MARK-TO-MARKET
Description of Processes and Methods
VERSION: ABRIL/2008
TABLE OF CONTENTS
1.
2
Introduction ................................................................................................................................. 5
1.1
Definition of Variables ........................................................................................................ 5
1.2
Construction of Curves (Yield Curves) .............................................................................. 5
1.3
MtM Methods ..................................................................................................................... 5
Definition of Variables................................................................................................................. 5
2.1.
Method for rate accumulation ............................................................................................ 6
2.2.
Linear Rate......................................................................................................................... 6
2.2.1. Exponential Rate ........................................................................................................... 6
2.2.2. Rate percentage accumulation...................................................................................... 6
3.
Construction of Curves (Yield Curves) ....................................................................................... 7
3.1
Methods ............................................................................................................................. 7
3.1.1 Bootstrapping................................................................................................................. 7
3.1.2 Interpolation ................................................................................................................... 8
3.1.3 Extrapolation.................................................................................................................. 8
4
3.2
Pre-Fixed, No Cash Curve (zero coupon curve – nominal yield curve)............................. 9
3.3
Pre-Fixed, Cash Curve ...................................................................................................... 9
3.4
SELIC Coupon Curve (brazilian financial market define nominal yield curve as ‘coupon’)9
3.5
IGP-M Coupon, Cash Curve ............................................................................................ 10
3.6
IGP-M Coupon, No Cash Curve ...................................................................................... 10
3.7
IPCA Coupon, Cash Curve .............................................................................................. 10
3.8
IPCA Coupon, No Cash Curve ........................................................................................ 10
3.9
INPC Coupon Curve ........................................................................................................ 11
3.10
Dollar Coupon, Cash Curve ............................................................................................. 11
3.11
Dollar Coupon, No Cash Curve ....................................................................................... 11
3.12
Euro Coupon Curve ......................................................................................................... 11
3.13
Yen Coupon Curve .......................................................................................................... 12
3.14
ANBID Coupon Curve...................................................................................................... 12
3.15
TJLP Coupon Curve ........................................................................................................ 12
3.16
TR Coupon Curve ............................................................................................................ 13
MtM Methods ............................................................................................................................ 13
4.1
Financial Treasury Bills (LFT) .......................................................................................... 13
4.2
Brazilian Treasury Bills (LTN) .......................................................................................... 13
4.3
Brazilian Treasury Notes (NTN) - Series B...................................................................... 13
4.4
Brazilian Treasury Notes (NTN) - Series C...................................................................... 14
4.5
Brazilian Treasury Notes (NTN) - Series D...................................................................... 15
4.6
Brazilian Treasury Notes (NTN) - Series F ...................................................................... 16
4.7
Brazilian Treasury Bonds (BTN) ...................................................................................... 16
4.8
Stocks .............................................................................................................................. 17
4.9
Stocks Forward ................................................................................................................ 18
4.10
Repurchase Agreement ................................................................................................... 18
4.11
Swaps .............................................................................................................................. 18
4.11.1
CDI leg..................................................................................................................... 19
4.11.2
US Dollar Leg .......................................................................................................... 19
4.11.3
Pre-Fixed Leg.......................................................................................................... 19
4.11.4
IGP-M Leg ............................................................................................................... 19
4.11.5
IPCA Leg ................................................................................................................. 20
4.11.6
Euro Leg .................................................................................................................. 20
4.11.7
Yen Leg ................................................................................................................... 20
4.11.8
Dollar leg with threshold and intermediate adjustment payment ............................ 21
4.11.9
Libra Sterling Leg .................................................................................................... 22
4.11.10 Libor Edge ............................................................................................................... 23
4.11.11 Index or Commodity Futures Leg............................................................................ 24
4.11.12 FIDC Leg ................................................................................................................. 24
4.11.13 CRI Leg ................................................................................................................... 24
4.11.14 NCE Leg .................................................................................................................. 24
4.11.15 Swap with Cash Flow .............................................................................................. 24
4.11.16 Leg with Double Index............................................................................................. 24
4.11.17 Swap linked to NTN-D............................................................................................. 25
4.12
Options............................................................................................................................. 26
4.12.1
Common Definitions ................................................................................................ 26
4.12.2
Black & Scholes Model............................................................................................ 26
4.12.3
Garman Model......................................................................................................... 27
4.12.4
Black Model............................................................................................................. 27
4.12.5
Barrier Options ........................................................................................................ 28
4.12.6
Global Bond Option ................................................................................................. 29
4.12.7
Nominal Interest Rate Volatility Option ................................................................... 29
4.12.8
Digital Options ......................................................................................................... 30
4.12.9
Asian Options .......................................................................................................... 30
4.12.10 Implied Volatility ...................................................................................................... 31
4.13
Subscription Right............................................................................................................ 32
4.14
Box Spread – Fixed Income Strategy with Options ......................................................... 33
4.15
CDB - Certificado de Depósito Bancário (Bank Certificate of Deposit) ........................... 33
4.15.1
CDB Indexed by the CDI Rate ................................................................................ 33
4.15.2
CDB Indexed by the SELIC Rate ............................................................................ 34
4.15.3
CDBs Indexed by Price Indexes (Inflation rate) ...................................................... 34
4.15.4
Pre-Fixed Rate CDB................................................................................................ 35
4.16
RDB - Recibo de Depósito Bancário (Bank Deposit Receipt) ......................................... 35
4.17
Promissory Notes............................................................................................................. 35
4.18
CCB – Cédula de Crédito Bancário (Bank Credit Note) .................................................. 35
4.18.1
CCB Indexed by the CDI Rate ................................................................................ 36
4.18.2
CCB Indexed by Price Indexes ............................................................................... 36
4.18.3
Pre-Fixed Rate CCB................................................................................................ 36
4.18.4
Assessment of the spread factor............................................................................. 36
4.19
CPR - Cédula do Produto Rural (Rural Product Certificate)............................................ 37
4.19.1
4.19.2
4.20
Pre-Fixed Rate CPR................................................................................................ 37
CPR Indexed by an Agricultural Commodity........................................................... 37
NDF – Non Deliverable Forward...................................................................................... 37
4.21
Debentures....................................................................................................................... 38
4.21.1
Debenture Indexed by the CDI Rate ....................................................................... 38
4.21.2
Debenture Indexed by the IGP-M Index.................................................................. 38
4.21.3
Debenture Indexed by the IPCA Index.................................................................... 39
4.21.4
Debenture Indexed by the INPC Index ................................................................... 40
4.21.5
Spread Factor Calculation....................................................................................... 40
1.3.1 4.21.6 Special Cases................................................................................................... 42
5
4.22
TDA - Título da Dívida Agrária (Agrarian Debt Bond)...................................................... 44
4.23
Nota de Crédito de Exportação (Export Credit Note) ...................................................... 44
4.24
CRI – Certificado de Recebíveis Imobiliários (Real Estate Receivables Certificate) ...... 44
4.25
LH – Letra Hipotecária (Mortgage-Backed Security) ....................................................... 45
4.26
LCI - Letra de Crédito Imobiliário (Real Estate Credit Bill) .............................................. 46
4.27
Certificado a Termo de Energia Elétrica (Electric Power Forward Certificate)................ 46
4.28
CVS.................................................................................................................................. 46
4.29
CPR - Cédula do Produto Rural (Rural Product Certificate)............................................ 47
4.30
Loan Indexed by the LIBOR Rate .................................................................................... 47
4.31
NCE – Nota de Crédito de Exportação (Export Credit Note)........................................... 48
4.32
Currency Forward ............................................................................................................ 49
Other Structured Transactions ................................................................................................. 49
5.1
Pre-Fixed Rate Structured Transaction ........................................................................... 49
5.2
Libor x Fixed Rate Swap linked to Libor .......................................................................... 50
5.2.1 Libor Leg...................................................................................................................... 50
5.2.2 Pre-Fixed Rate Leg...................................................................................................... 50
5.2.3 Data ............................................................................................................................. 52
6
Other procedures and methods................................................................................................ 52
6.1
Procedures for dates without data disclosure.................................................................. 52
6.2
Valuation in the Curve...................................................................................................... 52
6.3
SELIC Rate - Updating Procedure................................................................................... 53
1.
Introduction
This document presents the processes and methods used to mark to market the
assets held in trust by Citibank. The file content can be summarized as follows:
1.1
Definition of Variables
This topic presents the variables to be used throughout the text.
1.2
Construction of Curves (Yield Curves)
This topic presents the method and the data sources used to construct the different
market yield curves required to mark the assets to market.
1.3
MtM Methods
This topic analyzes the characteristics of each product and presents the method to
obtain the market value thereof from the curves detailed in the previous topic.
Next, each of such topics will be detailed. The purpose of this manual is to serve as
a reference to Citibank’s asset pricing process, so that there will be no exceptions to the
procedures set forth herein.
2
Definition of Variables
This chapter will address the definition of the variables to be used throughout this
document. In some cases, particularly for less common variables, the variables will be
defined according to the need.
t = date as of which the market value of the asset will be calculated;
t0 = date of issue (or base date, for securities that have one) of the asset;
ti = date of payment of the i-th coupon of the security;
tF = maturity date of the asset;
PU0 = value of issue of a security or derivative;
PUt = adjusted asset (value) amount up to date t;
C0 = coupon value (interest rate) of the security;
a0 = percentual of the financial index, upon the issue of the security;
(1 + Ind ) tt0 = adjustment index variation from date t0 to date t;
(1 + r) tt F = discount rate variation, according to the rate specification, from date t
to date tF;
CDI = CDI observed (or CDI projection/forecast, identical to the pre-fixed rate
forecast given by the Pre-Fixed, No Cash Curve) between the concerned dates;
Ind = security index variation between the concerned dates, without using
forecasts;
SELIC= SELIC rate, available from BACEN - The Central Bank of Brazil;
VFace = face value of the security;
VF = future value contracted under the transaction;
VPA = value of the active (asset) leg on the concerned date;
VPP = value of the passive (liability) leg on the concerned date;
at = issuer’s credit spread, in percentage, on the concerned date;
Ct = issuer’s credit spread on the concerned date;
PUi = non-amortized value up to the i-th interest payment;
Aj = value of the j-th amortization.
2.1.
Method for rate accumulation
Throughout this topic, we will always assume that we want to accumulate a rate T
between two certain dates, t0 and t. Such value will be represented by Tt0t .
2.2.
Linear Rate
In this case, the accumulation is given by:
Tt0t = Tt0t1 + Tt1t2 + ...Tt t−−21 + Tt t−1
2.2.1. Exponential Rate
In this case, the accumulation is given by:
(
) (
)
(
) (
)
Tt0t = 1 + Tt0t1 × 1 + Tt1t2 × ... × 1 + Tt t−−21 × 1 + Tt t−1 − 1
2.2.2. Rate percentage accumulation
The accumulation of rates that can be calculated after the alteration by a percentage
thereof, such as CDI, is made as follows:
(1 + αT )tt
t
0
= ∏ (1 + αT )i
i =t 0
3.
Construction of Curves (Yield Curves)
Throughout this chapter, we will describe the methods and sources of data used to
construct market curves. These curves will be applied to the assets the market value of
which is to be obtained.
3.1
Methods
This section presents the possible methods used to construct the reference curves
for the market assets.
3.1.1 Bootstrapping
The method known as bootstrapping is the most commonly used method to extract
market curves from the prices of securities that pay intermediate coupons. In the case of
the domestic market, this method applies to NTN-B, NTN-C and NTN-F, for example.
The method consists of the following steps:
•
Determine, from the prices thereof, the return rate of the security with the shortest
maturity;
•
From such rate and the price of the security with the next maturity date, determine
the rate for the next period, comprised between the maturity date of the security
with the shortest maturity and the maturity date of the concerned security;
•
Repeat the process, recursively, for the other securities;
•
The interest rate curve obtained is the market curve for the coupon index of the
concerned securities.
Suppose that, for the curve to be calculated, there are k securities, the maturity
dates of which are arranged in ascending order. In addition, suppose that the internal rates
of return (IRR) of each of such securities is given by R1, ..., Rk, and that their prices is given
by P1, ..., Pk.
The purpose of the model is to determine the interest rate for each security
maturity, incorporating the rates from previous maturities. The rates to be determined will
be called r1, ..., rk. The model will operate as follows:
•
The rate for the first period will be influenced by the security with the first
maturity date only. Thus, we can represent:
r1 = R1
•
After the second security, there will be influence from the first and the second
maturity dates. The rate r2 is determined as follows:
a
P2 = ∑
i −1
b
Fi
Fi
+
∑
ti
(1 + r1 )t i=1 (1 + r2 )tti
where:
a: number of flows (coupons) occuring up to the maturity of the first security;
b: number of flows (coupons) occuring between the maturity of the first security
and the maturity of the second security;
F1: value of the i-th flow.
By resolving this equation at the variable r2, it becomes possible to obtain the
desired rate.
•
The subsequent securities will be treated likewise, always taking into account all
the rates found until the security with the immediately preceding maturity date. Hence,
the maturity rates for each security are found on a recursive basis;
•
With the rates r1, ..., rk, making the exponential interpolation between the maturity
dates is all that takes to construct the curve. This interpolation is described in the next
topic.
3.1.2 Interpolation
The purpose of this method is to determine the value of an interest rate on a
specific date, provided that such rate has known values on dates before and after the
concerned date. Given that:
•
i: number of days from today to the known vertices immediately before x;
•
ratei: interest rate for the vertices i;
•
j: number of days from today to the known vertices immediately after x;
•
ratej: interest rate for the vertices j;
•
x: number of days counted from today, where i ≤ x ≤ j.
The Exponential Interpolation is:
 1 + rate j
rate x = (1 + ratei ) ⋅ 
 1 + ratei



x −i
j −i
−1
Likewise, the Linear Interpolation is:
rate x =
( j − x ) ⋅ rate + (x − i ) ⋅ rate
i
j
( j − i)
( j − i)
3.1.3 Extrapolation
The purpose of this method is to determine the value of an interest rate on a
specific date, supposing that only a previous rate or a subsequent rate is known. Given
that:
•
i: number of days from today to the penultimate vertices before x;
•
ratei: interest rate for the vertices i;
•
j: number of days from today to the last vertices before x;
•
ratej: interest rate for the vertices j;
•
x: number of days counted from today, where i ≤ j ≤ x.
The Exponential Extrapolation is:
 1 + rate j
rate x = (1 + rate j )⋅ 
 1 + ratei
x− j
 j −i
 − 1

Likewise, the Linear Extrapolation is:
 x− j
(rate j − ratei )
rate x = ratei + 
 j −i 
3.2
Pre-Fixed, No Cash Curve (zero coupon curve – nominal yield curve)
•
•
•
This curve must have the following characteristics:
Data Source: CETIP and BM&F;
Origin: the origin will be obtained according to the period:
DI Rate for the first business day;
Future DI Rate for the first six maturities of the contract;
DI x Pre-fixed swap rate, for longer vertices.
Method to obtain Vertices: Exponential Interpolation, based on 252 business days;
Interpolation method: Exponential, based on 252 business days;
Extrapolation method: Exponential, based on 252 business days.
3.3
Pre-Fixed, Cash Curve
•
•
•
•
•
•
•
This curve must have the following characteristics:
Data Source: ANDIMA;
Origin: indicative rates of LTN (Letras do Tesouro Nacional – Brazilian Treasury Bills)
and NTN-F (Notas do Tesouro Nacional – Brazilian Treasury Notes – Series F);
Method to obtain Vertices: Bootstrapping;
Interpolation method: Exponential, based on 252 business days;
Extrapolation method: Exponential, based on 252 business days.
3.4
SELIC Coupon Curve (brazilian financial market define nominal yield
curve as ‘coupon’)
This curve must have the following characteristics:
Data Source: ANDIMA;
Origin: indicative rates of LFT (Letras Financeiras do Tesouro – Financial Treasury
•
•
Bills);
•
Method to obtain Vertices: Exponential Interpolation, based on 252 business days;
•
Interpolation method: Exponential, based on 252 business days;
•
Extrapolation method: Exponential, based on 252 business days.
3.5
IGP-M Coupon, Cash Curve
This curve must have the following characteristics:
•
Data Source: ANDIMA;
•
Origin: indicative rates of NTN-C (Notas do Tesouro Nacional – Brasilian Treasury
Notes – Series C);
•
Method to obtain Vertices: Bootstrapping;
•
Interpolation method: Exponential, based on 252 business days;
•
Extrapolation method: Exponential, based on 252 business days.
The same method will be used to construct the IGP-D Coupon, Cash curve.
3.6
IGP-M Coupon, No Cash Curve
•
•
•
•
•
This curve must have the following characteristics:
Data Source: BM&F;
Origin: DI x IGP-M swap reference rates;
Method to obtain Vertices: Exponential Interpolation, based on 252 business days;
Interpolation method: Exponential, based on 252 business days;
Extrapolation method: Exponential, based on 252 business days.
The same method will be used to construct the IGP-DI Coupon, No Cash curve.
3.7
IPCA Coupon, Cash Curve
This curve must have the following characteristics:
•
Data Source: ANDIMA;
•
Origin: indicative rates of NTN-B (Notas do Tesouro Nacional – Brasilian Treasury
Notes – Series B);
•
Method to obtain Vertices: Bootstrapping;
•
Interpolation method: Exponential, based on 252 business days;
•
Extrapolation method: Exponential, based on 252 business days.
3.8
IPCA Coupon, No Cash Curve
•
•
•
•
•
This curve must have the following characteristics:
Data Source: BM&F;
Origin: DI x IPCA swap reference rates;
Method to obtain Vertices: Exponential Interpolation, based on 252 business days;
Interpolation method: Exponential, based on 252 business days;
Extrapolation method: Exponential, based on 252 business days.
3.9
INPC Coupon Curve
•
•
•
•
•
This curve must have the following characteristics:
Data Source: BM&F;
Origin: INPC swap reference rates;
Method to obtain Vertices: Exponential Interpolation, based on 252 business days;
Interpolation method: Exponential, based on 252 business days;
Extrapolation method: Exponential, based on 252 business days.
3.10
Dollar Coupon, Cash Curve
This curve must have the following characteristics:
•
Data Source: ANDIMA;
•
Origin: indicative rates of NTN-D (Notas do Tesouro Nacional – Brasilian Treasury
Notes – Series D);
•
Method to obtain Vertices: Bootstrapping;
•
Interpolation method: Linear, based on 360 calendar days;
•
Extrapolation method: Linear, based on 360 calendar days.
It is important to emphasize that this procedure generates a curve known as “dirty
coupon”, since it originates from rates that are based on the dollar of the day before the
day of its formation.
3.11
Dollar Coupon, No Cash Curve
This curve must have the following characteristics:
•
Data Source: BM&F;
•
Origin: DI x Dollar swap reference rate;
•
Method to obtain Vertices: Linear interpolation, based on 360 calendar days;
•
Interpolation method: Exponential, based on 252 business days;
•
Extrapolation method: Exponential, based on 252 business days.
It is important to emphasize that this procedure generates a curve known as “dirty
coupon”, since it originates from rates that are based on the dollar of the day before the
day of its formation.
3.12
Euro Coupon Curve
•
•
•
•
•
This curve must have the following characteristics:
Data Source: BM&F;
Origin: DI x Euro swap reference rates;
Method to obtain Vertices: Linear interpolation, based on 360 calendar days;
Interpolation method: Exponential, based on 252 business days;
Extrapolation method: Exponential, based on 252 business days.
3.13
•
•
•
•
•
Yen Coupon Curve
This curve must have the following characteristics:
Data Source: BM&F;
Origin: according to the vertices:
REAL x YEN Rate and Pre-Fixed, No Cash Curve for the first month;
“Dirty Yen Coupon” rate for longer vertices.
Method to obtain Vertices: Linear interpolation, based on 360 calendar days;
Interpolation method: Exponential, based on 252 business days;
Extrapolation method: Exponential, based on 252 business days.
In order to calculate the e-mail coupon in the first month:
S=
1+ r
−1
1+ y
where:
s = Yen coupon for a certain period;
r = pre-fixed rate obtained from the Pre-Fixed, No Cash Curve for the concerned
period;
y = projection of the Yen variation in the concerned period.
3.14
ANBID Coupon Curve
•
•
•
This curve must have the following characteristics:
Data Source: ANBID and BM&F;
Origin: the origin will be obtained according to the period:
ANBID rate for the vertices before the last known rate;
DI x ANBID swap rate, for longer vertices.
Method to obtain Vertices: Exponential Interpolation, based on 252 business days;
Interpolation method: Exponential, based on 252 business days;
Extrapolation method: Exponential, based on 252 business days.
3.15
TJLP Coupon Curve
•
•
•
•
•
•
•
This curve must have the following characteristics:
Data Source: BNDES and BM&F;
Origin: the origin will be obtained according to the period:
TJLP rate up to the expiration of its term;
TJLP x Pre-fixed swap rate, for longer vertices.
Method to obtain Vertices: Exponential Interpolation, based on 252 business days;
Interpolation method: Exponential, based on 252 business days;
Extrapolation method: Exponential, based on 252 business days.
3.16
TR Coupon Curve
•
•
•
This curve must have the following characteristics:
Data Source: BACEN and BM&F;
Origin: the origin will be obtained according to the period:
TR rate up to the expiration of its term;
DI x TR swap rate, for longer vertices.
Method to obtain Vertices: Exponential Interpolation, based on 252 business days;
Interpolation method: Exponential, based on 252 business days;
Extrapolation method: Exponential, based on 252 business days.
4
MtM Methods
•
•
This chapter will address the specification of methods to mark to market several
products of the Brazilian financial market.
4.1
Financial Treasury Bills (LFT)
The market value of a LFT on the concerned date is given by:
PU 0 × (1 + SELIC )t0
t
MtM t =
(1 + r )tt
F
where:
r = SELIC Coupon.
4.2
Brazilian Treasury Bills (LTN)
The market value of a LTN on the concerned date is given by:
MtM t =
VFace
(1 + r )ttF
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, Cash Curve.
4.3
Brazilian Treasury Notes (NTN) - Series B
The principal amount adjusted to date t is given by:
PU t = PU 0 × (1 + IPCA)t0
t
where:
(1+ IPCA)tt
(1 + IPCA)tt
a
0
a
= IPCA variation from the base date to the date of the last anniversary;
= pro rata IPCA variation from the date of the last anniversary to the
concerned date.
Suppose that there are n coupon payments until the maturity date, including the
last date, when there is the repayment of the principal. The value of the i-th payment (or ith coupon), where 1 ≤ i ≤ n -1, is given by:


Ci = PU t × (1 + C0 ) 2 − 1
1
*****
The value of the n-th payment is given by:
1
C n = PU t × (1 + C0 ) 2
The market value of the NTN-B on date t is given by:
n
MtM t = ∑
i =1
Ci
(1 + r )tti
where:
r = expected IPCA coupon, obtained from the IPCA Coupon, Cash Curve.
The data sources used for IPCA are:
•
•
4.4
IPCA: IBGE;
IPCA Forecast: ANDIMA.
Brazilian Treasury Notes (NTN) - Series C
The principal amount adjusted to date t is given by:
PU t = PU 0 × (1 + IGPM )t0
t
with:
(1 + IGPM )tt
= (1 + IGPM )t0a × (1 + IGPM )ta
t
0
t
where:
(1+ IGPM )tt
(1 + IGPM )tt
a
0
a
= IGP-M variation from the base date to the date of the last published;
= pro rata temporis IGP-M variation from the date of the last published to the
concerned date.
Suppose that there are n coupon payments until the maturity date, including the last date,
when there is the repayment of the principal. The value of the i-th payment (or i-th
coupon), where 1 ≤ i ≤ n − 1 , is given by:
1


Ci = PU t × (1 + C0 ) 2 − 1


The value of the n-th payment is given by:
1
C n = PU t × (1 + C0 ) 2
The market value of the NTN-C on date t is given by:
n
MtM t = ∑
i =1
Ci
(1 + r )tti
where:
r = expected IGP-M coupon, obtained from the IGP-M Coupon, Cash Curve.
The data sources used for IGP-M are:
•
•
4.5
IGP-M: FGV;
IGP-M Forecast/Estimated: ANDIMA.
Brazilian Treasury Notes (NTN) - Series D
The principal amount adjusted to date t is given by:
PU t = PU 0 × (1 + Dollar )t0
t
where:
Dollar = Commercial dollar, considering the average sale rates on the business day
immediately preceding the concerned date. This value can be obtained from ANDIMA, at
the Títulos Públicos (Public Bonds) section.
Suppose that there are n coupon payments until the maturity date, including the last date
when there is the repayment of the principal. The value of the i-th payment (or i-th
coupon), where 1 ≤ i ≤ n − 1 , is given by:
Ci = PU t ×
C0
2
The value of the n-th payment is given by:
 C 
C n = PU t × 1 + 0 
2 

The market value of the NTN-D on date t is given by:
n
MtM t = ∑
i =1
Ci
(1 + r )tti
where:
r = expected Dollar coupon, obtained from the Dollar Coupon, Cash Curve.
4.6
Brazilian Treasury Notes (NTN) - Series F
Suppose that there are n coupon payments until the maturity date, including the last date,
when there is the repayment of the principal. The value of the i-th payment (or i-th
coupon), where 1 ≤ i ≤ n − 1 , is given by:
Ci = PU 0 ×
C0
2
The value of the n-th payment is given by:
 C 
C n = PU 0 × 1 + 0 
2 

The market value of the NTN-F on date t is given by:
n
MtM t = ∑
i =1
Ci
(1 + r )tt1
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, Cash Curve.
4.7
Brazilian Treasury Bonds (BTN)
The Brazilian Treasury Bonds (Bônus do Tesouro Nacional) were issued on Jun. 01, 1989,
based on Law No. 7.777. Said securities were extinguished on Mar. 01, 1991 by Law No.
8.177. However, as the maturity date established upon the issue was of up to 25 years,
there are outstanding bonds in the market.
Due to of the lack of liquidity in the secondary market for such securities, the market price
of the BTNs will be determined according to the guarantee prices (preços de lastro - PU 550)
published on a daily basis by the Central Bank of Brazil. Next, we will show how such
prices can be obtained.
Data can be obtained from ANDIMA’s website, following the path below:
• “Preço Unitário” (“Unit Price”) Section;
• “Resolução 550” (“Resolution 550”) Subsection;
• “Último Valor” (“Last Value”) Link.
The price of the security must be obtained according to its code, which has a biunivocal
correspondence with the maturity thereof. The table below shows such correspondence.
Code
711726
711746
711786
711806
711836
711846
711866
711896
711906
711926
711956
711966
711986
712016
712026
712046
712076
712086
4.8
Issued on
11/15/1990
9/18/1990
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
Maturity
3/15/2005
3/15/2005
9/15/2005
9/15/2005
3/15/2005
3/15/2006
3/15/2006
3/15/2005
9/15/2006
9/15/2006
3/15/2006
3/15/2007
3/15/2007
9/15/2006
9/15/2007
9/15/2007
3/15/2007
3/15/2008
Code
712106
712136
712146
712166
712196
712206
712226
712256
712266
712286
712316
712326
712346
712376
712386
712406
712436
712446
Issued on
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
Maturity
3/15/2008
9/15/2007
9/15/2008
9/15/2008
3/15/2008
3/15/2009
3/15/2009
9/15/2008
9/15/2009
9/15/2009
3/15/2009
3/15/2010
3/15/2010
9/15/2009
9/15/2010
9/15/2010
3/15/2010
3/15/2011
Code
712466
712496
712506
712526
712556
712566
712586
712616
712626
712646
712676
712686
712706
712736
712746
712766
712796
712856
Issued on
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
11/15/1990
9/18/1990
12/15/1989
12/15/1989
Maturity
3/15/2011
9/15/2010
9/15/2011
9/15/2011
3/15/2011
3/15/2012
3/15/2012
9/15/2011
9/15/2012
9/15/2012
3/15/2012
3/15/2013
3/15/2013
9/15/2012
9/15/2013
9/15/2013
3/15/2013
9/15/2013
Stocks
The marking to market of shares must meet the following criteria:
•
•
Stocks (or Equities) must be valued at their average trade price as published on a
daily basis by BOVESPA;
Stocks the price of which was not published on a certain date will be valued at the
last available quotation. This rule is not valid for stocks that have already exceeded
the legal period for lack of liquidity, on which occasion they must be valued
according to the procedures set forth by the applicable laws.
4.9
Stocks Forward
The stock forward transaction is a purchase or sale transaction of a certain amount of
stocks, at a fixed price, to be settled within a certain period, counted from the date of the
transaction. The market value of such transaction is given by:

q St − NV0 , if St > NV0 / q
 
q 

MtM t = 

mín q S − NV0 / q ,0
  t
tF  

(
)
1
+
r

t t  
 


where:
q = quantity
NV0 = nominal value (forward price)
St = spot price
If the trade is sell forward stock, the market value will be:
MtM t =
VF
(1 + r )ttF
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve.
4.10
Repurchase Agreement
A repurchase agreement is a transaction where funds are loaned, against the provision of
guarantees to the lender. In general, the lender’s remuneration rate and the maturity date
of the transaction are previously agreed. The mark to market is calculated using the same
methods to operations of CDB pre-fixed and pos-fixed, described in section 4.15.
4.11
Swaps
Throughout this topic, we will present the pricing method for each leg of a swap. The
general rule to calculate the market value of this kind of transaction is always the same,
namely:
MtM t = VPA − VPP
Thus, we only have to know the pricing method for each leg in order to obtain the value of
the swap transaction. The next topics present these methods. It is worth emphasizing,
however, that, in the event of structured or swap transactions that do not follow the
standard established by BM&F, the method to be followed will be presented in a separate
chapter.
4.11.1 CDI leg
The market value of the CDI leg on the concerned date is given by:
PU 0 × (1 + α 0 CDI )t0 × (1 + α 0 CDI )t F
t
MtM t =
(1 + r )tt
t
F
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve.
4.11.2 US Dollar Leg
The market value of the Dollar Edge on the concerned date is given by:
PU 0 × (1 + C0 )t0F × (1 + Dollar )t0
t
MtM t =
(1 + r )tt
t
F
Dollar = Purchase or sale Dollar PTAX rate (according to swap specification) as available
from BACEN; r = expected Dollar coupon, obtained from the Dollar Coupon, No Cash
Curve.
4.11.3 Pre-Fixed Leg
The market value of the Pre-Fixed Edge on the concerned date is given by:
PU 0 × (1 + C0 )t0F
t
MtM t =
(1 + r )tt
F
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve.
4.11.4 IGP-M Leg
The market value of the IGP-M Edge on the concerned date is given by:
PU 0 × (1 + C0 )t0F × (1 + IGPM )t0
t
MtM t =
t
(1 + r )tt
F
where:
(1+ IGPM )tt
0
= IGP-M accumulated up to the concerned date, without using projections;
r = expected IGP-M coupon, obtained from the IGP-M Coupon, No Cash Curve.
4.11.5 IPCA Leg
The market value of the IPCA Edge on the concerned date is given by:
PU 0 × (1 + C0 )t0F × (1 + IPCA)t0
t
MtM t =
t
(1 + r )tt
F
where:
(1+ IPCA)tt
0
= IPCA accumulated up to the concerned date, without using projections;
r = expected IPCA coupon, obtained from the IPCA Coupon, No Cash Curve.
4.11.6 Euro Leg
PU 0 × (1 + C0 )t0F × (1 + Euro )t0
t
MtM t =
t
(1 + r )tt
F
Euro = Purchase or sale Euro PTAX rate (according to swap specification) as available from
BACEN; r = expected Euro coupon, obtained from the Euro Coupon Curve.
4.11.7
Yen Leg
The market value of the Yen Edge on the concerned date is given by:
PU 0 × (1 + C0 )t0F × (1 + Ien )t0
t
MtM t =
(1 + r )tt
t
F
where:
Yen = Purchase or sale Yen PTAX rate (according to swap specification) as available from
BACEN;
r = expected Yen coupon, obtained from the Yen Coupon Curve.
4.11.8 Dollar leg with threshold and intermediate adjustment payment
In this case, the dollar edge has a pre-fixed coupon established in the agreement, in
addition to a limit price for the Ptax rate upon the payment of the adjustments. The pricing
will be split into two parts.
Pre-Fixed Portion
The MtM of each semiannual, pre-fxed coupon will be calculated as follows:
T 
PU 0 ⋅  
2
PPi =
(1 + s )tti
where:
T: pre-fixed, annual rate, based on a 360-day year, as established in the agreement;
s: Dollar Coupon projection, obtained from the Dollar Coupon, No Cash Curve;
ti: date of the i-th payment.
On the last coupon payment date, the principal amount of the transaction will be repaid as
well, which will be given by:
T 
PU 0 ⋅  + 1
2 
PPF =
(1 + c )ttF
If n is the number of coupons remaining up to the maturity of the transaction, the total
amount of the pre-fixed portion, in dollars, is given by:
n
PPt = ∑ PPi + PPF
i =1
Sell Put (short Put)
The thresholds established in the agreement may be priced through the sell of a series of
puts, with strike and maturity dates adjusted according to the thresholds; this transaction
is known as cap. The cap value is given by the sum of the value of each of the puts that
comprise it.
For each of the puts that comprise the cap, the price is given by the method proposed by
Reiner and Rubinstein (1991) for the pricing of digital (or binary) options:
Pi = e − s⋅∆t ⋅ N (d )
where:
σ2 
S 
 ⋅ ∆t
1n  +  s − r −
2 
K 
d=
σ ⋅ ∆t
∆t: time for the expiration of the option, in years;
r: expected Interest Rate, obtained from the Pre-Fixed, No Cash Curve;
s: Dollar Coupon projection, obtained from the Dollar Coupon, No Cash Curve;
K: strike value;
σ: implied volatility of the transaction;
N(d): standard normal.
Thus, the MtM of the cap on date t will be given by:
n
capt = ∑ (PPi ⋅ P1 )
i =1
Accordingly, the value of the dollar leg is given by:
MtM tDOLLAR = PP1 − cap1
4.11.9
Libra Sterling Leg
The market value of the Pound Edge on the concerned date is given by:
PU 0 × (1 + C 0 )t0F × (1 + Libra )t0
t
MtM t =
t
(1 + r )tt
F
where:
Libra = Libra variation, as available from BACEN. As this agreement is not a standard
form, the type of quotation set forth in the agreement must be used.
r = expected Libra coupon. By virtue of the nonexistence of a future market for this rate,
this expectation will be calculated as follows:
(1 + r )tt
F
=
(1 + R )tt
F
(1 + Libra )tt
F
where:
(1 + R )tt = pre-fixed rate variation, obtained from the Pre-Fixed, No Cash Curve;
(1 + Libra )tt = projected Pound variation, obtained from the following equation:
F
F
(1 + Libra )tt
F
= (1 + LbUS $ )t F × (1 + Dollar )t F
t
t
where:
(1 + Dollar )tt
F
= projected Dollar variation between the concerned dates, obtained from the
equation;
(1 + Dollar ) =
tF
t
(1 + R )tt
F
(1 + CpDol )tt
F
with:
CpDol = expected Dollar coupon, obtained from the Dollar Coupon, No Cash Curve.
(1 + LbUS $ )tt
= projected Libra variation, in Dollars, between the concerned dates. This
F
variation can be obtained from the futures contracts traded at the Chicago Mercantile
Exchange (CME), through the following link at Bloomberg:
BPA <"Currency" button> CT <"Go" button>.
4.11.10
Libor Edge
The market value of the Libor Edge on the concerned date, considering that the notional
principal of the transaction is adjusted by the exchange rate variation of the U.S. dollar, is
given by:
PU 0 × (1 + Libor0 )t0 × (1 + Libor )tL × (1 + Libor )t FL × (1 + C0 )t0F
t
MtM =
t
(1 + r )tt
t
t
F
where:
tL = last date on which the Libor rate is known;
(1 + Libor0 )tt
= Libor variation observed on the date of the transaction, between dates t0
0
and t;
(1+ Libor0 )tt
L
= Libor variation observed on the date of the transaction, between dates t and
t L;
(1 + Libor )tt
F
L
= projected Libor variation, between dates tL and tF;
r = expected Dollar coupon, obtained from the Dollar Coupon, No Cash Curve.
It must be observed that both the Libor rate and the coupon of the transaction accrue on a
calendar-day basis, based on a 360-day year. The necessary data can be obtained from the
following sources:
• Libor: at the Bloomberg system, by typing:
US000
<"Index" button>
<"Go" button>.
Alternatively, said amounts can be found at BBA’s website: www.bba.org.uk.
• Projected Libor: at the Bloomberg system, by typing:
IRSB <"Go" button> 18 <"Go" button>.
4.11.11
Index or Commodity Futures Leg
The market value of an leg linked to financial index or commodity futures is given by the
present value, calculated in reais, of the futures contract on the date on which the swap is
to be marked to market.
4.11.12
FIDC Leg
The market value of an edge linked to the value of a FIDC (Receivables Investment Fund)
must be appraised according to the share in the respective fund.
4.11.13
CRI Leg
The market value of an edge linked to the value of a CRI (Real Estate Receivables
Certificate) must be appraised according to the market value of the respective security.
4.11.14
NCE Leg
The market value of an edge linked to the value of a NCE (Export Credit Note) must be
appraised according to the market value of the respective security.
4.11.15
Swap with Cash Flow
The method to calculate the market value of a swap with cash flow must be consistent
with the methods established for each individual edge. The cash flows must be treated on
an individual basis, in the same manner as the transactions already described above are
treated.
The market value of an edge of a swap with cash flow is given by the sum of the market
values of each of the individual flows.
4.11.16
Leg with Double Index
There are two possible cases for the edge of a swap with double index: the edge yields the
minimum value between two previously-established indexes, or the maximum value
between them. Below is a description of the method to be used in both cases, always
assuming that the indexes of the transaction are CDI and Dollar.
Minimum Value:
The market value of this swap edge on date t is given by:
MtM t = PU 0 × (1 + (α 0 − 1)CDI )tF × (1 + α 0 CDI )t0 −
t
t
PU 0
t
× (1 + C0 )t0F × PutValue
S0
where:
S0 = exchange rate value on the initial date of the transaction;
PutValue = price of a dollar put option, the strike of which is given by:
S 0 × (1 + α 0 CDI )t F × (1 + α 0 CDI )t0
t
X=
t
(1 + C0 )tt
F
0
The value of this option must be calculated according to the appropriate procedure for this
asset.
Maximum Value:
The market value of this swap edge on date t is given by the value of the CDI edge,
normally calculated as shown above, plus the value of a dollar call option. The exercise
price of this option is given by:
PU 0R $ × (1 + CDI )t0F
t
Strike =
PU 0US $ × (1 + C0 )t F
t
where:
PU 0R $ = principal amount of the operation, in Reais;
PU 0US $ = principal amount of the operation, in US Dollars.
4.11.17
Swap linked to NTN-D
In the event of an operation linked to a federal government bond, the marking to market
of the edge linked to CDI must be made from the percentage published on a daily basis by
ANDIM. The remaining part of the method follows the same standard established for the
respective transaction.
To obtain the percentage published by ANDIMA, the path to be followed is:
•
•
•
•
•
•
•
Mercado Financeiro (Financial Market) Section;
Marcação a Mercado (Mark-to-Market);
Títulos Públicos (Public Bonds);
Taxas Médias (Average Rates);
Último Valor ou Histórico (Last or Historical Value) (as applicable);
In case of Último Valor (Last Value): select "NTN-D/SW";
In case of Histórico (Historical Value): select the desired date and select "NTND/SW".
This percentage (at) must be used as a transaction value discount, so as to obtain:
PU 0 × (1 + α 0 CDI )t0 × (1 + α 0 CDI )t F
t
MtM t =
(1 + α t ⋅ r )tt
t
F
4.12
Options
This topic will describe the procedures to calculate option premiums. The organization of
this topic is based on the model used to calculate premium. It is worth highlighting that
the models below are adjusted to European options.
4.12.1 Common Definitions
Below we will examine the common definitions of the formulae used to calculated option
premiums and which will be used in this topic:
c = call option premium;
p = put option premium;
σ = implied volatility of the option;
t = time to maturity, in years;
N (.) = standard normal distribution, of average 0 and variance 1;
S = spot price of the underlying asset;
F = future price of the underlying asset;
X = exercise price;
H = barrier value, if applicable;
r = interest rate, in continuous form, projected from the Pre-Fixed, No Cash Curve, for the
option maturity. This rate is given by:
r = 1n(1 + R )
Where R is the interest rate projected from the Pre-Fixed, No Cash Curve for the option
maturity.
re = external interest rate, in continuous form, projected from the Coupon, No Cash Curve
of the asset under the option, or cost of carry of the base asset. This rate is given by:
re = 1n(1 + Re )
where Re is the external interest rate projected from the Coupon, No Cash Curve of the
asset under the option for the option maturity.
4.12.2 Black & Scholes Model
Application: Stocks, Ibovespa
The premium of an option, according to the Black & Scholes model, is given by:
c = S × N (d1 ) − X × e − rt × N (d 2 )
with:
p = X × e − rt × N (− d 2 ) − S × N (− d1 )
2
S  σ 
×t
1n  +  r +
2 
X 
d1 =
σ× t
d 2 = d1 − σ × t
It is worth emphasizing that, in the case of stock options, the following procedure must be
adopted in order to obtain the premium:
•
•
In the event that the stock present more than 50 trades a day at Bovespa, they will
be deemed liquid, and the premium to be used will be the premium published by
Bovespa;
Otherwise, the premium will be obtained from the described model.
4.12.3 Garman Model
Application: Currencies
The premium of an option, according to the Garman model, is given by:
c = e − ret × S × N (d1 ) − e − n × X × N (d 2 ) p = e − n × X × N (− d 2 ) − e − ret × S × N (− d1 )
with:
d1 =
S
1n
X
σ2 
 

×t
+
r
−
r
+
 
e
2 
 
σ× t
d 2 = d1 − σ × t
4.12.4 Black Model
Application: Futures
The premium of an option, according to the Black model, is given by:
c = [F × N (d1 ) − X × N (d 2 )]× e − rt
p = [X × N (− d 2 ) − F × N (− d1 )]× e − rt
with:
2
 F  σ 
 × t
1n  + 
X  2 
d1 =
σ× t
d 2 = d1 − σ × t
4.12.5 Barrier Options
In this section we will describe options with barriers. The formulae for these types
of options may be generalized, changing only by function of the type of barrier of the
option. Thus, initially, we will present the common definitions of the formulae; next, we
will specify the variables by virtue of the established type of barrier. Given that:
(
A = φ .S .e (re − r ) .N (φx1 ) − φ . X .e −rt .N φσ t
t
B = φ .S .e
C = φ .S .e
D = φ .S .e
( re − r )1
( re − r )1
( re − r )1
.(H / S )
2 ( µ +1)
2 ( µ +1)
.(H / S )
E = K .e
[
− rt
[N (ηx
F = K (H S )
2
µ +λ
(
)
.N (φx2 ) − φ . X .e .N φx2 − φσ t
− rt
)
(
.N (ηγ
)
t)
.N (ηγ 1 ) − φ . X .e .(H / S ) .N ηγ 1 − ησ t
2µ
− rt
.N (ηγ 2 ) − φ . X .e .(H / S )
2µ
− rt
)
(
2
− ησ
)]
1 )]
− ησ t − (H / S ) N ηy 2 − ησ 1
2µ
N (ηz ) + (H / S )
µ −λ
(
N ηz − 2ηλσ
where:
ln (S X )
+ (1 + µ )σ t
σ t
ln (H 2 SX )
yt =
+ (1 + µ )σ t
σ t
ln(H S )
z=
+ λσ t
σ t
ln (S H )
+ (1 + µ )σ t
σ t
ln (H S )
y2 =
+ (1 + µ )σ t
σ t
r − σ2 2
µ= e
2
x1 =
x2 =
(
)
σ
λ = µ2 +
2r
σ2
The pricing of the options must be made as follows, according to the option type:
Type
Down-and-in call
“In” Options
Up-and-in call
Down-and-in put
Up-and-in put
“Out” Options
Type
Down-and-out call
Case S > H
c=C+E
η = 1,φ = 1
c = A+ E
η = −1,φ = 1
p = B −C + D + E
η = 1,φ = −1
p = A− B + D + E
η = −1,φ = −1
CaseS > H
c = A−C + F
η = 1,φ = 1
Case S ≤ H
c = A− B + D + E
η = 1,φ = 1
c = B −C + D + E
η = −1,φ = 1
p = A+ E
η = 1,φ = −1
p=C+E
η = −1,φ = −1
CaseS ≤ H
c= B−D+F
η = 1,φ = 1
c=F
Up-and-in call
c = A− B+C − D+ F
η = −1,φ = 1
p = A− B +C − D + F
Down-and-in put
Up-and-in put
η = −1,φ = 1
p=F
η = 1,φ = −1
η = 1,φ = −1
p = B−D+F
η = −1,φ = −1
p= A−C + F
η = −1,φ = −1
4.12.6 Global Bond Option
For Global Bond options, the premium must be obtained directly from the
Bloomberg system, after 7:00 p.m., in order for the dealings in the market not to interfere
in the price obtained.
4.12.7 Nominal Interest Rate Volatility Option
The premium for these options must be calculated through the Black & Scholes
model. However, some particularities must be observed in the calculation of the model
input values. Said values must be calculated as follows:
S=
100.00
(1 + rFRA )
where:
rFRA = interest rate for the FRA interest rate period, given by:
(1 + R )tt
rFRA =
(1 + R )tt
AO
F
F
where:
R = interest rate projected from the Pre-Fixed, No Cash Curve;
t FAO = maturity date of the future asset under the option;
tF = maturity date of the option.
X=
100.000
(1 + rx )tt
AO
F
F
where:
rX = interest rate agreed under the transaction.
In addition, we must have:
r = 0;
t = term, in years, between the maturity of the option and the maturity of the
respective future asset.
4.12.8 Digital Options
These options have the following pay-off characteristics:
Call: it pays 0, if the value is below the strike; otherwise, it pays K;
Put: it pays K, if the value is below the strike; otherwise, it pays 0.
•
•
The market value is given by:
c = Ke − rt N (d )
p = Ke − rt N (− d )
where:
d=
( X )+ (− σ 2 )× t
2
In S
σ t
4.12.9 Asian Options
This type of option does not have a closed pricing formula. To estimate the
premium, we will use the approximation described in Turnbull & Wakeman, published in
1991. This approximation suggests the following premiums:
c = X × e − rT2 × N (− d 2 ) − S × e (b−r )T2 N (− d1 )
c = S × e (b−r )T2 N (d1 ) − X × e − rT2 × N (d 2 )
with:
σ2
S 
In  +  b + M
2
X 
d1 =
σ M × T2
σM =
e reT − e reτ
M1 =
re (T − τ )
where:

 xT2

d 2 = d1 − σ M × T2
In(M 2 )
− 2b
T
2e (2 re +σ )T
M2 =
2
re + σ 2 2r2 + σ 2 (T − τ )
2
(
)(
In(M 1 )
T
2
2e (2 re +σ )τ  1
e re (T −τ ) 
−
+

2 
2
re + σ 2 
re (T − τ )  2re + σ
b=
)
T2 = time, in years from the current date to the maturity date of the option;
T = time, in years from the start date to the maturity date of the option;
? = time, in years from the current date to the start date of the period for which the
average of the base asset under the option is to be calculated;
re = external interest rate, in continuous form, projected from the Coupon, No Cash
Curve of the asset under the option.
If the option is already within the period for which the average is to be calculated,
the strike price (X) must be replaced with:
X2 =
T − T2
T
X−
SM
T2
T2
where:
SM = average price of the asset during the period in which the average thereof is to
be calculated.
4.12.10
Implied Volatility
The implied volatility of an option is the volatility obtained from close-form
formula option. As the formula to price these derivatives are not invertible in the
‘volatility’ variable, numerical methods are generally used to obtain the volatility value.
As a standard, Microsoft Excel is used. To obtain the value, all variables that are
present in the pricing formula must be set, except for volatility, and such volatility must be
varied until the resulting premium is equal to the negotiated premium.
Due to the different characteristics of the options presented herein, we will briefly
describe the processes used to obtain the implied volatility for the different options.
•
Stock Option:
The average of the implied volatilities calculated every 15 minutes during
the trading hours of the market is used;
The implied volatility values are obtained from the option premium and the
price of the respective share asset at the same moment;
A source to obtain such data is the Bloomberg system.
•
Currency Option:
As a priority, the average of the implied volatilities published by at least 2
brokers that work with such kind of transaction is used;
If such values are not available, the volatility provided by the Bloomberg
system is used.
•
IDI Option:
The average of the implied volatilities published by at least 2 brokers that
work with such kind of transaction is used.
•
DI Option:
•
The volatility obtained from the premiums published by BM&F is used.
Option on Ibovespa Futures:
The implied volatility calculated from the trades recorded by a broker that
works with this kind of transaction is used.
There are some particular cases, which will be addressed separately, as follows:
Case 1: Illiquid Options
For illiquid options, which will be deemed here as barrier option and all options
that are part of a structured transaction (except for box strategy), the implied volatility of
the transaction will be used, to be obtained as already described in this document.
Case 2: Implied volatility not available
In the event that the implied volatility of an option is not available, but there are
volatilities for other options on the same base asset, the interpolation or extrapolation of
values is used. Once again, this case must be subdivided into two new sub-cases:
Case 2 - item A: Simple linear interpolation
In the event that there are at least 2 other options with the same base asset and the
same maturity of the concerned option, the volatilities of such options must be calculated
and applied in the linear interpolation (or extrapolation) used to estimate the volatility of
the concerned option.
The method to be followed is the same method used in the linear interpolation (or
extrapolation) of interest, it being observed that, instead of the time, the strike of the
option is used. Likewise, instead of the rates, the implied volatility of the options is used.
Case 2 - item B: Linear interpolation in two axes
In the event that it is not possible to find at least 2 options with the same maturity
of the concerned option, other options with other maturities must be used in the
interpolation. In this case, the idea is to keep the linear interpolation, but so as to
contemplate the plan determined by 3 options. The following steps are followed:
•
Options are chosen with maturity and strike as close as possible to the concerned
option;
•
The plan established by such options is determined, subject to the variables of
interest: time to maturity, implied volatility and strike;
•
From the equation determined for the plan, the volatility of the concerned option is
estimated.
4.13
Subscription Right
The right to subscribe for a share is a right to purchase new shares of the same
corporation and of the same type (preferred, common, etc.) for a certain price. This price,
in general, is lower than the market value of the share on the date on which the right is
announced.
This right is valid from the date of its announcement (that is, the date on which the
corporation grants the right to its shareholders) and the date on which the shareholder
chooses whether or not to exercise the right.
The right is negotiable on the stock market, and its quotation is published on a
daily basis by BOVESPA. Therefore, it must be marked to market preferably at its average
price (or closing price, based on the mark to market of the shares) of trade in the market.
In the event that such price is not available, the subscription right must be priced
as an option, with the following characteristics:
•
Strike: subscription value;
•
Maturity: subscription date;
•
Concerned asset: share to be subscribed for.
The other pricing parameters must follow the same procedure already described in
the calculation of option premium.
4.14
Box Spread – Fixed Income Strategy with Options
In principle, a box of options is a structured transaction that uses options, which
aims to achieve a result that is previously established at the very beginning of the
transaction. The market value of the transaction on the concerned date is given by:
MtM t =
FV
(1 + α t rt )tt F
where:
FV = Future Value
αt = spread rate
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve.
The spread rate, αt, will be defined through of a sample of spread rate observed in
the market. In cases of misbehavior market, the Citi will establish an alternative procedure
to calculate this spread and to reflect the market conditions, considering the principles of
mark to market methodology. If the spread factor is impossible to determine at the day, will
be adopted the same rate in the purchase time.
4.15
CDB - Certificado de Depósito Bancário (Bank Certificate of Deposit)
This topic describes mark-to-market methods for CDBs, according to their index.
The sub-items will be segmented according to the index of the security.
4.15.1 CDB Indexed by the CDI Rate
The market value of the security on the concerned date is given by:
PU 0 × (1 + α 0 CDI )t0 × (1 + α 0 CDI )t F
t
MtM t =
t
(1 + α1r )tt
F
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve.
In the event that the profitability of the security is given by CDI + Spread, the
mark-to-market is:
PU 0 × (1 + CDI )t0 × (1 + C0 )t0F
t
MtM t =
t
(1 + Ct )tt
F
The credit spread rate, αt or Ct, will be defined through of a sample of spread rate
observed in the market. In cases of misbehavior market, the Citi will establish an alternative
procedure to calculate this spread and to reflect the market conditions, considering the
principles of mark to market methodology. If the spread factor is impossible to determine at
the day, will be adopted the same rate in the purchase time.
4.15.2 CDB Indexed by the SELIC Rate
The market value of the security on the concerned date is given by:
PU 0 × (1 + α 0 SELIC )t0 × (1 + α 0 SELIC )t F
t
MtM t =
t
(1 + α1r )tt × (1 + s )tt
F
F
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve;
s = expected SELIC coupon, obtained from the Selic Coupon Curve;
In the event that the profitability of the security is given by SELIC + Spread, the
mark-to-market is:
PU 0 × (1 + SELIC )t0 × (1 + C0 )t0F
t
MtM t =
(1 + Ct )tt
t
F
In both cases, The credit spread rate will be defined as described at the prior item.
4.15.3 CDBs Indexed by Price Indexes (Inflation rate)
The market value of the security on the concerned date is given by:
PU 0 × (1 + Ind )t0 × (1 + C0 )t F
t
MtM t =
t
(1 + Ct )tt × (1 + r )tt
F
F
where:
r = expected coupon of the index, obtained from the Coupon, No Cash Curve of
such index.
The credit spread is given by the difference between the rate of issue of the security
and the coupon rate of the index of the security on the same date. If it is not possible to
determine the credit spread of the issuer on the concerned date, the same spread as of the
date of acquisition of the security will be used.
4.15.4 Pre-Fixed Rate CDB
The market value of the security on the concerned date is given by:
PU 0 × (1 + C0 )t F
t
MtM t =
(1 + r )tt × (1 + C1 )tt
F
F
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve;
The credit spread rate, Ct , will be defined as described at the prior item.
4.16
RDB - Recibo de Depósito Bancário (Bank Deposit Receipt)
The marking to market of these securities will be subject to the same method
adopted to mark Bank Certificates of Deposit to market.
4.17
Promissory Notes
The marking to market of these securities will be subject to the same method
adopted to mark Bank Certificates of Deposit to market.
4.18
CCB – Cédula de Crédito Bancário (Bank Credit Note)
This topic describes mark-to-market methods for CCBs, according to their index.
The sub-items will be segmented according to the index of the security.
4.18.1 CCB Indexed by the CDI Rate
The market value of this security must be calculated according to the method
established to calculate the market value of Bank Certificates of Deposit indexed by CDI.
In the event that the CCB presents intermediate payment flows, each of the flows must be
treated individually, according to the same method. In this case, the market value of the
security is given by the sum of the market values of each flow.
4.18.2 CCB Indexed by Price Indexes
The market value of this security must be calculated according to the method
established to calculate the market value of Bank Certificates of Deposit indexed by price
indexes. In the event that the CCB presents intermediate payment flows, each of the flows
must be treated individually, according to the same method. In this case, the market value
of the security is given by the sum of the market values of each flow.
4.18.3 Pre-Fixed Rate CCB
The market value of this security must be calculated according to the method
established to calculate the market value of pre-fixed rate Bank Certificates of Deposit. In
the event that the CCB presents intermediate payment flows, each of the flows must be
treated individually, according to the same method. In this case, the market value of the
security is given by the sum of the market values of each flow.
4.18.4 Assessment of the spread factor
According to the structure of the Brazilian market, it is impossible to assess, in each
moment, the correct spread factor based on the credit risk of the entity issuing the CCB.
Thus, in order to refine the pricing of the security, we will only use the spread obtained in
the moment of its acquisition. We will present below the suggested method to assess such
spread.
From the acquisition price of the security, it is possible to determine the rate at
which the security was purchased, thus equalling the acquisition price to the equation that
provides the price of the security, and having the market rate vary.
This must be done as follows:
n
P=∑
i =1
(1 + r )
tii
t
Fi
t
× (1 + α )ti
where:
P = acquisition value of the security;
Fi = value of the i-th flow of the security;
r = value of the discount rate of the security for the concerned period.
From this equation, the S factor is determined, which measures the credit risk of
the security. This value will remain constant throughout the maturity period of the
security.
4.19
CPR - Cédula do Produto Rural (Rural Product Certificate)
This topic describes mark-to-market methods for CPRs, according to their type:
pre-fixed rate or indexed by the value of some agricultural commodity. The sub-items will
be segmented according to the index of the security.
4.19.1 Pre-Fixed Rate CPR
The market value of this security must be calculated according to the method
established to calculate the market value of pre-fixed rate Bank Certificates of Deposit. In
the event that the CPR presents intermediate payment flows, each of the flows must be
treated individually, according to the same method. In this case, the market value of the
security is given by the sum of the market values of each flow.
4.19.2 CPR Indexed by an Agricultural Commodity
The market value of this security must be calculated from the value specificied in
the contract of the concerned agricultural commodity. The prices of such commodities can
be
obtained
from
ESALQ
–
USP,
through
the
link:
http://www.cepea.esalq.usp.br/indicador/.
4.20
NDF – Non Deliverable Forward
This topic describes the marking to market of the operation known as NDF. The
market value of the derivative on the concerned date is given by:
MtM t = ε
CtF
(1 + r )
tF
t
−ε
Ct
(1 + s )tt
F
where:
U = sign of the operation (“+” for purchase, “-“ for sale);
Ct = exchange rate on the concerned date, as specified in the agreement or as
obtained from the same source as described in the swap agreements;
Ct F = exchange rate agreed for the final date of the transaction;
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve;
s = expected coupon in the traded currency, obtained from the Coupon, No Cash
Curve.
4.21
Debentures
This topic describes the procedure to mark debentures to market. Initially, the
general process will be described. Then, the process to obtain the credit spreads for each
type of security will be described.
4.21.1 Debenture Indexed by the CDI Rate
Suppose that n interest payments and m amortizations remain until the maturity of
the security. The market value of the security on the concerned date is given by:
n
MtM t = ∑
PU i × (1 + α 0 CDI )t0 × (1 + α 0 CDI )ti0
t
(1 + α1r )tt
i =1
t
i
m
+∑
j =1
A j × (1 + α 0 CDI )t0 × (1 + α 0 CDI )t0j
t
t
(1 + α t r )tt
j
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve;
In the event that the profitability of the security is given by CDI + Spread, the
mark-to-market is:
n
MtM t = ∑
i =1
PU i × (1 + CDI )t0 × (1 + C 0 )ti0
t
t
(1 + Ct )tt
i
m
+∑
j =1
A j × (1 + CDI )t0 × (1 + C 0 )t0j
t
t
(1 + Ct )tt
j
4.21.2 Debenture Indexed by the IGP-M Index
Case 1: Monthly Updating of the IGP-M Index
Suppose that n interest payments and m amortizations remain until the maturity of
the security. The market value of the security on the concerned date is given by:
n
MtM t = ∑
PU i × (1 + IGPM )t0 × (1 + C 0 )ti0
t
(1 + S t )tt
i =1
t
i
m
+∑
j =1
A j × (1 + IGPM )t0 × (1 + C 0 )t0j
t
(1 + S t )tt
t
j
where:
(1+ IGPM )tt
0
= IGP-M accumulated up to the concerned date, without using
projections;
St = spread calculated for the security on date t.
Case 2: Annual Updating of the IGP-M Index
Suppose that n interest payments and m amortizations remain until the maturity of
the security. The value of the installments (referring to interest or amortizations) to be paid
before the next indexation of the security must be calculated as follows:
V × (1 + IGPM )t0a × (1 + C )tv0
t
MtM t =
(1 + S t )tt
t
v
where:
V = nominal value of the concerned installment;
ta = date of the last anniversary of the security, which indicates the last indexation;
tv = date of payment of the concerned installment;
C = interest rate that adjusts the concerned installment;
The value of the installments (referring to interest or amortizations) to be paid after
new indexations must be calculated as follows:
V × (1 + IGPM )t0a × (1 + IGPM )tba × (1 + C )tv0
t
MtM t =
t
(1 + S t )tt
t
v
where:
V = nominal value of the concerned installment;
ta = date of the last anniversary of the security, which indicates the last indexation;
tb = last date of anniversary before the payment of the concerned installment;
tv = date of payment of the concerned installment;
C = interest rate that adjusts the concerned installment;
St = spread calculated for the security on date t.
The forecast IGP-M between the anniversary dates ta and tb must be calculated
from the Pre-Fixed, No Cash Curve and the IGP-M Coupon, No Cash Curve. Said forecast
is given by:
(1 + r )tt
(1 + IGPM ) =
(1 + s )tt
tb
ta
b
a
b
a
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve;
s = expected IGP-M coupon, obtained from the IGP-M Coupon, No Cash Curve;
Finally, the market value of the security is given by the sum of the market values of
each of the installments that comprise it, calculated as shown in this section.
4.21.3 Debenture Indexed by the IPCA Index
Debentures indexed by IPCA must be treated in the same manner as debentures
indexed by IGP-M, both where the adjustment is made on a monthly basis as well as
where the adjustment is made on an annual basis.
In this case, the accumulated IGP-M values must be replaced with the accumulated
IPCA values. Likewise, the IGP-M Coupon values must be replaced with the IPCA
Coupon values from the IPCA Coupon, No Cash Curve.
4.21.4 Debenture Indexed by the INPC Index
Debentures indexed by INPC must be treated in the same manner as debentures
indexed by IGP-M, both where the adjustment is made on a monthly basis as well as
where the adjustment is made on an annual basis.
In this case, the accumulated IGP-M values must be replaced with the accumulated
INPC values. Likewise, the IGP-M Coupon values must be replaced with the INPC
Coupon values from the INPC Coupon, No Cash Curve.
4.21.5 Spread Factor Calculation
Two different methods are adopted to determine the spread factor to be used when
marking the security to market. The first one, called Method 1, will be preferably applied,
but it may not be appropriate for some securities. In the event that it is not possible to
apply Method 1, we will adopt an alternative method, which we will call Method 2. Said
methods will be detailed below.
Method 1
Method 1 consists of using the spreads assessed by ANDIMA to mark the security
to market. On a daily basis, ANDIMA publishes the spreads of several securities. Said
values are obtained from consultation with a pool of collaborators of said entity.
The spread to be used in the marking to market must be calculated as follows:
•
For the concerned date t, all spreads published by ANDIMA on the last 10 business
days before date t (or the number of days on which they are published) for the
concerned debenture must be obtained;
•
Be such values S t ,1 , S t , 2 ,..., S t ,n . The spread on date t, St, will be given by:
St =
1 n
∑ S t ,i
n i =1
The method is identical for both percentage spread and nominal rate spread. The
value obtained must be considered as the discount rate in the calculation of the MtM of the
debentures, as already shown.
In the event that ANDIMA fails to publish the rates for the debenture for 10
calendar days or more, Method 2 will be adopted to price such debenture.
Method 2
ANDIMA does not publish spreads for all securities traded in the secondary
market. Therefore, in some cases, Method 1 is not appropriate to calculate the market
value of the debentures. In view of this, we will adopt an alternative method for this
calculation.
Method 2 consists of estimating the credit spread from observations of trades in the
secondary market. To determine the spread (S) of a particular transaction, it must be
observed what is the type of the security that is being considered. Let us separate this
analysis into two cases:
Case 1: Adjustment by % Index
n
P=∑
Fi × (1 + α 0 Ind )ti
t
(1 + αr )tt
i =1
i
where:
P = negotiation value of the security;
Ind = projected value of the index between the concerned dates;
Fi = value of the i-th flow of the security;
r = value of the discount rate of the security for the concerned period.
Case 2: Adjustment by Index + Spread
n
P=∑
i =1
Fi × (1 + C 0 )ti
t
(1 + α )tt
i
where:
P = negotiation value of the security;
Fi = value of the i-th flow of the security;
From these equations, the S factor is determined, which measures the credit risk of
the security. The periodic calculation of alpha is a way of refining the pricing of the
debentures, because, in this manner, the market’s perception of the credit risk is taken into
account.
•
The spreads for the last 5 transactions with the security will be calculated,
provided that the earliest transaction has occurred within no more than 1 year;
•
The highest and the lowest values are excluded, in order to eliminate any outliers
from the sample;
•
The spread of the security will be the average of the remaining three values.
•
If the number of operations it’s less than 5 (five), so it will be considered all
transactions made during a year.
This value will be reviewed on a periodic basis, or every 1 month. In addition, the
method comprehends also some sub-cases, which will be described below:
Illiquidity
In the event that the security has not been traded in the secondary market for over
1 year, then, upon the updating of the spread factor, the alpha at the issue of the debenture
will be adopted (if such alpha is below 100%, then 100% will be adopted).
In the event that the security has not been traded for over 3 years, the curve value
of the security will be adopted, as we believe that the secondary market no longer reflects
the value of such security.
Renegotiation
When a date for renegotiation of the debenture has been set, such date will be
considered as the maturity thereof, because the conditions thereof may be changed
(change of index, rate, flow, etc.). Generally, the issuer allows the repurchase of the
debenture on the renegotiation dates.
It is worth emphasizing that, after the renegotiation date of a debenture, the
counting of transactions to calculate the alpha will be restarted, that is, transactions before
the renegotiation date will be disregarded in the subsequent calculations of the alpha
value.
Early Redemption
In case of early redemption of a debenture, that is, in the event that the issuer
thereof makes the payments before the maturity of the securities, the following procedure
will be observed:
•
After the date on which the early redemption is disclosed, the concerned debenture
will be appraised at the curve, and no longer at the market value.
•
Should the issuer pay any premium by virtue of the early redemption, such
premium must be calculated on a pro rata basis from the date of disclosure of the
early redemption to the date of actual payment.
Note: In cases of misbehavior market, the Citi will define an alternative procedure to
calculate this spread factor to reflect the market conditions, considering the principles of
mark to market methodology.
1.3.1 4.21.6 Special Cases
In this sub-topic, we will present cases that are addressed separately, differently
from the general methods already presented.
FGTR11
The Unitary Price (UP) from the curve, as published by the trustee, will be adopted.
This procedure will be adopted due to the fact that there is no payment schedule
established for this security.
CVRDA6
By virtue of recent trades in the secondary market that point to a very significant
discrepancy in the prices of trades with this security, the UP from the curve as published
by the trustee will be adopted.
IVSC11
By virtue of the current situation of the issuer of the security, and also the lack of
events for this debenture, the following procedure will be adopted:
•
In investment funds, the market value of the securities is zero;
•
In managed portfolios, the price is frozen since December 31, 2002.
FGUI12
According to the procedure adopted by the trustee of this security, the UP thereof
is frozen. Changes in the value will only be made by virtue of the payment of monthly
amortizations, if any.
VLGC11
By virtue of the current situation of the issuer of the security, the value thereof
should remain unchanged. The current value of the security is the same value observed on
January 17, 2002.
SULT13
Due to the fact that payments are overdue on the security since 2004, the UP from
the curve will be adopted, which must consider the following:
•
The principal amount of the security must be adjusted at the index thereof, that is,
Anbid – 6,7% per annum;
•
The overdue amounts corresponding to interest and amortization must be adjusted
at the index thereof until the date on which they should have been paid to the
debenture holders. After said date, such amounts must bear interest of 1% per
month.
LORZ12
By virtue of the current situation of the issuer of the security, the value thereof
should remain unchanged. The current value of the security is R$ 1,594.660734.
CEL Participações (CLPA11 to 92)
Due to the fact that the issuer is in default since 2001, said debentures will be
assigned a null value.
Feniciapar – FPAR11 and FPAR21
Due to the fact that the issuer is in default since 2003, said debentures will be
assigned a null value.
Hopi Hari – PQTM 11, 21, 31 and 41
In view of the constant changes in the schedule of the flows for such securities, and
the lack of a secondary market therefor, the value thereof will be calculated according to
the UP from the curve, as published by the trustee.
CP Cimentos – CPCM12
By virtue of the recent reorganization of the company’s debt and the consequent
reorganization of the payment flows, the market value of this security will be determined
as follows:
•
Eighty percent (80%) of the curve value of the security.
This method and the percentage used must be reviewed no more frequently than
monthly.
4.22
TDA - Título da Dívida Agrária (Agrarian Debt Bond)
The market value of the security on the concerned date is given by:
PU 0 × (1 + C 0 )ti0 × (1 + TR )t0
t
n
MtM t = ∑
t
(1 + r )tt
i =1
i
where:
r = expected TR coupon, obtained from the TR Coupon Curve.
4.23
Nota de Crédito de Exportação (Export Credit Note)
Suppose that the concerned promissory note is indexed by the dollar, and that its
payments are composed of the index plus a spread. Suppose also that there are n interest
payments and m amortizations until the maturity of the security.
The MtM of the Export Credit Note on the concerned date is given by:
n
MtM t = ∑
PU i × (1 + Dólar )t 0 × (1 + C0 )ti0
t
(1 + r )tt
i =1
t
i
m
+∑
Aj × (1 + Dólar )t 0
t
(1 + r )tt
j =1
j
where:
r = expected Dollar coupon, obtained from the Dollar Coupon, No Cash Curve;
Dollar = Dollar specified in the agreement, or Dollar PTAX available from BACEN;
4.24 CRI – Certificado de Recebíveis Imobiliários (Real Estate Receivables
Certificate)
Most of the CRIs have the structure that we call “hybrid”, that is, they have both
the IGP-M risk and the Pre-Fixed Rate risk. This occurs because these CRIs are monetarily
adjusted by the IGP-M on a yearly basis. This peculiarity must be taken into account when
pricing the asset.
Suppose that n interest payments and m amortizations remain until the maturity of
the security. The value of the installments (referring to interest or amortizations) to be paid
before the next indexation of the security must be calculated as follows:
V × (1 + IGPM )t0a × (1 + C )t0v
t
MtM t =
(1 + r )t × (1 + Ct )tt
tv
t
v
where:
V = nominal value of the concerned installment;
ta = date of the last anniversary of the security, which indicates the last indexation;
tv = date of payment of the concerned installment;
C = interest rate that adjusts the concerned installment;
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve;
The value of the installments (referring to interest or amortizations) to be paid after
new indexations must be calculated as follows:
V × (1 + IGPM )t0a × (1 + IGPM )tba (1 + C )t0v
t
MtM t =
t
(1 + r )t × (1 + Ct )tt
tv
t
v
where:
V = nominal value of the concerned installment;
ta = date of the last anniversary of the security, which indicates the last indexation;
tb = last date of anniversary before the payment of the concerned installment;
tv = date of payment of the concerned installment;
C = interest rate that adjusts the concerned installment;
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve;
The forecast IGP-M between the anniversary dates ta and tb must be calculated from
the Pre-Fixed, No Cash Curve and the IGP-M Coupon, No Cash Curve. Said forecast is
given by:
(1 + r )tt
(1 + IGPM ) =
(1 + s )tt
tb
ta
b
a
b
a
where:
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve;
s = expected IGP-M coupon, obtained from the IGP-M Coupon, No Cash Curve;
Finally, the market value of the security is given by the sum of the market values of
each of the installments that comprise it, calculated as shown in this section.
In the event that the CRI does not have the “hybrid” structure, then its pricing
must be considered as a particular case of this method, where the anniversary dates are
replaced with the current dates.
4.25
LH – Letra Hipotecária (Mortgage-Backed Security)
Suppose that the LH is indexed by the IGP-M index, and that n interest payments
and m amortizations remain until the maturity of the security.
The value of the security on the concerned date is given by:
n
MtM t = ∑
i =1
PU i × (1 + IGPM )t0 × (1 + C0 )ti0
t
(1 + r )tt × (1 + Ct )tt
i
t
i
m
+∑
j =1
A j × (1 + IGPM )t0
t
(1 + r )tt × (1 + Ct )tt
j
i
where:
(1+ IGPM )tt
0
= IGP-M accumulated up to the concerned date, without using
projections;
r = expected IGP-M coupon, obtained from the IGP-M Coupon, No Cash Curve.
4.26
LCI - Letra de Crédito Imobiliário (Real Estate Credit Bill)
Real Estate Credit Bills must be marked to market through the same method
proposed to price Mortgage-Backed Securities.
It must be observed that the LCIs may have indexes other than the IGP-M,
provided that they are price indexes. In such event, replace the index and the coupons
used with the respective index. In case of an index the coupon curve of which has not been
contemplated herein, it must be treated separately as a particular case.
4.27 Certificado a Termo de Energia Elétrica (Electric Power Forward
Certificate)
This topic presents the method proposed to price a CTEE - Electric Power Forward
Certificate. There is only one outstanding series of this security: the 9th issue. This issue
pays according to the highest of the following indexes: Bandeirante’s B-3 Tariff, or CDI +
2.0% per annum.
By virtue of the lack of liquidity for this security in the secondary market, and also
the difficulty to estimate a value for the power tariff, it is chosen to mark the security to
market exclusively according to the CDI-linked payments thereunder.
Thus, the market value of this security must be calculated in the same manner as
the market value of a CDB indexed by the CDI index, with payments made on a CDI +
Spread basis.
4.28
CVS
This topic presents the method adopted to price the following privatization
currencies: CVSA970101, CVSB970101, CVSC970101 and CVSD970101. In the marking-tomarket process, some characteristics must be taken into account.
•
Interest Rate: 0.50% per month (series A and C) or 3.12% per annum (series B and
D). Interest up to, but excluding, Jan. 01, 2005, are incorporated into the principal;
•
Indexation: on a monthly basis, on the anniversary date, based on the TR
(Referencial Rate) variation for the previous month, as published by the Central
Bank;
•
Amortization: monthly payments, starting on Jan. 01, 2009 up to the maturity date,
at the fixed rate of 0.4608%.
Incorporation of Interest
To incorporate interest into the principal, the following method must be adopted:
PU t = PU 0 × (1+ C0 )t0
t
where:
PUt = amount adjusted by the incorporation of interest up to date t;
Mark-to-Market
Suppose that n interest payments and m amortizations remain until the maturity of
the security. The value of the security on the concerned date is given by:
n
MtM t = ∑
i =1
PU t ,i × (1 + TR )t0 × (1 + C 0 )ti0
t
(1 + r )tt
t
i
m
+∑
A j × (1 + TR )t0
j =1
t
(1 + r )tt
j
where:
(1+ TR )tt
0
= TR accumulated up to the concerned date;
r = expected TR coupon, obtained from the TR Coupon Curve;
PUt,i = nominal value with the incorporation of interest, not amortized up to the ith interest payment.
4.29
CPR - Cédula do Produto Rural (Rural Product Certificate)
This topic presents the method adopted to price Rural Product Certificates indexed
by the Fattened Steer Price, the financial settlement and price of which are previously
established on the date the transaction is closed.
The market value of the transaction on the concerned date is given by:
MtM t =
(1 + r )
tF
t
P
t
× (1 + Ct )t F
where:
P = settlement price agreed on the date of issue;
r = expected pre-fixed rate, obtained from the Pre-Fixed, No Cash Curve.
4.30
Loan Indexed by the LIBOR Rate
This topic presents the pricing method of a loan agreement indexed by the Libor
rate, the base currency of which is the United States Dollar (USD). The hypothesis is
adopted that the loan is settled always at its curve price, regardless of the moment in
which it occurs.
The value of the loan on date t is given by:
Vt = V0,t × (1 + L + C0 )t0 + Emp
t
where:
V0,t = initial value of the transaction, net of amortizations up to date t;
(1 + L + C0 )tt
0
= variation of the Libor (L) rate compounded with the interest rate of
the transaction (C0), observed between dates t0 and t, calculated on a linear basis, based on
calendar days;
Emp = loan rate, given by:
Emp = 0,5% × V0,t
The value of the Libor rate can be obtained from the Bloomberg system, through
the link:
US000 <“Index” button> <“Go” button>.
4.31
NCE – Nota de Crédito de Exportação (Export Credit Note)
This topic presents the method to price a security with credit risk from its issuer.
The market value of the security on the concerned date is given by:
PU 0 × (1 + C0 )t0F × (1 + Dólar )t0
t
MtM t =
(1 + r )tt × (1 + Ct )tt
F
t
F
where:
r = expected Dollar Coupon, obtained from the Dollar Coupon, No Cash Curve.
In the event that the security has intermediate interest payments and/or
amortizations, each of the fllows must be considered on a separate basis, and the market
value of the security will be given by the sum of the market values of each flow.
From the acquisition price of the security, it is possible to determine the credit
spread at which the security was purchased, thus equalling the acquisition price to the
equation that provides the price of the security, and having the market rate vary. This
must be done as follows:
n
P=∑
i =1
(1 + r )
tii
t
Fi
t
× (1 + Ct )ti
where:
P = acquisition value of the security;
Fi = value of the i-th flow of the security;
r = expected Dollar Coupon, obtained from the Dollar Coupon, No Cash Curve.
4.32
Currency Forward
This topic presents the pricing method of a forward contract between currencies.
The market value of the transaction on the concerned date is given by:
MtM t =
PU 0
PU 0
−
M t × Ct M 0 × Ct
where:
Mt = value of the concerned currency, as compared to the base currency of the
transaction, on date t;
Ct = value of the base currency, in Reais, on date t.
The values of the concerned currency can be obtained from the following sources:
•
Bloomberg;
•
The Brazilian Central Bank.
5
Other Structured Transactions
This topic will focus on the method used for some structured transactions that have
very specific characteristics and cannot be priced from the methods described so far.
5.1
Pre-Fixed Rate Structured Transaction
This transaction consists of the following structures:
•
There are two players: a purchaser and a seller. The purchaser has a debt with the
seller;
•
An investor pays off the debt with the seller and becomes a creditor of the
purchaser;
•
In order to mitigate his credit risk, the investor carries insurance for the debt,
which secures receipt of all debts that the purchaser fails to pay.
Suppose that there are n monthly flows of amortizations and interest payments, on
dates t1 , t 2 ,..., t n . Be r the monthly interest rate of the transaction.
The curve value of the transaction on date t, with t i −1 ≤ t ≤ t i , is given by:
Vt = Vi −1 × (1 + r )ti −1
t
where:
Vt = value of the transaction on date t;
Vi-1 = debit balance of the transaction, calculated on date ti-1, after deducting the
payments made up to such date.
5.2
Libor x Fixed Rate Swap linked to Libor
This topic presents the method used to price a Libor x Fixed Rate swap, with
clauses that determine the fixed rate based on the Libor rate level. The agreement will be
priced in dollars. To translate the amounts into reais, the PTAX sale rate must be used.
5.2.1 Libor Leg
To price this edge, it will be assumed that there are no intermediate flow payments.
The MtM of the Libor edge on the concerned date is given by the following expression:
PU 0 × (1 + Libor0 )t0 × (1 + Libor0 )t L × (1 + Libor )t FL
t
MtM t =
(1 + Treasury )tt
t
t
F
where:
tL = last date on which the Libor rate is known;
(1+ Libor0 )tt
0
= Libor variation observed on the date of the transaction, between
dates t0 and t;
(1+ Libor0 )tt
L
= Libor variation observed on the date of the transaction, between
dates t and tL;
(1+ Libor0 )tt = projected Libor variation, between dates tL and tF;
(1 + Treasury )tt = projected Treasury variation, between dates t and tF.
F
L
F
It must be observed that the LIBOR rate accrues on a calendar-day basis, based on a
360-day year. Treasury also accrues on a calendar-day basis, but based on a 365-day year.
5.2.2 Pre-Fixed Rate Leg
The value of the pre-fixed rate is determined according to the following rule. Given
that:
Libor = Libor rate observed on the date of determination;
L = limit rate as established in the agreement;
C = fixed interest rate of the edge indexed by the Dollar;
T = fixed interest rate of the edge indexed by the Dollar as set forth in the
agreement;
The rule to determine the rate is given by:
•
•
If Libor ≥ L, then C = T;
If Libor < L, then C = T + (L - Libor).
Thus, it can be understood that this edge of the transaction is the composition of a
pre-fixed-rate transaction, with the purchase of a series of Libor puts, with strike at the
rate L. This transaction will be priced in two stages.
Pre-Fixed Portion
The MtM of the pre-fixed portion will be calculated as follows:
V0 × (1 + T )t0F
t
PPt =
(1 + Treasury )tt
F
where:
(1 + Treasury )tt
F
= projected Treasury variation, between dates t and tF.
Buy Put (long)
The purchase of a series of puts is known as floor. The floor value is given by the
sum of the value of each of the puts that comprise it. The value of such puts may be
obtained through the Black model, as adjusted to reflect the different start dates thereof.
For each of the puts that comprise the floor, the price is given by:
d
360 [K × N (− d 2 ) − Ft × N (− d1 )]
Pt =
d 
e rtc

1 + Ft ×

360 

V×
where:
2
 F  σ 

 × tc
ln  + 
K  2 
d1 =
σ × tc
d 2 = d1 − σ × tc
V = non-amortized value for each installment of the swap flows;
d = forward rate period;
tc = time, in years, to the maturity of the option (based on 365 calendar days);
R = Treasury projection, in % per annum, as of the date of maturity of the option;
r = ln(1+R);
Ft = forward Libor rate, that is, projected rate for the start date of the option up to
its maturity;
K = put strike value, given by L;
Y = implied volatility of the option, based on 365 days.
Finally, the value of this swap edge is given by:
Vt = PPt + Pt
5.2.3 Data
The necessary data can be obtained from the following sources:
•
Libor: at the Bloomberg system, by typing:
US000 <botão “Index”> <botão “Go”>.
•
Projected Libor: at the Bloomberg system, by typing:
IRSB <botão “Go”> 18 <botão “Go”>.
•
Treasury Curve: at the Bloomberg system, by typing:
USGG <botão “Index”>.
•
Libor Volatility: at the Bloomberg system, through the following path
IRSB <botão “Go”> 18 <botão “Go”>;
Due to the proximity of the data obtained to the events to be priced, the
interpolation of the volatilities obtained is dispensable. The volatility used will be the one
referring to the closest vertex to the maturity date of the flow.
6
Other procedures and methods
This topic will address some procedures and methods not directly used in the
marking-to-market process, but which are part of the pricing processes adopted by
Citibank.
6.1
Procedures for dates without data disclosure
In case of dates on which the data used (rates, prices, etc.) is not available (such as
on the days preceding the turn of the year), the procedure to be used is the following:
•
The projected rates must be kept unchanged, that is, it will be assumed that they
have not changed;
•
If the UP is used to determine the rate (e.g.: LTN or DI), it is necessary to
recalculate the UP in order to keep the same expectation for the rate given by such
price;
•
On such date, opening and closing quotes must coincide.
6.2
Valuation in the Curve
This topic presents the methods used by Citibank to value the securities according
to their curves (held-to-maturity definition).
Calculation at the TIR
An alternative to calculate the value in the curve of an asset is to use as adjustment
value the Internal Return Rate (TIR) of the security, obtained from the price observed on
the date of acquisition.
The internal return rate of the security can be obtained as follows:
n
P=∑
i =1
Fi
(1 + r )ttic
where:
P = acquisition price of the security;
Fi = value of the i-th flow of the security;
r = TIR of the security;
tc = security purchase date.
The value of r can be obtained from some convergence method, such as the Solver
function of the Microsoft Excel® software.
From this return rate, the value in the curve of the security, on any date t, is given
by:
Pt = P × (1 + Ind )tc × (1 + r )tc
t
t
Linear Calculation
This method is used by Citibank’s Drive system and carries out the linear
appropriation of the discount obtained on the date of purchase of the security. The
calculation will be described below. Suppose that the security is acquired on date tc. Given
that:
VNAt = adjusted nominal value of the security on date t;
P = acquisition price of the security;
Dt = discount of the security on date t, given by:
Dt = P − VNAt
The discount of the security on date t + 1, and on any date subsequent to the
concerned date (t + j), is given by:
 D
 ∆(t + j , t F )
Dt + j =  t ⋅ VNAt + j  ⋅
 VNAt
 ∆(t , t F )
where:
∆ (t, t F ) = number of business days between date t and date tF.
From the discount, it is possible to obtain the price on any date t + j. Such value is given
by:
Vt + j = VNAt + j + Dt + j
6.3
SELIC Rate - Updating Procedure
To update the SELIC rate, the following rules must be observed:
•
•
•
The SELIC rate used for the current day will be the average rate estimated by
ANDIMA, available at: www.andima.com.br;
On the day following such updating, the rate estimated by ANDIMA must be
replaced with the rate published by BACEN, which will already be available. The
purpose of this procedure is to prevent minor inaccuracies in ANDIMA’s estimates
from interfering, in the long term, with the marking of the assets to market;
In exceptional cases, such as high volatility in the fixed income market or no
estimate from ANDIMA, it is possible to use directly the rate published by
BACEN.