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Defined contribution and defined benefit retirement plans (Tables) | 12 Months Ended
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Dec. 31, 2022
Funded Status of Defined Benefit Plan | The funded status of our non-U.S. defined benefit pension plans is presented in the table below.
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | December 31,
​ |  | 2021 |  | 2022
​ | ​ | (In millions)
Change in projected benefit obligations (PBO): |  | ​ |  |  | ​ |
Benefit obligations at beginning of the year | ​ | $ | 846.7 | ​ | $ | 748.1
Service cost | ​ |  | 14.7 | ​ |  | 11.3
Interest cost | ​ |  | 8.2 | ​ |  | 10.5
Participant contributions | ​ |  | 2.0 | ​ |  | 1.7
Actuarial gains | ​ |  | (45.4) | ​ |  | (195.8)
Settlements | ​ |  | - | ​ |  | (1.2)
Change in currency exchange rates | ​ |  | (55.0) | ​ |  | (50.3)
Benefits paid | ​ |  | (23.1) | ​ |  | (21.5)
Benefit obligations at end of the year | ​ |  | 748.1 | ​ |  | 502.8
​ | ​ | ​ | ​ | ​ | ​ | ​
Change in plan assets: | ​ |  |  | ​ |  |
Fair value of plan assets at beginning of the year | ​ |  | 481.8 | ​ |  | 470.1
Actual return on plan assets | ​ |  | 17.7 | ​ |  | (49.5)
Employer contributions | ​ |  | 18.7 | ​ |  | 15.0
Participant contributions | ​ |  | 2.0 | ​ |  | 1.7
Settlements | ​ |  | - | ​ |  | (1.2)
Change in currency exchange rates | ​ |  | (27.0) | ​ |  | (31.0)
Benefits paid | ​ |  | (23.1) | ​ |  | (21.5)
Fair value of plan assets at end of year | ​ |  | 470.1 | ​ |  | 383.6
Funded status | ​ | $ | (278.0) | ​ | $ | (119.2)
​ | ​ | ​ | ​ | ​ | ​ | ​
Amounts recognized in the balance sheet: | ​ |  |  | ​ |  |
Noncurrent pension asset | ​ | $ | 7.6 | ​ | $ | 8.2
Noncurrent accrued pension costs | ​ |  | (285.6) | ​ |  | (127.4)
Total | ​ | $ | (278.0) | ​ | $ | (119.2)
​ | ​ | ​ | ​ | ​ | ​ | ​
Amounts recognized in accumulated other comprehensive loss: | ​ |  |  | ​ |  |
Actuarial losses | ​ | $ | 233.3 | ​ | $ | 85.4
Prior service cost | ​ |  | .4 | ​ |  | .4
Total | ​ | $ | 233.7 | ​ | $ | 85.8
​ | ​ | ​ | ​ | ​ | ​ | ​
Accumulated benefit obligations (ABO) | ​ | $ | 723.7 | ​ | $ | 488.0
​ The funded status of our U.S. defined benefit pension plan is presented in the table below.
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | December 31,
​ |  | 2021 |  | 2022
​ | ​ | (In millions)
Change in PBO: |  | ​ |  |  | ​ |
Benefit obligations at beginning of the year | ​ | $ | 19.4 | ​ | $ | 17.7
Interest cost | ​ |  | .5 | ​ |  | .4
Actuarial gains | ​ |  | (.6) | ​ |  | (3.9)
Settlements | ​ | ​ | (.5) | ​ | ​ | -
Benefits paid | ​ |  | (1.1) | ​ |  | (1.0)
Benefit obligations at end of the year | ​ |  | 17.7 | ​ |  | 13.2
​ | ​ | ​ | ​ | ​ | ​ | ​
Change in plan assets: | ​ |  |  | ​ |  |
Fair value of plan assets at beginning of the year | ​ |  | 16.1 | ​ |  | 15.9
Actual return on plan assets | ​ |  | .5 | ​ |  | (3.2)
Employer contributions | ​ |  | .4 | ​ |  | .3
Benefits paid | ​ |  | (1.1) | ​ |  | (1.0)
Fair value of plan assets at end of year | ​ |  | 15.9 | ​ |  | 12.0
Funded status | ​ | $ | (1.8) | ​ | $ | (1.2)
​ | ​ | ​ | ​ | ​ | ​ | ​
Amounts recognized in the balance sheet: | ​ |  |  | ​ |  |
Accrued pension costs: | ​ |  |  | ​ |  |
Current | ​ | $ | - | ​ | $ | -
Noncurrent | ​ |  | (1.8) | ​ |  | (1.2)
Total | ​ | $ | (1.8) | ​ | $ | (1.2)
​ | ​ | ​ | ​ | ​ | ​ | ​
Amounts recognized in accumulated other comprehensive loss - actuarial losses | ​ | $ | 9.8 | ​ | $ | 9.2
​ | ​ | ​ | ​ | ​ | ​ | ​
ABO | ​ | $ | 17.7 | ​ | $ | 13.2
​
Components of Net Periodic Benefit Cost | The components of our net periodic defined benefit pension cost for our non-U.S. defined benefit pension plans are presented in the table below. The amounts shown below for the amortization of prior service cost and recognized actuarial losses for 2020, 2021 and 2022 were recognized as components of our accumulated other comprehensive loss at December 31, 2019, 2020 and 2021, respectively, net of deferred income taxes.
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | Years ended December 31,
​ | 2020 |  | 2021 |  | 2022
​ | (In millions)
Net periodic pension cost (income): | ​ |  |  | ​ |  |  | ​ |
Service cost | $ | 13.3 | ​ | $ | 14.7 | ​ | $ | 11.3
Interest cost |  | 10.0 | ​ |  | 8.2 | ​ |  | 10.5
Expected return on plan assets |  | (8.7) | ​ |  | (11.2) | ​ |  | (11.0)
Amortization of prior service cost | ​ | .2 | ​ | ​ | .2 | ​ | ​ | .1
Recognized actuarial losses |  | 17.3 | ​ |  | 19.4 | ​ |  | 12.5
Settlements | ​ | - | ​ | ​ | - | ​ | ​ | .3
Total | $ | 32.1 | ​ | $ | 31.3 | ​ | $ | 23.7
​ The components of our net periodic defined benefit pension cost for our U.S. defined benefit pension plan is presented in the table below. The amounts shown below for recognized actuarial losses for 2020, 2021 and 2022 were recognized as components of our accumulated other comprehensive loss at December 31, 2019, 2020 and 2021 respectively, net of deferred income taxes.
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Years ended December 31,
​ |  | 2020 |  | 2021 |  | 2022
​ | ​ | (In millions)
Net periodic pension cost (income): |  | ​ |  |  | ​ |  |  | ​ |
Interest cost | ​ | $ | .6 | ​ | $ | .5 | ​ | $ | .4
Expected return on plan assets | ​ |  | (.6) | ​ |  | (.6) | ​ |  | (.6)
Recognized actuarial losses | ​ |  | .6 | ​ |  | .6 | ​ |  | .6
Settlements | ​ |  | - | ​ |  | (.5) | ​ |  | -
Total | ​ | $ | .6 | ​ | $ | - | ​ | $ | .4
​
Schedule of Defined Benefit Pension Plans Balances (for which Accumulated Benefit Obligation Exceeds Fair Value of Plan Assets) | 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: |  | ​ |  |  | ​ |
PBO | ​ | $ | 695.2 | ​ | $ | 403.5
ABO | ​ |  | 674.4 | ​ |  | 392.4
Fair value of plan assets | ​ |  | 409.4 | ​ |  | 276.0
​
Key Actuarial Assumptions Used | The weighted-average rate assumptions used in determining the actuarial present value of benefit obligations for our non-U.S. defined benefit pension plans as of December 31, 2021 and 2022 are presented in the table below.
​ | ​ | ​ | ​ | ​
​ | ​ | December 31,
Rate |  | 2021 |  | 2022
Discount rate |  | 1.5% | ​ | 3.9%
Increase in future compensation levels |  | 2.6% | ​ | 2.7%
​ The weighted-average rate assumptions used in determining the net periodic pension cost for our non-U.S. defined benefit pension plans for 2020, 2021 and 2022 are presented in the table below.
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Years ended December 31,
Rate |  | 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%
​ The weighted-average rate assumptions used in determining the net periodic pension cost for our U.S. defined benefit pension plan for 2020, 2021 and 2022 are presented in the table below. The impact of assumed increases in future compensation levels also does not have an effect on the periodic pension cost as the plan is frozen with regards to compensation.
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Years ended December 31,
Rate |  | 2020 |  | 2021 |  | 2022
Discount rate |  | 3.1% | ​ | 2.2% | ​ | 2.6%
Long-term return on plan assets |  | 4.5% | ​ | 4.0% | ​ | 4.0%
​
Pension Plans Defined Benefit [Member] |
Benefit Payments |
​ | ​ | ​ | ​
Years ending December 31, |  | Amount
​ | ​ | (In millions)
2023 | ​ | $ | 23.0
2024 | ​ |  | 24.7
2025 | ​ |  | 25.1
2026 | ​ |  | 26.4
2027 | ​ |  | 28.5
Next 5 years | ​ |  | 158.7
Changes in Benefit Obligations Recognized in Accumulated Other Comprehensive Income (Loss) | The table below details the changes in our consolidated 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): | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
Current year: |  | ​ |  |  | ​ |  |  | ​ |
Net actuarial (losses) gains | ​ | $ | (36.8) | ​ | $ | 52.5 | ​ | $ | 135.1
Amortization of unrecognized: | ​ |  |  | ​ |  |  | ​ |  |
Net actuarial losses | ​ |  | 17.9 | ​ |  | 20.0 | ​ |  | 13.0
Prior service cost | ​ |  | .2 | ​ |  | .2 | ​ |  | .1
Settlement loss | ​ | ​ | - | ​ | ​ | - | ​ | ​ | .3
Total | ​ | $ | (18.7) | ​ | $ | 72.7 | ​ | $ | 148.5
​
Composition of Pension Plan Assets | 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 | ​ | other | ​ | Significant | ​ | ​ | ​
​ | ​ | ​ | ​ | ​ | in active | ​ | observable | ​ | unobservable | ​ | Assets
​ | ​ | ​ | ​ | ​ | markets | ​ | inputs | ​ | inputs | ​ | measured at
​ |  | Total |  | (Level 1) |  | (Level 2) |  | (Level 3) |  | 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 | ​ |  | 5.6 | ​ |  | .4 | ​ |  | - | ​ |  | - | ​ |  | 5.2
Fixed income | ​ |  | 9.3 | ​ |  | - | ​ |  | - | ​ |  | - | ​ |  | 9.3
Cash and other | ​ |  | 1.0 | ​ |  | .6 | ​ |  | - | ​ |  | - | ​ |  | .4
Other | ​ |  | 18.1 | ​ |  | .4 | ​ |  | - | ​ |  | 17.7 | ​ |  | -
Total | ​ | $ | 486.0 | ​ | $ | 151.6 | ​ | $ | 9.2 | ​ | $ | 310.3 | ​ | $ | 14.9
​
​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | Fair Value Measurements at December 31, 2022
​ | ​ | ​ | ​ | ​ | Quoted | ​ | Significant | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | ​ | ​ | ​ | prices | ​ | other | ​ | Significant | ​ | ​ | ​
​ | ​ | ​ | ​ | ​ | in active | ​ | observable | ​ | unobservable | ​ | Assets
​ | ​ | ​ | ​ | ​ | markets | ​ | inputs | ​ | inputs | ​ | measured at
​ |  | Total |  | (Level 1) |  | (Level 2) |  | (Level 3) |  | 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 | ​ |  | 3.8 | ​ |  | .3 | ​ |  | - | ​ |  | - | ​ |  | 3.5
Fixed income | ​ |  | 7.0 | ​ |  | .1 | ​ |  | - | ​ |  | - | ​ |  | 6.9
Cash and other | ​ |  | 1.2 | ​ |  | .9 | ​ |  | - | ​ |  | - | ​ |  | .3
Other | ​ |  | 17.3 | ​ |  | .4 | ​ |  | - | ​ |  | 16.9 | ​ |  | -
Total | ​ | $ | 395.6 | ​ | $ | 111.1 | ​ | $ | 14.8 | ​ | $ | 259.0 | ​ | $ | 10.7
Roll forward of Change in Fair Value of Level 3 Assets | A rollforward of the change in fair value of Level 3 assets follows.
​ | ​ | ​ | ​ | ​ | ​ | ​
​ | ​ | December 31,
​ |  | 2021 |  | 2022
​ | ​ | (In millions)
Fair value at beginning of year | ​ | $ | 314.3 | ​ | $ | 310.3
Gain (loss) on assets held at end of year | ​ |  | 15.6 | ​ |  | (31.0)
Gain (loss) on assets sold during the year | ​ |  | .4 | ​ |  | (.7)
Assets purchased | ​ |  | 16.2 | ​ |  | 13.8
Assets sold | ​ |  | (14.8) | ​ |  | (14.9)
Transfers in | ​ | ​ | 3.5 | ​ | ​ | -
Currency exchange rate fluctuations | ​ |  | (24.9) | ​ |  | (18.5)
Fair value at end of year | ​ | $ | 310.3 | ​ | $ | 259.0

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