<?xml version="1.0" encoding="us-ascii"?><InstanceReport xmlns:xsd="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"><Version>2.4.0.8</Version><ReportLongName>2408404 - Disclosure - Fair Value Measurements - Reconciliation of the beginning and ending balances of the Level 3 assets and liabilities (Detail)</ReportLongName><DisplayLabelColumn>true</DisplayLabelColumn><ShowElementNames>false</ShowElementNames><RoundingOption>In Thousands, unless otherwise specified</RoundingOption><HasEmbeddedReports>false</HasEmbeddedReports><Columns><Column FlagID="0"><Id>1</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><CurrencyCode>USD</CurrencyCode><FootnoteIndexer /><hasSegments>false</hasSegments><hasScenarios>false</hasScenarios><MCU><KeyName /><CurrencySymbol>$</CurrencySymbol><contextRef><ContextID>D2013Q2YTD</ContextID><EntitySchema>http://www.sec.gov/CIK</EntitySchema><EntityValue>0000709804</EntityValue><PeriodDisplayName /><PeriodType>duration</PeriodType><PeriodStartDate>2013-01-01T00:00:00</PeriodStartDate><PeriodEndDate>2013-06-30T00:00:00</PeriodEndDate><Segments /><Scenarios /></contextRef><UPS><UnitProperty><UnitID>usd</UnitID><UnitType>Standard</UnitType><StandardMeasure><MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema><MeasureValue>USD</MeasureValue><MeasureNamespace>iso4217</MeasureNamespace><MeasureDescription /><MeasureType /></StandardMeasure><Scale>0</Scale></UnitProperty></UPS><CurrencyCode>USD</CurrencyCode><OriginalCurrencyCode>USD</OriginalCurrencyCode></MCU><CurrencySymbol>$</CurrencySymbol><Labels><Label Key="CalendarSupplement" Id="0" Label="6 Months Ended" /><Label Key="Calendar" Id="1" Label="Jun. 30, 2013" /></Labels></Column></Columns><Rows><Row FlagID="0"><Id>1</Id><IsAbstractGroupTitle>true</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>2</Level><ElementName>us-gaap_FairValueAssetsMeasuredOnRecurringBasisUnobservableInputReconciliationCalculationRollForward</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>na</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="" UnitID=""><Id>1</Id><IsNumeric>false</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>0</NumericAmount><RoundedNumericAmount>0</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:stringItemType</ElementDataType><SimpleDataType>string</SimpleDataType><IsTotalLabel>false</IsTotalLabel><UnitID>0</UnitID><Label>Fair Value, Assets Measured on Recurring Basis, Unobservable Input Reconciliation, Calculation [Roll Forward]</Label></Row><Row FlagID="0"><Id>2</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>3</Level><ElementName>us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationsRecurringBasisLiabilityValue</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>credit</BalanceType><PeriodType>instant</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>true</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>periodStartLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="I2012Q4" UnitID="usd"><Id>1</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>2804000</NumericAmount><RoundedNumericAmount>2804</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode>USD</CurrencyCode><CurrencySymbol>$</CurrencySymbol><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>true</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:monetaryItemType</ElementDataType><SimpleDataType>monetary</SimpleDataType><ElementDefenition>This element represents a liability measured at fair value using significant unobservable inputs (Level 3) which is required for reconciliation purposes of beginning and ending balances.</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 820

 -SubTopic 10

 -Section 50

 -Paragraph 2

 -Subparagraph (c)

 -URI http://asc.fasb.org/extlink&amp;oid=7578670&amp;loc=d3e19207-110258



</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>2</UnitID><Label>Balance, Beginning</Label></Row><Row FlagID="0"><Id>3</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>3</Level><ElementName>us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetPurchases</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>debit</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>true</IsReverseSign><PreferredLabelRole>negatedLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="D2013Q2YTD" UnitID="usd"><Id>1</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>39332000</NumericAmount><RoundedNumericAmount>39332</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:monetaryItemType</ElementDataType><SimpleDataType>monetary</SimpleDataType><ElementDefenition>Purchases that have taken place during the period in relation to assets measured at fair value and categorized within Level 3 of the fair value hierarchy.</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 820

 -SubTopic 10

 -Section 50

 -Paragraph 2

 -Subparagraph (c)(2)

 -URI http://asc.fasb.org/extlink&amp;oid=7578670&amp;loc=d3e19207-110258



</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>2</UnitID><Label>Purchases</Label></Row><Row FlagID="0"><Id>4</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>3</Level><ElementName>us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetSales</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>credit</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="D2013Q2YTD" UnitID="usd"><Id>1</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>-22926000</NumericAmount><RoundedNumericAmount>-22926</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:monetaryItemType</ElementDataType><SimpleDataType>monetary</SimpleDataType><ElementDefenition>Sales that have taken place during the period in relation to assets measured at fair value and categorized within Level 3 of the fair value hierarchy.</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 820

 -SubTopic 10

 -Section 50

 -Paragraph 2

 -Subparagraph (c)(2)

 -URI http://asc.fasb.org/extlink&amp;oid=7578670&amp;loc=d3e19207-110258



</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>2</UnitID><Label>Sales</Label></Row><Row FlagID="0"><Id>5</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>3</Level><ElementName>us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetGainLossIncludedInEarnings1</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>credit</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="D2013Q2YTD" UnitID="usd"><Id>1</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>1556000</NumericAmount><RoundedNumericAmount>1556</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:monetaryItemType</ElementDataType><SimpleDataType>monetary</SimpleDataType><ElementDefenition>Amount of gain (loss) recognized in earnings, arising from assets measured at fair value on a recurring basis using unobservable inputs (level 3).</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 820

 -SubTopic 10

 -Section 50

 -Paragraph 2

 -Subparagraph (c)(1)

 -URI http://asc.fasb.org/extlink&amp;oid=7578670&amp;loc=d3e19207-110258



</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>2</UnitID><Label>Realized gain on sale</Label></Row><Row FlagID="0"><Id>6</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>3</Level><ElementName>us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationRecurringBasisAssetPeriodIncreaseDecrease</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>na</BalanceType><PeriodType>duration</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>false</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>terseLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="D2013Q2YTD" UnitID="usd"><Id>1</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>186000</NumericAmount><RoundedNumericAmount>186</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode /><CurrencySymbol /><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>false</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:monetaryItemType</ElementDataType><SimpleDataType>monetary</SimpleDataType><ElementDefenition>Amount of increase (decrease) of assets measured at fair value on a recurring basis using unobservable inputs (level 3). Such increase (decrease) may be comprised of changes in fair value (gains or losses) whether realized or unrealized that have been included in earnings, changed net asset value, or other comprehensive income (a separate component of shareholders' equity); purchases, sales, issues, or settlements of assets; and net transfers in to or out of this category within the fair value hierarchy.</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 820

 -SubTopic 10

 -Section 50

 -Paragraph 2

 -Subparagraph (c)

 -URI http://asc.fasb.org/extlink&amp;oid=7578670&amp;loc=d3e19207-110258



Reference 2: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Statement of Financial Accounting Standard (FAS)

 -Number 157

 -Paragraph 32

 -Subparagraph c(1), c(2), c(3)

 -LegacyDoc This reference is SUPERSEDED by the Accounting Standards Codification effective for interim and annual periods ending after September 15, 2009.  This reference is included to help users transition from the previous accounting hierarchy and will be removed from future versions of this taxonomy.



</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>2</UnitID><Label>Change in fair value</Label></Row><Row FlagID="0"><Id>7</Id><IsAbstractGroupTitle>false</IsAbstractGroupTitle><LabelSeparator>

</LabelSeparator><Level>3</Level><ElementName>us-gaap_FairValueMeasurementWithUnobservableInputsReconciliationsRecurringBasisLiabilityValue</ElementName><ElementPrefix>us-gaap_</ElementPrefix><IsBaseElement>true</IsBaseElement><BalanceType>credit</BalanceType><PeriodType>instant</PeriodType><IsReportTitle>false</IsReportTitle><IsSegmentTitle>false</IsSegmentTitle><IsCalendarTitle>false</IsCalendarTitle><IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow><IsEquityAdjustmentRow>false</IsEquityAdjustmentRow><IsBeginningBalance>false</IsBeginningBalance><IsEndingBalance>true</IsEndingBalance><IsReverseSign>false</IsReverseSign><PreferredLabelRole>periodEndLabel</PreferredLabelRole><FootnoteIndexer /><Cells><Cell FlagID="0" ContextID="I2013Q2" UnitID="usd"><Id>1</Id><IsNumeric>true</IsNumeric><IsRatio>false</IsRatio><DisplayZeroAsNone>false</DisplayZeroAsNone><NumericAmount>20952000</NumericAmount><RoundedNumericAmount>20952</RoundedNumericAmount><NonNumbericText /><FootnoteIndexer /><CurrencyCode>USD</CurrencyCode><CurrencySymbol>$</CurrencySymbol><IsIndependantCurrency>false</IsIndependantCurrency><ShowCurrencySymbol>true</ShowCurrencySymbol><DisplayDateInUSFormat>false</DisplayDateInUSFormat></Cell></Cells><ElementDataType>xbrli:monetaryItemType</ElementDataType><SimpleDataType>monetary</SimpleDataType><ElementDefenition>This element represents a liability measured at fair value using significant unobservable inputs (Level 3) which is required for reconciliation purposes of beginning and ending balances.</ElementDefenition><ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef

 -Publisher FASB

 -Name Accounting Standards Codification

 -Topic 820

 -SubTopic 10

 -Section 50

 -Paragraph 2

 -Subparagraph (c)

 -URI http://asc.fasb.org/extlink&amp;oid=7578670&amp;loc=d3e19207-110258



</ElementReferences><IsTotalLabel>false</IsTotalLabel><UnitID>2</UnitID><Label>Balance, End</Label></Row></Rows><Footnotes /><IsEquityReport>false</IsEquityReport><ReportName>Fair Value Measurements - Reconciliation of the beginning and ending balances of the Level 3 assets and liabilities (Detail) (USD $)</ReportName><MonetaryRoundingLevel>Thousands</MonetaryRoundingLevel><SharesRoundingLevel>UnKnown</SharesRoundingLevel><PerShareRoundingLevel>UnKnown</PerShareRoundingLevel><ExchangeRateRoundingLevel>UnKnown</ExchangeRateRoundingLevel><HasCustomUnits>true</HasCustomUnits><IsEmbedReport>false</IsEmbedReport><IsMultiCurrency>false</IsMultiCurrency><ReportType>Sheet</ReportType><RoleURI>http://www.steelexcel.com/role/FairValueMeasurementsReconciliationOfBeginningAndEndingBalancesOfLevel3AssetsAndLiabilitiesDetail</RoleURI><NumberOfCols>1</NumberOfCols><NumberOfRows>7</NumberOfRows></InstanceReport>
