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Investment Securities (Tables) | 9 Months Ended
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Sep. 30, 2022
Investments, Debt and Equity Securities [Abstract] |
Pre-tax net investment income | The following table presents pre-tax net investment income for the three and nine months ended September 30, 2022 and 2021:
|  | Three Months Ended |  | Nine Months Ended
|  | September 30, |  | September 30,
Millions |  | 2022 |  | 2021 |  | 2022 |  | 2021
Fixed maturity investments |  | $ | 11.1 |  |  | $ | 8.2 |  |  | $ | 28.6 |  |  | $ | 21.9 |
Short-term investments |  | 6.7 |  |  | — |  |  | 7.2 |  |  | .5 |
Common equity securities |  | .1 |  |  | .1 |  |  | .2 |  |  | .1 |
Other long-term investments |  | 17.1 |  |  | 12.0 |  |  | 45.9 |  |  | 37.1 |
Amount attributable to TPC Providers |  | (.3) |  |  | (.2) |  |  | (.8) |  |  | (.8) |
Total investment income |  | 34.7 |  |  | 20.1 |  |  | 81.1 |  |  | 58.8 |
Third-party investment expenses |  | (.8) |  |  | (.6) |  |  | (1.5) |  |  | (1.6) |
Net investment income, pre-tax |  | $ | 33.9 |  |  | $ | 19.5 |  |  | $ | 79.6 |  |  | $ | 57.2 |
Schedule of net realized and unrealized investment gains and losses | The following table presents net realized and unrealized investment gains (losses) for the three and nine months ended September 30, 2022 and 2021:
|  | Three Months Ended September 30, |  | Nine Months Ended September 30,
Millions |  | 2022 |  | 2021 |  | 2022 |  | 2021
Realized investment gains (losses) |  |  |  |  |  |  |  |
Fixed maturity investments |  | $ | (4.9) |  |  | $ | 1.0 |  |  | $ | (10.8) |  |  | $ | 3.8 |
Short-term investments |  | (.1) |  |  | (.3) |  |  | (.4) |  |  | — |
Common equity securities |  | — |  |  | — |  |  | — |  |  | .4 |
Investment in MediaAlpha |  | — |  |  | — |  |  | — |  |  | 160.3 |
Other long-term investments |  | 3.4 |  |  | 9.2 |  |  | 24.1 |  |  | 0.9 |
Net realized investment gains (losses) |  | (1.6) |  |  | 9.9 |  |  | 12.9 |  |  | 165.4 |
Unrealized investment gains (losses) |  |  |  |  |  |  |  |
Fixed maturity investments |  | (58.8) |  |  | (7.9) |  |  | (192.9) |  |  | (27.0) |
Short-term investments |  | (2.1) |  |  | — |  |  | (2.6) |  |  | — |
Common equity securities |  | 4.3 |  |  | 1.6 |  |  | (6.3) |  |  | 7.3 |
Investment in MediaAlpha |  | (18.6) |  |  | (396.8) |  |  | (113.3) |  |  | (485.8) |
Other long-term investments |  | 27.5 |  |  | 28.4 |  |  | 40.8 |  |  | 112.3 |
Net unrealized investment gains (losses) |  | (47.7) |  |  | (374.7) |  |  | (274.3) |  |  | (393.2) |
Net realized and unrealized investment gains (losses), before amount attributable to TPC providers(1) |  | (49.3) |  |  | (364.8) |  |  | (261.4) |  |  | (227.8) |
Amount attributable to TPC Providers |  | 1.3 |  |  | (1.7) |  |  | 6.2 |  |  | (6.5) |
Net realized and unrealized investment gains (losses) |  | $ | (48.0) |  |  | $ | (366.5) |  |  | $ | (255.2) |  |  | $ | (234.3) |
Fixed maturity and short-term investments |  |  |  |  |  |  |  |
Net realized and unrealized investment gains (losses) |  | $ | (65.9) |  |  | $ | (7.2) |  |  | $ | (206.7) |  |  | $ | (23.2) |
Less: net realized and unrealized gains (losses) on investment securities sold during the period |  | .1 |  |  | (5.7) |  |  | (3.0) |  |  | (6.9) |
Net unrealized investment gains (losses) on investment securities held at the end of the period |  | $ | (66.0) |  |  | $ | (1.5) |  |  | $ | (203.7) |  |  | $ | (16.3) |
Common equity securities and investment in MediaAlpha |  |  |  |  |  |  |  |
Net realized and unrealized investment gains (losses) on common equity securities |  | $ | 4.3 |  |  | $ | 1.6 |  |  | $ | (6.3) |  |  | $ | 7.7 |
Net realized and unrealized investment gains (losses) from investment in MediaAlpha |  | (18.6) |  |  | (396.8) |  |  | (113.3) |  |  | (325.5) |
Total net realized and unrealized investment gains (losses) |  | (14.3) |  |  | (395.2) |  |  | (119.6) |  |  | (317.8) |
Less: net realized and unrealized gains (losses) on investment securities sold during the period |  | — |  |  | .4 |  |  | — |  |  | 20.3 |
Net unrealized investment gains (losses) on investment securities held at the end of the period |  | $ | (14.3) |  |  | $ | (395.6) |  |  | $ | (119.6) |  |  | $ | (338.1) |
(1) For the three months ended September 30, 2022 and 2021, includes $(22.6) and $(7.0) of realized and unrealized investment gains (losses) related to foreign currency exchange. For the nine months ended September 30, 2022 and 2021, includes $(44.4) and $(7.4) of realized and unrealized investment gains (losses) related to foreign currency exchange.
Net unrealized investment gains (losses) for Level 3 investments | The following table presents total gains included in earnings attributable to net unrealized investment gains for Level 3 investments for the three and nine months ended September 30, 2022 and 2021 for investments still held at the end of the period:
|  | Three Months Ended |  | Nine Months Ended
|  | September 30, |  | September 30,
Millions |  | 2022 |  | 2021 |  | 2022 |  | 2021
Other long-term investments |  | $ | 40.8 |  |  | $ | 26.5 |  |  | $ | 57.7 |  |  | $ | 75.5 |
Total net unrealized investment gains, pre-tax - Level 3 investments |  | $ | 40.8 |  |  | $ | 26.5 |  |  | $ | 57.7 |  |  | $ | 75.5 |
Investment holdings, equity securities, convertible fixed maturities and other long-term investments | The following tables present the cost or amortized cost, gross unrealized investment gains (losses), net foreign currency gains (losses) and carrying values of White Mountains’s fixed maturity investments as of September 30, 2022 and December 31, 2021:
|  | September 30, 2022
Millions |  | Cost or Amortized Cost |  | Gross Unrealized Gains |  | Gross Unrealized Losses |  | Net Foreign Currency Gains (Losses) |  | Carrying Value
U.S. Government and agency obligations |  | $ | 227.9 |  |  | $ | — |  |  | $ | (11.0) |  |  | $ | — |  |  | $ | 216.9 |
Debt securities issued by corporations |  | 1,068.8 |  |  | — |  |  | (91.2) |  |  | (3.1) |  |  | 974.5 |
Municipal obligations |  | 289.3 |  |  | — |  |  | (24.4) |  |  | — |  |  | 264.9 |
Mortgage and asset-backed securities |  | 278.3 |  |  | — |  |  | (37.6) |  |  | — |  |  | 240.7 |
Collateralized loan obligations |  | 169.7 |  |  | — |  |  | (7.6) |  |  | (4.8) |  |  | 157.3 |
Total fixed maturity investments |  | $ | 2,034.0 |  |  | $ | — |  |  | $ | (171.8) |  |  | $ | (7.9) |  |  | $ | 1,854.3 |
|  | December 31, 2021
Millions |  | Cost or Amortized Cost |  | Gross Unrealized Gains |  | Gross Unrealized Losses |  | Net Foreign Currency Gains (Losses) |  | Carrying Value
U.S. Government and agency obligations |  | $ | 212.1 |  |  | $ | .5 |  |  | $ | (1.1) |  |  | $ | — |  |  | $ | 211.5 |
Debt securities issued by corporations |  | 993.3 |  |  | 8.7 |  |  | (8.7) |  |  | (.4) |  |  | 992.9 |
Municipal obligations |  | 276.4 |  |  | 16.8 |  |  | (1.3) |  |  | — |  |  | 291.9 |
Mortgage and asset-backed securities |  | 277.2 |  |  | 2.9 |  |  | (2.5) |  |  | — |  |  | 277.6 |
Collateralized loan obligations |  | 136.5 |  |  | — |  |  | (.4) |  |  | (1.1) |  |  | 135.0 |
Total fixed maturity investments |  | $ | 1,895.5 |  |  | $ | 28.9 |  |  | $ | (14.0) |  |  | $ | (1.5) |  |  | $ | 1,908.9 |
The following tables present the cost or amortized cost, gross unrealized investment gains (losses), net foreign currency gains (losses), and carrying values of common equity securities, White Mountains’s investment in MediaAlpha and other long-term investments as of September 30, 2022 and December 31, 2021:
|  | September 30, 2022
Millions |  | Cost or Amortized Cost |  | Gross Unrealized Gains |  | Gross Unrealized Losses |  | Net Foreign Currency Gains (Losses) |  | Carrying Value
Common equity securities |  | $ | 324.2 |  |  | $ | 32.7 |  |  | $ | (8.7) |  |  | $ | (15.4) |  |  | $ | 332.8 |
Investment in MediaAlpha |  | $ | — |  |  | $ | 148.2 |  |  | $ | — |  |  | $ | — |  |  | $ | 148.2 |
Other long-term investments |  | $ | 1,390.7 |  |  | $ | 345.0 |  |  | $ | (90.0) |  |  | $ | (23.0) |  |  | $ | 1,622.7 |
|  | December 31, 2021
Millions |  | Cost or Amortized Cost |  | Gross Unrealized Gains |  | Gross Unrealized Losses |  | Net Foreign Currency Gains (Losses) |  | Carrying Value
Common equity securities |  | $ | 236.3 |  |  | $ | 16.1 |  |  | $ | — |  |  | $ | (1.3) |  |  | $ | 251.1 |
Investment in MediaAlpha |  | $ | — |  |  | $ | 261.6 |  |  | $ | — |  |  | $ | — |  |  | $ | 261.6 |
Other long-term investments |  | $ | 1,186.7 |  |  | $ | 239.0 |  |  | $ | (44.1) |  |  | $ | (3.8) |  |  | $ | 1,377.8 |
Schedule of fixed maturity investment holdings | The following table presents the cost or amortized cost and carrying values of White Mountains’s fixed maturity investments by contractual maturity as of September 30, 2022. Actual maturities could differ from contractual maturities because borrowers may have the right to call or prepay certain obligations with or without call or prepayment penalties.
|  | September 30, 2022
Millions |  | Cost or Amortized Cost |  | Carrying Value
Due in one year or less |  | $ | 207.5 |  |  | $ | 203.8 |
Due after one year through five years |  | 917.9 |  |  | 849.4 |
Due after five years through ten years |  | 354.8 |  |  | 309.3 |
Due after ten years |  | 105.8 |  |  | 93.8 |
Mortgage and asset-backed securities and collateralized loan obligations |  | 448.0 |  |  | 398.0 |
Total fixed maturity investments |  | $ | 2,034.0 |  |  | $ | 1,854.3 |
Fair value measurements by level, investment securities | The following tables present White Mountains’s fair value measurements for investments as of September 30, 2022 and December 31, 2021 by level. The major security types were based on the legal form of the securities. White Mountains has disaggregated its fixed maturity investments based on the issuing entity type, which impacts credit quality, with debt securities issued by U.S. government entities carrying minimal credit risk, while the credit and other risks associated with other issuers, such as corporations, municipalities or entities issuing mortgage and asset-backed securities vary depending on the nature of the issuing entity type. White Mountains further disaggregates debt securities issued by corporations by industry sector because investors often reference commonly used benchmarks and their subsectors to monitor risk and performance. Accordingly, White Mountains has further disaggregated this asset class into subclasses based on the similar sectors and industry classifications it uses to evaluate investment risk and performance against commonly used benchmarks, such as the Bloomberg Barclays U.S. Intermediate Aggregate.
|  | September 30, 2022
Millions |  | Fair Value |  | Level 1 |  | Level 2 |  | Level 3
Fixed maturity investments: |  |  |  |  |  |  |  |
U.S. Government and agency obligations |  | $ | 216.9 |  |  | $ | 216.9 |  |  | $ | — |  |  | $ | — |
Debt securities issued by corporations: |  |  |  |  |  |  |  |
Financials |  | 269.0 |  |  | — |  |  | 269.0 |  |  | — |
Consumer |  | 181.4 |  |  | — |  |  | 181.4 |  |  | — |
Industrial |  | 115.8 |  |  | — |  |  | 115.8 |  |  | — |
Healthcare |  | 114.3 |  |  | — |  |  | 114.3 |  |  | — |
Technology |  | 110.8 |  |  | — |  |  | 110.8 |  |  | — |
Utilities |  | 75.9 |  |  | — |  |  | 75.9 |  |  | — |
Communications |  | 49.1 |  |  | — |  |  | 49.1 |  |  | — |
Energy |  | 36.1 |  |  | — |  |  | 36.1 |  |  | — |
Materials |  | 22.1 |  |  | — |  |  | 22.1 |  |  | — |
Total debt securities issued by corporations |  | 974.5 |  |  | — |  |  | 974.5 |  |  | — |
Municipal obligations |  | 264.9 |  |  | — |  |  | 264.9 |  |  | — |
Mortgage and asset-backed securities |  | 240.7 |  |  | — |  |  | 240.7 |  |  | — |
Collateralized loan obligations |  | 157.3 |  |  | — |  |  | 157.3 |  |  | — |
Total fixed maturity investments |  | 1,854.3 |  |  | 216.9 |  |  | 1,637.4 |  |  | — |
Short-term investments |  | 1,196.4 |  |  | 1,196.4 |  |  | — |  |  | — |
Common equity securities (1) |  | 332.8 |  |  | 46.4 |  |  | 286.4 |  |  | — |
Investment in MediaAlpha |  | 148.2 |  |  | 148.2 |  |  | — |  |  | — |
Other long-term investments |  | 1,058.0 |  |  | — |  |  | 13.9 |  |  | 1,044.1 |
Other long-term investments — NAV (2) |  | 564.7 |  |  | — |  |  | — |  |  | — |
Total other long-term investments |  | 1,622.7 |  |  | — |  |  | 13.9 |  |  | 1,044.1 |
Total investments |  | $ | 5,154.4 |  |  | $ | 1,607.9 |  |  | $ | 1,937.7 |  |  | $ | 1,044.1 |
(1) Consist of investments in exchange traded funds (“ETFs”) and listed funds that predominantly invest in international equities. (2) Consists of private equity funds and hedge funds, a bank loan fund, Lloyd’s trust deposits and ILS funds for which fair value is measured at net asset value (“NAV”) using the practical expedient. Investments for which fair value is measured at NAV are not classified within the fair value hierarchy.
|  | December 31, 2021
Millions |  | Fair Value |  | Level 1 |  | Level 2 |  | Level 3
Fixed maturity investments: |  |  |  |  |  |  |  |
U.S. Government and agency obligations |  | $ | 211.5 |  |  | $ | 211.5 |  |  | $ | — |  |  | $ | — |
Debt securities issued by corporations: |  |  |  |  |  |  |  |
Financials |  | 264.2 |  |  | — |  |  | 264.2 |  |  | — |
Consumer |  | 178.1 |  |  | — |  |  | 178.1 |  |  | — |
Technology |  | 117.9 |  |  | — |  |  | 117.9 |  |  | — |
Industrial |  | 112.9 |  |  | — |  |  | 112.9 |  |  | — |
Healthcare |  | 112.8 |  |  | — |  |  | 112.8 |  |  | — |
Utilities |  | 70.9 |  |  | — |  |  | 70.9 |  |  | — |
Communications |  | 56.0 |  |  | — |  |  | 56.0 |  |  | — |
Energy |  | 48.0 |  |  | — |  |  | 48.0 |  |  | — |
Materials |  | 32.1 |  |  | — |  |  | 32.1 |  |  | — |
Total debt securities issued by corporations |  | 992.9 |  |  | — |  |  | 992.9 |  |  | — |
Municipal obligations |  | 291.9 |  |  | — |  |  | 291.9 |  |  | — |
Mortgage and asset-backed securities |  | 277.6 |  |  | — |  |  | 277.6 |  |  | — |
Collateralized loan obligations |  | 135.0 |  |  | — |  |  | 135.0 |  |  | — |
Total fixed maturity investments |  | 1,908.9 |  |  | 211.5 |  |  | 1,697.4 |  |  | — |
Short-term investments |  | 465.9 |  |  | 465.9 |  |  | — |  |  | — |
Common equity securities (1) |  | 251.1 |  |  | — |  |  | 251.1 |  |  | — |
Investment in MediaAlpha |  | 261.6 |  |  | 261.6 |  |  | — |  |  | — |
Other long-term investments |  | 895.3 |  |  | — |  |  | 4.7 |  |  | 890.6 |
Other long-term investments — NAV (2) |  | 482.5 |  |  | — |  |  | — |  |  | — |
Total other long-term investments |  | 1,377.8 |  |  | — |  |  | 4.7 |  |  | 890.6 |
Total investments |  | $ | 4,265.3 |  |  | $ | 939.0 |  |  | $ | 1,953.2 |  |  | $ | 890.6 |
(1) Consist of investments in listed funds that predominantly invest in international equities. (2) Consists of private equity funds and hedge funds, a bank loan fund, Lloyd’s trust deposits and ILS funds for which fair value is measured at NAV using the practical expedient. Investments for which fair value is measured at NAV are not classified within the fair value hierarchy.
Debt securities issued by corporations, credit ratings | The following table presents the credit ratings of debt securities issued by corporations held in White Mountains’s investment portfolio as of September 30, 2022 and December 31, 2021:
|  | Fair Value at
Millions |  | September 30, 2022 |  | December 31, 2021
AAA |  | $ | 11.2 |  |  | $ | 12.0 |
AA |  | 90.5 |  |  | 85.0 |
A |  | 524.0 |  |  | 490.4 |
BBB |  | 343.0 |  |  | 396.8 |
Other |  | 5.8 |  |  | 8.7 |
Debt securities issued by corporations (1) |  | $ | 974.5 |  |  | $ | 992.9 |
(1) Credit ratings are based upon issuer credit ratings provided by Standard & Poor’s Financial Services LLC (“Standard & Poor’s”), or if unrated by Standard & Poor’s, long-term obligation ratings provided by Moody’s Investor Service, Inc.
Mortgage-backed, asset-backed securities | The following table presents the fair value of White Mountains’s mortgage and asset-backed securities and collateralized loan obligations as of September 30, 2022 and December 31, 2021:
|  | September 30, 2022 |  | December 31, 2021
Millions |  | Fair Value |  | Level 2 |  | Level 3 |  | Fair Value |  | Level 2 |  | Level 3
Mortgage-backed securities: |  |  |  |  |  |  |  |  |  |  |  |
Agency: |  |  |  |  |  |  |  |  |  |  |  |
FNMA |  | $ | 114.1 |  |  | $ | 114.1 |  |  | $ | — |  |  | $ | 125.4 |  |  | $ | 125.4 |  |  | $ | — |
FHLMC |  | 74.5 |  |  | 74.5 |  |  | — |  |  | 90.5 |  |  | 90.5 |  |  | — |
GNMA |  | 28.9 |  |  | 28.9 |  |  | — |  |  | 40.1 |  |  | 40.1 |  |  | — |
Total agency (1) |  | 217.5 |  |  | 217.5 |  |  | — |  |  | 256.0 |  |  | 256.0 |  |  | — |
Non-agency: Residential |  | .4 |  |  | .4 |  |  | — |  |  | .5 |  |  | .5 |  |  | — |
Total non-agency |  | .4 |  |  | .4 |  |  | — |  |  | .5 |  |  | .5 |  |  | — |
Total mortgage-backed securities |  | 217.9 |  |  | 217.9 |  |  | — |  |  | 256.5 |  |  | 256.5 |  |  | — |
Other asset-backed securities: |  |  |  |  |  |  |  |  |  |  |  |
Credit card receivables |  | 11.9 |  |  | 11.9 |  |  | — |  |  | 12.3 |  |  | 12.3 |  |  | — |
Vehicle receivables |  | 10.9 |  |  | 10.9 |  |  | — |  |  | 8.8 |  |  | 8.8 |  |  | — |
Total other asset-backed securities |  | 22.8 |  |  | 22.8 |  |  | — |  |  | 21.1 |  |  | 21.1 |  |  | — |
Total mortgage and asset-backed securities |  | 240.7 |  | 240.7 |  | — |  |  | 277.6 |  |  | 277.6 |  |  | — |
Collateralized loan obligations |  | 157.3 |  |  | 157.3 |  |  | — |  |  | 135.0 |  |  | 135.0 |  |  | — |
Total mortgage and asset-backed securities and collateralized loan obligations |  | $ | 398.0 |  |  | $ | 398.0 |  |  | $ | — |  |  | $ | 412.6 |  |  | $ | 412.6 |  |  | $ | — |
(1) Represents publicly traded mortgage-backed securities which carry the full faith and credit guaranty of the U.S. Government (i.e., GNMA) or are guaranteed by a government sponsored entity (i.e., FNMA, FHLMC).
Schedule of carrying values for other long-term investments | The following table presents the carrying values of White Mountains’s other long-term investments as of September 30, 2022 and December 31, 2021:
|  | Fair Value at
Millions |  | September 30, 2022 |  | December 31, 2021
Kudu’s Participation Contracts |  | $ | 813.2 |  |  | $ | 669.5 |
PassportCard/DavidShield |  | 132.0 |  |  | 120.0 |
Elementum Holdings, L.P. |  | 45.0 |  |  | 45.0 |
Other unconsolidated entities (1) |  | 37.0 |  |  | 34.4 |
Total unconsolidated entities |  | 1,027.2 |  |  | 868.9 |
Private equity funds and hedge funds |  | 239.8 |  |  | 153.8 |
Bank loan fund |  | 155.7 |  |  | 163.0 |
Lloyd’s trust deposits |  | 117.8 |  |  | 113.8 |
ILS funds |  | 49.4 |  |  | 51.9 |
Private debt investments |  | 13.0 |  |  | 14.1 |
Other |  | 19.8 |  |  | 12.3 |
Total other long-term investments |  | $ | 1,622.7 |  |  | $ | 1,377.8 |
(1) Includes White Mountains’s non-controlling equity interests in certain private common equity securities, convertible preferred securities, limited liability company units and Simple Agreement for Future Equity (“SAFE”) investments.
Other long-term investments | The following table presents the fair value of investments and unfunded commitments in private equity funds and hedge funds by investment objective and sector as of September 30, 2022 and December 31, 2021:
|  | September 30, 2022 |  | December 31, 2021
Millions |  | Fair Value |  | Unfunded Commitments |  | Fair Value |  | Unfunded Commitments
Private equity funds |  |  |  |  |  |  |  |
Aerospace/Defense/Government |  | $ | 102.9 |  |  | $ | 45.1 |  |  | $ | 69.8 |  |  | $ | 11.8 |
Financial services |  | 77.9 |  |  | 54.9 |  |  | 67.7 |  |  | 29.3 |
Real estate |  | 4.3 |  |  | 2.5 |  |  | 4.3 |  |  | 2.9 |
Total private equity funds |  | 185.1 |  |  | 102.5 |  |  | 141.8 |  |  | 44.0 |
Hedge funds |  |  |  |  |  |  |  |
Long/short equity financials and business services |  | 49.3 |  |  | — |  |  | — |  |  | — |
European small/mid cap |  | 5.4 |  |  | — |  |  | 12.0 |  |  | — |
Total hedge funds |  | 54.7 |  |  | — |  |  | 12.0 |  |  | — |
Total private equity funds and hedge funds included in other long-term investments |  | $ | 239.8 |  |  | $ | 102.5 |  |  | $ | 153.8 |  |  | $ | 44.0 |
Fair value of private equity funds subject to lock-up periods | The following table presents investments in private equity funds that were subject to lock-up periods as of September 30, 2022:
Millions |  | 1 – 3 years |  | 3 – 5 years |  | 5 – 10 years |  | >10 years |  | Total
Private equity funds — expected lock-up period remaining |  | $27.5 |  | $19.0 |  | $135.4 |  | $3.2 |  | $185.1
Rollforward of fair value investments by level | The following table presents the changes in White Mountains’s fair value measurements for Level 3 investments for the nine months ended September 30, 2022 and 2021:
Level 3 Investments |  |  |  |
Millions |  |  |  |  |  |  |  |  |  |  | Other Long-term Investments |  |  |  | Other Long-term Investments |  |
Balance at December 31, 2021 |  |  |  |  |  |  |  |  |  |  | $ | 890.6 |  |  | Balance at December 31, 2020 |  | $ | 614.2 |  |  |  |  |
Net realized and unrealized gains |  |  |  |  |  |  |  |  |  |  | 57.3 |  |  | Net realized and unrealized gains |  | 79.0 |  |  |  |  |
Amortization/accretion |  |  |  |  |  |  |  |  |  |  | — |  |  | Amortization/accretion |  | — |  |  |  |  |
Purchases |  |  |  |  |  |  |  |  |  |  | 127.9 |  |  | Purchases |  | 143.6 |  |  |  |  |
Sales |  |  |  |  |  |  |  |  |  |  | (31.7) |  |  | Sales |  | (31.9) |  |  |  |  |
Effect of Ark Transaction |  |  |  |  |  |  |  |  |  |  | — |  |  | Effect of Ark Transaction |  | 9.6 |  |  |  |  |
Transfers in |  |  |  |  |  |  |  |  |  |  | — |  |  | Transfers in |  | — |  |  |  |  |
Transfers out |  |  |  |  |  |  |  |  |  |  | — |  |  | Transfers out |  | — |  |  |  |  |
Balance at September 30, 2022 |  |  |  |  |  |  |  |  |  |  | $ | 1,044.1 |  |  | Balance at September 30, 2021 |  | $ | 814.5 |  |  |  |  |
Schedule of significant unobservable inputs used in estimating the fair value of investment securities | The following tables present significant unobservable inputs used in estimating the fair value of White Mountains’s other long-term investments, classified within Level 3 as of September 30, 2022 and December 31, 2021. The tables below exclude $42.9 million and $46.7 million of Level 3 other long-term investments generally valued based on recent or expected transaction prices. The fair value of investments in private equity funds and hedge funds, bank loan funds, Lloyd’s trust deposits and ILS funds are generally estimated using the NAV of the funds.
$ in Millions |  | September 30, 2022
Description |  | Valuation Technique(s) (1) |  | Fair Value (2) |  | Unobservable Inputs
|  |  |  |  |  | Discount Rate (6) |  | Terminal Cash Flow Exit Multiple (x) or Terminal Revenue Growth Rate (%) (6)
Kudu’s Participation Contracts (3)(4)(5) |  | Discounted cash flow |  | $813.2 |  | 17% - 24% |  | 7x - 16x
PassportCard/DavidShield |  | Discounted cash flow |  | $132.0 |  | 24% |  | 4%
Elementum Holdings, L.P. |  | Discounted cash flow |  | $45.0 |  | 18% |  | 4%
Private debt investments |  | Discounted cash flow |  | $11.0 |  | 10% |  | N/A
(1) Key inputs to the discounted cash flow analysis generally include projections of future revenue and earnings, discount rates and terminal exit multiples or growth rates. (2) Includes the net unrealized investment gains (losses) associated with foreign currency; foreign currency effects based on observable inputs. (3) Since Kudu’s Participation Contracts are not subject to corporate taxes within Kudu Investment Management, LLC, pre-tax discount rates are applied to pre-tax cash flows in determining fair values. The weighted average discount rate and weighted average terminal cash flow exit multiple applied to Kudu’s Participation Contracts is 20% and 9.7x. (4) As of September 30, 2022, three of Kudu’s Participation Contracts with a total fair value of $312.4 were valued using a probability weighted expected return method, which was based on a discounted cash flow analysis and an expected sale transaction. (5) In 2022, Kudu deployed a total of $97.9 into new and existing Kudu Participation Contracts, including Gramercy Funds Management, GenTrust, EC Management Services, and TK Partners. (6) Increases (decreases) to the discount rates in isolation would result in lower (higher) fair value measurements, while increases (decreases) to the terminal cash flow exit multiples or terminal revenue growth rates in isolation would result in higher (lower) fair value measurements.
$ in Millions |  | December 31, 2021
Description |  | Valuation Technique(s) (1) |  | Fair Value (2) |  | Unobservable Inputs
|  |  |  |  |  | Discount Rate (6) |  | Terminal Cash Flow Exit Multiple (x) or Terminal Revenue Growth Rate (%) (6)
Kudu’s Participation Contracts (3)(4)(5) |  | Discounted cash flow |  | $669.5 |  | 18% - 23% |  | 7x - 13x
PassportCard/DavidShield |  | Discounted cash flow |  | $120.0 |  | 23% |  | 4%
Elementum Holdings, L.P. |  | Discounted cash flow |  | $45.0 |  | 17% |  | 4%
Private debt investments |  | Discounted cash flow |  | $9.4 |  | 8% |  | N/A
(1) Key inputs to the discounted cash flow analysis generally include projections of future revenue and earnings, discount rates and terminal exit multiples or growth rates. (2) Includes the net unrealized investment gains (losses) associated with foreign currency; foreign currency effects based on observable inputs. (3) Since Kudu’s Participation Contracts are not subject to corporate taxes within Kudu Investment Management, LLC, pre-tax discount rates are applied to pre-tax cash flows in determining fair values. The weighted average discount rate and weighted average terminal cash flow exit multiple applied to Kudu’s Participation Contracts is 20% and 10x. (4) In 2021, Kudu deployed a total of $223.4 into new and existing Kudu Participation Contracts, including TIG Advisors, TK Partners, Third Eye Capital Management, Douglass Winthrop Advisors, Granahan Investment Management and Radcliffe Capital Management. (5) As of December 31, 2021, one of Kudu’s Participation Contracts with a total fair value of $78.8 was valued using a probability weighted expected return method, which was based on a discounted cash flow analysis and an expected sale transaction. (6) Increases (decreases) to the discount rates in isolation would result in lower (higher) fair value measurements, while increases (decreases) to the terminal cash flow exit multiples or terminal revenue growth rates in isolation would result in higher (lower) fair value measurements.

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