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Defined Contribution and Defined Benefit Retirement | 12 Months Ended
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Dec. 31, 2022
Compensation And Retirement Disclosure [Abstract] |
Defined Benefit Pension Plans | Note 11 – Defined contribution and defined benefit retirement: Defined contribution plans. Certain of our subsidiaries maintain various defined contribution pension plans for our employees worldwide. Defined contribution plan expense approximated $6.6 million in 2020, $7.8 million in 2021 and $8.0 million in 2022. Defined benefit plans. Kronos and NL sponsor various defined benefit pension plans worldwide. The benefits under our defined benefit plans are based upon years of service and employee compensation. Our funding policy is to contribute annually the minimum amount required under ERISA (or equivalent foreign) regulations plus additional amounts as we deem appropriate. We recognize an asset or liability for the over or under funded status of each of our individual defined benefit pension plans on our Consolidated Balance Sheets. Changes in the funded status of these plans are recognized either in net income, to the extent they are reflected in periodic benefit cost, or through other comprehensive income (loss). In accordance with applicable U.K. pension regulations, we entered into an agreement in March 2021 for the bulk annuity purchase, or “buy-in” with a specialist insurer of defined benefit pension plans. Following the buy-in, individual policies will replace the bulk annuity policy in a “buy-out” which is expected to be completed in 2023. The buy-out is expected to be completed with existing plan funds. At the completion of the buy-out we will remove the assets and liabilities of the U.K. pension plan from our Consolidated Financial Statements and a final plan settlement gain or loss (which we are currently unable to estimate) will be included in net periodic pension cost. At December 31, 2022 the U.K. plan had a benefit obligation of $7.8 million, plan assets of $9.3 million and a pension plan asset of $1.5 million was recognized in our Consolidated Balance Sheet. We expect to contribute the equivalent of approximately $18 million to all of our defined benefit pension plans during 2023. Benefit payments to plan participants out of plan assets are expected to be the equivalent of: ​
​ | ​ | ​ | ​
Years ending December 31, |  | Amount
​ | ​ | (In millions)
2023 | ​ | $ | 26.5
2024 | ​ |  | 28.0
2025 | ​ |  | 28.4
2026 | ​ | ​ | 29.6
2027 | ​ |  | 31.7
Next 5 years | ​ | ​ | 173.2
​ The funded status of our U.S. defined benefit pension plans is presented in the table below.
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Years ended December 31,
​ |  | 2021 |  | 2022
​ | ​ | (In millions)
Change in projected benefit obligations (“PBO”): | ​ | ​ | ​ | ​ | ​ | ​
Balance at beginning of the year | ​ | $ | 63.2 | ​ | $ | 58.0
Interest cost | ​ |  | 1.3 | ​ |  | 1.4
Actuarial gains | ​ |  | (1.8) | ​ |  | (11.8)
Settlements | ​ | ​ | (.5) | ​ | ​ | —
Benefits paid | ​ |  | (4.2) | ​ |  | (4.2)
Balance at end of the year | ​ | $ | 58.0 | ​ | $ | 43.4
​ | ​ | ​ | ​ | ​ | ​ | ​
Change in plan assets: | ​ | ​ | ​ | ​ | ​ | ​
Fair value at beginning of the year | ​ | $ | 53.3 | ​ | $ | 52.4
Actual return on plan assets | ​ |  | 1.7 | ​ |  | (10.7)
Employer contributions | ​ |  | 1.6 | ​ |  | 1.6
Benefits paid | ​ |  | (4.2) | ​ |  | (4.2)
Fair value at end of the year | ​ | $ | 52.4 | ​ | $ | 39.1
Funded status | ​ | $ | (5.6) | ​ | $ | (4.3)
​ | ​ | ​ | ​ | ​ | ​ | ​
Amounts recognized in the Consolidated Balance Sheets: | ​ | ​ | ​ | ​ | ​ | ​
Accrued pension costs: | ​ | ​ | ​ | ​ | ​ | ​
Current | ​ | $ | (.1) | ​ | $ | (.1)
Noncurrent | ​ |  | (5.5) | ​ |  | (4.2)
Total | ​ |  | (5.6) | ​ |  | (4.3)
Accumulated other comprehensive loss - actuarial losses | ​ |  | 33.2 | ​ |  | 32.2
Total | ​ | $ | 27.6 | ​ | $ | 27.9
Accumulated benefit obligations (“ABO”) | ​ | $ | 58.0 | ​ | $ | 43.4
​ The total net underfunded status of our U.S. defined benefit pension plans decreased from $5.6 million at December 31, 2021 to $4.3 million at December 31, 2022 due to the change in our PBO during 2022 exceeding the change in our plan assets during 2022. The decrease in our PBO in 2022 was primarily attributable to higher actuarial gains due to the increase in discount rates from year end 2021. The decrease in our plan assets in 2022 was primarily attributable to negative plan asset returns in 2022. The components of our net periodic defined benefit pension cost for U.S. plans are presented in the table below. The amounts shown below for the amortization of recognized actuarial losses for 2020, 2021 and 2022 were recognized as components of our accumulated other comprehensive income (loss) at December 31, 2019, 2020 and 2021, respectively, net of deferred income taxes and noncontrolling interest.
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Years ended December 31,
​ |  | 2020 |  | 2021 |  | 2022
​ | ​ | (In millions)
Net periodic pension cost for U.S. plans: | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
Interest cost | ​ | $ | 1.9 | ​ | $ | 1.3 | ​ | $ | 1.4
Expected return on plan assets | ​ |  | (2.1) | ​ |  | (2.1) | ​ |  | (2.0)
Recognized net actuarial losses | ​ |  | 2.1 | ​ |  | 2.1 | ​ |  | 1.9
Settlements | ​ | ​ | — | ​ | ​ | (.5) | ​ | ​ | —
Total | ​ | $ | 1.9 | ​ | $ | .8 | ​ | $ | 1.3
​ Information concerning our U.S. defined benefit pension plans (for which the ABO of all of the plans exceeds the fair value of plan assets as of the indicated date) is presented in the table below.
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | December 31,
​ |  | 2021 |  | 2022
​ | ​ | (In millions)
Plans for which the ABO exceeds plan assets: | ​ | ​ | ​ | ​ | ​ | ​
Projected benefit obligations | ​ | $ | 58.0 | ​ | $ | 43.4
Accumulated benefit obligations | ​ |  | 58.0 | ​ |  | 43.4
Fair value of plan assets | ​ |  | 52.4 | ​ |  | 39.1
​ The discount rate assumptions used in determining the actuarial present value of the benefit obligation for our U.S. defined benefit pension plans as of December 31, 2021 and 2022 are 2.6% and 5.3%, respectively. The impact of assumed increases in future compensation levels does not have an effect on the benefit obligation as the plans are frozen with regards to compensation. The weighted-average rate assumptions used in determining the net periodic pension cost for our U.S. defined benefit pension plans for 2020, 2021 and 2022 are presented in the table below. The impact of assumed increases in future compensation levels does not have an effect on the periodic pension cost as the plans are frozen with regards to compensation.
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Years ended December 31, |
​ |  | 2020 |  | 2021 |  | 2022 |
Discount rate |  | 3.1% | ​ | 2.2% | ​ | 2.6% | ​
Long-term return on plan assets |  | 4.5% | ​ | 4.0% | ​ | 4.0% | ​
​ Variances from actuarially assumed rates will result in increases or decreases in accumulated pension obligations, pension expense and funding requirements in future periods. The funded status of our non-U.S. defined benefit pension plans is presented in the table below.
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Years ended December 31,
​ |  | 2021 |  | 2022
​ | ​ | (In millions)
Change in PBO: | ​ | ​ | ​ | ​ | ​ | ​
Balance at beginning of the year | ​ | $ | 855.8 | ​ | $ | 758.1
Service cost | ​ |  | 14.7 | ​ |  | 11.3
Interest cost | ​ |  | 8.3 | ​ |  | 10.6
Participants’ contributions | ​ |  | 2.0 | ​ |  | 1.7
Actuarial gains | ​ |  | (43.9) | ​ |  | (198.6)
Settlements | ​ | ​ | — | ​ | ​ | (1.4)
Change in currency exchange rates | ​ |  | (55.2) | ​ |  | (51.2)
Benefits paid | ​ |  | (23.6) | ​ |  | (21.9)
Balance at end of the year | ​ | $ | 758.1 | ​ | $ | 508.6
​ | ​ | ​ | ​ | ​ | ​ | ​
Change in plan assets: | ​ | ​ | ​ | ​ | ​ | ​
Fair value at beginning of the year | ​ | $ | 494.8 | ​ | $ | 481.5
Actual return on plan assets | ​ |  | 16.7 | ​ |  | (52.5)
Employer contributions | ​ |  | 18.7 | ​ |  | 15.0
Participants' contributions | ​ |  | 2.0 | ​ |  | 1.7
Settlements | ​ | ​ | — | ​ | ​ | (1.2)
Change in currency exchange rates | ​ |  | (27.1) | ​ |  | (32.1)
Benefits paid | ​ |  | (23.6) | ​ |  | (21.9)
Fair value at end of the year | ​ | $ | 481.5 | ​ | $ | 390.5
Funded status | ​ | $ | (276.6) | ​ | $ | (118.1)
​ | ​ | ​ | ​ | ​ | ​ | ​
Amounts recognized in the Consolidated Balance Sheets: | ​ | ​ | ​ | ​ | ​ | ​
Noncurrent pension asset | ​ | $ | 9.0 | ​ | $ | 9.3
Noncurrent accrued pension costs | ​ |  | (285.6) | ​ |  | (127.4)
Total | ​ |  | (276.6) | ​ |  | (118.1)
Accumulated other comprehensive loss: | ​ | ​ | ​ | ​ | ​ | ​
Actuarial losses | ​ |  | 238.3 | ​ |  | 90.0
Prior service cost | ​ |  | .4 | ​ |  | .4
Total | ​ |  | 238.7 | ​ |  | 90.4
Total | ​ | $ | (37.9) | ​ | $ | (27.7)
ABO | ​ | $ | 733.8 | ​ | $ | 493.9
​ The total net underfunded status of our non-U.S. defined benefit pension plans decreased from $276.6 million at December 31, 2021 to $118.1 million at December 31, 2022 due to the change in our PBO during 2022 exceeding the change in plan assets during 2022. The decrease in our PBO in 2022 was primarily attributable to higher actuarial gains due to the increase in discount rates from year end 2021 and favorable foreign currency fluctuations, primarily from the strengthening of the U.S. dollar relative to the euro. The decrease in our plan assets in 2022 was primarily attributable to the net effects of negative plan asset returns in 2022, unfavorable currency fluctuations, primarily from the strengthening of the U.S. dollar relative to the euro, and employer contributions. ​ The components of our net periodic pension benefit cost for our non-U.S. plans are presented in the table below. The amounts shown below for the amortization of prior service cost and recognized net actuarial losses for 2020, 2021 and 2022 were recognized as components of our accumulated other comprehensive income (loss) at December 31, 2019, 2020 and 2021, respectively, net of deferred income taxes and noncontrolling interest. ​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Years ended December 31,
​ |  | 2020 |  | 2021 |  | 2022
​ | ​ | (In millions)
Net periodic pension cost for non-U.S. plans: | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
Service cost | ​ | $ | 13.3 | ​ | $ | 14.7 | ​ | $ | 11.3
Interest cost | ​ |  | 10.1 | ​ |  | 8.3 | ​ |  | 10.6
Expected return on plan assets | ​ |  | (9.0) | ​ |  | (11.4) | ​ |  | (11.1)
Recognized net actuarial losses | ​ |  | 17.3 | ​ |  | 19.5 | ​ |  | 12.8
Amortization of prior service cost | ​ |  | .2 | ​ |  | .2 | ​ |  | .1
Settlements | ​ |  | — | ​ |  | — | ​ |  | .4
Total | ​ | $ | 31.9 | ​ | $ | 31.3 | ​ | $ | 24.1
​ Information concerning certain of our non-U.S. defined benefit pension plans (for which the ABO exceeds the fair value of plan assets as of the indicated date) is presented in the table below.
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | December 31,
​ |  | 2021 |  | 2022
​ | ​ | (In millions)
Plans for which the ABO exceeds plan assets: | ​ | ​ | ​ | ​ | ​ | ​
Projected benefit obligations | ​ | $ | 695.2 | ​ | $ | 403.5
Accumulated benefit obligations | ​ |  | 674.4 | ​ |  | 392.4
Fair value of plan assets | ​ |  | 409.4 | ​ |  | 276.0
​ The key actuarial assumptions used to determine our non-U.S. benefit obligations as of December 31, 2021 and 2022 are as follows:
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | ​ | December 31, |
​ | ​ |  | 2021 |  | 2022 |
Discount rate | ​ |  | 1.5% | ​ | 3.9% | ​
Increase in future compensation levels | ​ |  | 2.6% | ​ | 2.7% | ​
​ A summary of our key actuarial assumptions used to determine non-U.S. net periodic benefit cost for 2020, 2021 and 2022 are as follows:
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | ​ | Years ended December 31, |
​ | ​ |  | 2020 |  | 2021 | ​ | 2022 |
Discount rate | ​ |  | 1.4% | ​ | 1.0% | ​ | 1.5% | ​
Increase in future compensation levels | ​ |  | 2.6% | ​ | 2.6% | ​ | 2.6% | ​
Long-term return on plan assets | ​ |  | 2.0% | ​ | 2.4% | ​ | 2.5% | ​
​ Variances from actuarially assumed rates will result in increases or decreases in accumulated pension obligations, pension expense and funding requirements in future periods. The amounts shown for all of our periodic defined benefit plans for actuarial losses and prior service cost at December 31, 2021 and 2022 have not been recognized as components of our periodic defined benefit pension cost as of those dates. These amounts will be recognized as components of our periodic defined benefit cost in future years. These amounts, net of deferred income taxes and noncontrolling interest, are recognized in our accumulated other comprehensive income (loss) at December 31, 2021 and 2022. We expect approximately $3.8 million and $.1 million of the unrecognized actuarial losses and prior service cost, respectively, will be recognized as components of our periodic defined benefit pension cost in 2023. The table below details the changes in other comprehensive income (loss) during 2020, 2021 and 2022. ​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ |  | Years ended December 31,
​ | ​ | 2020 | ​ | 2021 | ​ | 2022
​ | (In millions)
Changes in plan assets and benefit obligations recognized in other comprehensive income (loss): |  | ​ |  |  | ​ |  |  | ​ |
Net actuarial gains (losses) | ​ | $ | (37.7) | ​ | $ | 50.7 | ​ | $ | 134.1
Amortization of unrecognized: | ​ |  |  | ​ |  |  | ​ |  |
Net actuarial losses | ​ |  | 19.4 | ​ |  | 21.6 | ​ |  | 14.7
Prior service cost | ​ |  | .2 | ​ |  | .2 | ​ |  | .1
Settlements | ​ |  | — | ​ |  | — | ​ |  | .4
Total | ​ | $ | (18.1) | ​ | $ | 72.5 | ​ | $ | 149.3
​ In determining the expected long-term rate of return on plan asset assumptions, we consider the long-term asset mix (e.g. equity vs. fixed income) for the assets for each of our plans and the expected long-term rates of return for such asset components. In addition, we receive third-party advice about appropriate long-term rates of return. Such assumed asset mixes are summarized below:
| ● | In Germany, the composition of our plan assets is established to satisfy the requirements of the German insurance commissioner. Our German pension plan assets represent an investment in a large collective investment fund established and maintained by Bayer AG in which several pension plans, including our German pension plans and Bayer’s pension plans, have invested. Our plan assets represent a very nominal portion of the total collective investment fund maintained by Bayer. These plan assets are a Level 3 in the fair value hierarchy because there is not an active market that approximates the value of our investment in the Bayer investment fund. We estimate the fair value of the Bayer plan assets based on periodic reports we receive from the managers of the Bayer fund and using a model we developed with assistance from our third-party actuary that uses estimated asset allocations and correlates such allocation to similar asset mixes in fund indexes quoted on an active market. We periodically evaluate the results of our valuation model against actual returns in the Bayer fund and adjust the model as needed. The Bayer fund periodic reports are subject to audit by the German pension regulator.
| ● | In Canada, we currently have a plan asset target allocation of 10 - 20% to equity securities and 80 - 90% to fixed income securities. We expect the long-term rate of return for such investments to approximate the applicable equity or fixed income index. The Canadian assets are Level 1 inputs because they are traded in active markets.
| ● | In Norway, we currently have a plan asset target allocation of 15% to equity securities, 62% to fixed income securities, 14% to real estate and the remainder primarily to other investments and liquid investments such as money markets. The expected long-term rate of return for such investments is approximately 7%, 4%, 6% and 4%, respectively. The majority of Norwegian plan assets are Level 1 inputs because they are traded in active markets; however approximately 17% of our Norwegian plan assets are invested in real estate and other investments not actively traded and are therefore a Level 3 input.
| ● | In the U.S. we currently have a plan asset target allocation of 34% to equity securities, 59% to fixed income securities, and the remainder is allocated to multi-asset strategies. The expected long-term rate of return for such investments is approximately 7%, 5% and 4%, respectively (before plan administrative expenses). Approximately 90% of our U.S. plan assets are invested in funds that are valued at net asset value (NAV) and not subject to classification in the fair value hierarchy.
| ● | We also have plan assets in Belgium and the United Kingdom. The Belgian plan assets are invested in certain individualized fixed income insurance contracts for the benefit of each plan participant as required by the local regulators and are therefore a Level 3 input. The United Kingdom plan assets are invested primarily in insurance contracts and are a Level 3 input.
We regularly review our actual asset allocation for each plan, and will periodically rebalance the investments in each plan to more accurately reflect the targeted allocation and/or maximize the overall long-term return when considered appropriate. The composition of our pension plan assets by asset category and fair value level at December 31, 2021 and 2022 is shown in the tables below.
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Fair Value Measurements at December 31, 2021
​ |  | ​ | ​ |  | Quoted |  | Significant |  | ​ | ​ |  | ​ | ​
​ | ​ | ​ | ​ | ​ | Prices in | ​ | Other | ​ | Significant | ​ | ​ | ​
​ | ​ | ​ | ​ | ​ | Active | ​ | Observable | ​ | Unobservable | ​ | Assets
​ | ​ | ​ | ​ | ​ | Markets | ​ | Inputs | ​ | Inputs | ​ | measured
​ | ​ | Total | ​ | (Level 1) | ​ | (Level 2) | ​ | (Level 3) | ​ | at NAV
​ | ​ | (In millions)
Germany | ​ | $ | 282.9 | ​ | $ | — | ​ | $ | — | ​ | $ | 282.9 | ​ | $ | —
Canada: | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  |
Local currency equities | ​ |  | .2 | ​ |  | .2 | ​ |  | — | ​ |  | — | ​ |  | —
Non local currency equities | ​ |  | 21.9 | ​ |  | 21.9 | ​ |  | — | ​ |  | — | ​ |  | —
Local currency fixed income | ​ |  | 89.3 | ​ |  | 89.3 | ​ |  | — | ​ |  | — | ​ |  | —
Cash and other | ​ |  | .8 | ​ |  | .8 | ​ |  | — | ​ |  | — | ​ |  | —
Norway: | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  |
Local currency equities | ​ |  | 3.1 | ​ |  | 3.1 | ​ |  | — | ​ |  | — | ​ |  | —
Non local currency equities | ​ |  | 5.9 | ​ |  | 5.9 | ​ |  | — | ​ |  | — | ​ |  | —
Local currency fixed income | ​ |  | 25.1 | ​ |  | 15.9 | ​ |  | 9.2 | ​ |  | — | ​ |  | —
Non local currency fixed income | ​ |  | 6.7 | ​ |  | 6.7 | ​ |  | — | ​ |  | — | ​ |  | —
Real estate | ​ |  | 9.1 | ​ |  | — | ​ |  | — | ​ |  | 9.1 | ​ |  | —
Cash and other | ​ |  | 7.0 | ​ |  | 6.4 | ​ |  | — | ​ |  | .6 | ​ |  | —
U.S.: | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  |
Equities | ​ |  | 18.5 | ​ |  | 1.2 | ​ |  | — | ​ |  | .1 | ​ |  | 17.2
Fixed income | ​ |  | 30.6 | ​ |  | — | ​ |  | — | ​ |  | — | ​ |  | 30.6
Cash and other | ​ |  | 3.2 | ​ |  | 1.9 | ​ |  | — | ​ |  | — | ​ |  | 1.3
Other | ​ |  | 29.6 | ​ |  | 1.8 | ​ |  | — | ​ |  | 27.8 | ​ |  | —
Total | ​ | $ | 533.9 | ​ | $ | 155.1 | ​ | $ | 9.2 | ​ | $ | 320.5 | ​ | $ | 49.1
​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Fair Value Measurements at December 31, 2022
​ |  | ​ | ​ |  | Quoted |  | Significant |  | ​ | ​ |  | ​ | ​
​ | ​ | ​ | ​ | ​ | Prices in | ​ | Other | ​ | Significant | ​ | ​ | ​
​ | ​ | ​ | ​ | ​ | Active | ​ | Observable | ​ | Unobservable | ​ | Assets
​ | ​ | ​ | ​ | ​ | Markets | ​ | Inputs | ​ | Inputs | ​ | measured
​ | ​ | Total | ​ | (Level 1) | ​ | (Level 2) | ​ | (Level 3) | ​ | at NAV
​ | ​ | (In millions)
Germany | ​ | $ | 234.0 | ​ | $ | — | ​ | $ | — | ​ | $ | 234.0 | ​ | $ | —
Canada: | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  |
Local currency equities | ​ |  | .1 | ​ |  | .1 | ​ |  | — | ​ |  | — | ​ |  | —
Non local currency equities | ​ |  | 11.0 | ​ |  | 11.0 | ​ |  | — | ​ |  | — | ​ |  | —
Local currency fixed income | ​ |  | 72.9 | ​ |  | 72.9 | ​ |  | — | ​ |  | — | ​ |  | —
Cash and other | ​ |  | .6 | ​ |  | .6 | ​ |  | — | ​ |  | — | ​ |  | —
Norway: | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  |
Local currency equities | ​ |  | 2.3 | ​ |  | 2.3 | ​ |  | — | ​ |  | — | ​ |  | —
Non local currency equities | ​ |  | 4.7 | ​ |  | 4.7 | ​ |  | — | ​ |  | — | ​ |  | —
Local currency fixed income | ​ |  | 21.8 | ​ |  | 7.0 | ​ |  | 14.8 | ​ |  | — | ​ |  | —
Non local currency fixed income | ​ |  | 8.4 | ​ |  | 8.4 | ​ |  | — | ​ |  | — | ​ |  | —
Real estate | ​ |  | 7.8 | ​ |  | — | ​ |  | — | ​ |  | 7.8 | ​ |  | —
Cash and other | ​ |  | 2.7 | ​ |  | 2.4 | ​ |  | — | ​ |  | .3 | ​ |  | —
U.S.: | ​ |  |  | ​ |  |  | ​ |  |  | ​ |  | ​ | ​ |  |
Equities | ​ |  | 12.4 | ​ |  | 1.0 | ​ |  | — | ​ |  | — | ​ |  | 11.4
Fixed income | ​ |  | 22.9 | ​ |  | .2 | ​ |  | — | ​ |  | — | ​ |  | 22.7
Cash and other | ​ |  | 3.8 | ​ |  | 2.8 | ​ |  | — | ​ |  | — | ​ |  | 1.0
Other | ​ |  | 24.2 | ​ |  | 1.5 | ​ |  | — | ​ |  | 22.7 | ​ |  | —
Total | ​ | $ | 429.6 | ​ | $ | 114.9 | ​ | $ | 14.8 | ​ | $ | 264.8 | ​ | $ | 35.1
​ A rollforward of the change in fair value of Level 3 assets follows.
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Years ended December 31,
​ |  | 2021 |  | 2022
​ | ​ | (In millions)
Fair value at beginning of year | ​ | $ | 314.8 | ​ | $ | 320.5
Gain (loss) on assets held at end of year | ​ |  | 15.2 | ​ |  | (31.0)
Gain (loss) on assets sold during the year | ​ |  | .4 | ​ |  | (3.6)
Assets purchased | ​ |  | 16.2 | ​ |  | 13.8
Assets sold | ​ |  | (14.8) | ​ |  | (15.5)
Transfers in (out) | ​ | ​ | 13.9 | ​ | ​ | (.1)
Currency exchange rate fluctuations | ​ |  | (25.2) | ​ |  | (19.3)
Fair value at end of year | ​ | $ | 320.5 | ​ | $ | 264.8
​

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