﻿<?xml version="1.0" encoding="utf-8"?>
<InstanceReport xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xsd="http://www.w3.org/2001/XMLSchema">
  <Version>2.2.0.7</Version>
  <hasSegments>false</hasSegments>
  <ReportName>Transactions with AIG</ReportName>
  <ReportLongName>1100 - Disclosure - Transactions with AIG</ReportLongName>
  <DisplayLabelColumn>true</DisplayLabelColumn>
  <ShowElementNames>false</ShowElementNames>
  <RoundingOption />
  <HasEmbeddedReports>false</HasEmbeddedReports>
  <Columns>
    <Column>
      <LabelColumn>false</LabelColumn>
      <Id>1</Id>
      <Labels>
        <Label Id="1" Label="9 Months Ended" />
        <Label Id="2" Label="Sep. 30, 2010" />
      </Labels>
      <CurrencyCode>USD</CurrencyCode>
      <FootnoteIndexer />
      <hasSegments>false</hasSegments>
      <hasScenarios>false</hasScenarios>
      <Segments />
      <Scenarios />
      <Units>
        <Unit>
          <UnitID>USD</UnitID>
          <UnitType>Standard</UnitType>
          <StandardMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema>
            <MeasureValue>USD</MeasureValue>
            <MeasureNamespace>iso4217</MeasureNamespace>
          </StandardMeasure>
          <Scale>0</Scale>
        </Unit>
        <Unit>
          <UnitID>USDPerShare</UnitID>
          <UnitType>Divide</UnitType>
          <NumeratorMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema>
            <MeasureValue>USD</MeasureValue>
            <MeasureNamespace>iso4217</MeasureNamespace>
          </NumeratorMeasure>
          <DenominatorMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/instance</MeasureSchema>
            <MeasureValue>shares</MeasureValue>
            <MeasureNamespace>xbrli</MeasureNamespace>
          </DenominatorMeasure>
          <Scale>0</Scale>
        </Unit>
        <Unit>
          <UnitID>Shares</UnitID>
          <UnitType>Standard</UnitType>
          <StandardMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/instance</MeasureSchema>
            <MeasureValue>shares</MeasureValue>
            <MeasureNamespace>xbrli</MeasureNamespace>
          </StandardMeasure>
          <Scale>0</Scale>
        </Unit>
      </Units>
      <CurrencySymbol>$</CurrencySymbol>
    </Column>
  </Columns>
  <Rows>
    <Row>
      <Id>2</Id>
      <Label>Transactions with AIG</Label>
      <Level>0</Level>
      <ElementName>trh_TransactionsWithRelatedPartyAbstract</ElementName>
      <ElementPrefix>trh</ElementPrefix>
      <IsBaseElement>false</IsBaseElement>
      <BalanceType>na</BalanceType>
      <PeriodType>duration</PeriodType>
      <ShortDefinition>No definition available.</ShortDefinition>
      <IsReportTitle>false</IsReportTitle>
      <IsSegmentTitle>false</IsSegmentTitle>
      <IsSubReportEnd>false</IsSubReportEnd>
      <IsCalendarTitle>false</IsCalendarTitle>
      <IsTuple>false</IsTuple>
      <IsAbstractGroupTitle>true</IsAbstractGroupTitle>
      <IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow>
      <IsEquityAdjustmentRow>false</IsEquityAdjustmentRow>
      <IsBeginningBalance>false</IsBeginningBalance>
      <IsEndingBalance>false</IsEndingBalance>
      <IsReverseSign>false</IsReverseSign>
      <PreferredLabelRole />
      <IsEPS>false</IsEPS>
      <FootnoteIndexer />
      <Cells>
        <Cell>
          <Id>1</Id>
          <ShowCurrencySymbol>false</ShowCurrencySymbol>
          <IsNumeric>false</IsNumeric>
          <IsRatio>false</IsRatio>
          <DisplayZeroAsNone>false</DisplayZeroAsNone>
          <NumericAmount>0</NumericAmount>
          <RoundedNumericAmount>0</RoundedNumericAmount>
          <NonNumbericText />
          <NonNumericTextHeader />
          <FootnoteIndexer />
          <hasSegments>false</hasSegments>
          <hasScenarios>false</hasScenarios>
          <DisplayDateInUSFormat>false</DisplayDateInUSFormat>
        </Cell>
      </Cells>
      <OriginalInstanceReportColumns />
      <ElementDataType>xbrli:stringItemType</ElementDataType>
      <SimpleDataType>string</SimpleDataType>
      <ElementDefenition>No definition available.</ElementDefenition>
      <IsTotalLabel>false</IsTotalLabel>
    </Row>
    <Row>
      <Id>3</Id>
      <Label>Transactions with AIG</Label>
      <Level>1</Level>
      <ElementName>us-gaap_RelatedPartyTransactionsDisclosureTextBlock</ElementName>
      <ElementPrefix>us-gaap</ElementPrefix>
      <IsBaseElement>true</IsBaseElement>
      <BalanceType>na</BalanceType>
      <PeriodType>duration</PeriodType>
      <ShortDefinition>No definition available.</ShortDefinition>
      <IsReportTitle>false</IsReportTitle>
      <IsSegmentTitle>false</IsSegmentTitle>
      <IsSubReportEnd>false</IsSubReportEnd>
      <IsCalendarTitle>false</IsCalendarTitle>
      <IsTuple>false</IsTuple>
      <IsAbstractGroupTitle>false</IsAbstractGroupTitle>
      <IsEquityPrevioslyReportedAsRow>false</IsEquityPrevioslyReportedAsRow>
      <IsEquityAdjustmentRow>false</IsEquityAdjustmentRow>
      <IsBeginningBalance>false</IsBeginningBalance>
      <IsEndingBalance>false</IsEndingBalance>
      <IsReverseSign>false</IsReverseSign>
      <PreferredLabelRole>terselabel</PreferredLabelRole>
      <IsEPS>false</IsEPS>
      <FootnoteIndexer />
      <Cells>
        <Cell>
          <Id>1</Id>
          <ShowCurrencySymbol>false</ShowCurrencySymbol>
          <IsNumeric>false</IsNumeric>
          <IsRatio>false</IsRatio>
          <DisplayZeroAsNone>false</DisplayZeroAsNone>
          <NumericAmount>0</NumericAmount>
          <RoundedNumericAmount>0</RoundedNumericAmount>
          <NonNumbericText>&lt;table style="font-size:10pt; font-family:'Times New Roman',times,serif;"&gt;
&lt;tr&gt;
&lt;td&gt;
&lt;p style="MARGIN: 0in 0in 0pt"&gt;&lt;b&gt;&lt;font style="FONT-WEIGHT: bold; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;10. Transactions with AIG&lt;/font&gt;&lt;/b&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;&amp;nbsp;&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;(a)&amp;nbsp;Secondary Public Offerings of the Company&amp;#146;s Common Stock by the AIG Group&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;&amp;nbsp;&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt; TEXT-INDENT: 0.5in"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;Prior to June&amp;nbsp;10, 2009, AIG beneficially owned approximately 59% of the Company&amp;#146;s outstanding shares. On June&amp;nbsp;10, 2009, AIG and American Home Assurance Company (&amp;#147;AHAC&amp;#148;), a wholly owned subsidiary of AIG, consummated a secondary public offering (the &amp;#147;June&amp;nbsp;2009 Offering&amp;#148;) of 29.9 million issued and outstanding shares of the Company&amp;#146;s common stock owned by AIG and AHAC. According to the Form&amp;nbsp;13F filed on February&amp;nbsp;18, 2010 by AIG, as of December&amp;nbsp;31, 2009, the AIG Group had sole voting authority over 9.2 million shares of the Company&amp;#146;s common stock, representing approximately 13.8% of the Company&amp;#146;s outstanding shares as of December&amp;nbsp;31, 2009.&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;&amp;nbsp;&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt; TEXT-INDENT: 0.5in"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;In connection with the June&amp;nbsp;2009 Offering, TRH entered into a Master Separation Agreement (the &amp;#147;MSA&amp;#148;) with AIG and AHAC on May&amp;nbsp;28, 2009. The MSA set forth TRH&amp;#146;s agreements with AIG and AHAC regarding the orderly separation of TRH from AIG, AHAC and their subsidiaries (the &amp;#147;Separation&amp;#148;) and governs certain aspects of TRH&amp;#146;s relationship with AIG, AHAC and their subsidiaries on a going forward basis, including their waiver of certain rights they may have under intercompany agreements and insurance agreements. TRH also entered into a transition services agreement (the &amp;#147;TSA&amp;#148;) with AIG, pursuant to which the AIG Group would continue to provide various services to TRH for a limited period of time following the Separation while TRH transitioned away from receiving such services from the AIG Group by either outsourcing the services or bringing the services in-house. TRH has completed the transition of most services from the AIG Group. As part of the transition process, TRH entered into subleases with the AIG Group for office space in New York, Chicago and Toronto.&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;&amp;nbsp;&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt; TEXT-INDENT: 0.5in"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;On March&amp;nbsp;15, 2010, AHAC consummated another secondary public offering (the &amp;#147;March&amp;nbsp;2010 Offering&amp;#148;, and collectively with the June&amp;nbsp;2009 Offering, the &amp;#147;Secondary Offerings&amp;#148;) of 8.5 million issued and outstanding shares of the Company&amp;#146;s common stock owned by AHAC. The Company repurchased two million shares of its common stock from AHAC in the March&amp;nbsp;2010 Offering pursuant to a stock offering agreement for an aggregate purchase price of approximately $105 million. TRH did not receive any proceeds from the Secondary Offerings. Immediately following the March&amp;nbsp;2010 Offering, the AIG Group, including AHAC, beneficially owned 0.7 million shares of the Company&amp;#146;s common stock (excluding shares held by certain mutual funds that are advised or managed by subsidiaries of AIG), representing approximately 1.1% of the Company&amp;#146;s then outstanding shares. As a result of its reduced ownership percentage, the AIG Group is no longer considered a related party after March&amp;nbsp;15, 2010.&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;&amp;nbsp;&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;(b)&amp;nbsp;Reinsurance Assumed from the AIG Group&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;&amp;nbsp;&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt; TEXT-INDENT: 0.5in"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;After March&amp;nbsp;15, 2010, the AIG Group ceased to be a related party of TRH, as discussed above. Gross premiums written originated by the AIG Group and ceded to TRH from contracts that were entered into while the AIG Group was a related party totaled approximately $44.3&amp;nbsp;million (4.2 %) and $72.1&amp;nbsp;million (6.9%) in the third quarter of 2010 and 2009, respectively, and $159.9&amp;nbsp;million (5.1%) and $215.5&amp;nbsp;million (6.7%) in the first nine months of 2010 and 2009, respectively. These amounts exclude (a)&amp;nbsp;premiums assumed that initially were insured by AIG subsidiaries as a result of TRH&amp;#146;s marketing efforts and then ceded to TRH by prearrangement; (b)&amp;nbsp;amounts assumed from an AIG subsidiary and ceded in an equal amount to other AIG subsidiaries; and (c)&amp;nbsp;all premiums from contracts that were effective after March&amp;nbsp;15, 2010. The amount of premiums assumed that initially were insured by AIG subsidiaries as a result of TRH&amp;#146;s marketing efforts and then ceded to TRH by prearrangement are not material. (See Note&amp;nbsp;12 for the amount of premiums assumed from an AIG subsidiary and ceded in an equal amount to other AIG subsidiaries.)&lt;/font&gt;&lt;/p&gt;
&lt;p style="MARGIN: 0in 0in 0pt; TEXT-INDENT: 0.5in"&gt;&lt;font style="FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman" size="2"&gt;&amp;nbsp;&lt;/font&gt;&lt;/p&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;
</NonNumbericText>
          <NonNumericTextHeader>10. Transactions with AIG
&amp;nbsp;
(a)&amp;nbsp;Secondary Public Offerings of the Company&amp;#146;s Common Stock by the AIG Group
&amp;nbsp;
Prior to June&amp;nbsp;10, 2009,</NonNumericTextHeader>
          <FootnoteIndexer />
          <hasSegments>false</hasSegments>
          <hasScenarios>false</hasScenarios>
          <DisplayDateInUSFormat>false</DisplayDateInUSFormat>
        </Cell>
      </Cells>
      <OriginalInstanceReportColumns />
      <ElementDataType>us-types:textBlockItemType</ElementDataType>
      <SimpleDataType>textblock</SimpleDataType>
      <ElementDefenition>This element may be used for the entire related party transactions disclosure as a single block of text. Disclosure may include:  the nature of the relationship(s), a description of the transactions, the amount of the transactions, the effects of any change in the method of establishing the terms of the transaction from the previous period, stated interest rate, expiration date, terms and manner of settlement per the agreement with the related party, and amounts due to or from related parties. If the entity and one or more other entities are under common ownership or management control and this control affects the operating results or financial position, disclosure includes the nature of the control relationship even if there are no transactions between the entities. Disclosure may also include the aggregate amount of current and deferred tax expense for each statement of earnings presented where the entity is a member of a group that files a consolidated tax return, the amount of any tax related balances due to or from affiliates as of the date of each statement of financial position presented, the principal provisions of the method by which the consolidated amount of current and deferred tax expense is allocated to the members of the group and the nature and effect of any changes in that method. Examples of related party transactions include transactions between (a) a parent company and its subsidiary; (b) subsidiaries of a common parent; (c) and entity and its principal owners; and (d) affiliates.</ElementDefenition>
      <ElementReferences>Reference 1: http://www.xbrl.org/2003/role/presentationRef
 -Publisher SEC
 -Name Regulation S-X (SX)
 -Number 210
 -Section 04
 -Paragraph b
 -Article 3A

Reference 2: http://www.xbrl.org/2003/role/presentationRef
 -Publisher SEC
 -Name Regulation S-X (SX)
 -Number 210
 -Section 08
 -Paragraph k
 -Article 4

Reference 3: http://www.xbrl.org/2003/role/presentationRef
 -Publisher FASB
 -Name Statement of Financial Accounting Standard (FAS)
 -Number 57
 -Paragraph 1-4

</ElementReferences>
      <IsTotalLabel>false</IsTotalLabel>
    </Row>
  </Rows>
  <Footnotes />
  <NumberOfCols>1</NumberOfCols>
  <NumberOfRows>2</NumberOfRows>
  <HasScenarios>false</HasScenarios>
  <MonetaryRoundingLevel>UnKnown</MonetaryRoundingLevel>
  <SharesRoundingLevel>UnKnown</SharesRoundingLevel>
  <PerShareRoundingLevel>UnKnown</PerShareRoundingLevel>
  <HasPureData>false</HasPureData>
  <SharesShouldBeRounded>true</SharesShouldBeRounded>
</InstanceReport>
