the curve has
the 훽푥 credit profile level parameter, which is differentiated by credit profile and
the 훽퐶푟é푑푖푡표 slope factor common to all profiles. This formulation aims to prevent the curves’ temporal structure from intersecting.
1.2 Model Estimation
1.2.1 Streamlining method
Since credit spread curves are not directly observable, the estimation problem is to find a discount curve for each credit profile group that minimizes the differential between the theoretical price and the traded price. The weighted least squares method (MQP) is used to jointly estimate all credit curves:
푛푥 푥 2 min ∑ ∑(푊푖(푃푖,푡푒ó푟푖푐표 − 푃푖,푚푒푟푐푎푑표)/푃푖,푚푒푟푐푎푑표) (2) 훽퐴,훽퐵,훽퐶,훽퐷,훽퐶푟é푑𝑖푡표 푥∈{퐴,⋯,퐷} 푖=1
Where:
• 푛푥 is the number of debentures used in 푥 credit profile estimate;
푥 푚 푥 푥 • 푃푖,푡푒ó푟푖푐표 = ∑푘=1 퐹퐶푖,푘 ∗ 퐷퐹푖,푘, where 퐹퐶푖,푘 and 퐷퐹푖,푘 are the cash flow and the discount function for the 푘th flow, respectively. The discount function can be broken down as follows:
푥 푚 1 푃푖,푡푒ó푟푖푐표 = ∑푘=1 퐹퐶푖,푘 ∗ 휏𝑖,푘 ((1+푅(휏𝑖,푘))(1+푟푥(휏𝑖,푘)))
Where:
푅(휏푘) is the discount curve (DI or IPCA) and 푟푥 is the credit spread for the 푥 credit profile to which the 𝑖 debenture belongs, as per equation (1). The
휏푖,푘 term is the maturity of each annualized flow;
• 푃푚푒푟푐푎푑표 is the price observed in the market (see section 1.2.4);
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1 • 푊푖 = ⁄ , where Duration is expressed in years: 퐷푢푟푎푡𝑖표푛푖
∑푚𝑖 퐹퐶 ∗ 휏 퐷푢푟푎푡𝑖표푛 = 푘=1 푖,푘 푖,푘 푖 푚𝑖 ∑푘=1 퐹퐶푖,푘
1.2.1 Spacing between credit curves
Since the liquidity of debentures is concentrated on credit profile A, the parameter
of this group (훽퐴) is used as a reference to obtain the other 훽푥. More specifically, in order to preserve the economic sense of the credit curves level parameters, the spacing between the curves is estimated using the order 1 autoregressive
vector model (VAR) with the temporal series of the 훽푥 level parameters. Through these temporal series the following spacing rates are built:
퐵 퐶 퐶− ( ) ( ) ( ) 퐴 퐴 퐴 푦푡 = 훽퐴,푡 − 훽퐵,푡, 푦푡 = 훽퐴,푡 − 훽퐶,푡, 푦푡 = 훽퐴,푡 − 훽퐶−,푡.
The VAR model is defined from the spacing rates as follows:
퐵 퐵 퐶 퐶− ( ) ( ) ( ) ( ) 퐴 퐴 퐴 퐴 푦푡 = 푎퐵 + 휙퐵,퐵푦푡−1 + 휙퐵,퐶푦푡−1 + 휙퐵,퐶−푦푡−1 + 휖퐵,푡 퐶 퐵 퐶 퐶− ( ) ( ) ( ) ( ) 퐴 퐴 퐴 퐴 (4) 푦푡 = 푎퐶 + 휙퐶,퐵푦푡−1 + 휙퐶,퐶푦푡−1 + 휙퐶,퐶−푦푡−1 + 휖퐶,푡 퐷 퐵 퐶 퐶− ( ) ( ) ( ) ( ) 퐴 퐴 퐴 퐴 푦푡 = 푎퐷 + 휙퐷,퐵푦푡−1 + 휙퐷,퐶푦푡−1 + 휙퐷,퐷푦푡−1 + 휖퐶−,푡
Where:
푞 ( ) 퐴 • 휙푝,푞 is the participation of the 푦푡 spacing in the description of the 푝 ( ) 퐴 푦푡 spacing;
푝 ( ) 퐴 • 푎푝 is the intercept of the 푦푡 spacing;
• 휖 is the residual matrix of the model.
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Once the VAR model 푎푝 and 휙푝,푞 parameters are estimated and with the 푥 ( ) 푥 ℎ 퐴 ( ) ∑푡=1 푦 calculation of the 푦̅ 퐴 = 푡−1 averages for a number of ℎ observations, the ℎ
remaining 훽푥,푡 for 푥 = 퐵, 퐶, 퐶 − can be estimated, as follows:
푥 퐵 퐶 퐶− ( ) ( ) ( ) ( ) 퐴 퐴 퐴 퐴 훽푥,푡 = 훽퐴,푡 − 푦푡 = 훽퐴,푡 − (푎푥 + 휙푥,퐵푦̅ + 휙푥,퐶푦̅ + 휙푥,퐶−푦̅ ) (5)
This avoids some economic inconsistencies observed in estimation on days when a group is practically illiquid.
1.2.2 Tax Incentive
Incentivized debentures tend to have a lower spread level than debentures without tax incentives. This factor may also distort the effect observed in the issuer's fundamentals (credit profile). Thus, it is necessary to differentiate between incentivized and non-incentivized debentures when estimating the credit 푖푛푐푒푛푡 profile spread. That is, for each 푥 credit profile group, the spread curve for 푟푥 푛ã표−푖푛푐푒푛푡 incentivized debentures and 푟푥 non-incentivized debentures is estimated.
1.2.3 Outliers treatment
In traded prices, it is common to find some values that present an implied rate of return in the trade that does not match the spreads observed for other debentures of the same credit profile. These outliers in the implied spreads on debenture prices can be explained for several reasons. Some of them are listed below.
• Credit profiles may be lagged due to the time lag between the market’s perception and the updating and disclosure of economic and financial information that allows the credit profile to be adjusted.
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• Inconsistencies in the (퐹퐶푘) cash flow, existence of premium events, amortization, or interest rate extended or canceled.
• The liquidity of a debenture may generate distorted traded prices from its fundamentals (credit profile).
Two distinct independent filters are evaluated for outlier detection and at the end of the process the filter that generates the smallest spread variation over the previous day is chosen. One of the filters is named endogenous, as it defines statistics based only on the day’s residuals, and the other is named exogenous, as it defines statistics based on residuals from previous dates.
The endogenous filter considers an outlier if the debenture meets the criterion of the following standardized statistics:
휀 − 푚푒푑(휀 , ⋯ , 휀 ) | 푖 1 푛 | > 1,5 (7) 휎(휀1, ⋯ , 휀푛)
Where:
2 • 휀푖 = (푊푖(푃푡푒ó푟푖푐표 − 푃푛푒𝑔ó푐푖표)/푃푛푒𝑔ó푐푖표) is the residual resulting from the adjustment process of the 𝑖th debenture;
• 푃푚푒푟푐푎푑표 and 푃푡푒ó푟푖푐표, as described in equation (2);
• 푚푒푑(휀1, ⋯ , 휀푛) is the residual median of all debentures traded on the reference date;
• 푛 is the number of debentures traded on the reference date;
• 휎(휀1, ⋯ , 휀푛) is the residual sample standard deviation on the reference date;
• The |⋯ | > 1,5 criterion is justified as an adjustment of the empirical residual distribution to a normal distribution.
The exogenous filter considers a debenture as an outlier if
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휀푖 > resí푑푢표훼
Where:
• resí푑푢표훼 is the 훼 level percentile of the 휖{푡,푖푡} sample.
• 휖{푡,푖푡} is the set containing the 휀푖푡 residuals for each 푡 date and each 𝑖푡 debenture used in the final estimation on 푡 date for 푡 = 1, . . . , 푇, where 푇 is the size of the window used.
1.2.4 Market price
For estimation of the average traded price, a spread filter implied in the price of each trade is applied with the purpose of removing those trades with spreads that are not compatible with the issuance trading profile. If the implied spread of the j
trade on the t date is expressed by ρ푡,푗, it is obtained by numerically solving the following equality:
푚 퐹퐶푘 푃푡,푗 = ∑ 휏푘 푘=1 ((1 + 푅푡(휏푘))(1 + ρ푡,푗))
Where:
• 푃푡,푗 is the j trade price on the 푡 date;
• 퐹퐶푘 are the debenture’s cash flows;
• 휏푘 is the time in business days to the 푘 flow;
• 푅푡 is the discount curve.
For a historical window, the implied spreads on trade prices between different
participants are collected for each debenture. In this history we obtain the ρ훼 and
ρ1−훼 quantiles and with them a price interval is defined.
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푚 퐹퐶푘 푚 퐹퐶푘 푃푚푖푛 = ∑푘=1 휏푘 and 푃푚푎푥 = ∑푘=1 휏푘 ((1+푅푡(휏푘))(1+ρ훼)) ((1+푅푡(휏푘))(1+ρ1−훼))
Therefore, for the calculation day all trades within this interval are considered. The weighted average number of trades in the interval defines the
푃푚푒푟푐푎푑표 market price of the debenture on the calculation date.
1.3 Estimation algorithm
The estimation of all spread curves for credit profiles is performed in two phases.
I) Daily procedure – T+0 estimation of the A profile incentivized curve parameters and the non-incentivized A profile parameters and use of the estimated parameters to obtain other profile curves through temporal series models.
II) Periodical procedure – Estimation and extraction of temporal series of spacing between credit profile curves.
Daily procedure
• 1st Step: The market price of the debentures traded on the reference date is calculated based on trade prices (according to section 1.2.4), by grouping them by credit profile and classifying them as incentivized or non-incentivized debentures.
nd • 2 Step: The (푊푖) Duration for each traded debenture is calculated.
• 3rd Step: The Nelson-Siegel model is estimated by least squares weighted for (A) reference credit profile with incentivized debentures. To preserve the economic sense of the curve, the λ non-negative and β non-positive constraints, specifically 휆 ≥ 0.0001 and 훽 ≤ 0, are used.
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• 4th Step: The estimated model residuals are used to generate the standardized statistics (see item 1.2.3).
• 5th Step: The outliers filter is applied. For the endogenous outliers model, debentures with standardized statistics greater than 1.5 are excluded. The model is estimated again and a second analysis of endogenous outliers is applied.
h • 6t Step: The (훽퐴) level and (훽푐푟é푑푖푡표) slope estimated parameters are collected.
푥 ( ) th 퐴 • 7 Step: The (푦푡 ) spacing parameters are collected.
• 8th Step: The final formulation obtained by other credit profiles from the reference curve level is as follows:
푥 ( ) 푖푛푐푒푛푡 푖푛푐푒푛푡 퐴 훽푥 = 훽퐴 − 푦푡
Steps 3 to 8 are executed for issuances of non-incentivized debentures by obtaining the curves of each credit profile for those issuances.
• 9th Step: The Nelson-Siegel formula is used with the estimated 푛ã표−푖푛푐푒푛푡 푖푛푐푒푛푡 푖푛푐푒푛푡 훽푥,푡 , 훽푥 , 훽퐶푟é푑푖푡표 parameters to generate the 푟푥 푛ã표−푖푛푐푒푛푡 푟푥 curves in the vertices for calculation.
Periodic procedure
• 1st Step: For each 푡 day within an ℎ day window, steps 1 through 6 of the daily procedure are performed using all the incentivized and non-incentivized debentures credit profiles.
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nd • 2 Step: Estimates are obtained for the (훽푥) level parameters and 푥 ( ) 퐴 spacing between all credit profiles and the (푦푡 ) benchmark profile as formalized in item 1.2.1.
2 SYSTEMATIC BIAS CORRECTION MODEL FOR TRADED DEBENTURES
The systematic bias correction premium is modeled to better adjust the debenture trade prices by capturing its own and individual characteristics. The premium comes as a shock applied to the spread curve for the debenture credit profile.
The premium is estimated when the debenture has a minimum of trades within
the calculation period. For the 푃푚푒푟푐푎푑표 day’s market price (calculated according to section 1.2.4), the implied premium is calculated by the equation
푚 퐹퐶푘 푃푚푒푟푐푎푑표 = ∑ 휏푘 푘=1 ((1 + 푅(휏푘))(1 + 푟푥(휏푘))(1 + 푝푟ê푚𝑖표))
On the dates when the debentures are not traded, the last calculated premium will be used.
For non-traded debentures, a premium is defined from the trade premium for debentures with similar characteristics. Issuances that do not provide sufficient liquidity for the price to be calculated according to equation (8) and debentures whose issuances do not present similar characteristics with liquidity have their estimated premium on the trades executed in the calculation period.
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CHANGE LOG
Effective: July 1st, 2019
1st version: July 1st, 2019
Areas in charge of this document at B3:
Responsible Area
Draft Fixed Income Pricing and Pricing Modeling
Review Pricing
Approval Risk Management
Updates
Version Change Reason Date
1 Original Version - July 1st, 2019
Addition of treatment to incorporate interest into 2 Sections 1 and 2 September 1st, 2019 the calculation of reference price.
Addition of estimate for 3 Section 2 – Annex debentures premium September 12, 2019 with little liquidity.
Reference to document describing the 4 Section 1 - Annex methodology for October 24, 2019 assigning credit risk profiles.
Spread curves for non- incentivized debenture issuances are now 5 Sections 1.2.2 and 1.3 - Annex March 31st, 2020 generated from trades executed for those issuances.
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