Investment Securities

6 Months Ended

Jun. 30, 2022

Investments, Debt and Equity Securities [Abstract]

Investment Securities

Investment Securities

White Mountains’s portfolio of investment securities held for general investment purposes consists of fixed maturity investments, short-term investments, common equity securities, its investment in MediaAlpha and other long-term investments, which are classified as trading securities. Trading securities are reported at fair value as of the balance sheet date.  Net realized and unrealized investment gains (losses) on trading securities are reported in pre-tax revenues.

White Mountains’s fixed maturity investments are generally valued using industry standard pricing methodologies. Key inputs include benchmark yields, benchmark securities, reported trades, issuer spreads, bids, offers, credit ratings and prepayment speeds. Income on mortgage and asset-backed securities is recognized using an effective yield based on anticipated prepayments and the estimated economic life of the securities. When actual prepayments differ significantly from anticipated prepayments, the estimated economic life is recalculated and the remaining unamortized premium or discount is amortized prospectively over the remaining economic life.

Realized investment gains (losses) resulting from sales of investment securities are accounted for using the specific identification method. Premiums and discounts on all fixed maturity investments are amortized or accreted to income over the anticipated life of the investment. Short-term investments consist of interest-bearing money market funds, certificates of deposit and other securities, which at the time of purchase, mature or become available for use within one year.  Short-term investments are carried at fair value, which approximated amortized cost, as of June 30, 2022 and December 31, 2021.

Other long-term investments consist primarily of unconsolidated entities, including Kudu’s Participation Contracts, private equity funds and hedge funds, a bank loan fund, Lloyd’s trust deposits, insurance-linked securities (“ILS”) funds and private debt investments.

Net Investment Income

White Mountains’s net investment income is comprised primarily of interest income associated with White Mountains’s fixed maturity investments and short-term investments, dividend income from common equity securities, distributions from its investment in MediaAlpha and distributions from other long-term investments.

The following table presents pre-tax net investment income for the three and six months ended June 30, 2022 and 2021:

	Three Months Ended				Six Months Ended
	June 30,				June 30,
Millions	2022		2021		2022		2021
Fixed maturity investments	$	9.4	$	7.2	$	17.6	$	13.7
Short-term investments	.4		(1.1)		.5		.5
Common equity securities	.1		—		.1		—
Other long-term investments	15.6		11.3		28.8		25.1
Amount attributable to TPC Providers	—		.3		(.5)		(.6)
Total investment income	25.5		17.7		46.5		38.7
Third-party investment expenses	(.4)		(.6)		(.8)		(1.0)
Net investment income, pre-tax	$	25.1	$	17.1	$	45.7	$	37.7

Net Realized and Unrealized Investment Gains (Losses)

The following table presents net realized and unrealized investment gains (losses) for the three and six months ended June 30, 2022 and 2021:

	Three Months Ended June 30,				Six Months Ended June 30,
Millions	2022		2021		2022		2021
Realized investment gains (losses)
Fixed maturity investments	$	(4.8)	$	.9	$	(5.9)	$	2.8
Short-term investments	(.1)		.6		(.3)		.3
Common equity securities	—		.4		—		.4
Investment in MediaAlpha	—		—		—		160.3
Other long-term investments	2.0		10.5		20.7		(8.3)
Net realized investment gains (losses)	(2.9)		12.4		14.5		155.5
Unrealized investment gains (losses)
Fixed maturity investments	(54.6)		8.2		(134.1)		(19.2)
Short-term investments	(.7)		1.0		(.5)		—
Common equity securities	(11.6)		4.6		(10.6)		5.6
Investment in MediaAlpha	(113.5)		113.0		(94.7)		(89.0)
Other long-term investments	(37.0)		36.9		13.4		83.9
Net unrealized investment gains (losses)	(217.4)		163.7		(226.5)		(18.7)
Net realized and unrealized investment gains (losses), before    amount attributable to TPC providers(1)	(220.3)		176.1		(212.0)		136.8
Amount attributable to TPC Providers	2.7		(3.5)		4.8		(4.8)
Net realized and unrealized investment gains (losses)	$	(217.6)	$	172.6	$	(207.2)	$	132.0
Fixed maturity and short-term investments
Net realized and unrealized investment gains (losses)	$	(60.2)	$	10.7	$	(140.8)	$	(16.1)
Less: net realized and unrealized gains (losses) on investment           securities sold during the period	(2.1)		(.3)		(3.1)		(1.3)
Net unrealized investment gains (losses) on investment    securities held at the end of the period	$	(58.1)	$	11.0	$	(137.7)	$	(14.8)
Common equity securities and investment in MediaAlpha
Net realized and unrealized investment gains (losses) on common    equity securities	$	(11.6)	$	5.0	$	(10.6)	$	6.0
Net realized and unrealized investment gains (losses) from    investment in MediaAlpha	(113.5)		113.0		(94.7)		71.3
Total net realized and unrealized investment gains (losses)	(125.1)		118.0		(105.3)		77.3
Less: net realized and unrealized gains (losses) on investment          securities sold during the period	—		—		—		19.9
Net unrealized investment gains (losses) on investment    securities held at the end of the period	$	(125.1)	$	118.0	$	(105.3)	$	57.4

(1) For the three months ended June 30, 2022 and 2021, includes $(24.5) and $(0.2) of realized and unrealized investment gains (losses) related to foreign currency exchange. For the six months ended June 30, 2022 and 2021, includes $(21.8) and $(0.4) of realized and unrealized investment gains (losses) related to foreign currency exchange.

The following table presents total gains included in earnings attributable to net unrealized investment gains for Level 3 investments for the three and six months ended June 30, 2022 and 2021 for investments still held at the end of the period:

	Three Months Ended				Six Months Ended
	June 30,				June 30,
Millions	2022		2021		2022		2021
Other long-term investments	$	(10.2)	$	32.3	$	17.0	$	49.0
Total net unrealized investment gains, pre-tax - Level 3 investments	$	(10.2)	$	32.3	$	17.0	$	49.0

Investment Holdings

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 June 30, 2022 and December 31, 2021:

	June 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	$	233.0	$	—	$	(7.2)	$	—	$	225.8
Debt securities issued by corporations	1,043.5		.5		(67.4)		(2.4)		974.2
Municipal obligations	291.8		1.0		(12.5)		—		280.3
Mortgage and asset-backed securities	268.5		—		(23.9)		—		244.6
Collateralized loan obligations	150.8		—		(6.0)		(3.0)		141.8
Total fixed maturity investments	$	1,987.6	$	1.5	$	(117.0)	$	(5.4)	$	1,866.7

	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 table presents the cost or amortized cost and carrying values of White Mountains’s fixed maturity investments by contractual maturity as of June 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.

	June 30, 2022
Millions	Cost or Amortized Cost		Carrying Value
Due in one year or less	$	205.1	$	202.2
Due after one year through five years	892.5		843.7
Due after five years through ten years	361.1		330.0
Due after ten years	109.6		104.4
Mortgage and asset-backed securities and    collateralized loan obligations	419.3		386.4
Total fixed maturity investments	$	1,987.6	$	1,866.7

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 June 30, 2022 and December 31, 2021:

	June 30, 2022
Millions	Cost or Amortized Cost		Gross Unrealized Gains		Gross Unrealized Losses		Net Foreign Currency Gains (Losses)		Carrying Value
Common equity securities	$	274.3	$	20.9	$	(6.7)	$	(9.9)	$	278.6
Investment in MediaAlpha	$	—	$	166.9	$	—	$	—	$	166.9
Other long-term investments	$	1,264.2	$	295.6	$	(81.1)	$	(10.0)	$	1,468.7

	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

Fair Value Measurements

Fair value measurements are categorized into a hierarchy that distinguishes between inputs based on market data from independent sources (observable inputs) and a reporting entity’s internal assumptions based upon the best information available when external market data is limited or unavailable (unobservable inputs). Quoted prices in active markets for identical assets or liabilities have the highest priority (Level 1), followed by observable inputs other than quoted prices, including prices for similar but not identical assets or liabilities (Level 2) and unobservable inputs, including the reporting entity’s estimates of the assumptions that market participants would use, having the lowest priority (Level 3). See Note 19 — “Fair Value of Financial Instruments.”

As of June 30, 2022 and December 31, 2021, White Mountains used quoted market prices or other observable inputs to determine fair value for approximately 64% and 68% of the investment portfolio.

Fair Value Measurements by Level

The following tables present White Mountains’s fair value measurements for investments as of June 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.

	June 30, 2022
Millions	Fair Value		Level 1		Level 2		Level 3
Fixed maturity investments:
U.S. Government and agency obligations	$	225.8	$	225.8	$	—	$	—
Debt securities issued by corporations:
Financials	263.5		—		263.5		—
Consumer	176.8		—		176.8		—
Technology	117.9		—		117.9		—
Industrial	112.7		—		112.7		—
Healthcare	105.9		—		105.9		—
Utilities	75.1		—		75.1		—
Communications	57.8		—		57.8		—
Energy	40.3		—		40.3		—
Materials	24.2		—		24.2		—
Total debt securities issued by corporations	974.2		—		974.2		—
Municipal obligations	280.3		—		280.3		—
Mortgage and asset-backed securities	244.6		—		244.6		—
Collateralized loan obligations	141.8		—		141.8		—
Total fixed maturity investments	1,866.7		225.8		1,640.9		—
Short-term investments	245.1		245.1		—		—
Common equity securities (1)	278.6		—		278.6		—
Investment in MediaAlpha	166.9		166.9		—		—
Other long-term investments	947.1		—		13.8		933.3
Other long-term investments — NAV (2)	521.6		—		—		—
Total other long-term investments	1,468.7		—		13.8		933.3
Total investments	$	4,026.0	$	637.8	$	1,933.3	$	933.3

(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 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.

Investments Held on Deposit or as Collateral

As of June 30, 2022 and December 31, 2021, investments of $461.5 million and $479.5 million were held in trusts required to be maintained in relation to HG Global’s reinsurance agreements with BAM.

BAM and one of the Other Operating Businesses are required to maintain deposits with certain insurance regulatory agencies in order to maintain their insurance licenses. The fair value of such deposits, which represent state deposits and are included within the investment portfolio, totaled $4.8 million as of June 30, 2022 and December 31, 2021.

Lloyd’s trust deposits are generally required of Lloyd's syndicates to protect policyholders in non-U.K. markets and are pledged into Lloyd’s trust accounts to provide a portion of the capital needed to support obligations at Lloyd’s. As of June 30, 2022 and December 31, 2021, Ark held Lloyd’s trust deposits with a fair value of $113.4 million and $113.8 million

The underwriting capacity of a member of Lloyd’s must be supported by providing a deposit (“Funds at Lloyd’s”) in the form of cash, securities or letters of credit in an amount determined by Lloyd’s. The amount of such deposit is calculated for each member through the completion of an annual capital adequacy exercise. These requirements allow Lloyd’s to evaluate that each member has sufficient assets to meet its underwriting liabilities plus a required solvency margin. As of June 30, 2022 and December 31, 2021, the fair value of Ark’s Funds at Lloyd’s investment deposits totaled $304.8 million and $342.8 million.

As of June 30, 2022 and December 31, 2021, Ark had $49.8 million and $50.0 million of short-term investments pledged as collateral under an uncommitted standby letter of credit. See Note 7 — “Debt.”

Debt Securities Issued by Corporations

The following table presents the credit ratings of debt securities issued by corporations held in White Mountains’s investment portfolio as of June 30, 2022 and December 31, 2021:

	Fair Value at
Millions	June 30, 2022		December 31, 2021
AAA	$	11.6	$	12.0
AA	87.2		85.0
A	502.3		490.4
BBB	366.4		396.8
Other	6.7		8.7
Debt securities issued by corporations (1)	$	974.2	$	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 and Asset-backed Securities and Collateralized Loan Obligations

The following table presents the fair value of White Mountains’s mortgage and asset-backed securities and collateralized loan obligations as of June 30, 2022 and December 31, 2021:

	June 30, 2022						December 31, 2021
Millions	Fair Value		Level 2		Level 3		Fair Value		Level 2		Level 3
Mortgage-backed securities:
Agency:
FNMA	$	109.0	$	109.0	$	—	$	125.4	$	125.4	$	—
FHLMC	79.2		79.2		—		90.5		90.5		—
GNMA	32.2		32.2		—		40.1		40.1		—
Total agency (1)	220.4		220.4		—		256.0		256.0		—
Non-agency: Residential	.4		.4		—		.5		.5		—
Total non-agency	.4		.4		—		.5		.5		—
Total mortgage-backed securities	220.8		220.8		—		256.5		256.5		—
Other asset-backed securities:
Credit card receivables	12.0		12.0		—		12.3		12.3		—
Vehicle receivables	11.8		11.8		—		8.8		8.8		—
Total other asset-backed securities	23.8		23.8		—		21.1		21.1		—
Total mortgage and asset-backed securities	244.6		244.6		—		277.6		277.6		—
Collateralized loan obligations	141.8		141.8		—		135.0		135.0		—
Total mortgage and asset-backed securities and collateralized loan obligations	$	386.4	$	386.4	$	—	$	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).

As of June 30, 2022, White Mountains’s investment portfolio included $141.8 million of collateralized loan obligations that are within the senior tranches of their respective fund securitization structures. All of White Mountains’s collateral loan obligations were rated AAA or AA as of June 30, 2022.

Investment in MediaAlpha

Following the MediaAlpha IPO, White Mountains’s investment in MediaAlpha is accounted for at fair value based on the publicly traded share price of MediaAlpha’s common stock and is presented as a separate line item on the balance sheet.

At the June 30, 2022 closing price of $9.85 per share, the fair value of White Mountains’s investment in MediaAlpha was $166.9 million. See Note 2 — “Significant Transactions.”

Other Long-Term Investments

The following table presents the carrying values of White Mountains’s other long-term investments as of June 30, 2022 and December 31, 2021:

	Fair Value at
Millions	June 30, 2022		December 31, 2021
Kudu’s Participation Contracts	$	727.0	$	669.5
PassportCard/DavidShield	132.0		120.0
Elementum Holdings, L.P.	45.0		45.0
Other unconsolidated entities (1)	12.2		34.4
Total unconsolidated entities	916.2		868.9
Private equity funds and hedge funds	200.1		153.8
Bank loan fund	154.4		163.0
Lloyd’s trust deposits	113.4		113.8
ILS funds	51.6		51.9
Private debt investments	13.0		14.1
Other	20.0		12.3
Total other long-term investments	$	1,468.7	$	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.

Private Equity Funds and Hedge Funds

White Mountains invests in private equity funds and hedge funds, which are included in other long-term investments. The fair value of these investments is generally estimated using the NAV of the funds. As of June 30, 2022, White Mountains held investments in 16 private equity funds and one hedge fund. The largest investment in a single private equity fund or hedge fund was $35.4 million and $31.3 million as of June 30, 2022 and December 31, 2021.

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 June 30, 2022 and December 31, 2021:

	June 30, 2022				December 31, 2021
Millions	Fair Value		Unfunded Commitments		Fair Value		Unfunded Commitments
Private equity funds
Aerospace/Defense/Government	$	107.0	$	43.5	$	69.8	$	11.8
Financial services	80.0		55.6		67.7		29.3
Real estate	4.7		2.5		4.3		2.9
Total private equity funds	191.7		101.6		141.8		44.0
Hedge funds
European small/mid cap	8.4		—		12.0		—
Total hedge funds	8.4		—		12.0		—
Total private equity funds and hedge funds    included in other long-term investments	$	200.1	$	101.6	$	153.8	$	44.0

 

Investments in private equity funds are generally subject to a lock-up period during which investors may not request a redemption. Distributions prior to the expected termination date of the fund may be limited to dividends or proceeds arising from the liquidation of the fund’s underlying investments. In addition, certain private equity funds have the option to extend the lock-up period.

The following table presents investments in private equity funds that were subject to lock-up periods as of June 30, 2022:

Millions	1 – 3 years	3 – 5 years	5 – 10 years	>10 years	Total
Private equity funds — expected lock-up period remaining	$29.8	$21.0	$128.8	$12.1	$191.7

Investors in private equity funds are generally subject to indemnification obligations outside of the capital commitment period and prior to the winding up of the fund. As of June 30, 2022 and December 31, 2021, White Mountains is not aware of any indemnification claims relating to its investments in private equity funds. 

Redemption of investments in most hedge funds is subject to restrictions, including lock-up periods where no redemptions or withdrawals are allowed, restrictions on redemption frequency and advance notice periods for redemptions. Amounts requested for redemptions remain subject to market fluctuations until the redemption effective date, which generally falls at the end of the defined redemption period. White Mountains’s hedge fund investment is subject to a perpetual two-year restriction on redemption frequency from the initial investment in the fund and a 90 days advanced notice period requirement.

Bank Loan Fund

White Mountains’s other long-term investments include a bank loan fund with a fair value of $154.4 million as of June 30, 2022. The fair value of this investment is estimated using the NAV of the fund. The bank loan fund’s investment objective is to provide, on an unleveraged basis, high current income consistent with preservation of capital and low duration. The bank loan fund primarily invests in a broad portfolio of U.S. dollar-denominated, non-investment grade, floating-rate senior secured loans and may invest in other financial instruments, such as secured and unsecured corporate debt, credit default swaps, reverse repurchase agreements, synthetic indices and cash and cash equivalents.

The investment in the bank loan fund is subject to restrictions on redemption frequency and advance notice periods for redemptions. Amounts requested for redemptions remain subject to market fluctuations until the redemption effective date, which generally falls at the end of the defined redemption period. White Mountains may redeem all or a portion of its bank loan fund investment as of any calendar month-end upon 15 calendar days advanced written notice.

Lloyd’s Trust Deposits

White Mountains’s other long-term investments include Lloyd’s trust deposits, which consist of non-U.K. deposits and Canadian comingled pooled funds. The Lloyd’s trust deposits invest primarily in short-term government securities, agency securities and corporate bonds held in trusts that are managed by Lloyd's of London. These investments are generally required of Lloyd's syndicates to protect policyholders in non-U.K. markets and are pledged into Lloyd’s trust accounts to provide a portion of the capital needed to support obligations at Lloyd’s. The fair value of the Lloyd’s trust deposits is generally estimated using the NAV of the funds. As of June 30, 2022, White Mountains held Lloyd’s trust deposits with a fair value of $113.4 million.

Insurance-Linked Securities Funds

White Mountains’s other long-term investments include ILS fund investments. The fair value of these investments is generally estimated using the NAV of the funds. As of June 30, 2022, White Mountains held investments in ILS funds with a fair value of $51.6 million.

Investments in ILS funds are generally subject to restrictions, including lock-up periods where no redemptions or withdrawals are allowed, non-renewal clauses, restrictions on redemption frequency and advance notice periods for redemptions. From time to time, natural catastrophe, liquidity, market or other events will occur that make the determination of fair value for underlying investments in ILS funds less certain due to the potential for loss development. In such circumstances, the impacted investments may be subject to additional lock-up provisions.

ILS funds are typically subject to monthly and annual restrictions on redemptions and advance redemption notice period requirements that range between 30 and 90 days. Amounts requested for redemption remain subject to market fluctuations until the redemption effective date, which generally falls at the end of the defined redemption period.

One of the ILS funds in White Mountains’s portfolio requires shareholders to provide advance redemption notice on or before September 15 of each calendar year. Amounts requested for redemption in this fund remain subject to market fluctuation until the underlying investment has fully matured or been commuted, which may be up to a period of three years from the start of each calendar year.

Rollforward of Level 3 Investments

Level 3 measurements as of June 30, 2022 and 2021 consist of securities for which the estimated fair value has not been determined based upon quoted market price inputs for identical or similar securities. The following table presents the changes in White Mountains’s fair value measurements for Level 3 investments for the six months ended June 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	16.6		Net realized and unrealized gains	48.9
Amortization/accretion	—		Amortization/accretion	—
Purchases	57.8		Purchases	13.2
Sales	(31.7)		Sales	(11.5)
Effect of Ark Transaction	—		Effect of Ark Transaction	9.6
Transfers in	—		Transfers in	—
Transfers out	—		Transfers out	—
Balance at June 30, 2022	$	933.3	Balance at June 30, 2021	$	674.4

Fair Value Measurements — Transfers Between Levels - Six months ended June 30, 2022 and 2021

Transfers between levels are recorded using the fair value measurement as of the end of the quarterly period in which the event or change in circumstance giving rise to the transfer occurred.

During the six months ended June 30, 2022 and 2021, there were no fixed maturity investments or other long-term investments classified as Level 3 measurements in the prior period that were transferred to Level 2 measurements.

During the six months ended June 30, 2022 and 2021, there were no fixed maturity investments or other long-term investments classified as Level 2 measurements in the prior period that were transferred to Level 3 measurements.

Significant Unobservable Inputs

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 June 30, 2022 and December 31, 2021. The tables below exclude $18.3 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	June 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	$727.0	17% - 24%	7x - 15x
PassportCard/DavidShield	Discounted cash flow	$132.0	23%	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 11x.

(4) As of June 30, 2022, two of Kudu’s Participation Contracts with a total fair value of $185.9 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 $52.8 into new and existing Kudu Participation Contracts, including Gramercy Funds Management 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.