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Price bonds with embedded options by Black-Derman-Toy interest-rate tree

`[`

calculates price for bonds with embedded options
from a Black-Derman-Toy interest-rate tree and
returns exercise probabilities in
`Price`

,`PriceTree`

]
= optembndbybdt(`BDTTree`

,`CouponRate`

,`Settle`

,`Maturity`

,`OptSpec`

,`Strike`

,`ExerciseDates`

)`PriceTree`

.

`optembndbybdt`

computes prices of
vanilla bonds with embedded options, stepped
coupon bonds with embedded options, amortizing
bonds with embedded options, and sinking fund
bonds with call embedded option. For more
information, see More About.

`[`

adds optional name-value pair arguments.`Price`

,`PriceTree`

]
= optembndbybdt(___,`Name,Value`

)

Create a `BDTTree`

with the following data:

ZeroRates = [ 0.035;0.04;0.045]; Compounding = 1; StartDates = ['jan-1-2007';'jan-1-2008';'jan-1-2009']; EndDates = ['jan-1-2008';'jan-1-2009';'jan-1-2010']; ValuationDate = 'jan-1-2007';

Create a `RateSpec`

.

RateSpec = intenvset('Rates', ZeroRates, 'StartDates', ValuationDate, 'EndDates', ... EndDates, 'Compounding', Compounding, 'ValuationDate', ValuationDate)

`RateSpec = `*struct with fields:*
FinObj: 'RateSpec'
Compounding: 1
Disc: [3x1 double]
Rates: [3x1 double]
EndTimes: [3x1 double]
StartTimes: [3x1 double]
EndDates: [3x1 double]
StartDates: 733043
ValuationDate: 733043
Basis: 0
EndMonthRule: 1

Create a `VolSpec`

.

Volatility = 0.10 * ones (3,1); VolSpec = bdtvolspec(ValuationDate, EndDates, Volatility)

`VolSpec = `*struct with fields:*
FinObj: 'BDTVolSpec'
ValuationDate: 733043
VolDates: [3x1 double]
VolCurve: [3x1 double]
VolInterpMethod: 'linear'

Create a `TimeSpec`

.

TimeSpec = bdttimespec(ValuationDate, EndDates, Compounding);

Build the `BDTTree`

.

BDTTree = bdttree(VolSpec, RateSpec, TimeSpec)

`BDTTree = `*struct with fields:*
FinObj: 'BDTFwdTree'
VolSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 1 2]
dObs: [733043 733408 733774]
TFwd: {[3x1 double] [2x1 double] [2]}
CFlowT: {[3x1 double] [2x1 double] [3]}
FwdTree: {[1.0350] [1.0406 1.0495] [1.0447 1.0546 1.0667]}

To compute the price of an American callable bond that pays a 5.25% annual coupon, matures in Jan-1-2010, and is callable on Jan-1-2008 and 01-Jan-2010.

BondSettlement = 'jan-1-2007'; BondMaturity = 'jan-1-2010'; CouponRate = 0.0525; Period = 1; OptSpec = 'call'; Strike = [100]; ExerciseDates = {'jan-1-2008' '01-Jan-2010'}; AmericanOpt = 1; PriceCallBond = optembndbybdt(BDTTree, CouponRate, BondSettlement, BondMaturity,... OptSpec, Strike, ExerciseDates, 'Period', 1,'AmericanOp', 1)

PriceCallBond = 101.4750

Create a `BDTTree`

with the following data:

ZeroRates = [ 0.025;0.03;0.045]; Compounding = 1; StartDates = ['jan-1-2007';'jan-1-2008';'jan-1-2009']; EndDates = ['jan-1-2008';'jan-1-2009';'jan-1-2010']; ValuationDate = 'jan-1-2007';

Create a `RateSpec`

.

RateSpec = intenvset('Rates', ZeroRates, 'StartDates', ValuationDate, 'EndDates', ... EndDates, 'Compounding', Compounding, 'ValuationDate', ValuationDate);

Build the BDT tree with the following data.

Volatility = 0.10 * ones (3,1); VolSpec = bdtvolspec(ValuationDate, EndDates, Volatility)

`VolSpec = `*struct with fields:*
FinObj: 'BDTVolSpec'
ValuationDate: 733043
VolDates: [3x1 double]
VolCurve: [3x1 double]
VolInterpMethod: 'linear'

TimeSpec = bdttimespec(ValuationDate, EndDates, Compounding); BDTTree = bdttree(VolSpec, RateSpec, TimeSpec)

`BDTTree = `*struct with fields:*
FinObj: 'BDTFwdTree'
VolSpec: [1x1 struct]
TimeSpec: [1x1 struct]
RateSpec: [1x1 struct]
tObs: [0 1 2]
dObs: [733043 733408 733774]
TFwd: {[3x1 double] [2x1 double] [2]}
CFlowT: {[3x1 double] [2x1 double] [3]}
FwdTree: {[1.0250] [1.0315 1.0385] [1.0614 1.0750 1.0917]}

Define the callable bond instruments.

Settle = '01-Jan-2007'; Maturity = {'01-Jan-2008';'01-Jan-2010'}; CouponRate = {{'01-Jan-2008' .0425;'01-Jan-2010' .0450}}; OptSpec='call'; Strike= [86;90]; ExerciseDates= {'01-Jan-2008';'01-Jan-2010'};

Price the instruments.

[Price, PriceTree]= optembndbybdt(BDTTree, CouponRate, Settle, Maturity, OptSpec, Strike,... ExerciseDates, 'Period', 1,'AmericanOp', 1)

`Price = `*2×1*
86
90

`PriceTree = `*struct with fields:*
FinObj: 'BDTPriceTree'
tObs: [0 1 2 3]
PTree: {1x4 cell}
ProbTree: {[1] [0.5000 0.5000] [0.2500 0.5000 0.2500] [1x3 double]}
ExTree: {1x4 cell}
ExProbTree: {1x4 cell}
ExProbsByTreeLevel: [2x4 double]

Examine the output `PriceTree.ExTree`

. `PriceTree.ExTree`

contains the exercise indicator arrays. Each element of the cell array is an array containing `1`

's where an option is exercised and `0`

's where it is not.

PriceTree.ExTree{4}

`ans = `*2×3*
0 0 0
1 1 1

There is no exercise for instrument 1 and instrument 2 is exercised at all nodes.

PriceTree.ExTree{3}

`ans = `*2×3*
0 0 0
0 0 0

There is no exercise for instrument 1 or instrument 2.

PriceTree.ExTree{2}

`ans = `*2×2*
1 1
1 0

There is exercise for instrument 1 at all nodes and instrument 2 is exercised at some nodes.

Next view the probability of reaching each node from the root node using `PriceTree.ProbTree`

.

PriceTree.ProbTree{2}

`ans = `*1×2*
0.5000 0.5000

PriceTree.ProbTree{3}

`ans = `*1×3*
0.2500 0.5000 0.2500

PriceTree.ProbTree{4}

`ans = `*1×3*
0.2500 0.5000 0.2500

Then view the exercise probabilities using `PriceTree.ExProbTree`

. `PriceTree.ExProbTree`

contains the exercise probabilities. Each element in the cell array is an array containing 0's where there is no exercise, or the probability of reaching that node where exercise happens.

PriceTree.ExProbTree{4}

`ans = `*2×3*
0 0 0
0.2500 0.5000 0.2500

PriceTree.ExProbTree{3}

`ans = `*2×3*
0 0 0
0 0 0

PriceTree.ExProbTree{2}

`ans = `*2×2*
0.5000 0.5000
0.5000 0

View the exercise probabilities at each tree level using `PriceTree.ExProbsByTreeLevel`

. `PriceTree.ExProbsByTreeLevel`

is an array with each row holding the exercise probability for a given option at each tree observation time.

PriceTree.ExProbsByTreeLevel

`ans = `*2×4*
1.0000 1.0000 0 0
1.0000 0.5000 0 1.0000

Price the following single stepped callable bonds using the following data: The data for the interest rate term structure is as follows:

Rates = [0.035; 0.042147; 0.047345; 0.052707]; ValuationDate = 'Jan-1-2010'; StartDates = ValuationDate; EndDates = {'Jan-1-2011'; 'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'}; Compounding = 1; % Create RateSpec RS = intenvset('ValuationDate', ValuationDate, 'StartDates', StartDates,... 'EndDates', EndDates,'Rates', Rates, 'Compounding', Compounding); % Instrument Settle = '01-Jan-2010'; Maturity = {'01-Jan-2013';'01-Jan-2014'}; CouponRate = {{'01-Jan-2012' .0425;'01-Jan-2014' .0750}}; OptSpec='call'; Strike=100; ExerciseDates='01-Jan-2012'; %Callable in two years % Build the tree % Assume the volatility to be 10% Sigma = 0.1; BDTTimeSpec = bdttimespec(ValuationDate, EndDates, Compounding); BDTVolSpec = bdtvolspec(ValuationDate, EndDates, Sigma*ones(1, length(EndDates))'); BDTT = bdttree(BDTVolSpec, RS, BDTTimeSpec); % The first row corresponds to the price of the callable bond with maturity % of three years. The second row corresponds to the price of the callable bond % with maturity of four years. PBDT= optembndbybdt(BDTT, CouponRate, Settle, Maturity ,OptSpec, Strike,... ExerciseDates, 'Period', 1)

`PBDT = `*2×1*
100.0945
100.0297

A corporation issues a three year bond with a sinking fund obligation requiring the company to sink 1/3 of face value after the first year and 1/3 after the second year. The company has the option to buy the bonds in the market or call them at $98. The following data describes the details needed for pricing the sinking fund bond:

Rates = [0.1;0.1;0.1;0.1]; ValuationDate = 'Jan-1-2011'; StartDates = ValuationDate; EndDates = {'Jan-1-2012'; 'Jan-1-2013'; 'Jan-1-2014'; 'Jan-1-2015'}; Compounding = 1; % Create RateSpec RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates',... StartDates, 'EndDates', EndDates, 'Rates', Rates, 'Compounding', Compounding); % Build the BDT tree % Assume the volatility to be 10% Sigma = 0.1; BDTTimeSpec = bdttimespec(ValuationDate, EndDates); BDTVolSpec = bdtvolspec(ValuationDate, EndDates, Sigma*ones(1, length(EndDates))'); BDTT = bdttree(BDTVolSpec, RateSpec, BDTTimeSpec); % Instrument % The bond has a coupon rate of 9%, a period of one year and matures in % 1-Jan-2014. Face decreases 1/3 after the first year and 1/3 after the % second year. CouponRate = 0.09; Settle = 'Jan-1-2011'; Maturity = 'Jan-1-2014'; Period = 1; Face = { ... {'Jan-1-2012' 100; ... 'Jan-1-2013' 66.6666; ... 'Jan-1-2014' 33.3333}; }; % Option provision OptSpec = 'call'; Strike = [98 98]; ExerciseDates ={'Jan-1-2012', 'Jan-1-2013'}; % Price of non-sinking fund bond. PNSF = bondbybdt(BDTT, CouponRate, Settle, Maturity, Period)

PNSF = 97.5131

Price of the bond with the option sinking provision.

PriceSF = optembndbybdt(BDTT, CouponRate, Settle, Maturity,... OptSpec, Strike, ExerciseDates,'Period', Period, 'Face', Face)

PriceSF = 96.8364

This example shows how to price an amortizing callable bond and a vanilla callable bond using a BDT lattice model.

Create a `RateSpec`

.

Rates = [0.025;0.028;0.030;0.031]; ValuationDate = 'Jan-1-2018'; StartDates = ValuationDate; EndDates = {'Jan-1-2019'; 'Jan-1-2020'; 'Jan-1-2021'; 'Jan-1-2022'}; Compounding = 1; RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates',... StartDates, 'EndDates', EndDates, 'Rates', Rates, 'Compounding', Compounding);

Build a BDT tree and assume a volatility of 5%.

Sigma = 0.05; BDTTimeSpec = bdttimespec(ValuationDate, EndDates); BDTVolSpec = bdtvolspec(ValuationDate, EndDates, Sigma*ones(1, length(EndDates))'); BDTT = bdttree(BDTVolSpec, RateSpec, BDTTimeSpec);

Define the callable bond.

CouponRate = 0.05; Settle = 'Jan-1-2018'; Maturity = 'Jan-1-2021'; Period = 1; Face = { {'Jan-1-2019' 100; 'Jan-1-2020' 70; ... 'Jan-1-2021' 50}; }; OptSpec = 'call'; Strike = [97 95 93]; ExerciseDates ={'Jan-1-2019' 'Jan-1-2020' 'Jan-1-2021'};

Price a callable amortizing bond using the BDT tree.

BondType = 'amortizing'; Pcallbonds = optembndbybdt(BDTT, CouponRate, Settle, Maturity, OptSpec, Strike, ExerciseDates, 'Period', Period,'Face',Face,'BondType', BondType)

Pcallbonds = 99.5122

`BDTTree`

— Interest-rate tree structurestructure

Interest-rate tree structure, specified by using `bdttree`

.

**Data Types: **`struct`

`CouponRate`

— Bond coupon rate positive decimal value

Bond coupon rate, specified as an `NINST`

-by-`1`

decimal
annual rate or `NINST`

-by-`1`

cell
array, where each element is a `NumDates`

-by-`2`

cell
array. The first column of the `NumDates`

-by-`2`

cell
array is dates and the second column is associated rates. The date
indicates the last day that the coupon rate is valid.

**Data Types: **`double`

| `cell`

`Settle`

— Settlement dateserial date number | date character vector

Settlement date for the bond option, specified as a `NINST`

-by-`1`

vector
of serial date numbers or date character vectors.

**Note**

The `Settle`

date for every bond is set to
the `ValuationDate`

of the BDT tree. The bond argument `Settle`

is
ignored.

**Data Types: **`double`

| `char`

`Maturity`

— Maturity dateserial date number | date character vector

Maturity date, specified as an `NINST`

-by-`1`

vector
of serial date numbers or date character vectors.

**Data Types: **`double`

| `char`

`OptSpec`

— Definition of option character vector with value

`'call'`

or `'put'`

| cell array of character vectors with values `'call'`

or `'put'`

Definition of option, specified as a `NINST`

-by-`1`

cell
array of character vectors.

**Data Types: **`char`

`Strike`

— Option strike price valuesnonnegative integer

Option strike price value, specified as a `NINST`

-by-`1`

or `NINST`

-by-`NSTRIKES`

depending
on the type of option:

European option —

`NINST`

-by-`1`

vector of strike price values.Bermuda option —

`NINST`

by number of strikes (`NSTRIKES`

) matrix of strike price values. Each row is the schedule for one option. If an option has fewer than`NSTRIKES`

exercise opportunities, the end of the row is padded with`NaN`

s.American option —

`NINST`

-by-`1`

vector of strike price values for each option.

**Data Types: **`double`

`ExerciseDates`

— Option exercise datesserial date number | date character vector

Option exercise dates, specified as a `NINST`

-by-`1`

, `NINST`

-by-`2`

,
or `NINST`

-by-`NSTRIKES`

using serial
date numbers or date character vectors, depending on the type of option:

For a European option, use a

`NINST`

-by-`1`

vector of dates. For a European option, there is only one`ExerciseDates`

on the option expiry date.For a Bermuda option, use a

`NINST`

-by-`NSTRIKES`

vector of dates.For an American option, use a

`NINST`

-by-`2`

vector of exercise date boundaries. The option can be exercised on any date between or including the pair of dates on that row. If only one non-`NaN`

date is listed, or if`ExerciseDates`

is a`NINST`

-by-`1`

vector, the option can be exercised between`ValuationDate`

of the stock tree and the single listed`ExerciseDates`

.

**Data Types: **`double`

| `char`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`Price = optembndbybdt(BDTTree,CouponRate,Settle,Maturity,OptSpec,Strike,ExerciseDates,'Period',1,'AmericanOp',1)`

`'AmericanOpt'`

— Option type`0`

European/Bermuda (default) | integer with values `0`

or `1`

Option type, specified as the comma-separated pair consisting of
`'AmericanOpt'`

and
`NINST`

-by-`1`

positive integer flags with values:

`0`

— European/Bermuda`1`

— American

**Data Types: **`double`

`'Period'`

— Coupons per year`2`

per year (default) | vectorCoupons per year, specified as the comma-separated pair consisting of
`'Period'`

and a
`NINST`

-by-`1`

vector.

**Data Types: **`double`

`'Basis'`

— Day-count basis`0`

(actual/actual) (default) | integer from `0`

to `13`

Day-count basis, specified as the comma-separated pair consisting of
`'Basis'`

and a
`NINST`

-by-`1`

vector of integers.

0 = actual/actual

1 = 30/360 (SIA)

2 = actual/360

3 = actual/365

4 = 30/360 (PSA)

5 = 30/360 (ISDA)

6 = 30/360 (European)

7 = actual/365 (Japanese)

8 = actual/actual (ICMA)

9 = actual/360 (ICMA)

10 = actual/365 (ICMA)

11 = 30/360E (ICMA)

12 = actual/365 (ISDA)

13 = BUS/252

For more information, see Basis.

**Data Types: **`double`

`'EndMonthRule'`

— End-of-month rule flag`1`

(in effect) (default) | nonnegative integer with values `0`

or `1`

End-of-month rule flag, specified as the comma-separated pair consisting of
`'EndMonthRule'`

and a
nonnegative integer using a
`NINST`

-by-`1`

vector. This rule applies only when
`Maturity`

is an end-of-month
date for a month having 30 or fewer days.

`0`

= Ignore rule, meaning that a bond coupon payment date is always the same numerical day of the month.`1`

= Set rule on, meaning that a bond coupon payment date is always the last actual day of the month.

**Data Types: **`double`

`'IssueDate'`

— Bond issue dateserial date number | date character vector

Bond issue date, specified as the comma-separated pair consisting of
`'IssueDate'`

and a
`NINST`

-by-`1`

vector using serial date numbers or date character
vectors.

**Data Types: **`double`

| `char`

`'FirstCouponDate'`

— Irregular first coupon dateserial date number | date character vector

Irregular first coupon date, specified as the comma-separated pair consisting of
`'FirstCouponDate'`

and a
`NINST`

-by-`1`

vector using serial date numbers date or date
character vectors.

When `FirstCouponDate`

and `LastCouponDate`

are
both specified, `FirstCouponDate`

takes precedence
in determining the coupon payment structure. If you do not specify
a `FirstCouponDate`

, the cash flow payment dates
are determined from other inputs.

**Data Types: **`double`

| `char`

`'LastCouponDate'`

— Irregular last coupon dateserial date number | date character vector

Irregular last coupon date, specified as the comma-separated pair consisting of
`'LastCouponDate'`

and a
`NINST`

-by-`1`

vector using serial date numbers or date character
vectors.

In the absence of a specified `FirstCouponDate`

,
a specified `LastCouponDate`

determines the coupon
structure of the bond. The coupon structure of a bond is truncated
at the `LastCouponDate`

, regardless of where it falls,
and is followed only by the bond's maturity cash flow date. If you
do not specify a `LastCouponDate`

, the cash flow
payment dates are determined from other inputs.

**Data Types: **`char`

| `double`

`'StartDate'`

— Forward starting date of paymentsserial date number | date character vector

Forward starting date of payments (the date from which a bond cash flow is considered),
specified as the comma-separated pair consisting
of `'StartDate'`

and a
`NINST`

-by-`1`

vector using serial date numbers or date character
vectors.

If you do not specify `StartDate`

, the effective
start date is the `Settle`

date.

**Data Types: **`char`

| `double`

`'Face'`

— Face value`100`

(default) | `NINST`

-by-`1`

vector | `NINST`

-by-`1`

cell arrayFace or par value, specified as the
comma-separated pair consisting of
`'Face'`

and a
`NINST`

-by-`1`

vector or a
`NINST`

-by-`1`

cell array where each element is a
`NumDates`

-by-`2`

cell array where the first column is dates and the
second column is associated face value. The date
indicates the last day that the face value is valid.

**Note**

Instruments without a
`Face`

schedule are treated as
either vanilla bonds or stepped coupon bonds with
embedded options.

**Data Types: **`double`

`'BondType'`

— Type of underlying bond`'vanilla'`

for scalar `Face`

values,
`'callablesinking'`

for scheduled
`Face`

values (default) | cell array of character vectors with values
`'vanilla'`

,`'amortizing'`

,
or `'callablesinking'`

| string array with values
`"vanilla"`

,
`"amortizing"`

, or
`"callablesinking"`

Type of underlying bond, specified as the
comma-separated pair consisting of
`'BondType'`

and a
`NINST`

-by-`1`

cell array of character vectors or string array
specifying if the underlying is a vanilla bond, an
amortizing bond, or a callable sinking fund bond.
The supported types are:

`'vanilla`

' is a standard callable or puttable bond with a scalar`Face`

value and a single coupon or stepped coupons.`'callablesinking'`

is a bond with a schedule of`Face`

values and a sinking fund call provision with a single or stepped coupons.`'amortizing'`

is an amortizing callable or puttable bond with a schedule of`Face`

values with single or stepped coupons.

**Data Types: **`char`

| `string`

`'Options'`

— Derivatives pricing optionsstructure

Derivatives pricing options, specified as the comma-separated pair consisting of
`'Options'`

and a structure that
is created with `derivset`

.

**Data Types: **`struct`

`Price`

— Expected prices of embedded option at time `0`

matrix

Expected price of the embedded option at time `0`

,
returned as a `NINST`

-by-`1`

matrix.

`PriceTree`

— Structure containing trees of vectors of instrument prices and exercise probabilities for each
nodestructure

Structure containing trees of vectors of instrument prices, a vector of observation times for each node, and exercise probabilities. Values are:

`PriceTree.PTree`

contains the clean prices.`PriceTree.tObs`

contains the observation times.`PriceTree.ExTree`

contains the exercise indicator arrays. Each element of the cell array is an array containing`1`

's where an option is exercised and`0`

's where it isn't.`PriceTree.ProbTree`

contains the probability of reaching each node from root node.`PriceTree.ExProbTree`

contains the exercise probabilities. Each element in the cell array is an array containing`0`

's where there is no exercise, or the probability of reaching that node where exercise happens.`PriceTree.ExProbsByTreeLevel`

is an array with each row holding the exercise probability for a given option at each tree observation time.

A vanilla coupon bond is a security representing an obligation to repay a borrowed amount at a designated time and to make periodic interest payments until that time.

The issuer of a bond makes the periodic interest payments until the bond matures. At maturity, the issuer pays to the holder of the bond the principal amount owed (face value) and the last interest payment. A vanilla bond with an embedded option is where an option contract has an underlying asset of a vanilla bond.

A step-up and step-down bond is a debt security with a predetermined coupon structure over time.

With these instruments, coupons increase (step up) or decrease (step down) at specific times during the life of the bond. Stepped coupon bonds can have options features (call and puts).

A sinking fund bond is a coupon bond with a sinking fund provision.

This provision obligates the issuer to amortize portions of the principal prior to maturity, affecting bond prices since the time of the principal repayment changes. This means that investors receive the coupon and a portion of the principal paid back over time. These types of bonds reduce credit risk, since it lowers the probability of investors not receiving their principal payment at maturity.

The bond may have a sinking fund call option provision allowing the issuer to retire the sinking fund obligation either by purchasing the bonds to be redeemed from the market or by calling the bond via a sinking fund call, whichever is cheaper. If interest rates are high, then the issuer buys back the requirement amount of bonds from the market since bonds are cheap, but if interest rates are low (bond prices are high), then most likely the issuer is buying the bonds at the call price. Unlike a call feature, however, if a bond has a sinking fund call option provision, it is an obligation, not an option, for the issuer to buy back the increments of the issue as stated. Because of this, a sinking fund bond trades at a lower price than a non-sinking fund bond.

Amortizing callable or puttable bonds work
under a scheduled `Face`

.

An amortizing callable bond gives the issuer the right to call
back the bond, but instead of paying the
`Face`

amount at maturity, it
repays part of the principal along with the coupon payments.
An amortizing puttable bond, repays part of the principal
along with the coupon payments and gives the bondholder the
right to sell the bond back to the issuer.

`bdtprice`

| `bdttree`

| `cfamounts`

| `instoptembnd`

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