MARK TO MARKET MANUAL
JULY/2015
07/01/2015
i
TABLE OF CONTENTS
I - INTRODUCTION ........................................................................................... IV
A-
MARK TO MARKET PROCESS AT BB DTVM ................................................. iv
B-
GENERAL PRINCIPLES ................................................................................... xi
C-
METHODOLOGY OF THE PRACTICES ......................................................... xiii
II – APPLICATIONS ........................................................................................... 1
A.
TERM STRUCTURE OF PRE-FIXED EXCHANGE RATES ............................... 1
B-
FEDERAL GOVERNMENT BONDS – PRE-FIXED ............................................ 3
1.
LTN – Brazilian Treasury Bills .......................................................................... 3
2.
NTN – F - BRAZILIAN TREASURY NOTES Series – F ................................... 4
C-
GOVERNMENT FEDERAL BONDS – POS-FIXED ............................................ 6
3.
LFT – Financial Treasury Bills .......................................................................... 6
4.
NTN-B – Brazilian Treasury Notes – Series B .................................................. 7
5.
NTN-C – Brazilian Treasury Notes – Series C.................................................. 9
6.
NTN-D - Brazilian Treasury Notes – Series D ................................................ 10
D-
PRIVATE SECURITIES ................................................................................... 13
7.
Debentures .................................................................................................... 13
8.
Pre-fixed Assets: CDB/RDB and Other Private Securities .............................. 16
9.
Pos-fixed Assets Indexed to SELIC Rate or DI: CDB/RDB ............................. 17
10.
Term deposit with Special Guarantee from FGC ........................................... 23
10.1.
DPGE Pos-Fixed – Indexed to SELIC rate or DI ...................................... 23
10.2.
DPGE Pre-fixed ....................................................................................... 27
11.
Pos-fixed Assets Indexed to SELIC or DI: CCB rate....................................... 28
12.
Pos-fixed Assets Indexed to SELIC or DI: Financial Bill - LF .......................... 31
13.
Pos-fixed Assets Indexed to SELIC or DI Rate: NCE ..................................... 39
14.
Other Pos-fixed Assets Indexed to SELIC or DI Rate ..................................... 43
15.
Mortgage Bills – LH........................................................................................ 45
16.
Private securities indexed to the IPCA – CDB, DPGEs and Financial Bill - LF 47
E-
ASSETS NEGOTIATED ABROAD .................................................................. 49
ii
17.
ADR – American Depositary Receipt ............................................................. 49
18.
Fixed Income – Corporate Bonds, Treasury Bonds, Global, CLN, etc. ........... 49
F-
VARIABLE INCOME AND FUTURES .............................................................. 50
19.
Shares and BDRs – Brazilian Depositary Receipts ........................................ 50
20.
Subscription Rights and Receipts of Shares .................................................. 50
21.
Rent (or loans) of Shares ............................................................................... 51
22.
Futures BM&F/BOVESPA ............................................................................. 51
23.
SWAP ............................................................................................................ 52
24.
Options .......................................................................................................... 53
24.1.
Liquid Options ......................................................................................... 53
24.2.
Low Liquidity Options............................................................................... 53
⇒
BLACK & SCHOLES MODEL ..................................................................... 53
24.3.
⇒
MERTON (1973) E REINER E RUBINSTEIN (1991) MODEL ..................... 55
24.4.
⇒
Futures Options ....................................................................................... 59
BLACK MODEL .......................................................................................... 59
24.5.
⇒
25.
Barrier Options ........................................................................................ 55
Foreign Currency Options ........................................................................ 62
GARMAN-KOHLHAGEN MODEL ............................................................... 62
Exotic Options ................................................................................................ 64
25.1.
QUANTO Adjustment .............................................................................. 64
25.2.
Asian Options .......................................................................................... 66
26.
Synthetic Operations ...................................................................................... 68
G - FIXED EARNINGS OPERATION – SHARE TERM .............................................. 70
27.
Share Term .................................................................................................... 70
H – REPURCHASE AGREEMENTS .......................................................................... 71
28.
Repurchase Agreements ............................................................................... 71
I – SHARES OF FUNDS ............................................................................................. 72
29.
Shares of Investment Funds .......................................................................... 72
iii
I - INTRODUCTION
A - MARK TO MARKET PROCESS AT BB DTVM
1 Overview of the Process
The mark to market process – MtM – the portfolios of investment funds at BB
DTVM is performed in compliance with the general principles emanating from
CVM Instruction 438, 12/07/2006, of the Securities Commission and their later
alterations and recommendations provided for in the Self Regulating Code
issued by ANBIMA.
1.1
Considering that the Brazilian Capital Market has low liquidity, the prices
of the assets at which the agents are prepared to buy/sell, are not
always found easily.
1.2
In order to become these practices as transparent as possible we shall
detail the hierarchy of the mark to market process formulated by BB
DTVM in accordance with the Mark to Market Directives formulated by
ANBIMA.
1.3
In the hierarchy of the processes we have:
1.3.1
1.3.2
1.3.3
1.3.4
1.3.5
1.3.6
1.4
gathering of the prices;
handling of the gathered prices;
validating of the data/prices handled;
applying of the prices to the portfolios;
validating of the price application to the portfolios; and
dynamic supervision of the methodology.
Gathering of Prices – the primary sources of which the processes and
methodologies are composed are obtained by downloading files from
the following institutions:
-
ANBIMA
–
Federal
Government
Bonds/Private
Securities
(Debentures)
-
BM&F/BOVESPA – Derivatives, adjustments to the Futures Market
and ETTJ
-
BM&F/BOVESPA – Shares, BDRs and Options
-
CETIP – Private Equity
-
Various
Providers
of
Market
Prices
(CMA,
BROADCAST,
BLOOMBERG, Others) – International equity
iv
1.5
Dealing with gathered prices – We describe ahead a detailed
methodology for each asset, where some are treated directly from the
source in the form of unitary pricing - UP, others are calculated in
accordance with their rates or quota, in consideration of the standard
methodologies used by the market for each asset, albeit:
1.5.1 Capitalization period – daily, monthly, semester or annual;
1.5.2 Capitalization – linear or exponential;
1.5.3 Number of days counted – 252 working days of the year, days
360, 30/360 consecutive days;
1.5.4 of payments - month, semester, annual, etc.
1.6
Validating the handled prices - final edition of the handled prices is made
by way of a comparative analysis between the daily yield, in search of
the adherence of the yield used by the market.
1.7
Applying the prices to the portfolios – Prices calculated in accordance to
rates are made available to the team that is responsible for processing
the portfolios on the Board Solutions Wholesale – DISAT, by way of their
own software, whereas prices used for portfolios are gathered by the
processing team directly from the primary source.
1.8
Validating the application of prices to portfolios – The Processing
Division from DISAT makes an analysis of price deviation, checking its
coherence with the market of the referred Asset. The external auditors
run periodic tests to validate the pricing and the internal auditors run
periodic evaluates the process of mark to market with a focus on risk.
1.9
Dynamic supervision of the methodology – Follow-up of the processes in
the order shown is performed by the division that is responsible for the
mark to market process.
2 Methodological Aspects
2.1 The team responsible for processing funds and portfolios gather, on a
daily basis, the Ups, rates or quotas, in accordance with each case, by
way of their own applications and closure of the markets.
2.2 For funds that work with opening quotas, the quota is calculated by the
Methodologies Team from DISAT based on the earnings projection
observed for each asset that makes up the portfolio of the fund on dzero updated for d+1, date of the quota.
2.2.1 The opening quota shall be updated in accordance with the
following:
v
a - estimated appreciation of the assets in the portfolio,
observed in the markets on D0;
b – Weighing the appreciation of the assets, with respect to the
percentage of Net Equity.
cab = ( 1 + i ) × cd − 0
Where:
• Cab = opening quota D+1
• i = index for updating quota
• Cd-0 = opening quota on D0
2.2.2 Alternatively, in case of inoperable system, communication
failure, the opening quota may be updated by the DI effective on
the day, in accordance with the following:
c ab
1


252
i



= 
+ 1  × c
 100   d − 0


Where:
• Cab = opening quota D+1
• i = rate of DI of the day, seen on D0,
• Cd-0 = opening quota D0;
3 Organizational Structures involved in the process
3.1 The organizational structure at BB DTVM includes a directorship made
up of the Chairman, Two Senior Directors, and Seven Senior
Managers, who participate in the forums and committees detailed
below in item 3.5.
3.2 The MTM process is conducted by the Board Solutions Wholesale –
DISAT, Banco do Brasil S.A., through the Funds Controller Division,
composed of a division manager, two team managers, who are
responsible for the balance sheets of the funds and a team of
analysts, who are responsible for assessing, testing and proposing
methodologies for mark to market and pricing of financial assets.
3.3 The Funds Controller Division is linked to the Fiduciary Services
Management - DISAT, therefore segregated from the Senior
Managers of Fund Management, owned by BBDTVM.
3.4 The methodologies are developed in accordance with the
characteristics of each asset or group of assets and the models are
vi
then forwarded to the BBDTVM directors through Notes awaiting
approval
3.5 The Statute of the BB DTVM foresees that the distributor should adopt
a group decision at all levels. The existing forums and committees are
listed below:
3.5.1 Forums
Strategic Forum;
Amplified Forum;
Executive Forum;
Top-Down Analysis;
Bottom-Up Analysis.
3.5.2 Committees
Asset Management Committee;
Allocation Committee;
Product Committee;
Credit and Governance Committee;
Risk Committee;
Asset Pricing Committee;
Information Technology Committee;
Performance Evaluation Committee;
Management Committee.
3.5.3 Asset Management Subcommittees.
Subcommittee on Management of Fixed Income;
Subcommittee on Management of Shares;
Subcommittee Management of Multi-Market and Offshore;
Allocation Subcommittee.
3.5.4 The Committee Regulation for Asset Pricing are provided below,
taking into consideration what it truly has a jurisdiction to discuss
on to Mark to Market subjects
Article 1 – MEMBERS – The Asset Pricing Committee shall be
made up of:
a) Full Members:
I. Executive Manager of Management Funds;
II. Division Manager of Fiduciary Services;
III. Division Manager of Management Structured Funds;
and
IV. Division Manager of Credit Analysis;
b) Alternate Members: appointed by the holders.
vii
c) Members Invited – Permanent participation, without
voting rights:
I. Division Manager of Market Operations;
II. Compliance Manager;
III. Division Manager for Modeling Market Risk and
Liquidity Funds;
IV. Division Manager of Fund Controllers - DISAT; and
V. Business Consultant from Division Fund Controllers DISAT.
Paragraph 1 – The full members may only be substituted by the
respective alternate members in case of vacation, sabbatical,
paid leave, attendance allowance, time off, training or business
trips.
Paragraph 2 – The members invited in his absence will be
represented by officials of the respective Divisions indicated by
them.
Article 2 – RESPONSIBILITIES:
I. To
analyze
and
propose
to
the
Directors
inclusions/alterations/updates of asset pricing methodologies
to the fund portfolios;
II. Accompany and analyze the market conditions, and
propose adjustments/modifications to the asset prices, in
consequence of moments of crisis on the market or low
liquidity;
III. Assess and validate the information on prices gathered and
supplied by the Market Operations Division;
IV. Assess and validate the ratings information used in asset
pricing, supplied by the Credit Analysis Division;
V. Validate
the
baseline
prices
used
Interpolation/extrapolation of rates, for asset pricing;
for
VI. Assess and validate price ranges to be used in asset
pricing;
VII. Assess and improve the cutting points in the samples
used for the price ranges;
VIII. Define materials that may be submitted to the members of
the Committee for deliberation, using mechanisms for
consultations/response by e-mail;
viii
IX. Other subjects pertinent to asset pricing.
Sole paragraph – The responsibility of the members of the
Committee for omission in the execution of their duties is
solidarity, and yet shall excuse the member who may duly
given reason for the diversions in the minutes of the
Committee Meeting.
Article 3 - THE MEETINGS – The Committee shall hold ordinary
meetings on the last Thursday of each month, or extraordinary
meetings whenever determined by its Coordinator.
Article 4 - THE QUORUM – The minimum quorum for the
installation of the Committee is 03 (three) members, the
presence of the Coordinator.
Article 5 - CRITERIA FOR DELIBERATION – the Committee's
decisions shall be approved by:
I. In quorum, unanimously; and
II. Above the minimum quorum, by a simple majority.
Article 6 - THE COORDINATION – coordination shall be
exercised by the Executive Manager of Management Funds. In
his absence, the coordination shall be exercised by one of the
other permanent members of the Committee, indicated by him.
Article 7 - GUESTS – May participate yet without right to vote.
Employees may offer background information to help in the
decisions of the subject on the agenda.
Article 8 – SECRECY – The materials are confidential by nature
and those that are discussed by the committee shall be kept in
secret by the members present at the meeting, as well as those
to whom the aforesaid has come to their knowledge.
Article 9 - THE MINUTES – The minutes shall be written in a
clear and concise manner. The documents used for background
information for the decisions shall be attached and kept for a
minimum of 10 (ten) years in the Administrative Division.
Article 10 – SECRETARIAT – The Committee shall establish a
Secretariat, whose activities are carried out by one of its
members, who will be responsible: only collect the signatures of
participants, as well as keeping the minutes and filed documents
relating to activities.
I. Call and ask staff to stakeholders;
ix
II. Prepare and distribute meeting agendas and copies of
documents, with at least two days prior to the date of
realization, except where expressly authorized by the
Coordinator; and
III. Prepare minutes of meetings and collect the signatures of
its members.
Sole Paragraph – In the Administrative Division to keep fit in
their custody the documents concerning the activities of the
Committee.
Article 11 - GENERAL PROVISIONS – The senior directors shall
remove any existing doubts related to this regulation, as well as
provide modifications that may be judged necessary and decide
on pending cases..
3.6 Since approved, the methodologies for calculation will be sent to the
information technology division, to develop systems/applications, if
necessary, for the purposes of making the processes operational thus
requiring less manual interference.
3.7 Once having been placed in production, the methodology is applied
uniformly to all assets owned by BB DTVM, whose funds
contemplated in the Contract for Qualified Services to the Capital
Markets and controllership services whose assets are in charge of
Funds Controller Division - DISAT, including exclusive funds. Except
for the referred procedure: funds under custody of other assets or
financial institutions, funds whose shareholders have decided to
classify their assets in accordance with item 1.2.2.3 from CVM
Instruction 438, 12/07/2008, – Securities kept up to maturity.
x
B-
GENERAL PRINCIPLES
These principles are considered to be guidelines for establishing the policies of
MTM and are used as directives for the processes and practice of Mark to
Market, and should be applied in a coherent manner, that is, the way in which
one is applied shall not interfere in the application of another, in order to
perform the MARK TO MARKET – MTM.
1. Best practices
The process and methodology of MTM must follow the best practices of the
market.
2. Scope
Taking the view that the principal purpose of Mark to Market is to avoid the
transfer of wealth between various shareholders of an investment fund,
these directives cover all non-exclusive and non-restrictive funds, that is,
those in which the aforesaid transfer of wealth needed to be avoided. Under
these measures, for the purposes of these directives, and exclusive funds is
that which is determined exclusively to one investor and restricted fund is
that which is destined to a group of determined investors, who have
amongst themselves family ties, partnerships or belong to the same
economic group, or in writing determined this condition. In case an
investment fund lose its characteristic of exclusive or restrictive fund, then
the general rules of other funds shall be immediately applied to it.
3. Commitment
The administrator of the fund should be committed to guaranteeing that the
prices reflect the market, and whenever this should prove to be impossible,
take all necessary measures to estimate what would be the market prices of
assets for which these would be effectively negotiated.
4. Equality
The preponderant criteria used in the process of choosing the methodology,
data source and/or any decision for MTM must be treated equally among
the shareholders.
5. Frequency
The MTM should have as a minimum frequency the periodic calculation of
quotas.
6. Formality
xi
The administrator of the fund should have a formal process for handling
MTM. For this purpose, the methodology should be defined in the MTM
manual and the institution should have a department or person responsible
for the execution, quality of both the process and methodology, as well as
safekeeping of documentation that contains the justification for the
decisions taken.
7. Objectivity
The information on prices and/or factors to be used in the MTM process
should be obtained preferably from external independent sources.
8. Consistency
If the administrator is responsible for the pricing of all his funds, a same
asset cannot have a different price in any of the funds. In case of hiring the
service from qualified provider:
8.1. In analogous form, if one or more of the funds of the same
administrator, the pricing comes under the responsibility of an
outsourced service provider, under these funds, a same asset may not
have a different price when the same MTM manual is used; and
8.2. The outsourced service provider may not adopt for the same assets,
even when in different funds and from different administrators,
different prices when using the same MTM manual, outsourced
service provider, neither adopt for the same assets, even in different
funds and with different administrators, different pricing, when using
the same MTM Manual, to determine consistency in the performance
of his duty.
9. Transparency
The methodologies of mark to market must be both published and available.
xii
C-
METHODOLOGY OF THE PRACTICES
1. Should be used as a primary source for prices:
•
Federal Government Bonds: ANBIMA
•
Shares, BDRs, share options, share term: BM&F/BOVESPA
•
Illiquid options: mathematical models, statistics, Black & Scholes, etc.
•
Futures
contracts,
BM&F/BOVESPA
•
Private securities: ANBIMA (Debentures), mathematical models and
statistics, in the absence of a consistent secondary market.
•
International securities: CMA, BROADCAST and BLOOMBERG.
swaps,
commodities
(agricultural):
2. Contingency or alternative form of pricing
2.1.
Federal Government Bonds
For the purposes of a contingency, or even as an alternative form of
pricing, the unit pricing of government bonds are calculated daily and
based on the rentability observed on d-1 and updated for d+0, as
explained below:
2.1.1. Pos-fixed Assets -is used as an alternative for pricing fixed earning
assets, Pos-fixed, when the primary source is not available, in
order of preference:
a - Updating of the Ups on D+0, based on the earning ability
observed in the primary source between D-2 and D-1.
b - when it is impossible to check the earning ability at the primary
source, between D-2 and D-1, use the respective up dating
rate on day 1 of the index, or, TMS, CDI, etc, as shown in the
sample below:
1
 i
 252
fat = 
+ 1
 100 
Where:
•
•
fat = update factor
i = rate expressed in one year by the asset index
xiii
2.1.2. Pre-fixed Assets - have the following alternatives for pricing in
order of use, if the primary source is not be available:
a - are priced by taking the term structure rate described in this
manual, adding the premium seen at the last auction held by
the Brazil Central Bank.
b - secondarily, if it is not be possible to use the ETTJ the prices
are adjusted on D+0 based on the ETTJ for the previous day (
d-1 ).
2.1.3. Pos-fixed assets indexed to the price index -have the following
alternatives for pricing in order of use, should the primary source
not be available:
a - update of the Ups on D+0, based on the earnings observed
that the primary source between D-2 and D-1.
b - Secondarily, should it not be possible to check the earnings at
the primary source between D-2 and D-1, the Internal Rate of
Return published by the primary source on D-1is use the as a
parameter.
2.2.
Assets with primary source indicators from the BM&F/BOVESPA
For contingency purposes, or even as an alternative form of pricing for
assets that have as their primary source indicators from the
BM&F/BOVESPA, information from D-1 shall be repeated in D-0
2.2.1. Whenever there is a stop in trading (Circuit Breaker), information
supplied by the BM&F/Bovespa is used.
3. Shares, BDRs, options on liquid shares and Share Term
The BM&F/BOVESPA itself reveals the quotas traded on the stock market.
In case of illiquid options mathematical and statistics models are used for
the attribution of price, among we list: Black & Scholes or Black, as the case
may be.
4. Futures, Swaps and Commodities Contracts
The derivative operations are carried out in specialized clearing, and the
Stock and Futures Market /BOVESPA are the greater daily trading.
xiv
5. Assets in Default
All assets are marked for the market, for which their respective criteria are
observed. As a consequence the value of the asset corresponds to that
which would be obtained should it be sold in the same market.
Once the assets accompany the movements of the market, their prices shall
reflect the perception of risk of the economic agents.
In the case of observing the non-payment of installments, amortization or
principal, then the following stages shall prevail:
5.1. The processes relative to the recuperation of credit are made directly
by the senior director, with help from the credit analysis division and
the respective administrators.
5.2. Once default of payment installments, amortization or principal is
detected,
extrajudicial
negotiations
begins
between
the
representatives of the BB DTVM and the respective debtors.
5.3. In parallel with these negotiations with the Senior Directors shall
determine provisions for the operation value.
5.4. Should the extrajudicial negotiations prove to be unsuccessful then
judicial proceedings will be brought against the debtor(s), to
recuperate part or the entire investment amount.
5.5. In case of recuperation the provision shall be reverted, otherwise, the
operation shall be accounted as a loss and removed from the fund.
6. Occurrences of systemic and market risk
Administering financial assets, like all other financial economic activities,
are subject to unexpected events that may cause oscillations in the value of
the assets impacting significantly on the equity of the funds.
Certain occurrences, even though they may in some way be known, that is,
classified by the doctrine, are in most cases unpredictable. In the case of
moratorium between emerging countries, external crisis in the energy sector
– oil, exchange rate crisis in strong currencies – such as the Dollar and the
Euro, American economic recession, amongst others. These factors may ––
or not – cause strong impacts on the Brazilian economy and consequently
on the financial assets.
Each crisis has its own characteristic and reflexes making it impossible to
plan actions and procedures, and as the same can only be determined
case-by-case. The decisions, depending on the subject, are taken at the
following Forums:
xv
6.1. Strategic Forum
6.2. Increased Forum
6.3. Executive Forum
xvi
II – APPLICATIONS
A.
TERM STRUCTURE OF PRE-FIXED EXCHANGE RATES
1.
The term structure of exchange rates was developed to create
parameters for pre-fixed pricing of fixed earning assets.
2.
The referred to structure consists of creating an interest curve, based on
trading on the Interest Rate Future Markets at the BM&F/BOVESPA.
3.
In the absence of intermediary points on the curve is used the
interpolation exponential method; in the case of extrapolation one
repeats the last forward in the construction of the curve.
4.
The contracts of the DI are traded by an express yearly percentage rate
and its liquidity occurs at the final value of R$100.000,00. The UP – unit
price, seen daily, is obtained using the interest rates of the contract,
discounting the same from the 100.000 amount, as shown below:

 100.000
PU = 
bd
 i

+ 1 252
 100
(
)






Where:
•
•
•
•
5.
PU = unit price;
100.000 = vlr. at maturity of contract;
i = interest rates;
bd = business days until the maturity of the contract.
The interest rates of a contract, from the UP, can be calculated in
accordance with the following formula :
  100.000 
i =  

  PU 
252
bd

−1 x100

1
Where:
•
•
•
•
i = Interest rates per year (base 252 du);
PU = Unit price traded on the market;
100.000 = Base value of contract1;
bd = business days.
6.
Having seen the necessity to obtain rates for periods where there is no
trading, we use the mathematical concept available known as the
interpolation technique.
7.
We adopt the exponential interpolation criteria using the following
formula:



100 . 000 , 00
PUy = 

 100 . 000 , 00  UP 1


UP 1
 UP 2

 bd 1 


  bd 2 










Where:
•
•
•
•
•
PUY = UP to be obtained by way of interpolation;
pu 1 = UP with liquidity, before the vertex without liquidity;
pu 2 = vertex with posterior liquidity UP1;
bd1 = number of business days before the maturity of the UPy and
the PU1;
bd2 = number of business days between the maturity of the UP2 and
the PU1
8.
Having obtained the PU, apply the formula in item 5, to extract the
respective interest rates.
9.
Source:
primary: future interest market at the BM&F/BOVESPA.
Secondary: Term Structure for Interest Rates released in the
ANBIMA table.
1
Presently the value of the Future Contracts for DI is R$100,000.00, in accordance with clause 2, defined
by the BM&F.
2
B-
1.
FEDERAL GOVERNMENT BONDS – PRE-FIXED
LTN – Brazilian Treasury Bills
1.1. Characteristics:
1.1.1. term: defined by the Brazilian Treasury, when issuing the bonds;
1.1.2. mode: nominative
1.1.3. nominal value: multiples of R$1.000,00 (thousand Reais);
1.1.4. earnings: defined by the discount on the nominal value
1.1.5. redemption: for the nominal value, on date of maturity
1.2. The Brazilian Treasury Bills are not accounted for on a daily basis with unit
prices – PU – they are calculated by of discounted cash flow, using as a
discount factor the interest rates priced in the market by the maturity of the
bond
1.3. Methodology
The Brazilian Treasury Bills - LTN are priced in accordance with the
following formula:
PUMtM =
V .R
( i100 + 1)
BD
252
Where:
•
•
•
•
PUMtM = UP of the market
V.R. = Value at maturity – VR = 1.000,00
i = pre-market interest rates for maturity
BD = business days accumulated between the present date until the
maturity of the asset
1.4. Source
Primary: ANBIMA table.
Secondary: Structural Terms of Interest Rates, detailed above, with
respective spread.
3
2.
NTN – F - BRAZILIAN TREASURY NOTES Series – F
2.1. Characteristics:
2.1.1. term: defined by the Brazilian Treasury, when bond is issued;
2.1.2. interest rates: defined by the Brazilian Treasury, when issuing, in the
yearly percentages, calculated at nominal value;
2.1.3. mode: nominative;
2.1.4. nominal value: multiples of R$ 1.000,00 (thousand Reais);
2.1.5. yield: defined by the setting concession on the nominal value;
2.1.6. Interest payments: every semester, with adjustment of the term
during the first period of flow, should it be the case. The first taxes to
be paid shall be at full tax rate for the first six months, independent
of the date the bond was issued;
2.1.7. redemption: at the nominal value, on the date of its maturity.
2.2. The Brazilian Treasury Notes – series F - are accounted daily with the unit
price – PU – calculated by discounted of cash flow, used as a factor for
internal tax discount the internal rate of return for the maturity of the bond.
2.3. Methodology
The calculation model of the market for this asset brings the cash flow to
present value, taking into consideration the given internal rate of return TIR, as we can see, considering the interest for six months at 10% per
year:
10
(
+ 1) − 1
1
100
COT = ∑
+
(TIR100 + 1)
(TIR100 + 1)
1
n
i =1
2
BDi
BDn
252
252
VP = COT × VN
Where:
•
•
•
•
•
BDi – number of business days between the present date and the
date of payment of the ith interest;
BDn – number of business days between the present date and the
redemption;
TIR –interest rate of return;
COT – quota, percentage of the nominal value;
VP – present value;
4
•
VN = 1.000,00;
2.4. Source:
Primary: the PUs daily released in Table ANBIMA, calculated
according to the indicative rate.
Secondary: structure term of interest rates, adjusted in accordance
with the spreads.
5
C-
3.
3.1.
GOVERNMENT FEDERAL BONDS – POS-FIXED
LFT – Financial Treasury Bills
Characteristics:
3.1.1. term: defined by the Brazilian Treasury, when the bonds are
issued;
3.1.2. mode: nominative;
3.1.3. nominal value on database: multiples of R$ 1.000,00 (one
thousand Reais);
3.1.4. yield: average adjusted rate of daily financing provided by the
Special System for Settlement And Custody - SELIC for federal
government bonds, released by the Brazilian central bank,
calculated at its nominal value;
3.1.5. redemption: by the nominal value, plus the respective yield, from
the base date of the bond.
3.2.
The Financial Treasury Bills are accounted daily with unit price – PU –
calculated using discounted cash flows, using as discount factor the
premium/discount priced by the market for the maturity of the bond.
3.3.
Methodology
The LFT are priced in accordance with the following formula:
PUMtM =
PU _ 238
( i100 + 1)
BD
252
Where:
• PUMtM = unit price marked by the market.
• PU_238 = UP there released by BACEN by way of the nominated table
Resolution 238. This PU is defined as the PU at issue, corrected by the
average accumulated SELIC rate.
• i = annual premium rate or discount applied to the maturity of the asset.
• BD = accumulated business days between the present date until the
maturity of the asset.
3.4.
Source
Primary: the Pus daily released in the Table ANBIMA, calculated
according to the indicated rate.
6
4.
4.1.
Secondary: the variations between the PUs of D-2 and D-1.
NTN-B – Brazilian Treasury Notes – Series B
Characteristics:
4.1.1. term: defined by the Brazilian Treasury, when the bonds are
issued;
4.1.2. interest rates: defined by the Federal Treasury, on issue, in yearly
percentages, calculated on the updated nominal value.
4.1.3. mode: nominative;
4.1.4. nominal value at base date: multiples of R$ 1.000,00 (one
thousand Reais);
4.1.5. update of nominal value: by the variation in the Ample Consumer
Price Index - IPCA for the previous month, released by the
Brazilian Institute for Geography Statistics Foundation (Fundação
Instituto Brasileiro de Geografia e Estatística) - IBGE, from the
base date of bond;
4.1.6. Interest payments: every semester, with adjustment term on the
first period of flow, whenever applicable. The first coupon of
interest to be paid shall include the full rate defined for the six
months period, independently of the date the bond was issued;
4.1.7. Redemption of the principal: in lump sum on date of maturity.
4.2.
The Brazilian Treasury Notes – series B - are accounted daily with the
unit price – PU – calculated using discounted cash flow, using the
internal rate of return– TIR priced by the market for the maturity of the
bond.
4.3.
Methodology
The calculation model of the market of this asset bring the cash flow to
the present value, taking into consideration a given internal tax rate on
return - TIR, as we see:
( i100 + 1)
1
n
COT = ∑
i =1
2
(TIR100 + 1)
−1
BDi
252
+
1
(TIR100 + 1)
BDn
252
Where:
7
•
•
•
•
•
•
•
•
•
•
•
•
BDi – number of business days between the present date and the
date of payment ith of interest;
BDn – number of business days between the present and redemption
dates;
TIR – internal rate of return;
COT – quota, percentage of corrected nominal value;
VNA – nominal value updated by the last 15th day of the month –
parity;
BDn – business days between the present date and the last 15th of
the month;
BDt – business days between the last 15th and the next 15 of the
month, in relation to the present date;
VNA corrected – nominal value updated and corrected for the present
date;
VP – present value;
i – interest rates of the asset;
FatIPCA - IPCA variation factor between the base date and the last
15th previous to the present date;
IPCAforecast – expected IPCA for the period.
VNA = R $ 1 .000 ,00 * ( FatIPCA )
VNAcorrect ed = VNA * (1 + IPCAforeca st )
BD n
BDt
VP = quota * VNACorrected
4.4.
Source
Primary: the PUs released daily in the Table ANBIMA, calculated on
the indicated rates.
IPCA forecast: it is used the average checked by the commission for
accompanying macroeconomics from ANBIMA and may be used,
alternatively, the ample consumer price index – IPC-A, defined by the
Macro-Economic Distributor Division.
Secondary: are used the rates released by ANBIMA on D-1.
8
5.
5.1.
NTN-C – Brazilian Treasury Notes – Series C
Characteristics:
5.1.1. term: defined by the Brazilian Treasury, when the bonds are
issued;
5.1.2. interest rates: defined by the Brazilian Treasury, on issue, in yearly
percentages, calculated on the updated nominal value;
5.1.3. mode: nominative;
5.1.4. nominal value on base date: multiples of R$ 1.000,00 (one
thousand Reais);
5.1.5. update of nominal value: by the variation in the Market Price Index
- IGP-M for the previous month, released by the Fundação Getúlio
Vargas, from the base date of the bond;
5.1.6. Payment of interest: every semester, with term adjustment in the
first period of flow, whenever applicable. The first interest coupon
to be paid contains full tax rate for the six months, independently
of the date the bond was issued;
5.1.7. Redemption of principal: in lump sum, on the date of its maturity.
5.2.
The Brazilian Treasury Notes– series C - are accounted daily by unit
price – PU –calculated by discounted cash flow, using the internal rate of
return – TIR priced by the market for the maturity of the bond.
5.3.
Methodology
The calculation model of the market for this asset bring the cash flow at
the present value, taking into consideration the given internal rate of
return - TIR , as we see:
( i100 + 1)
1
n
COT = ∑
i =1
2
(TIR100 + 1)
−1
BDi
252
+
1
(TIR100 + 1)
BDn
252
Where:
• BDi – number of business days between the present date and the date
of payment of the ith interest;
• BDn – number of business days between the present and redemption
date;
• TIR – internal rate of return;
• COT – quota, percentage of adjusted nominal value;
• VNA – nominal value updated to the last day 1 – parity;
9
• BDn – business days between the present date up to the last day 1;
• BDt – business days between the last day 1 and the next day 1, in
relation to the present date;
• VNA cor – nominal value updated to the present date;
• VP – present value;
• i – interest rates of the asset;
• FatIGP-M - variation factor in the IGP-M between the base date and the
last day 1 before the present date;
• IGP-Mforecast – expected IGP-M for the period.
VNA = R$1.000,00 * ( FatIGP − M )
VNAcor = VNA * (1 + IGP − Mforecast
)
BD n
BD t
VP = cot*VNAcor
5.4.
6.
6.1.
Source
Primary: the PUs released daily in the Table ANBIMA, calculated on
the indicated rates.
IGP-M: it is used the average checked by the Commission for
Accompanying Macroeconomics from ANBIMA, for the month
(except for the following day after the release of the final IGP-M,
whenever the index itself is used for the current month, published by
the FGV), alternatively, the IGP-M defined by the Macro-Economic
Distributing Division may be used.
Secondary: the rates published by ANBIMA in D-1.
NTN-D - Brazilian Treasury Notes – Series D
Characteristics:
6.1.1. term: defined by the Brazilian Treasury, when the bonds are
issued;
6.1.2. interest rates: defined by the Brazilian Treasury, at issue, as a
percentage per year, calculated on the nominal updated amount;
6.1.3. mode: nominative;
6.1.4. nominal value at the base date: multiples of R$ 1.000,00
(thousands of Reais);
6.1.5. update of nominal value: by the variation of the quotation for sale
of the US dollar on the free-rates exchange market, published by
10
the Brazilian Central Bank, considering the average rate of the day
immediately prior to the base date or the date of maturity of the
security;
6.1.6. payment of interest: every six months, with adjustments in the
term in the first period of influence, whenever applicable. The first
interest coupon to be paid shall consider the full rate defined for
six months, independently of the date of issue of the security;
6.1.7. redemption of principal: in one lump sum on the maturity date.
6.2.
The Brazilian Treasury Notes – series D – are accounted daily with unit
prices – PU – calculated by discounted cash flow using the internal rate
of return - TIR priced by the market for maturity of the title.
6.3.
Methodology
The model for calculation of the market for this asset translated into
bringing cash flows to the present value taking into account a given
internal rate of return - TIR, as we can see:
( i100 + 1)
1
n
COT = ∑
i =1
2
(TIR100 + 1)
−1
BDi
252
+
1
(TIR100 + 1)
BDn
252
Where:
•
•
•
•
BDi – number of business days between present date and date of
payment of the i-th interest;
BDn – number of business days between present date and
redemption;
TIR – internal rate of return;
COT – quota, percentage of corrected nominal value;
VP – present value;
•
i – interest rates for asset
•
 ptaxD − 1 

VNA = 1000 × 
 ptax(e) 
Where:
•
•
VNA – Updated nominal value;
PtaxD-1 – dollar quotation of D-1, informed by BACEN,
11
•
PTAX transaction, sales point, commercial dollar;
Ptax(e) – dollar quotation for working Day immediately before the
issue date of the asset, informed by the BACEN (Central Bank),
PTAX transaction, sales point, commercial dollar.
VP = COT × VNA
Where:
•
•
•
6.4.
VP – present value;
COT – quotation, percentage of corrected nominal value;
VNA – updated nominal value.
Source
Primary: the PUs released daily in the ANBIMA table, calculated on
the indicated rates.
PTAX – published daily by SISBACEN.
Secondary: the Internal Return Rates are obtained in the futures
market of Exchange Coupons of BM&F.
12
D - PRIVATE SECURITIES
7.
Debentures
7.1.
As marking to market of Primary Source – MtM of the Debentures, used
to price daily unit of calculation purposes quotes provided by Market
Table Secondary ANBIMA Debentures. In the case of the Debenture not
included in that Table, we use the model of the Secondary Source for
marking to market. For cases in which the ANBIMA fails to inform the
price of any asset is used the last quotation published, up to a maximum
period of 15 working days. Thereafter, the debenture also happens to be
priced by the criteria of the model Secondary Source.
7.2.
As a Secondary Source of mark to market – MtM of the Debentures,
used in the model Credit Curves ANBIMA, extracted from the debentures
Percentage DI, DI + Spread and IPCA + Spread priced daily by this
association, since they reflect zero-spread structures code on the
sovereign curve for different risk levels. The tool sets a benchmark for
the pricing of private credit assets.
7.3.
The Debentures are classified by credit risk. However, ANBIMA, Credit
Curves are only calculated for the AAA, AA and A risks (disregard the
signal variations within the same scale, that is, there is no distinction
between classifications: neutral, plus and minus).
7.4.
Currently the ANBIMA not available curves for the BBB and lower rating
for lack of Debentures with these ratings in daily pricing. So until that
Association pass to make it available for calculating the mark to market
credit spreads for new ratings* (BBB and B) performed the following:
a) Created a TREND curve (Function Trend Excel) rates of all higher
ratings (AAA, AA and A);
b) This curve is a series of linear trend.
7.5.
Considering all adjustments applied by ANBIMA on the model of credit
curve, such as risk spreads are only average, adjustments in the level of
slope and curvature of curves, lack of standardization in the
nomenclature of notes issued by rating agencies, elimination debentures
with clauses of early repurchase, insertion of synthetic securities short
term, heterogeneity of issuers and also, comparative studies of the model
rates with rates of debentures of de daily pricing of that Association, we
made a shift of credit curves, as shown below:
13
Table 1 – Table of equivalence used in the MtM of the Debentures
TABLE OF EQUIVALENCE OF CLASSIFICATION RISK OF DEBENTURES
Credit Curve ANBIMA
AAA** AAA AA
Risk Debentures in the Model
AAA
AA
A
A
BBB*
B*
BBB
B
C
**Credit synthetic curve created from the difference between AAA and
AA Credit risk curve ANBIMA less the AAA risk.
7.6.
Within each risk group in their respective maturity dates, daily calculate
the average of all Credit Curve rates ANBIMA the last 15 working day.
7.6.1. The Pricing Committee of Assets of BB DTVM can change the
number of working days catch ANBIMA Credit curve to compose
the average.
7.7.
Known average calculate the standard deviation of credit curves ANBIMA
the last three (3) months. The Upper and Lower Limit correspond to +/- 1
(one) Standard deviation, as well as an indicative range in spread (Fees
Acceptance Tunnel). These intervals are transformed into DI Percentage
(rate “DI” from the DI Future Interest Rate Curve BM&FBovespa).
7.7.1. The Pricing Committee of Assets of BB DTVM at any time can
apply a modifier (Mod) on the average Credit Curve ANBIMA,
aiming to adjust rates in extreme market situations, as well as, you
can also change the standard deviation.
7.7.2. The Pricing Committee of Assets of BB DTVM can differentiate he
individual rating of any issuer.
7.8.
Daily Debentures are separated by maturity and risk contemplating the
timing and risks of Credit Curve ANBIMA.
7.9.
All Debentures whose rates are within the Acceptance Tunnel are
marked to market by the hiring rate or the last marking to market rate
used. The Debentures that their rates are out of Acceptance Tunnel, are
marked to market with the closest limit Acceptance Tunnel.
7.10. Determination of Tunnel Acceptance of Exchange Rate and consequent
Mark to Market with modifier (Mod):
iMtM
Mod
= [(
iµ
+ / - Mod ) + / - σ ]
Where:
• iMtM Mod – mark to market reference rate with the modifier
(Mod)
14
•
•
iµ
σ
– average frequency rate
– standard deviation of the frequency rates
• Modifier (Mod) – determined by the Asset Pricing
Committee.
7.11. The unit price curve (PU PAR) is the value of the Debenture t a given
date, expressed in national currency. Their calculation is performed by
updating the title of the issue price as standardization of ANBIMA
calculation methodology.
7.12. For Linked Debentures to the IPCA, plus the credit curve ANBIMA, the
mark to market model uses the Term structure of Interest Rates
Estimated – ETTJ ANBIMA where sovereign zero-coupon yield curves
are extracted from the rates of government securities and fixed rate
pegged to the IPCA, plus the implicit inflation cornering.
7.12.1. In contingency situation will be used ETTJ IPCA Estimated
structure of ANBIMA T-1.
7.13. The MtM rates of each Debenture, from the Credit ANBIMA curves (plus
ETTJ the rete estimated IPCA ANBIMA for the Debentures in IPCA), are
transformed into Quotation (% of PU PAR) by the Bloomberg system.
7.13.1. For Debentures that the Bloomberg system does not support the
calculation, the MtM Quotation shall be determined by the Pricing
Committee of BB DTVM through the Asset Analysis Crusade
information with other Debentures, considering the time, risk and
similar indexes.
7.13.2. For Debentures that may not behave the Bloomberg system and
the calculation does not apply to use of Analysis Crusade
(debentures nonstandard or containing uncertainties on financial
events) may take the form of calculation of MtM Quotation
defined Pricing Committee Assets BB DTVM.
7.14. The final PU will be given by:
Where:
PU MtM = C * PUPAR
•
•
•
PUMtM = UP marked to market;
C = Quotation (% of PU PAR);
PUPAR = Unit Price Curve
15
7.15. For new acquisitions, its starting price record in the system will be the PU
or purchase quotation. The effects of marking to market only occur on
T+1.
7.16. Unable to apply the MtM model of the Debentures will be used in
emergency character / contingency, the Quotations defined in T-1.
7.17. Source
8.
Primary: Quotations published in the ANBIMA Debentures table.
Absence of publishing of quotations by ANBIMA shall result in the
use of the latest quotation for a maximum period of 15 days, and
after this period the model described in item 7.2 shall be adopted.
Secondary: If there is no publishing of the quote in ANBIMA table,
the model described in item 7.2 onwards shall be used.
Pre-fixed Assets: CDB/RDB and Other Private
Securities
8.1.
The pre-fixed assets are traded taking as a base an additional spread
over the interest rates negotiated at the BM&F/BOVESPA, as a premium
and/or credit risk spread.
8.2.
The credit risk spread and/or premium is defined by the difference when
buying, between the acquisition rate and the equivalent rate at 100 % of
the interest curve seen at BM&F/BOVESPA.
8.3.
Thus, starting from the principle that the credit risk spread is defined in
the act of acquisition the asset, its present value shall be calculated by
the following equation:
VP =
VR
[( i100 + 1)+ S ](
n
252
)
Where:
•
•
•
•
•
8.4.
VP = present value;
VR = redemption value;
i = interest rate;
n = business days until maturity;
S = spread.
If some private security presents a credit spread lower than the credit
spread for Brazilian treasury bonds – LTN, of the same duration, for the
effects of pricing, the greater spread shall prevail.
16
8.5.
The CDB classified at the CETIP with condition of anticipated
redemption, subject to the present market conditions prevailing at the
time of its request was made, may be accounted on its acquisition curve.
8.6.
The issuers is of the CDBs are classified in accordance with their credit
risk, by the team responsible for analysis of companies and financial
institutions. The reassessment is performed periodically.
8.7.
If the Credit Analysis Division note an alteration of the credit risk of an
issuer, in that his rating had been changed, the Controller Division,
through its methodology team, shall change the issuer to the new
category, with the premium level to this new category. The administrator
shall employ his best efforts in order to identify the alterations to the
credit risk of the assets, imputing to the prices such alterations.
8.8.
Reclassification of the issuer to another category of risk shall lead to an
alteration to his spread.
9.
Pos-fixed Assets Indexed to SELIC Rate or DI:
CDB/RDB
9.1.
Due to the absence of a secondary asset market that could provide
volume, rate and issuing, a mark to market model was developed for the
CDBs which use as a base and parameters trading over the last 15 days
in the investment funds of the BB Resources Management DTVM S.A.,
auctions and/or consulting made on the market and primary registered
issues on the CETIP.
9.2.
Initially the nominal corrected value was calculated, through the following
steps, for daily updates of the unitary prices of a CDB/RDB purchased at
a percentage of the CDI/SELIC rate:
Step 1: check the correction factor or accumulated factor
(
)
n
( 1252 ) 


accumulatedfactor = ∏  1 + Ti
− 1 * ∆C  + 1
100

i =1  

Where:
•
•
•
•
π = product;
Ti = SELIC RATE or DI day (% per annum);
∆C = percentage of the SELIC pro rata DI;
i = SELIC RATE or DI;
17
Step 2: checking of the corrected nominal value
VNC = VI * accumulatedfactor
Where:
•
•
9.3.
VNC = corrected nominal value;
VI = initial value.
For trading at 100 % of the SELIC rate or DI increased by the yearly
percentage; the updated daily calculation is performed through the
following formula:
Step 1: checking the correction factor or accumulated factor
(
n

accumulatedfactor = ∏  1 + Ti
100
i =1 
)
1
252
(
* 1+ I
)
100
1
252


Where:
• π = product;
• Ti = SELIC Pro rata DI (% p.a.);
• I = annual purchased rate;
Step 2: checking of corrected nominal value
VNC = VI * accumulated_factor
Where:
• VNC = corrected nominal value;
• VI = initial value.
9.4.
After the constant updates in the previous item the following shall
proceed:
9.4.1. Common CDBs (Classified as without anticipated redemption
clause and/or daily liquidity with clause of redemption at market
rate).
1. The CDBs are classified by groups of issuers who have equivalent
rating:
• Risk AAA
18
•
•
•
•
•
•
•
Risk AA+
Risk AA
Risk A+
Risk A
Risk BBB
Risk B
Risk C
2. They are separated by maturity, contemplating the following
periods:
•
•
•
•
•
•
•
Up to 120 consecutive days;
From 121 to 365 consecutive days;
From 366 to 720 consecutive days;
From 721 to 1086 consecutive days;
From 1087 to 1440 consecutive days;
From 1441 to 1825 consecutive days;
Longer than 1825 consecutive days.
3. Thus, within each risk group and within their respective maturity
group, weekly the average is calculated weighed by frequency of
all rates over the last 15 days, as seen below:
3.1.
Effective trading rates at BBDTVM;
3.2.
auctions (firm offers) and/or consultations made by the
Markets Operation Division (weekly minimum period for
receipt of rates, via Notes); and
3.3.
Primary issue rates registered at Cetip (excluding
information with respect to training performed by the
BBDTVM).
3.3.1. The accepted Cetip rates are from the financial
institutions, which have their rates calculated by
BBDTVM and who do not have the clause “yes” for
anticipated redemption.
4.
For the purposes of expurgating information that may twist the
results, there will be defined, by management area, minimum
rates, which are not conceptually accepted by the BBDTVM funds,
for the exclusion of outlier’s rates. The Information should be
forwarded, via notes, to the pricing team monthly, or in a lesser
period, when there may be some fact on the market that justifies
5.
Once the average by frequency is known, the standard deviation is
calculated. Both the upper and lower limit correspond to +/- 1
(one) standard deviation, creating an indicative interval (Tunnel of
Rate Acceptance).
19
6.
All CDBs whose rates are within the tunnel of acceptance are
marked to market by their own purchase rate or by the last mark to
market rate used. The CDBs whose rates are outside this Tunnel
of Acceptance, are marked to market by the closest limit of the
Tunnel of Acceptance.
6.1. In the absence of a Tunnel of Acceptance for some grouping
(rating and term), the Asset Pricing Committee, shall
determine the Tunnel of Acceptance based on existing
Tunnels and/or previous Tunnels.
6.2. The Asset Pricing Committee from the BBDTVM may at any
moment apply a modifier (Mod) over the frequency average,
for the purposes of adjusting the rates in situations where
there are extreme market conditions.
6.3. Checking the Tunnel of Rate Acceptance and consequent
Mark to Market rate with a modifier (Mod):
iMtM
Mod
= [(
iµ
+ / - Mod ) + / - σ ]
Where:
• iMtM Mod – mark to market reference rate with a
modifier (Mod)
• iµ – average frequency rate
• σ – standard deviation of the frequency rates
• Modifier (Mod) – determined by the Asset Pricing
Committee
6.4. For banks that have credit risk calculated between AAA and
BBB, but with limits are suspended, canceled or without
limitation, the fees will be used the upper limit of the Tunnel
Acceptance Rates of their respective risks.
6.5. Determination of the mark to market rates for the ratings (B
and C):
a) Creates a curve of the rates of upper Tunnel
Acceptance Rate Risk BBB, where the values to be
extrapolated to the risk B and C are based on
calculations of the exponential growth trend Excel;
b) This curve is a series of growth where the first initial
value is multiplied by the increment to get the next
value in the series. The resulting product and each
subsequent product are then multiplied by the
increment to generate the series;
20
c) The rates for the risk B and C are unique to each and
every term and every week the rates will be
recalculated.
9.4.2.
CDBs Subordinates
1. The CDBs subordinates are classified by the inherent credit risk of
each issuer.
2. Thus, for each issuer is calculate the rate curve based on all the
rates observed for the subordinate CDBs over the last 15 (fifteen)
days, as shown below:
2.1 Rates effectively traded at the BBDTVM;
2.2 Consultations made by the Market Operations Division;
2.3 Rates of primary emissions registered at the Cetip (excluding
information reference to effective trading by the BBDTVM).
2.3.1. Accepted Cetip rates, are from the financial institutions
that detained ratings calculated by the BBDTVM.
3. For CDBs whose term are less than or equal to 720 consecutive
days, due to the absence of issued subordinate CDBs with the
same period, use as “vertex” the rates of common CDBs of the
same Issuer and duration.
4. Daily, the rates of the vertex of the curve will be interpolated,
obtaining the individual interpolated rates for each maturity to be
used by the mark to market of the CDBs.
4.1
Checking the Interpolation Rate (Linear Interpolation):
iI = (i2 − i1 ) ×
(TI − T1 ) + i
(T2 − T1 ) 1
Where:
•
iI = Interpolated Rate
i1 = Previous Vertex Rate
i2 = Posterior Vertex Rate
•
TI
•
•
= Desired Term (in working days)
21
T1 = Previous Vertex Term (in working days)
T2 = Posterior Vertex Term (in working days)
•
•
5. With the absence of recent rates, the Asset Pricing Committee
shall determine the rates to be used on the “vertex” of this curve.
6. The Asset Pricing Committee of the BBDTVM may apply a
modifier (Mod) for the purposes of adjusting the rates in extreme
situations of the market, as seen below:
iMtM
Mod
=(
i
I
+ / - Mod )
Where:
• iMtM Mod – rate reference to the mark to market with Basis
point (Mod)
• i I – interpolated rate
• Modifier (Mod) – determined by the Asset Pricing
Committee
9.4.3. Calculation of the Mark to Market PU
a)
The mark to market PU of the common CDBs and subordinate
CDBs correspond to the following:
PUMtM = vnc ×
1
 


i  252
 


 − 1 × ∆c 
1 + 1 +


100

 




wd
1
 


i  252
 


 − 1 × iMtM Mod 
1 + 1 +


  100 




wd
Where:
• PU MtM = mark to market PU
• vnc = corrected nominal value
• iMtM Mod = verified mark to market rate with a Basis
point (Mod)
• bd = business days up to maturity
• i = curve preceding the Selic Swap X Pre (% per annum)
or DI X Pre (% per annum) Of the BM&FBovespa
• ∆C = percentage of the SELIC rate or purchased DI
Note.: On the date of acquisition of the CDB its initial price
registered on the system will be at the buying PU. The effect of the
mark to market may only occur on D+1.
22
9.5.
The CDBs with daily liquidity, with redemption clause at purchased
rate, shall be updated daily at the purchase rate.
9.6.
The responsibility for attributing the value to be used as modifier (Basis
Point) of the average rate and/or informed rate is of the Asset Pricing
Committee from the BBDTVM, whose deliberation is communicated to
the quote Fund Controller Division.
9.7.
The CDB classified at the CETIP with daily liquidity, with redemption
clause at the rate of the market, shall be submitted to the model
provided in item 9.4.
9.8.
Classification of the issuers of CDBs and other assets, with regard to
their risk note – rating is made by the Credit Analysis Division. Its
reassessment is periodic.
9.9.
If the Credit Analysis Division notes an alteration in the credit risk of an
issuer, that its rating has been moved, the Fund Controller Division shall
place the issuer in this new category, with the level of premium of this
new category. The administrator shall employ his best efforts to identify
alterations to the credit risks of the assets, imputing to the prices such
alterations.
10. Term deposit with Special Guarantee from FGC
10.1. DPGE Pos-Fixed – Indexed to SELIC rate or DI
10.1.1.
Contracts for Term deposits with the Special Guarantee from
FGC - DPGE have the following characteristics
a. A minimum term of twelve months and a maximum term of
thirty months for deposits, with no redemption, total or partial,
before the maturity date;
b. Be the object of a specific record, up to redemption, in asset
systems administered by recording and financial liquidation
entities, duly authorized by the Brazilian Central Bank;
c. They are formed with a single holder, person or entity,
identified by their respective federal registration number
(CPF) or federal corporate number (CNPJ);
d. There is no renegotiation of the original remuneration agreed
to;
23
e. The sum of DPGE I and DPGE II have guarantee by the
Credit Guarantee Fund – FGC limited to R$ 20 million per
holder, including principal plus interest
10.1.2.
Due to the absence of an active secondary market that provides
volume, rate and emitter, we use the mark to market model
described below.
10.1.3.
Initially calculate the corrected nominal value, which takes place
through the following steps to update the daily unit prices of
term deposits with special guarantee from FGC – (DPGE)
contracted to a percentage of the CDI/Selic:
Step 1: Finding the correction factor or accumulated factor
(
)
n
( 1252 ) 


acumulated _ factor = ∏   1 + i
− 1 * ∆C  + 1
100


i =1  
Where:
• ∏ = product (Pi)
• i = SELIC or DI rate day (% p.a.)
• ∆C = percentage of SELIC or DI rate contracted
Step 2: finding corrected nominal value
vnc = vi × accumulated _ factor
Where:
• vnc = corrected nominal value
• vi = initial value
10.1.4.
To negotiations conducted at 100% of fees SELIC or DI plus a
percentage per year, calculating the daily update is performed
through the following steps:
Step 1: Finding the correction factor or accumulated factor
(
n

accumulated _ factor = ∏  1 + i
100
i =1 
)
1
252
(
* 1+ I
)
100
1
252


Where:
• ∏ = product (Pi)
• i = SELIC or DI rate day (% p.a.)
• ∆C = percentage of SELIC or DI rate contracted
24
Step 2: finding corrected nominal value
vnc = vi × accumulated _ factor
Where:
• vnc = corrected nominal value
• vi = initial value
10.1.5.
After the constant updates of the above procedure is the
following:
1. For effects of classification, all DPGEs are considered as
Risk AAA.
2. They are separated by maturity, contemplating the following
periods:
•
•
•
•
•
Up to 540 consecutive days;
From 541 to 900 consecutive days;
From 901 to 1260 consecutive days;
From 1261 to 1620 consecutive days;
Longer than 1620 consecutive days.
3. Thus, within their respective maturity dates, weekly the
average is calculated weighed by frequency of all rates over the
last 15 days, as seen below:
3.1.
Effective trading rates at BBDTVM;
3.2.
Auctions / Quotes electronic order (firm offers) and/or
consultations made by the Markets Operation
Division; and
3.3.
Primary issue rates registered at Cetip (excluding
information with respect to training performed by the
BBDTVM).
4. For the purposes of expurgating information that may twist
the results (rates of institutions that has no intention of
delivering, but send minimum rates), there will be defined, by
Asset Pricing Committee, minimum rates, for the exclusion of
outlier’s rates.
5. Once the average by frequency is known, the standard
deviation (SD) is calculated. Both the upper and lower limit
corresponds to Confidence Interval (CI) of the t distribution,
according to the formula below.
25

CI =  µ +

− t *δ
n
;µ +
+ t *δ 

n 
Where:
CI – confidence interval
µ - average per frequency (after removal of outliers)
t - statistic t be found based on the significance level (5%)
and the degree of freedom that depends on the size of the
frequency (Note.: The significance level is set by the Asset
Pricing Committee, which can be changed depending on
conditions market)
δ - standard deviation
n - number of observations
6. In the event that there is only one fee or just repeated fees
and these rates are consistent, the Asset Pricing Committee will
set the SD to be used. The Upper Limit and Lower CI match a t
distribution, for the number of observations, as described in the
previous item.
7. All DPGE whose rates are within the CI are marked to
market by their own purchase rate or by the last mark to market
rate used. The DPGE whose rates are outside this CI, are
marked to market by the closest limit of the CI.
7.1
In the absence of the CI for some term, the Asset
Pricing Committee shall determine the CI based on
existing CI and/or previous CI.
7.2
The Asset Pricing Committee from the BBDTVM may
at any moment apply a modifier (Mod) over the
frequency average, for the purposes of adjusting the
rates in situations where there are extreme market
conditions.
7.3
Determination of Mark to Market rate with a modifier
(Mod):
iMtM
Mod
= [( IC + / - Mod )]
Where:
• iMtM Mod – mark to market reference rate with a
modifier (Mod)
• IC – Confidence Interval
• Modifier (Mod) – determined by the Asset Pricing
Committee
26
10.1.6.
The calculation of PU Mark to Market DPGE correspond to the
following:
PUMtM = vnc ×
1
 


i  252 
 

 − 1 × ∆c 
1 + 1 +

  100 


1
 

i  252 
 
 − 1 × iM
1 + 1 +

  100 


bd





bd
Where:
•
•
•
PU MtM = UP marked to market
vnc = corrected nominal value
= rate found for mark to market, that is, ∆C + Spread
•
•
•
(credit and/or liquidity risk)
bd = business days to maturity
i = average Selic or DI rate (% per annum)
∆C = percentage of SELIC or DI rate purchased
iM
Note: On the date of acquisition of the DPGE its initial price
registered on the system will be at the buying PU. The effect of the
mark to market may only occur on D+1.
10.2. DPGE Pre-fixed
10.2.1.
For effects of classification, all DPGEs shall be considered Risk
AAA.
10.2.2.
Pricing the pre-fixed DPGE is as follows:
VP
=








vf

i

 100

+ 1 

bd
252








Where:
•
•
•
VP = present value
vf = future value
bd = business days to maturity
27
•
10.2.3.
i = interest rate obtained by the Swap DI rate vs. PRE from
BM&FBovespa, and shall be interpolated by the system if
there be no specific maturity rate for the maturity of the
asset.
For effects of contingency, the interest curve obtained by the
Swap DI rate vs. PRE of BM&F/Bovespa the information from
D-1 shall be repeated in D-0.
At moments of suspension of trading (Circuit Breaker), we shall
use the information provided by BM&F/Bovespa.
11. Pos-fixed Assets Indexed to SELIC or DI: CCB rate
11.1. Bank Credit Papers – CCB indexed to the TMS or CDI are updated daily
by the following formula:
11.1.1. Contracting a percentage of the indexer:
Step 1: finding the correction factor or the accumulated factor
(
)
n
( 1252 ) 


accumulated _ factor = ∏  1 + i
− 1 * ∆C  + 1
100

i =1 

Where:
•
•
•
∏ = product
i = SELIC or DI Day rate (% p.a.)
∆C = percentage of SELIC or DI rate purchased
Step 2: finding the corrected nominal value
vnc = vi × accumulated _ factor
Where:
•
•
vnc = corrected nominal value
vi = initial value
11.1.2. Contracting at 100% of an indexer plus an annual rate:
Step 1: finding correction factor or accumulated factor
(
n

acumulated _ factor = ∏  1 + i
100
i =1 
)
1
252
(
* 1+ I
)
100
1
252


28
Where:
•
•
•
∏ = product
i = SELIC or DI rate (% p.a.)
I = purchased annual rate
Step 2: finding the corrected nominal value
vnc = vi × acumulated _ factor
Where:
•
•
vnc = nominal corrected value
vi = initial value
11.2. The Securities Commission – CVM determines in item 1.2.1.3-a
(Evaluation Criteria and Appropriation Accounting) of the COFI that in the
absence of a negotiating market for a given asset, its accounting shall be
by the amount that can be obtained for another asset that is similar, at a
minimum, in nature, maturity, risk and indexing.
11.3. Although theoretically, the risk of a CCB is not very different from a
debenture, there are usually less transactions taking place. In the
absence of an active market for CCBs, we use a Basket of Debenture
Rates published by ANBIMA (Indicative rate for D-1), to determine the
market rates of CCBs, in accordance with the model described below:
11.3.1. Each CCB has its own characteristics defined at issue, such as
Rate, Maturity, Indexing etc.
11.3.2. They are classified according to the rating (Risk) of the issuer,
provided periodically by the Credit Analysis Division of this
DTVM:
Risk A
Risk B
Risk C
11.3.3. Each debenture is also classified according to the risk provided
by the Credit Analysis Division of this DTVM.
11.3.4. The debentures will be checked if it meets the following
prerequisites to compose the Basket Rates that will be used for
mark to market of the CCB:
a)
The risk of the debenture must have the same classification
as the CCB;
b)
The maturity term for the debenture may not be less than
the CCB and shall be at the maximum 365 calendar days
from the maturity date of the CCB.
29
11.3.5. The average rate of the Debentures Basket shall be calculated
for mark to market for each CCB.
Finding average rate:
n
∑ i Debentures
iM =
i =1
n
n
N
Where:
•
•
iM – average rate of Basket of Debentures
i n - ANBIMA indicative rate for Debenture n
•
Debentures n - Debentures that make up the
Debentures Basket
N – Total Nº of observations (quantity of Debentures
that fulfilled the prerequisites)
•
11.3.6. The minimum quantity to make up the Basket of Debentures
shall be 02 (two) debentures.
11.3.7. In the absence of the minimum quantity of Debentures to make
up the 1st Basket of Debentures for a given CCB, we will use the
purchased rate for mark to market, until it is possible to make up
the Basket of Debentures.
11.3.8. Should the quantity of Debentures be less than 02 (two), the
latest average rate found shall be used for marking to market
until it is possible to make up a new Basket of Debentures.
11.3.9. The minimum period for calculating the average rate shall be
daily.
11.3.10. Thus the PU marked to market for the asset shall correspond to
the following:
PUMtM = vnc ×
1
 


i  252 
 

1
+
1
+
−
1
×
∆
c






100


 


bd
1
 


i  252 
 

1
+
1
+
−
1
×
iM






100


 




bd
Where:
•
•
•
•
PU MtM = UP marked to market
vnc = corrected nominal value
iM = average rate found for marking to market
bd = business days to maturity
30
•
•
i = average Selic or DI rate (% per annum)
∆C = percentage of SELIC or DI rate purchased
11.4. On the date of contracting the CCB its initial price recorded in the system
shall be by the PU for purchase. The effect of mark to market shall only
occur on D+1.
11.5. At each modification of the rating of the CCBs and/or the Debentures that
make up the Basket, the Fund Controller Division shall make the
alteration to the pricing program.
12. Pos-fixed Assets Indexed to SELIC or DI: Financial Bill
- LF
12.1. The Financial Bills - LF are classified into six groups of emitters holding
equivalent rating, as the equivalency table demonstrated below:
EQIVALENCE TABLE OF RATING LF
Credit Risk
AAA
AA+
Risk LF
AAA
AA+
AA
AA-
AA
A+
A+
A
A-
A
BBB+
BBB
BBB-
BBB
12.2. They are separated by maturity, including the following deadlines:
• Up to 120 consecutive days;
• From 121 to 365 consecutive days;
• From 366 to 540 consecutive days;
• From 541 to 900 consecutive days;
• From 1261 to 1620 consecutive days;
• Longer than 1620 consecutive days.
12.3. Thus, within each risk group and within their respective ranges of
maturity, we calculate, weekly, the average weighted by frequency of all
rates over the last 15 days, as follows:
12.3.1.
Effective trading rates at BBDTVM;
12.3.2.
Auctions (firm offers) and/ or consultations carried out by the
Market Operations Division (weekly minimum frequency to
receiving rates, via Notes); and
12.3.3.
Emission rates of primary operations recorded in Cetip,
excluding the information regarding the business effected by
BBDTVM, from financial institutions holding the rating
calculated by BBDTVM
.
31
12.4. In order to purge information that may distort the result, the area of
management will determine minimum rates, which are not conceptually
accepted by BBDTVM funds, for the exclusion of outliers rates. The
information should be sent, via notes, to the Pricing Team, whenever
there is some fact on the market that justifies it.
12.5. Once the average by frequency is known, the standard deviation is
calculated. The Upper and Lower limit correspond to +/- 1 (one) standard
deviation, creating a indication interval (Tunnel of Rate Acceptance).
12.6. All LF whose rates are within the Tunnel of Acceptance are marked to
market by their own purchase rate or by the last mark to market rate
used. The LF whose rates are out of that Tunnel of Acceptance, are
marked to market by the closest limit of the Tunnel of Acceptance.
12.7. In the absence of a Tunnel Acceptance for some grouping (rating and
term), the Asset Pricing Committee, will determine the Tunnel of
Acceptance based on the existing tunnels and/or previous Tunnels.
12.8. Given the restriction of a minimum term of 24 months for the issuance of
LF, to the tunnels acceptance below of 541consecuive days, will be used
the rates of tunnels of CDB with approximate duration, plus spread
defined by the Committee.
12.9. The Asset Pricing Committee of BBDTVM anytime can apply a modifier
(Mod) on the average by frequency, in order to adjust the rates in
extreme market situations.
12.10. Determination the Tunnel of Rate Acceptance and consequent Mark to
Market rate with the modifier (Mod):
iMtM
Mod
= [(
iµ
+ / - Mod ) + / - σ ]
Where:
• iMtM Mod – mark to market reference rate with the modifier
(Mod)
• iµ – average frequency rate
•
σ
– standard deviation of the frequency rates
• Modifier (Mod) – determined by the Asset Pricing
Committee.
12.11. If there is degradation of the issuers rating for LF classified as risk BBB
to an inferior risk, a mark to market risk for the issuer itself shall be
adopted. Thus, there will be a check over the last 15 (fifteen) days to see
whether there were trading by this issuer at Cetip. If so, assets will be
priced at the rate observed in Cetip, if not, the Pricing Committee will
determine a modifier to be applied to the last asset pricing rate.
32
12.12. The LF are updated daily by the following formulas:
12.12.1.
Hiring a percentage of the índex:
Step 1: determination of the correction factor or cumulative factor
(
)
n
( 1252 ) 


accumulated _ factor = ∏  1 + i
− 1 * ∆C  + 1
100

i =1 

Where:
•
∏ = product
•
i = SELIC rate or DI day (% per annum);
•
∆C = percentage of SELIC rate or DI
Step 2: determination of the nominal value corrected
nvc = iv × accumulated _ factor
Where:
• nvc = nominal value correct
• iv = initial value
12.12.2.
Hiring 100% of an index plus an annual rate:
Step 1: determination of the correction factor or accumulated
factor
(
n

accumulated _ factor = ∏  1 + i
100
i =1 
)
1
252
(
* 1+ I
)
100
1
252


Where:
• ∏ = product
• i = SELIC rate or DI day (% per annum);
• I = contracted annual rate
Step 2: determination of the nominal value corrected
33
nvc = iv × accumulated _ factor
Where:
• nvc = nominal value corrected
• iv = initial value
12.13. Calculation of PU Mark to Market:
12.13.1. The PU marked to market of LF correspond to the following:
PUMtM = vnc ×
1
 


i  252 
 

 − 1 × ∆c 
1 + 1 +


100


 




bd
1
 


i  252 
 

 − 1 × iMtM Mod 
1 + 1 +


100


 




bd
Where:
•
PU MtM = PU marked to market
•
nvc = nominal value correct
•
iMtM
•
•
•
Mod
= rate found for mark to market with the modifier
(Mod)
bd = business days to maturity
i = curve from Swap Selic X Pre (% per annum) or DI X Pre
(% py) from BM&FBovespa with Duration about 30 days
(moving average of the last 10 business days)
∆C = Selic rate percentage or contracted DI
Note: On the date of acquisition of LF its starting price of record in
the system is by purchase PU. The effect of marking to market
only occurs on D +1.
12.14. The Subordinated Financial Bills – LFS are a kind of bank debenture
and were created to allow stretching of bank liabilities, we use the
marking to market the model curves Credit ANBIMA, extracted from the
debentures Percentage DI, DI + Spread and IPCA + spread priced daily
by this association, since they reflect structures of zero-coupon spread
on sovereign curve for different levels of risk. The tool is a reference to
the asset pricing private credit.
12.15. The LFS are classified by credit risk. However, the curves Credit
ANBIMA are only calculated for AAA, AA, A (disregard the signal
variations within the same scale, ie, there is no distinction between
classifications: neutral, plus and minus) risks.
34
12.16. Considering the increased risk due to the subordinate clause to the LFS,
the rates of risk immediately above the curve Credit ANBIMA are used.
Therefore, it is necessary to match them in order to obtain new
classification ratings, as follows:
Table 1 – Table of equivalence used in the Mark to Market of LFS
TABLE OF EQUIVALENCE OF RISK CLASSIFICATION OF LFS
Curve Credit ANBIMA
Risk LFS
AAA
-
AA
A
AAA AA
BBB*
B
C
A
BBB
B/C
* Currently not available ANBIMA curves for BBB and lower rating for lack of
debentures with these ratings is daily pricing. Thus, until the ANBIMA pass to
make them available for determination of credit spreads of mark to Market for
new ratings (BBB, B e C) performed the following:
a) It creates a curve TREND (Trend Function Excel) rates of all
higher ratings (AAA, AA e A);
b) This curve is a series of linear trend.
12.17. Daily the LFS are separated by maturity contemplating the timing of the
Curves Credit ANBIMA.
12.18. Thus within each risk group in their respective maturity dates, daily
computes the average of all fees Curve Credit ANBIMA the last 15
working days.
12.18.1. The Pricing Committee of Assets of BB DTVM can change the
number of working days catch ANBIMA Credit curve to
compose the average.
12.19. Known average calculate the standard deviation of credit curves ANBIMA
the last three (3) months. The Upper and Lower Limit +/- correspond to 2
(two) Standard deviation, thus creating an indicative range in Spread
(Tunnel Acceptance of Fees). These intervals are transformed into
Percentage of DI (rate “DI” from Swap DI X Pre (% py) of BM&FBovespa
Duration approximately 30 calendar days – moving average of the last 10
days).
12.19.1. The Committee Asset Pricing BB DTVM anytime you can apply
a modifier (Mod) on average, aiming to adjust rates in extreme
market situations, as well as, you can also change the standard
deviation.
12.19.2. The Committee Asset Pricing BB DTVM can differentiate the
individual rating of any issuer.
12.20. All LFS whose rates are within the Tunnel Acceptance are marked to
market itself by hiring rate or the last mark to market rate use. The LFS
35
that their rates are out of this Tunnel Acceptance are marked to market
by the nearest boundary of the Tunnel Acceptance.
12.21. Determination of Tunnel Acceptance of Exchange Rate and consequent
Mark to Market with modifier (Mod):
iMtM
Mod
= [(
iµ
+ / - Mod ) + / - σ ]
Where:
• iMtM Mod – mark to market reference rate with the modifier
(Mod)
• iµ – average frequency rate
•
σ
– standard deviation of the frequency rates
• Modifier (Mod) – determined by the Asset Pricing
Committee.
12.22. The LFS are updated daily by the following formulas:
12.22.1.
Hiring a percentage of the index:
Step 1: determination of the correction factor or cumulative factor
(
)
n
( 1252 ) 


accumulated _ factor = ∏  1 + i
− 1 * ∆C  + 1
100


i =1 
Where:
•
∏ = product
•
i = SELIC rate or DI day (% per annum);
•
∆C = percentage of SELIC rate or DI
Step 2: determination of the nominal value corrected
nvc = iv × accumulated _ factor
Where:
• nvc = nominal value correct
• iv = initial value
36
12.22.2.
Hiring 100% of an index plus an annual rate:
Step 1: determination of the correction factor or accumulated
factor
(
n

accumulated _ factor = ∏  1 + i
100
i =1 
)
1
252
(
* 1+ I
)
100
1
252


Where:
• ∏ = product
• i = SELIC rate or DI day (% per annum);
• I = contracted annual rate
Step 2: determination of the nominal value corrected
nvc = iv × accumulated _ factor
Where:
• nvc = nominal value corrected
• iv = initial value
12.23. Calculation of PU Mark to Market:
12.23.1.
Periodic payment of income – Calculation of PU Mark to
Market of the asset is the sum of the payments of interest and
principal, discounted (discounted to present value) by Mark to
Market rate plus the modifier (Mod), according to the following
formulas:
a) Calculation of interest payments for asset based compensation
percentage of DI:
(
)
( 1 252 ) 


Payment _ of _ int erest = [VNIx ( accumulate d _ factor − 1)] X [   1 + i
− 1 * ∆ C  + 1] bd
100



b) Calculation of interest payments for asset referenced in DI +
Spread:
(

Payment _ of _ int erest = [VNIx(accumulated _ factor − 1)] X  1 + i
100

)
1
252
(
* 1+ I
)
1
100
252
c) Calculation of Unit Price Mark to Market:
37


bd







n 
 Interest _ Payment i + Pr incipal _ Paying i 
PUMtM = ∑ 

bd i


i =1 

252
 
  iMtM Mod



+
1

 




100


 
 




Where:
•
•
•
•
•
•
•
•
12.23.2.
PU MtM = PU marked to market
VNI = Initial Value
iMtM Mod = rate found for mark to market with the modifier
(Mod)
bd = business days between the reference date and the next
interest payment and / or business days between the dates of
each event of future interest payments
bdi = business day to maturity
i = curve from Swap Selic X Pre (% per annum) or DI X Pre
(% py) from BM&FBovespa with Duration about 30 days
(moving average of the last 10 business days)
∆C = Selic rate percentage or contracted DI
I = annual contracted rate
Payment of income at the end – For the calculation of PU
Mark to Market of the asset is used the following formula:
PUMtM = vnc ×
1
 


i  252 
 

1
+
1
+
−
1
×
∆
c






100


 




bd
1
 


i  252 
 

1
+
1
+
−
1
×



iMtM Mod 


100


 




bd
Where:
•
PU MtM = PU marked to market
•
vnc = nominal value correct
•
iMtM
•
•
•
Mod
= rate found for mark to market with the modifier
(Mod)
bd = business days to maturity
i = curve from Swap Selic X Pre (% per annum) or DI X Pre
(% py) from BM&FBovespa with Duration about 30 days
(moving average of the last 10 business days)
∆C = Selic rate percentage or contracted DI
38
12.24. For new issues, at the date of acquisition of the asset, its initial price of
the system will record the PU purchase. The effects of marking to market
only occur in D+1.
13. Pos-fixed Assets Indexed to SELIC or DI Rate: NCE
13.1. As a Secondary Source of mark to market – MtM of Credit Notes Export NCE, used in the model Credit Curves ANBIMA, extracted from the
debentures Percentage DI, DI + Spread and IPCA + Spread priced daily
by this association, since they reflect zero-spread structures code on the
sovereign curve for different risk levels. The tool sets a benchmark for
the pricing of private credit assets.
13.2. The Credit Notes Export - NCE are classified by credit risk. However,
ANBIMA, Credit Curves are only calculated for the AAA, AA and A risks
(disregard the signal variations within the same scale, that is, there is no
distinction between classifications: neutral, plus and minus).
13.3. Currently the ANBIMA not available curves for the BBB and lower rating
for lack of Debentures with these ratings in daily pricing. So until that
Association pass to make it available for calculating the mark to market
credit spreads for new ratings* (BBB and B) performed the following:
c) Created a TREND curve (Function Trend Excel) rates of all higher
ratings (AAA, AA and A);
d) This curve is a series of linear trend.
Table 1 – Table of equivalence used in the MtM of the Credit Notes Export
TABLE OF EQUIVALENCE OF CLASSIFICATION RISK OF NCE
Credit Curve ANBIMA
AAA
AA
A
BBB*
B*
C*
Risk NCE in the Model
AAA
AA
A
BBB
B
C
13.4. Within each risk group in their respective maturity dates, daily calculate
the average of all Credit Curve rates ANBIMA the last 15 working day.
13.4.1.
The Pricing Committee of Assets of BB DTVM can change
the number of working days catch ANBIMA Credit curve to
compose the average.
13.5. Known average calculate the standard deviation of credit curves ANBIMA
the last three (3) months. The Upper and Lower Limit correspond to +/- 1
39
(one) Standard deviation, as well as an indicative range in spread (Fees
Acceptance Tunnel). These intervals are transformed into DI Percentage
(rate “DI” from the DI Future Interest Rate Curve BM&FBovespa).
13.5.1.
The Pricing Committee of Assets of BB DTVM at any time
can apply a modifier (Mod) on the average Credit Curve ANBIMA,
aiming to adjust rates in extreme market situations, as well as, you
can also change the standard deviation.
13.5.2.
The Pricing Committee of Assets of BB DTVM can
differentiate he individual rating of any issuer.
13.6. Daily the NCE are separated by maturity and risk contemplating the
timing and risks of Credit Curve ANBIMA.
13.7. All NCE whose rates are within the Acceptance Tunnel are marked to
market by the hiring rate or the last marking to market rate used. The
NCE that their rates are out of Acceptance Tunnel, are marked to market
with the closest limit Acceptance Tunnel.
13.8. Determination of Tunnel Acceptance of Exchange Rate and consequent
Mark to Market with modifier (Mod):
iMtM
Mod
= [(
iµ
+ / - Mod ) + / - σ ]
Where:
• iMtM Mod – mark to market reference rate with the modifier
(Mod)
• iµ – average frequency rate
•
σ
– standard deviation of the frequency rates
• Modifier (Mod) – determined by the Asset Pricing
Committee.
13.9. The NCE are updated daily by the following formulas:
13.9.1.
Hiring a percentage of the index:
Step 1: determination of the correction factor or cumulative factor
(
)
n
( 1252 ) 


accumulated _ factor = ∏  1 + i
− 1 * ∆C  + 1
100

i =1 

Where:
•
∏ = product
40
•
i = SELIC rate or DI day (% per annum);
•
∆C = percentage of SELIC rate or DI
Step 2: determination of the nominal value corrected
nvc = iv × accumulated _ factor
Where:
• nvc = nominal value correct
• iv = initial value
13.9.2.
Hiring 100% of an index plus an annual rate:
Step 1: determination of the correction factor or accumulated
factor
(
n

accumulated _ factor = ∏  1 + i
100
i =1 
)
1
252
(
* 1+ I
)
100
1
252


Where:
• ∏ = product
• i = SELIC rate or DI day (% per annum);
• I = contracted annual rate
Step 2: determination of the nominal value corrected
nvc = iv × accumulated _ factor
Where:
• nvc = nominal value corrected
• iv = initial value
41
13.10. Calculation of PU Mark to Market:
13.10.1.
Periodic payment of income – Calculation of PU Mark to
Market of the asset is the sum of the payments of interest and
principal, discounted (discounted to present value) by Mark to
Market rate plus the modifier (Mod), according to the following
formulas:
a) Calculation of interest payments for asset based compensation
percentage of DI:
(
)
( 1 252 ) 


Payment _ of _ int erest = [VNIx ( accumulate d _ factor − 1)] X [   1 + i
− 1 * ∆ C  + 1] bd
100



b) Calculation of interest payments for asset referenced in DI +
Spread:
(

Payment _ of _ int erest = [VNIx(accumulated _ factor − 1)] X  1 + i
100

)
1
252
(
* 1+ I
)
1
100
252
c) Calculation of Unit Price Mark to Market:







n 
 Interest _ Payment i + Pr incipal _ Paying i 
PUMtM = ∑ 

bd i


i =1 

252




  iMtM Mod


 
+
1

 




100
  
 




Where:
•
•
•
•
•
•
•
•
13.10.2.
PU MtM = PU marked to market
VNI = Initial Value
iMtM Mod = rate found for mark to market with the modifier
(Mod)
bd = business days between the reference date and the next
interest payment and / or business days between the dates of
each event of future interest payments
bdi = business day to maturity
i = curve from Swap Selic X Pre (% per annum) or DI X Pre
(% py) from BM&FBovespa with Duration about 30 days
(moving average of the last 10 business days)
∆C = Selic rate percentage or contracted DI
I = annual contracted rate
Payment of income at the end – For the calculation of PU
Mark to Market of the asset is used the following formula:
42


bd
PUMtM = vnc ×
1
 


i  252 
 

1
+
1
+
−
1
×
∆
c






100

 




bd
1
 


i  252 
 

1
+
1
+
−
1
×



iMtM
Mod 


100

 




bd
Where:
•
PU MtM = PU marked to market
•
vnc = nominal value correct
•
iMtM
•
•
•
Mod
= rate found for mark to market with the modifier
(Mod)
bd = business days to maturity
i = curve from Swap Selic X Pre (% per annum) or DI X Pre
(% py) from BM&FBovespa with Duration about 30 days
(moving average of the last 10 business days)
∆C = Selic rate percentage or contracted DI
13.11. For new acquisitions, its starting price record in the system will be the PU
or purchase quotation. The effects of marking to market only occur on
T+1.
14. Other Pos-fixed Assets Indexed to SELIC or DI Rate
14.1. In the absence of an active secondary market that provides volume, rates
and issuers we use the credit risk spread and/or premium defined in the
act of purchase of the asset to price it
14.2. The Corrected Nominal Value is calculated following these steps for an
asset purchased at a percentage of the CDI/SELIC rate:
Step 1: finding the correction factor or the accumulated factor
(
)
n
( 1252 ) 


accumulated _ factor = ∏  1 + Ti
− 1 * ∆C  + 1
100

i =1  

Where:
•
•
•
•
π = product;
Ti = SELIC or DI Day rate (% per annum);
∆C = percentage of SELIC or DI rate;
i = SELIC or DI rate;
43
Step 2: finding the corrected nominal value
VNC = VI * accumulated factor
Where:
•
•
VNC = corrected nominal value;
VI = initial value.
14.3. For negotiations effected at 100% of the SELIC or DI rate with the
addition of an annual percentage, the daily updating calculation is
effected by the formula:
Step 1: finding the correction factor or the accumulated factor
(
n

accumulated _ factor = ∏  1 + Ti
100
i =1 
)
1
252
(
* 1+ I
)
100
1
252


Where:
• π = product;
• Ti = SELIC or DI rate (% per annum);
• I = annual rate purchased;
Step 2: finding the corrected nominal value
VNC = VI * accumulated factor
Where:
• VNC = corrected nominal value
• Corrected nominal value;
• VI = initial value.
14.4. Classification of issuers for risk - rating is done by the Credit Analysis
Division. Its reevaluation is done periodically.
14.5. If the Credit Analysis Division finds an alteration in the credit risk of an
issuer, in which its rating has changed, the Fund Controller Division shall
place the issuer in the new category, with the Premium level of this
category. The administrator shall use the best efforts to identify the credit
risk alterations for the assets, imputing prices to these alterations.
14.6. Reclassification of the issuer of other private securities into other risk
categories shall result in alteration to the spread.
44
15. Mortgage Bills – LH
15.1. Characteristics:
15.1.1.
Time term: defined when effecting the operation;
15.1.2.
Modality: negotiable;
15.1.3.
Form of launching: public offer or direct launching in favor of the
interested party;
Nominal value at base date: defined at purchase;
15.1.4.
15.1.5.
15.1.6.
15.1.7.
15.1.8.
15.1.9.
Updating nominal value: variation of IGP-M from the base date
of the security2;
Interest: defined as a percentage per annum when effecting the
operation based on 360 days;
Payment of Interest: periodicity is defined when launching the
asset;
Withdrawal of principal: in one lump sum, on the date of
maturity, by the nominal value indexed by the IGP-M, or in
accordance with the criteria defined when launching the asset;
Updating the nominal value: by the General Price Index Market - IGP-M of the previous month, published by the
Fundação Getúlio Vargas (FGV), from the base date of the
security
15.2. Methodology
15.2.1.
Updating the nominal value:
 fIGP − M 1 

vna = vn * 
 fIGP − M 


Where:
•
•
•
•
15.2.2.
vna = corrected nominal value;
vn = nominal issue value;
fIGP-M1 = month factor of the IGP-M published by the FGV;
fIGP-M = IGP-M factor in issuing the paper, published by the
FGV.
Updating of nominal value outside its base date or anniversary:
Vna = Vn(IGP − Mp ) BDt
BD2
2
There are Mortgage Bills with nominal values updated by other indexes. In this manual we are dealing
only with assets indexed to the price index.
45
Where:
•
•
•
•
•
15.2.3.
Vna = Present nominal value;
Vn = Nominal value on the date of its anniversary;
IGP-Mp = IGP-M index to be used;
BDt = business days - total in month;
BD2 = business days from the 1st of the month to the present
day.
Calculation of interest coupons, base of 360 days:
NMeses*30


 i
 360 

x= 
+ 1
−1
 100 



Where:
• x = +interest to be paid on the date;
• i = interest rate on asset;
• Nmeses = number of months between payments of
subsequent interest.
15.2.4.
Calculation of market price:
 n

cj
100


PULH =  ∑
+
* vna
dctj
dctn 365 
 j =1 (1 + TIR) 365 (1 + TIR )



Where:
• PU LH = market price of Mortgage Bill;
• Cj = interest coupons to be paid on the date;
• Dctj/dctn = calendar days between the Day of calculation and
the date of payment of the coupon j-th (1 ≤ j ≤ n);
• i = internal return rate.
15.3. Source:
IGP-M: it is used the average calculated by the ANBIMA
Macroeconomic Accompaniment Commission for the month,
excepting the day following publishing of the final IGP-M, in which
case the index for the current month is used, published by the FGV.
TIR: are used the TIRs of the NTN-C for equivalent maturities with
the addition of the Credit Risk spread, checked with the ANBIMA
Table, in the column for indicative rates.
46
16. Private securities indexed to the IPCA – CDB, DPGEs
and Financial Bill - LF
16.1. Due to the absence of an active secondary market that provides volume,
rate and emitter for private securities indexed to the IPCA (CDB, DPGEs
and Financial Bill), we elaborated model of mark to market based on the
Term Structure of Interest Rates - ETTJ provided by ANBIMA, where the
sovereign yield curve zero-coupon are extracted from the pre-fixed
exchange rate government federal bonds and linked to the IPCA, in
addition to the implied inflation in curve.
16.2. Initially calculate the Nominal Value Updated:
 fIPC − A1 

VNA = VNE * 
 fIPC − A 


Where:
•
•
•
•
VNA = Nominal Value Updated;
VNE = Face Value Issue;
fIPC-A1 = month IPC-A factor released by IBGE;
fIPC-A = IPCA factor in the issued of the role released by
IBGE, or new upgraded factor pro-rata business days, in
case of issued in different day that released by the IBGE.
16.3. If necessary, updates the Nominal Value to outside the base date or
anniversary, as follows:
VNA = VN (IPC − Ap )
DU 2
DUt
Where:
• VNA = Nominal Value Updated;
• VN = Nominal Value at the date of the anniversary;
• IPC-Ap = IPCA index to be used (last index);
• DUT = Number of business days ranging from the 15th of the
previous month to the 15th of the current month;
• DU2 = Number of business days ranging from the 15th of the
previous month to the current day
16.4. The calculation of PU PAR corresponds to the VNA plus interest of the
paper, according to the following formulas:
 Interest

PUPAR = VNA 
+ 1
 100

47
n

 
N
i
 
  
Interest =  
− 1 X 100



100



 

 Interest 
Interest_p ayment = (VNA )X 

 100 
Where:
• i = Interest as a percentage per year;
• N = number of days representative of the rate, that may
take the values 360 or 365 consecutive days or 252
business days;
• n = When "N" is equal to 360 or 365 days, "n" will assume
the number of days between the date of the next event and
the date of the previous event. When "N" is equal to 252
days, "n" will assume the number of days between the date
of the next event and the date of the previous event.
16.5. The calculation of PU Mark to Market of the asset is the sum of the
payments of principal and interest, discounted (discounted to present
value) for the interpolated Term Structure of Interest Rates - ETTJ IPCA,
provided by ANBIMA, plus the Credit Spread, according to the following
formula:






n 
Interest_payment i + Pr incipal _ paymenti 
PUMtM = ∑ 

du i
 
i =1  
252
 Interpolated rate + Spread

  
+ 1   
100
  
  
 
 
Where:
• PU MtM = PU mark to market
• du = business days to maturity
• Interpolated rate = Exponential Interpolation of ETTJ IPCA
• Spread = Credit Spread (date of issue)
16.6. For new issues, at the acquisition date of the asset, its initial register
price in the system will be the purchase PU. The effects of mark to
market only occur on D +1.
16.7. If there is change in issuer risk, the Committee of Asset Pricing of
BBDTVM reassess its respective Credit Spread
16.8. In a contingency situation will be used to structure ETTJ Estimated,
provided by ANBIMA on D-1.
48
E-
ASSETS NEGOTIATED ABROAD
17. ADR – American Depositary Receipt
17.1. Mark to market of assets negotiated in the New York Stock Exchange
(NYSE) shall take place through the so-called “latest price” published by
the systems, but not necessarily meaning the last effective transaction.
17.2. We use as a parameter of the prices captured and time to capture
international asset prices, the closing of the NYSE. If there is no trading
we repeat the quotes recorded on the last working day;
17.3. On the 24th and 31st of December, prices will be captured in Bloomberg,
at 14h (GMT).
17.4. If there is no trading is kept the “last price” as published by the systems
on the previous day.
17.5. The exchange rate for conversion to foreign currencies to the domestic
currency portfolio of foreign debt investment funds is performed using the
Ptax800, sales closing price.
17.6. The exchange rate for conversion to foreign currencies into the national
currency in mark-to-market portfolio of investment funds is performed
using the Exchange Rate Reference T2 calculated by BM&FBovespa.
17.7. Source: CMA, BROADCAST, BM&FBovespa and BLOOMBERG.
18. Fixed Income – Corporate Bonds, Treasury Bonds,
Global, CLN, etc.
18.1. The mark to market of fixed income assets traded abroad is done taking
as a basis offers of quotations published by CMA, BROADCAST and
BLOOMBERG.
18.2. We use as a parameter of the prices captured and time to capture
international asset prices, the closing of the NYSE. If there is no trading
we repeat the quotes recorded on the last working day;
18.3. On the 24th and 31st of December, prices will be captured in Bloomberg,
at 14h (GMT).
49
18.4. The exchange rate for conversion to foreign currencies to the domestic
currency portfolio of foreign debt investment funds is performed using the
Ptax800, sales closing price.
18.5. The exchange rate for conversion to foreign currencies into the national
currency in mark-to-market portfolio of investment funds is performed
using the Exchange Rate Reference T2 calculated by BM&FBovespa.
18.6. Source: CMA, BROADCAST, BM&FBovespa and BLOOMBERG.
F-
VARIABLE INCOME AND FUTURES
19. Shares and BDRs – Brazilian Depositary Receipts
19.1. Mark to market of assets traded in the stock market shall be by the
closing price published by BM&FBOVESPA.
19.2. The Brazilian Depositary Receipt – BDR (level II and III) is marked to
market at the closing price published by BM&FBOVESPA.
19.3. If there is no negotiation is held the previous day’s closing price, or in
accordance with current law.
19.4. For BDR Sponsored Level I, the mark to market will be given by the
reference value established by BM&FBOVESPA.
19.5. There is no disclosure of the reference value, is held the last available
value.
19.6. Source: BM&FBOVESPA
20. Subscription Rights and Receipts of Shares
20.1. In the period between the day on which the shares become exsubscription and the day for the beginning of trading of the right, the price
of the Subscription Rights will be calculated using the Black & Scholes
model, the same as a call option, as the following parameters:
C= S 0 ⋅ N (d 1 ) − E ⋅ e r⋅t ⋅ N (d 2 )
d1 =
ln(S 0 / E ) + (r + ∂ 2 / 2).t
∂. t
50
d2 =
ln(S 0 / E ) + (r − ∂ 2 / 2).t
∂. t
= d1 - ∂ .
t
Where:
•
•
•
•
C - value of subscription rights
r - Interest rate without risk
S0 - Price (closing) of the asset to which the right relates
E - Exercise price, which will be the subscription price of the share
to which it refers
• t - remaining period to the date of subscription right
• N (d1) N (d2) - value of the cumulative normal distribution
evaluated at point d1 and d2
• σ - volatility of the returns of the underlying asset, which the right
refers.
20.2. From the moment the Subscription Rights starts to be traded, the market
price shall be the closing price of the business conducted and published
by BM & F Bovespa.
20.3. If, in the negotiation period, the right has become illiquid, will be
maintained the closing price of the previous day.
20.4. The market price of the Subscription Receipt is the closing share price at
which the receipt refers.
20.5. Source: BM&F/BOVESPA
21. Rent (or loans) of Shares
21.1. For pricing of rent (or loan) shares the contract rate is used, because the
operation has guarantee from CBLC and the possibility of early
redemption to the contracted rate.
22. Futures BM&F/BOVESPA
22.1. Future contracts traded in the BM&F/BOVESPA have standardized
characteristics of value, maturity, minimum contracts and adjustment
prices. For purposes of daily updating, quotes or values are used for the
day´s adjustment price.
22.2. For contingency purpose, or even as an alternative form of pricing, for
assets that have BM&F/BOVESPA as their main source of indexes, the
information of D-1 shall be repeated for D-0.
51
22.3. At moments of suspension of trading (Circuit Breaker), we will use the
information supplied by BM&FBovespa itself.
22.4. Source: BM&F/BOVESPA
23. SWAP
23.1. The operation consists in an exchange of rentability referenced to a given
notional value between two parts through daily adjustments in cash flow.
23.2. The flows are updated by daily adjustments, keeping a direct correlation
with the business updated to the market. For example, a swap of pre
rates against CDIs the rentabilities adjust to one another, and maintain
their value positions in the market.
23.3. Adjustment of positions is provided daily by BM&F/BOVESPA.
23.4. For effects of contingency, or even as an alternative way of pricing, for
assets that have BM&F/BOVESPA as the primary source of indexes, the
information on D-1 shall be repeated on D-0.
23.5. At moments of suspension of trading (Circuit Breaker), we will use the
information supplied by BM&FBovespa itself.
23.6. Source: BM&F/BOVESPA.
23.7. Below we describe the methodology for specific calculations for swap
operations – Exchange leg:
Step 1: Calculation of Principal
 CotCurrencyDt0 

Pa = VE x 
 CotCurrencyDtE 
Where:
•
•
•
•
Pa= updated principal
VE = value of issue in Reais
CotCurrencyDt0 = quotation of Currency x R$ for the day of
calculation
CotCurrencyDtE = quotation Currency x R$ date of issue
Step 2: Calculation of interest
52
 coupon   BD0
J =
 X 
 100   BDE

 X Pa

Where:
•
•
•
•
J = interest
BD0 = business days between date of issue and date of
calculation
WDE = business days between date of issue and maturity of
contract
 CDC 
coupon = agreedrate (% per annun.) X 
 , where CDC is the
 360 
number of calendar days of the contract
•
Pa = updated principal
Step 3: Calculation of Foreign Exchange leg
Exchange leg = Principal + Interest
24. Options
24.1. Liquid Options
Closing price quotes from BM&F/BOVESPA are used for options with
liquidity.
Source: BM&F/BOVESPA.
24.2. Low Liquidity Options
For options with low liquidity we use the Black & Scholes model for
pricing, where the source of calculation of volatility is the historic series3
of prices of shares from BM&F/BOVESPA.
⇒ BLACK & SCHOLES MODEL
C=S 0 . N ( d1 ) – E . e- r.t . N ( d2 )
3
- equation 1
For index options will be used the implied volatility, found through the smile curve of the options that
were quoted on the stock exchange. In the impossibility of constructing the curve, will be used volatility
captured at Bloomberg.
53
P=E . e- r.t . N (-d2) - S 0 . N (-d1) ou P = C - S 0 + E . e-i.t
d1 =
d2 =
ln(S 0 / E ) + (r + ∂ 2 / 2).t
- equation 3
∂. t
ln(S 0 / E ) + (r − ∂ 2 / 2).t
∂. t
- equation 2
= d1 - ∂ .
t
-
equation 4
Variables involved in the model
• r – interest rate without risk
• C or P – current value of option
• S0 – current price of share to which the option refers
• E - exercise price of the option
• t – remaining price after option
• ∂ – volatility of returns from asset that is the object of the option
• N(d1) , N(d2) – value of normal accumulated distribution, evaluated at
point d1 , d2
Calculation of historical volatility ∂
Based on a sample of 21 latest prices of the asset we define the relation
ln
pi
p i −1
Where:
•
Pi – closing price from stock market at date i
•
Pi-1 – closing price from stock market at date i – 1
Then, the standard deviation is calculated for the results, multiplied by
252 giving the annualized volatility.
54
The function N(d) is the accumulated probability function of a normal
standardized variable, that is, the probability that a variable with a
standard normal distribution, φ (0 , 1), is less than d.
24.3. Barrier Options
\We use the model developed by Merton, Reiner and Rubinstein.
⇒ MERTON (1973) E REINER E RUBINSTEIN (1991) MODEL
Compiled by ESPEN GAARDER HAUG, The Complete Guide To
Option Pricing Formulas – Second Edition, 2007.
Exotic Options with Barriers
Conventions:
Options with knock in barriers – The option only exists if the barrier
is reached during the life of the contract.
Options with knock out barriers – the option no longer exists if the
barrier is reached during the life of the contract, which might or
might not have a rebate.
When a barrier is defined with a value above the current price of the
asset, it is called Up and In (UI) and when reaching the barrier the
right becomes available. Up and out, (UO) in this case, when the
barrier is reached the right no longer exists.
Similarly, when the barrier is established at an amount lower than
the current price of the asset, it is said that the option is Down and
IN (DI) or Down and OUT” (DO).
Rebate: the amount to be paid to one of the parties when the barrier
is reached, represented by a pre rate, not to be confused with the
asset in question.
Volatility: historic volatility, annualized, as described in the Black &
Scholes model, with the use of a sample of 21 days.
(
)
(
)
A = φSe ( b − r )T N (φx1) − φXe _ rT N φx1 − φσ T ,
B = φSe (b − r )T N (φx 2 ) − φXe _ rT N φx 2 − φσ T ,
55
C = φSe
( b − r )T
H
 
S 
H 
D = φSe (b − r )T  
S
2 ( µ +1)
2 ( µ +1)
N (ηy1) − φXe
_ rT
H
 
S 
2µ
H 
N (ηy 2 ) − φXe _ rT  
S
2µ
(
)
(
)
N (ηy1) − ησ T ,
N (ηy 2 ) − ησ T ,
2µ


H
E = Ke − rT  N ηx 2 − σ T −   N (ηy 2 ) − ησ T ,
S 


(
)
(
)
µ −λ
 H  µ + λ

H 
F = K   N (η .z ) +   N (η .z ) − 2ηλσ T ,
S
 S 

(
)
Where:
ln (S X )
• x1 =
σ T
• x2 =
• y1 =
• y2 =
• z=
ln (S H )
σ T
+ (1 + µ )σ T
(
ln H 2 SX
σ T
ln (H S )
σ T
ln (H S )
• µ =
+ (1 + µ )σ T
σ T
) + (1 + µ )σ
T
+ (1 + µ )σ T
+ λσ T
b−σ 2 2
σ2
• λ = µ2 +
2r
σ2
Where:
•
•
•
•
•
S = Spot price of the asset
X = exercise price of the option
H = barrier price of the option
T = option expiration
K = value of rebate
56
•
•
•
•
σ = volatility of the option
b = cost of carrying
r = pre-fixed rate
n, Φ = parameters for specific models for each kind of option which
can be 1 or -1
Options with Knock-in barrier clauses:
Options with barrier clauses are divided into Knock-in and Knock-out. In
the first the right to exercise only begins to exist if the price of the asset S
reaches the price of the barrier before the maturity date T.
Knock-in options are classified as in-and-down (S > H), that is, spot price
greater than barrier or in-and-up (S < H), that is, spot price lower than the
barrier. See following:
Knock-in-and-down – indicates that the price of the asset on the
date of launching is above the price of the barrier, that is S > H. The
payoff of the option is given by:
Call: pay-off = Max(S – X; 0) if S ≤ H before maturity T and pay-off –
K (rebate), otherwise;
Put: pay-off = Max(X – S; 0) if S ≤ H before maturity T and pay-off
– K (rebate), otherwise;
The formulae for calculating the premiums for these options are obtained
from the combination of the variables previously described, such as A, B,
C, D, E, and F, as established by the model, that is:
Call
in – and – down (X > H)
C+E
η = 1, Φ = 1
Call
in – and – down (X < H)
A–B+D+E
η = 1, Φ = 1
Put
in – and – down (X > H)
B–C+D+E
η = 1, Φ = -1
Put
in – and – down (X < H)
A+E
η = 1, Φ = -1
Knock-in-and-up – indicates that the price of the asset on the date
of launching the option is below the barrier price, that is, S < H. The
payoff for the option is given by:
Call: pay-off = Max(S – X; 0) if S ≥ H before maturity T and pay-off –
K (rebate), otherwise;
Put: pay-off = Max(X – S; 0) if S ≥ H before maturity T and pay-off
– K (rebate), otherwise;
57
The formulae for calculation of the premiums are given by the sum of the
following variables:
Call
in – and –up (X > H)
=A+E
η = -1, Φ = 1
Call
in – and – up (X < H)
=B–C+D+E
η = -1, Φ = 1
Put
in – and – up (X > H)
=A–B+D+E
η = -1, Φ = -1
Put
in – and – up (X < H)
=C+E
η = -1, Φ = -1
Options with Knock-out clauses:
Options with Knock-out barrier clauses are very similar to traditional
options, except that no longer exist if the underlying asset price S
reaches the knock-out barrier before the maturity date. Similarly to the
options with Knock-in barriers, there is a prerogative of a rebate (K),
which is paid if the option no longer exist before maturity. The Knock-out
barrier may be either out-and-down or out-and-up. The payoff and the
premium are calculated as shown below:
Knock-out-and-down – indicates that the price of the asset on the
date of launching of the option is above the barrier price, that is, S >
H. The payoff for the option is given by:
Call: pay-off = Max (S – X; 0) if S > H before maturity T and pay-off =
K (rebate),otherwise;
Put: pay-off = Max (X – S; 0) if S > H before maturity T and pay-off =
K (rebate),otherwise;
The formulas for calculating the premium for these options are obtained
by a combination of variables described above, such as A, B, C, D, E,
and F, as established by the model below:
Call
out – and – down (X > H)
Call out – and
– down (X < H)
=A–C+F
η = 1, Φ = 1
=B–D+F
η = 1, Φ = 1
Put
out – and – down (X > H)
=A–B+C–D+F
η = 1, Φ = -1
Put
out – and – down (X < H)
=F
η = 1, Φ = -1
58
Knock-out-and-up – indicates that the price of the asset on the date
of launching of the option is below the price of the barrier, that is, S <
H. The payoff for the option is given by:
Call: pay-off = Max (S – X; 0) if S < H before maturity T and pay-off =
K (rebate),otherwise;
Put: pay-off = Max (X – S; 0) if S < H before maturity T and pay-off
= K (rebate), otherwise;
The formulae for calculation of the premium are given by application of
the sum of the following variables:
Call
out – and – up (X > H)
Call out – and
– up (X < H)
= F
η = -1, Φ = 1
=A–B+C–D+F
η = -1, Φ = 1
Put
out – and – up (X > H)
=B–D+F
η = -1, Φ = -1
Put
out – and – up (X < H)
=A–C+F
η = -1, Φ = -1
24.4. Futures Options
Index Options, Dollar Options, IDI Options and DI Futures Options. Used
the averages of trades published by BM&F/Bovespa for the options that
have liquidity.
In the absence of trading on the day, will be used the Reference
Premium daily disclosed by BM&F/Bovespa for these assets.
If the BM&F Bovespa disclose the Premium Reference after limit time
for dispatch the prices to Funds Processing area, the options are
priced through the Black model (1976), detailed below:
⇒ BLACK MODEL
Call
Put
c = e − rT [Fn ⋅ N (d1 ) − K ⋅ N (d 2 )]
p = e − rT [K ⋅ N (− d 2 ) − Fn ⋅ N (− d1 )]
59
With
d1 =
ln( Fn / K ) + (σ 2 / 2).T
d2 =
σ T
ln( Fn / K ) − (σ 2 / 2).T
σ T
= d1 − σ T
Where:
•
c – price of a purchase option;
•
p – price of a European sales option;
•
K – exercise price or rate for exercising expressed in PU in the
case of options on DI futures;
•
rd – pre interpolated rate (interest rate without risk);
•
r – rate of continuous time defined as being equal to ln (1+ rd);
•
σ – implied volatility;
•
T – time in years nbd
•
nbd – number of business days between the date of D0 and the
(
252
);
maturity of the option;
•
N(.) – Accumulated distribution function of the standard normal;
•
Fn
– Price of adjustment of futures contracts as bellow:
F1
- Price of adjustment of index futures contract of maturity
equal to the maturity of the option;
F2 - Price of adjustment of futures contract for the dollar at
maturity equal to maturity of the option;
F3 -
Price of adjustment of futures contract for IDI, where:
F3 =
100.000
*s
a
Where:
•
s: IDI on the date of calculation
60
•
a: price of adjustment of futures contract for DI of 1 day
of maturity equal to maturity of the option
F4
- Forward rate expressed in PU between the maturity of the
option on the Future DI and the term according to the kind of
option.
Where:
F4 =
100.000
a
b
Where:
•
a: price of adjustment of the DI futures contract
of the 1 day, of maturity equal to maturity of the
option.
•
b: price of adjustment of the DI futures contract
of the 1 day, of maturity equal to maturity plus
the term of the agreement, according to the type
of option.
24.4.1 For the Option on the DI futures we have:
a)
The underlying asset of the option is the forward rate of
three (type A), six (type B) or twelve months (type C)
from the maturity of the option.
b)
Given that a European CALL for DI futures is equivalent
to a European PUT of futures PU for DI; a European
PUT for futures rate of DI is equivalent to European
CALL for PU of futures DI, the price of a CALL is
determined by PUT for futures rate of DI and vice-versa.
24.4.2 The volatility to be applied to Black model, will be the implied
volatility derived from the D-1 Premium Reference of BM&F
Bovespa.
24.4.3 In the absence of Premium Reference published by the BM&F
Bovespa, or if the data is considered inconsistent/distorted,
the volatility is calculated using a basket of implied volatilities
(Smile Volatility) of the options that had business listed on
stock
61
24.4.4 In the absence of data for the construction of smile volatility or,
if the data are distorted or incomplete, should be used the
market volatility captured on the Bloomberg terminal.
24.4.5 Failing to apply the proposed models, shall be used in an
emergency/contingency situation, the price set on the day
before.
24.4.6 Exceptionally, in extreme market circumstances, the Pricing
Committee of BBDTVM may establish additional criteria for the
use of secondary methods, in order to better reflect the reality
of the market, recording such criteria in the minutes.
24.5. Foreign Currency Options
For mark to market of options on foreign currencies, will be used prices
disclosed daily by BM&F/Bovespa.
In the absence of trading on the day, will be used the Reference
Premium daily disclosed by BM&F/Bovespa for these assets.
If the BM&F Bovespa disclose the Premium Reference after limit time
for dispatch the prices to Funds Processing area, the options are
priced through the Garman-Kohlhagen model (1983), detailed below:
⇒ GARMAN-KOHLHAGEN MODEL
Call
c = e − rcT ⋅ S ⋅ N (d1 ) − K ⋅ e − rT ⋅ N (d 2 )
Put
p = K ⋅ e − rT ⋅ N (− d 2 ) − e − rcT ⋅ S ⋅ N (− d1 )
With
d1 =
ln( S / K ) + (r − rc + σ 2 / 2).T
σ T
62
d2 =
ln(S / K ) − (r − rc + σ 2 / 2).T
σ T
= d1 − σ T
Where:
•
c – call option premium (purchase option);
•
p – put option premium (sales option);
•
e – irrational number called Euler number of value of
approximately 2.718281828459045;
•
K – exercise price of the option;
•
S – spot price of the underlying asset;
•
r – interest rate without risk (pre-interpolated rate);
•
rc –external interest rate free of risk;
•
σ – volatility of asset;
•
T – time in years for maturity of contract;
•
nbd – number of business days between date of D0 and maturity
of the option;
•
N(.) – Accumulated distribution function of standard normal;
24.5.1 The volatility to be applied to Garman-Kohlhagen model will be
the implied volatility derived from the D-1 Premium Reference
of BM&F/Bovespa.
24.5.2 In the absence of Premium Reference published by the
BM&F/Bovespa,
or
if
the
data
is
considered
inconsistent/distorted, the volatility is calculated using a basket
of implied volatilities (Smile Volatility) of the options that had
business listed on stock.
24.5.3 In the absence of data for the construction of smile volatility or,
if the data are distorted or incomplete, should be used the
market volatility captured on the Bloomberg terminal.
24.5.4 Failing to apply the proposed models shall be used in an
emergency/contingency situation, the price set on the day
before.
63
24.5.5 Exceptionally, in extreme market circumstances, the Pricing
Committee of BBDTVM may establish additional criteria for the
use of secondary methods, in order to better reflect the reality
of the market, recording such criteria in the minutes.
25. Exotic Options
25.1.
QUANTO Adjustment
Described in detail by Shreve (2004) and Hull (2000) – captures the fact
that an option is referenced to an index quoted in a foreign currency,
while its settlement is made in another currency, in this case the
domestic currency, the Real. The Quanto considers the currency in which
the object of the option is quoted and the currency in which the option is
quoted.
The Quanto Adjustment takes into consideration the following variables
in its calculation: Implied Volatility of the underlying Asset (External
Index); Implied Volatility of the Exchange Rate - Real x foreign exchange
- (settlement); the correlation between the returns of the Index and the
Exchange Rate – Real x Foreign exchange and the number of days to
maturity of the option, in accordance with the formula below:
Quanto Adjustment = exp(σ * ϕ *η * t )
Where:
•
•
•
•
σ = Implied Volatility of the underlying Asset (External Index)
ϕ = Implied Volatility of the Exchange Rate Real x Foreign exchange
η = Correlation between the return index and the exchange rate
t = number of days to the maturity of the Option
The Implied Volatilities used for the purposes of this calculation may not
be directly observed, as such, shall be calculated through trial and error4
(Solver tool in Excel). We detail below the Implied Volatility calculation:
a)
4
The historical Volatilities are captured on Bloomberg, through the
function GV (Graph Volatility) function, from the respectives
international indexes and exchange rate between Real x Foreign
Course “Qualifying Professionals for the Derivatives Market – BM&F – Mar2007”.
64
currency, for a period up to the maturity term of the operation (the
term depends on the conditions of the market at the moment of
calculation);
b)
The Implied volatility is then regarded as the volatility that balances
the premium supplied by the calculation agent 5 with the theoretical
premium calculated by the pricing model of the Option, obtained
through the application of the Excel Solver tool in the Historic
Volatilities..
The Correlation, also called the Correlation Coefficient, indicates the
force and direction of the linear relationship between the variables weekly
return on underlying asset and weekly return on exchange rate Real x
Foreign currency. Its value is always between -1 and 1, where 0 means
no correlation. The positive value means a positive relation between the
variables, while the negative value means the very opposite direction (on
average). The higher the value of the correlation (positive or negative),
the stronger the Association.
The equation for the Correlation Coefficient is:
ρ x, y =
Cov( X , Y )
σ x ⋅σ y
Where X and Y are the average samples of the variable matrixes; Cov is
the co-variance and σ x e σ y are the standard deviations of the samples.
The calculation of the Correlation is made directly by Excel Correl
Function.
The Return is calculated on a weekly basis in order to expurgate
time differences at the close of the indexes and exchange rates
Real x Foreign currency.
Obs.: The Market only consider the absolute number of correlation
for the calculation of the Quanto.
The result of the Spot or Forward (depending on the Option Model used)
of the International index, corrected by the Quanto Adjustment, is
obtained by the formula below:
QS = S * Ajuste Quanto
Where:
5
•
QS = Spot or Adjusted Forward
•
S = Spot or Forward price of the International index.
Agent defined under contract, responsible for calculating the operation.
65
The result obtained for the QS will be the new value of the index to be
employed on the pricing model for the options referred to more than one
currency.
25.2. Asian Options
Asian options averages may be calculated in three different ways:
geometrically (Geometric Average-Rate Options); arithmetically
(Arithmetic Average-Rate Options); and discrete arithmetic (Discrete
Arithmetic Average-Rate Options).
This manual deals with the discrete arithmetic of pricing (Discrete
Arithmetic Average-Rate Options), together with the Quanto Adjustment,
whenever necessary.
Discrete Asian Approximation (Levy, 1997 e Haug, Haug and
Margrabe, 2003)
Call
Put
c A ≈ e −rT .[FA N (d1 ) − X .N (d 2 )]
p A ≈ e −rT X .N .(− d 2 ) − [FA .N (− d1 )]
With
d1 =
ln( FA / X ) + Tσ A2 / 2
σA T
d 2 = d1 − σ A T
Where:
•
•
•
•
•
•
•
•
•
cA = price of the European Asiatic call
pA = price of the European Asiatic put
X = strike
SA = average up to the present moment
S = spot
r = pre interpolated rate (interest rate without risk);
b = cost of carrying
N(.) = Accumulated distribution function of the normal standard
ln = natural logarithm base e
66
• e = is an irrational number, known as the Euler number, with
approximate value equal to 2,718281828459045
• n = number of averages to be calculated
• m = number of averages already calculated
T − t1
• h=
n −1
• T = calendar days to maturity of the option (in years)
• t1 = calendar days to run before the next average (in years)
• σ = implied volatility
• σ A = adjusted volatility
σA =
(
)
ln E [ AT2 ] − 2 ln (E [ AT ])
T
• FA is defined as E[AT]
• E [ AT ] =
S bt1 1 − e bhn
e
n
1 − e bh
2
2
S 2 e (2b +σ )t1 1 − e (2b +σ )hn
2
+

• E[ A ] =
2
(2b +σ )h 1 − e (b +σ 2 )h
n2
 1 − e
2
T
 1 − e bhn 1 − e (2b +σ )hn

−
 1 − e bh 1 − e (2b +σ 2 )h

2
When b=0 we have:
E [ AT ] = S
S 2 eσ
E[ A ] =
n2
2
t1
2
T
1 − eσ 2hn
2
+

2
2
σ h
1 − eσ h
 1 − e
σ hn

n − 1− e 2

1 − eσ h

2




If are in the averaging period, m>0, the price of the strike should
be substituted for
X =
nX − mS A
m
n−
n−m
n−m
n
X , certainly the call will be exercised and the
m
put will end out of the money. That is, the value of the PUT will be
zero, while the value of the CALL shall be:
Moreover, if S A >
∧

c A = e − rT  S A − X 


Where:
67




∧
S A = SA
m
n−m
+ E [ A]
n
n
When we have only one price to be calculated before maturity of
the option, the value can be calculated using the Black-ScholesMerton (BSM) weighing by time remaining to maturity and adjusting
∧
the price of the strike ( X ). In this case, the value of the Asian
option call is:
 ∧
1
c A = c BSM  S , X , T , r , b, σ 

n
Where:
• CBSM is the general formula for the BSM of the call
c BSM = Se (b − r )T N (d1) − Xe − rT N (d 2)
∧
• X =n.X – (n-1)SA
The value of the Asian PUT is:
 ∧
1
p A = p BSM  S , X , T , r , b, σ 

n
Where:
• pBSM is the general formula for the BSM for the put
p BSM = Xe − rT N (−d 2) − Se (b − r ) )T N (−d1)
With
(
)
S
ln  + b + σ 2 / 2 T
X
d1 =
σ T
d 2 = d1 − σ T
26. Synthetic Operations
26.1. Synthetic operations present individualized treatment at their legs. Each
asset must be submitted to its own pricing methodology.
26.2. Each asset involved in the operation has its own form of pricing as
described in this manual, independent of whether it's part of synthetic
operation or not.
68
26.3. Market rates are obtained by way of an interest curve, demonstrated in
item Term Structure of Interest Rates, provided above.
26.4. Source: ANBIMA, BM&F/BOVESPA.
69
G - FIXED EARNINGS OPERATION – SHARE TERM
27. Share Term
27.1. The share term operations are considered fixed income and are similar to
a pre-fixed asset, once the return rate is known at the time of the
operation, as shown in the following formula:
252

 
 vf  bd  
i=  
− 1 × 100
 vi   


Where:
•
•
•
•
I = rate of return
vf = future value
vi = initial value
bd=- business days from settlement to maturity
27.2. Throughout the duration of the asset, the operation risk is the interest
rates, once their profitability has been defined until the end, because its
similarity to a pre-fixed operation.
27.3. The pricing of this asset is calculated using the following formula:
VP
=








vf

i

 100

+ 1 

bd
252








Where:
•
•
•
•
VP = present value
vf = future value
i = interest rates obtained by the Swap rate DI vs. PRE of the
BM&F/Bovespa, which should be Interpolated by the system, if there
is no final rate for the asset.
bd = business days until maturityo
27.4. For purposes of the contingency, the interest curve is obtained through
the Swap rate DI vs. PRE of the BM&F/Bovespa which shall be repeated
on D-0 information from D-1.
27.4.1. Whenever there is a break in trading (Circuit Breaker), we use
the informations supplied by the BM&F/Bovespa itself.
70
H – REPURCHASE AGREEMENTS
28. Repurchase Agreements
28.1. The repo operations are backed by public or private securities and are
characterized by the purchase and sale of securities with resale
commitment assumed by the buyer in conjunction with repurchase
agreements entered into by the seller.
28.2. May be registered in the SELIC or CETIP (depending on the type of
asset).
28.3. In SELIC are four types of repos:
Type 1 – Settlement of the repurchase / resell at a date predetermined
(conventional operation);
Type 2 – Settlement of the repurchase / resell at any time during a given
period, at the discretion of either party, as previously agreed between
them;
Type 3 – Settlement of the repurchase / resell at a date certain deadline
or within the sole discretion of the purchaser; and
Type 4 – Settlement of the repurchase / resell at a date certain deadline
or within the sole discretion of the seller.
28.4. In CETIP can type “repo” (sale of securities with repurchase
agreements), “with repo reverse repo” (sale of securities with repurchase
agreements and simultaneous purchase with resale commitment) and
“reverse repo” (buying securities with resale commitment).
28.5. The rate used to price of these is the committed rate of operation.
71
I – SHARES OF FUNDS
29. Shares of Investment Funds
29.1. For funds that have shares traded is used the closing price of the
business. In case there is no business for 90 days, we will use the value
of the shares issued by the administrator.
29.2. For funds that do not have publicly traded shares is used the value of the
shares released by the administrator.
BBDTVM Asset Management – Distribuidora de Títulos e Valores Mobiliários S.A.
Funds Controller Division – Board Solutions Wholesale – DISAT – Banco do Brasil S.A.
Rio de Janeiro (RJ)
º
º
Praça XV de Novembro, 20 – 2 e 3 andar – Centro – CEP 20010-010
#: 55-21-3808-7500 – Fax: # 55-21-3808-7600
bbdtvm@bb.com.br
São Paulo (SP)
º
Av. Paulista 2.300 – 4 andar – Conjunto 42 – Cerqueira César – CEP 01310-300
# 55 11 2149-4300 – Fax. # 55-11-2149-43100
bbdtvmsp@bb.com.br
www.bb.com.br/bbdtvm
72