2012 OASDI Trustees Report

C. FISCAL YEAR HISTORICAL DATA AND
PROJECTIONS THROUGH 2021

Tables VI.C1, VI.C2, and VI.C3 contain details of the fiscal year 2011 operations of the OASI, DI, and the combined OASI and DI Trust Funds, respectively. Fiscal year 2011 is the most recent fiscal year for which complete information is available. These tables are similar to the calendar year operations tables in section III.A. Please see that section for a description of the various items of income and outgo.

 

Table VI.C1.—Operations of the OASI Trust Fund, Fiscal Year 2011

[In millions]

Total assets, September 30, 2010					$2,398,377
Receipts:
	Net payroll tax contributions:
		Payroll tax contributions		$496,821
		Payments from the General Fund of the Treasury for payroll tax contributions subject to refund		-1,790
			Net payroll tax contributions		495,031
	Reimbursements from the general fund:
		Reduction in payroll tax contributions due to P.L. 111-312		67,433
		Reduction in payroll tax contributions due to P.L. 111-147		1,559
		Reimbursements directed by P.L. 110-246		7
		Adjustment of previous determinations of costs attributable to noncontributory wage credits for military service performed before 1957		-113
		Reimbursement for the costs of payments to uninsured persons who attained age 72 before 1968		a
		Payroll tax credits due to P.L. 98-21		a
			Net general fund reimbursements		68,886
	Income based on taxation of benefit payments:
		Withheld from benefit payments to nonresident aliens		161
		All other, not subject to withholding		21,013
			Total income from taxation of benefits		21,174
	Investment income and interest adjustments:
		Interest on investments		107,411
		Net Interest adjustments b		6
			Net investment income and interest adjustments		107,418
	Gifts				1
Total receipts					692,510
Disbursements:
	Benefit payments:
		Monthly benefits and lump-sum death benefits		591,535
		Reimbursement from the general fund for unnegotiated checks		-59
		Payment for costs of vocational rehabilitation services for disabled beneficiaries		1
			Net benefit payments		591,477
	Financial interchange with the Railroad Retirement “Social Security Equivalent Benefit Account”				4,110
	Administrative expenses:
		Costs incurred by:
			Social Security Administration	2,945
			Department of the Treasury	712
		Offsetting receipts from sales of supplies, materials, etc.		-5
		Miscellaneous reimbursements from the general fund c		-7
			Net administrative expenses		3,645
Total disbursements					599,232
Net increase in assets					93,278
Total assets, September 30, 2011					2,491,654

---

a

Between -$0.5 and $0.5 million.

b

Includes: (1) interest on adjustments in the allocation of administrative expenses between the trust fund and the general fund account for the Supplemental Security Income program; (2) interest arising from the revised allocation of administrative expenses among the trust funds; and (3) interest on certain reimbursements to the trust fund.

c

Reimbursements for costs incurred in performing certain legislatively mandated activities not directly related to administering the OASI program.

Note: Totals do not necessarily equal the sums of rounded components.

 

Table VI.C2.—Operations of the DI Trust Fund, Fiscal Year 2011 

[In millions]

Total assets, September 30, 2010					$186,946
Receipts:
	Net payroll tax contributions:
		Payroll tax contributions		$84,335
		Payments from the General Fund of the Treasury for payroll tax contributions subject to refund		-304
			Net payroll tax contributions		84,031
	Reimbursements from the general fund:
		Reduction in payroll tax contributions due to P.L. 111-312		11,482
		Reduction in payroll tax contributions due to P.L. 111-147		265
		Reimbursements directed by P.L. 110-246		1
		Adjustment of previous determinations of costs attributable to noncontributory wage credits for military service performed before 1957		-3
		Payroll tax credits due to P.L. 98-21		a
			Net general fund reimbursements		11,745
	Income based on taxation of benefit payments:
		Withheld from benefit payments to nonresident aliens		4
		All other, not subject to withholding		1,874
			Total income from taxation of benefits		1,878
	Investment income and interest adjustments:
		Interest on investments		8,567
		Interest adjustmentsb		4
			Total investment income and interest adjustments		8,571
Total receipts					106,225
Disbursements:
	Benefit payments:
		Monthly benefits		127,980
		Reimbursement from the general fund for unnegotiated checks		-31
		Payment for costs of vocational rehabilitation services for disabled beneficiaries		41
			Net benefit payments		127,990
	Financial interchange with the Railroad Retirement “Social Security Equivalent Benefit Account”				465
	Administrative expenses:
		Costs incurred by:
			Social Security Administration	2,877
			Department of the Treasury	133
		Demonstration projects		27
		Miscellaneous reimbursements from the general fund c		-3
			Total administrative expenses		3,034
Total disbursements					131,489
Net increase in assets					-25,264
Total assets, September 30, 2011					161,682

---

a

Between -$0.5 and $0.5 million.

b

Includes: (1) interest on adjustments in the allocation of administrative expenses between the trust fund and the general fund account for the Supplemental Security Income program; (2) interest arising from the revised allocation of administrative expenses among the trust funds; and (3) interest on certain reimbursements to the trust fund.

c

Reimbursements for costs incurred in performing certain legislatively mandated activities not directly related to administering the DI program.

Note: Totals do not necessarily equal the sums of rounded components.

 

Table VI.C3.—Operations of the Combined OASI and DI Trust Funds, Fiscal Year 2011 

[In millions]

Total assets, September 30, 2010					$2,585,323
Receipts:
	Net payroll tax contributions:
		Payroll tax contributions		$581,156
		Payments from the General Fund of the Treasury for payroll tax contributions subject to refund		-2,094
			Net payroll tax contributions		579,062
	Reimbursements from the general fund:
		Reduction in payroll tax contributions due to P.L. 111-312		78,915
		Reduction in payroll tax contributions due to P.L. 111-147		1,824
		Reimbursements directed by P.L. 110-246		8
		Adjustment of previous determinations of costs attributable to noncontributory wage credits for military service performed before 1957		-116
		Reimbursement for the costs of payments to uninsured persons who attained age 72 before 1968		a
		Payroll tax credits due to P.L. 98-21		a
			Net general fund reimbursements		80,631
	Income based on taxation of benefit payments:
		Withheld from benefit payments to nonresident aliens		166
		All other, not subject to withholding		22,887
			Total income from taxation of benefits		23,053
	Investment income and interest adjustments:
		Interest on investments		115,978
		Net Interest adjustmentsb		10
			Net investment income and interest adjustments		115,988
	Gifts				1
Total receipts					798,735
Disbursements:
	Benefit payments:
		Monthly benefits and lump-sum death benefits		719,515
		Reimbursement from the general fund for unnegotiated checks		-90
		Payment for costs of vocational rehabilitation services for disabled beneficiaries		42
			Net benefit payments		719,467
	Financial interchange with the Railroad Retirement “Social Security Equivalent Benefit Account”				4,574
	Administrative expenses:
		Costs incurred by:
			Social Security Administration	5,822
			Department of the Treasury	845
		Offsetting receipts from sales of supplies, materials, etc.		-5
		Demonstration projects		27
		Miscellaneous reimbursements from the general fundc		-10
			Net administrative expenses		6,680
Total disbursements					730,721
Net increase in assets					68,014
Total assets, September 30, 2011					2,653,336

---

a

Between -$0.5 and $0.5 million.

b

Includes: (1) interest on adjustments in the allocation of administrative expenses between the trust funds and the general fund account for the Supplemental Security Income program; (2) interest arising from the revised allocation of administrative expenses among the trust funds; and (3) interest on certain reimbursements to the trust funds.

c

Reimbursements for costs incurred in performing certain legislatively mandated activities not directly related to administering the OASI and DI programs.

Note: Totals do not necessarily equal the sums of rounded components.

Tables VI.C4, VI.C5, and VI.C6 show estimates of the operations and status of the OASI, DI, and the combined OASI and DI Trust Funds, respectively, during fiscal years 2007-21. Since 1976, the fiscal year for the U.S. Government is the 12-month period ending September 30.

 

Table VI.C4.—Operations of the OASI Trust Fund, Fiscal Years 2007-21

[Dollar amounts in billions]

Fiscal year		Income						Cost					Assets
		Total	Net pay- roll tax contri- butions	GF reim- burse- mentsa	Taxa- tion of benefits	Net interest		Total	Benefit pay- ments	Admin- istra- tive costs	RRB inter- change		Net increase during year	Amount at end of year	Trust fund ratio b
Historical data:
	2007	$663.4	$553.4	c	$16.7	$93.3		$488.6	$481.8	$3.2	$3.6		$174.8	$1,967.0	367
	2008	692.9	573.7	c	16.4	102.7		509.9	503.0	3.3	3.6		183.0	2,150.1	386
	2009	697.3	571.2	c	19.0	107.1		551.5	544.5	3.4	3.7		145.8	2,295.8	390
	2010	682.4	552.0	$0.7	21.1	108.6		579.9	572.5	3.5	3.9		102.5	2,398.4	396
	2011	692.5	495.0	68.9	21.2	107.4		599.2	591.5	3.6	4.1		93.3	2,491.7	400
Intermediate:
	2012	729.3	500.7	94.8	28.6	105.3		635.7	628.4	3.4	4.0		93.5	2,585.2	392
	2013	749.6	593.7	23.7	29.0	103.2		675.0	667.9	3.3	3.9		74.6	2,659.8	383
	2014	792.9	657.0	-.1	31.9	104.2		716.5	708.8	3.4	4.2		76.5	2,736.2	371
	2015	837.2	694.7	-.1	35.4	107.2		762.0	754.1	3.5	4.3		75.2	2,811.5	359
	2016	894.3	743.5	.1	39.2	111.5		810.6	802.9	3.6	4.1		83.7	2,895.1	347
	2017	946.3	786.1	.1	43.5	116.7		863.9	855.5	3.8	4.6		82.4	2,977.5	335
	2018	1,004.8	834.1	c	47.7	123.0		922.3	913.6	3.9	4.7		82.5	3,060.0	323
	2019	1,058.9	877.8	c	52.0	129.1		986.1	977.2	4.0	4.9		72.7	3,132.8	310
	2020	1,109.3	916.3	c	56.7	136.3		1,056.2	1,047.0	4.2	5.1		53.1	3,185.8	297
	2021	1,168.0	965.4	c	61.7	141.0		1,129.7	1,120.6	4.3	4.9		38.3	3,224.1	282
Low-cost:
	2012	731.2	502.2	95.1	28.6	105.3		635.6	628.2	3.4	4.0		95.6	2,587.3	392
	2013	766.6	609.0	24.6	28.9	104.2		673.5	666.4	3.3	3.8		93.2	2,680.4	384
	2014	814.9	677.3	-.1	31.6	106.1		710.3	702.7	3.4	4.2		104.6	2,785.0	377
	2015	864.7	719.6	-.1	34.8	110.4		749.2	741.5	3.5	4.2		115.5	2,900.5	372
	2016	926.9	772.4	.1	38.3	116.1		791.1	783.6	3.6	3.9		135.8	3,036.2	367
	2017	980.3	814.8	.1	42.1	123.3		837.2	829.1	3.7	4.4		143.0	3,179.3	363
	2018	1,037.4	860.5	c	45.8	131.1		886.5	878.2	3.8	4.5		150.9	3,330.2	359
	2019	1,091.8	902.2	c	49.5	140.0		940.1	931.6	3.9	4.6		151.7	3,481.9	354
	2020	1,143.1	939.6	c	53.6	149.9		998.3	989.5	4.0	4.7		144.8	3,626.7	349
	2021	1,204.5	988.1	c	57.8	158.5		1,058.5	1,049.9	4.1	4.5		145.9	3,772.6	343
High-cost:
	2012	726.0	498.0	94.2	28.6	105.2		636.0	628.6	3.4	4.0		90.1	2,581.7	392
	2013	727.1	574.0	22.5	29.1	101.5		676.7	669.5	3.3	3.9		50.4	2,632.1	382
	2014	767.6	634.1	-.1	32.2	101.5		722.8	715.1	3.4	4.3		44.8	2,676.9	364
	2015	803.5	664.1	-.1	35.9	103.6		774.5	766.5	3.5	4.4		29.0	2,705.9	346
	2016	856.8	709.8	.1	40.2	106.7		832.0	824.1	3.7	4.2		24.8	2,730.7	325
	2017	909.1	753.0	.1	45.1	111.0		896.6	887.9	3.8	4.8		12.5	2,743.2	305
	2018	970.0	803.0	c	50.0	116.9		967.6	958.6	4.0	5.0		2.3	2,745.6	283
	2019	1,029.5	853.3	c	55.1	121.1		1,046.4	1,036.9	4.2	5.3		-16.8	2,728.8	262
	2020	1,083.2	898.3	c	60.7	124.2		1,131.7	1,121.9	4.3	5.5		-48.5	2,680.3	241
	2021	1,141.2	950.9	c	66.7	123.6		1,221.3	1,211.5	4.5	5.4		-80.1	2,600.1	219

---

a

Includes reimbursements from the General Fund of the Treasury to the OASI Trust Fund for: (1) the cost of noncontributory wage credits for military service before 1957; (2) the cost of benefits to certain uninsured persons who attained age 72 before 1968; (3) the cost of payroll tax credits provided to employees in 1984 and self-employed persons in 1984-89 by Public Law 98-21; (4) the cost in 2009-17 of excluding certain self-employment earnings from SECA taxes under Public Law 110-246; and (5) payroll tax revenue forgone under the provisions of Public Laws 111-147, 111-312, and 112-96.

b

The “Trust fund ratio” column represents assets at the beginning of a year (which are identical to assets at the end of the prior year shown in the “Amount at end of year” column) as a percentage of cost for the year.

c

Between -$50 million and $50 million.

Note: Totals do not necessarily equal the sums of rounded components.

 

Table VI.C5.—Operations of the DI Trust Fund, Fiscal Years 2007-21

[Dollar amounts in billions]

Fiscal year		Income						Cost					Assets
		Total	Net pay- roll tax contri- butions	GF reim- burse- mentsa	Taxa- tion of benefits	Net interest		Total	Benefit pay- ments	Admin- istra- tive costs	RRB inter- change		Net increase during year	Amount at end of year	Trust fund ratio b
Historical data:
	2007	$108.4	$94.0	c	$1.4	$13.1		$96.8	$94.0	$2.4	$0.4		$11.6	$213.6	209
	2008	109.8	97.4	c	1.4	11.0		107.2	104.2	2.5	.4		2.7	216.2	199
	2009	109.7	97.0	c	1.8	10.8		118.1	115.1	2.6	.4		-8.5	207.8	183
	2010	105.5	93.7	$0.1	1.7	9.9		126.3	122.9	2.9	.5		-20.8	186.9	164
	2011	106.2	84.0	11.7	1.9	8.6		131.5	128.0	3.0	.5		-25.3	161.7	142
Intermediate:
	2012	109.4	85.0	16.1	1.1	7.2		138.8	135.2	3.1	.5		-29.4	132.3	116
	2013	112.9	100.8	4.0	2.5	5.6		146.0	142.3	3.1	.5		-33.1	99.2	91
	2014	118.2	111.6	c	2.7	4.0		151.8	147.9	3.4	.5		-33.6	65.6	65
	2015	123.5	118.0	c	3.0	2.5		157.8	153.7	3.6	.5		-34.3	31.3	42
	2016	d	126.3	c	3.3	d		163.6	159.3	3.8	.5		d	d	19
	2017	d	133.5	c	3.6	d		169.3	164.8	4.1	.5		d	d	d
	2018	d	141.6	c	3.9	d		175.4	170.7	4.3	.4		d	d	d
	2019	d	149.1	c	4.2	d		182.1	177.1	4.6	.4		d	d	d
	2020	d	155.6	c	4.6	d		189.6	184.4	4.8	.4		d	d	d
	2021	d	163.9	c	5.0	d		198.9	193.5	5.1	.3		d	d	d
Low-cost:
	2012	109.7	85.3	16.1	1.1	7.2		137.2	133.7	3.1	.5		-27.5	134.2	118
	2013	115.7	103.4	4.2	2.4	5.7		142.0	138.4	3.1	.5		-26.3	107.9	94
	2014	122.0	115.0	c	2.6	4.4		144.8	140.9	3.4	.5		-22.8	85.1	74
	2015	128.5	122.2	c	2.8	3.5		147.4	143.3	3.6	.5		-18.9	66.2	58
	2016	137.0	131.2	c	3.0	2.8		149.9	145.6	3.8	.4		-12.9	53.3	44
	2017	143.9	138.4	c	3.3	2.2		152.5	148.0	4.0	.4		-8.6	44.6	35
	2018	151.5	146.1	c	3.5	1.9		155.5	150.9	4.3	.4		-4.1	40.6	29
	2019	158.7	153.2	c	3.7	1.8		159.2	154.4	4.5	.4		-.6	40.0	25
	2020	165.2	159.6	c	3.9	1.7		163.5	158.5	4.7	.3		1.7	41.7	24
	2021	173.9	167.8	c	4.2	1.9		169.1	163.8	5.0	.3		4.8	46.6	25
High-cost:
	2012	108.9	84.6	16.0	1.1	7.2		140.4	136.9	3.1	.5		-31.6	130.1	115
	2013	109.2	97.5	3.8	2.6	5.4		150.3	146.7	3.1	.5		-41.1	89.0	87
	2014	113.9	107.7	c	2.9	3.4		159.7	155.7	3.4	.5		-45.7	43.3	56
	2015	d	112.8	c	3.2	d		169.5	165.4	3.6	.5		d	d	26
	2016	d	120.5	c	3.6	d		179.6	175.3	3.8	.5		d	d	d
	2017	d	127.9	c	4.1	d		189.7	185.1	4.1	.5		d	d	d
	2018	d	136.4	c	4.5	d		200.0	195.2	4.4	.5		d	d	d
	2019	d	144.9	c	4.9	d		210.9	205.7	4.7	.4		d	d	d
	2020	d	152.5	c	5.4	d		222.1	216.7	5.0	.4		d	d	d
	2021	d	161.5	c	5.9	d		235.0	229.4	5.3	.4		d	d	d

---

a

Includes reimbursements from the General Fund of the Treasury to the DI Trust Fund for: (1) the cost of noncontributory wage credits for military service before 1957; (2) the cost of payroll tax credits provided to employees in 1984 and self-employed persons in 1984-89 by Public Law 98-21; (3) the cost in 2009-17 of excluding certain self-employment earnings from SECA taxes under Public Law 110-246; and (4) payroll tax revenue forgone under the provisions of Public Laws 111-147, 111-312, and 112-96.

b

The “Trust fund ratio” column represents assets at the beginning of a year (which are identical to assets at the end of the prior year shown in the “Amount at end of year” column) as a percentage of cost for the year.

c

Between -$50 million and $50 million.

d

The DI Trust Fund becomes exhausted in fiscal years 2016 and 2015 under the intermediate and the high-cost assumptions, respectively. Accordingly, certain trust fund operation values from the year of trust fund exhaustion through 2021 are not meaningful under present law.

Note: Totals do not necessarily equal the sums of rounded components.

 

Table VI.C6.—Operations of the Combined OASI and DI Trust Funds,
Fiscal Years 2007-21 

[Dollar amounts in billions]

Fiscal year		Income						Cost					Assets
		Total	Net pay-roll tax contri- butions	GF reim- burse- mentsa	Taxa- tion of benefits	Net interest		Total	Benefit pay- ments	Admin- istra- tive costs	RRB inter- change		Net increase during year	Amount at end of year	Trust fund ratio b
Historical data:
	2007	$771.8	$647.4	c	$18.0	$106.4		$585.3	$575.8	$5.5	$4.0		$186.5	$2,180.6	341
	2008	802.7	671.2	c	17.8	113.7		617.0	607.2	5.8	4.0		185.7	2,366.3	353
	2009	807.0	668.2	c	20.8	118.0		669.7	659.6	6.0	4.1		137.3	2,503.6	353
	2010	788.0	645.8	$0.9	22.8	118.5		706.3	695.4	6.4	4.4		81.7	2,585.3	354
	2011	798.7	579.1	80.6	23.1	116.0		730.7	719.5	6.7	4.6		68.0	2,653.3	354
Intermediate:
	2012	838.7	585.7	110.8	29.7	112.4		774.5	763.7	6.4	4.4		64.1	2,717.5	343
	2013	862.5	694.5	27.8	31.5	108.8		821.0	810.2	6.4	4.3		41.5	2,759.0	331
	2014	911.2	768.5	-.1	34.6	108.1		868.3	856.7	6.9	4.7		42.9	2,801.9	318
	2015	960.7	812.7	-.1	38.4	109.7		919.8	907.8	7.1	4.8		40.9	2,842.8	305
	2016	1,024.8	869.7	.1	42.5	112.4		974.3	962.3	7.5	4.5		50.5	2,893.3	292
	2017	1,082.8	919.6	.1	47.1	116.0		1,033.2	1,020.3	7.9	5.1		49.5	2,942.8	280
	2018	1,148.1	975.7	c	51.6	120.7		1,097.7	1,084.3	8.2	5.2		50.4	2,993.1	268
	2019	1,208.2	1,026.8	c	56.2	125.1		1,168.2	1,154.3	8.6	5.3		40.0	3,033.1	256
	2020	1,263.6	1,071.9	c	61.3	130.4		1,245.9	1,231.4	9.0	5.4		17.7	3,050.8	243
	2021	1,329.0	1,129.3	c	66.7	133.0		1,328.6	1,314.0	9.4	5.2		.4	3,051.2	230
Low-cost:
	2012	840.9	587.5	111.2	29.7	112.5		772.8	761.9	6.4	4.4		68.1	2,721.4	343
	2013	882.4	712.4	28.7	31.3	109.9		815.5	804.8	6.4	4.3		66.9	2,788.3	334
	2014	936.9	792.3	-.1	34.2	110.5		855.1	843.6	6.9	4.7		81.7	2,870.1	326
	2015	993.2	841.8	-.1	37.6	113.9		896.6	884.7	7.2	4.7		96.6	2,966.7	320
	2016	1,063.9	903.6	.1	41.3	118.9		941.0	929.2	7.5	4.4		122.9	3,089.5	315
	2017	1,124.1	953.2	.1	45.4	125.5		989.7	977.1	7.8	4.8		134.4	3,223.9	312
	2018	1,188.9	1,006.6	c	49.3	133.0		1,042.1	1,029.0	8.1	4.9		146.9	3,370.8	309
	2019	1,250.5	1,055.5	c	53.2	141.8		1,099.4	1,085.9	8.4	5.0		151.1	3,521.9	307
	2020	1,308.3	1,099.2	c	57.5	151.7		1,161.8	1,148.0	8.8	5.1		146.5	3,668.4	303
	2021	1,378.4	1,155.9	c	62.0	160.4		1,227.6	1,213.7	9.1	4.8		150.8	3,819.2	299
High-cost:
	2012	834.9	582.6	110.2	29.7	112.4		776.4	765.5	6.4	4.4		58.5	2,711.9	342
	2013	836.3	671.5	26.3	31.6	106.8		827.0	816.2	6.4	4.4		9.2	2,721.1	328
	2014	881.5	741.7	-.1	35.1	104.9		882.5	870.8	6.9	4.8		-1.0	2,720.1	308
	2015	920.7	776.8	-.1	39.2	104.8		944.0	932.0	7.1	4.9		-23.3	2,696.8	288
	2016	979.6	830.4	.1	43.9	105.2		1,011.7	999.4	7.5	4.7		-32.1	2,664.7	267
	2017	1,035.9	880.8	.1	49.2	105.9		1,086.3	1,073.0	7.9	5.3		-50.3	2,614.4	245
	2018	1,101.6	939.4	c	54.5	107.6		1,167.7	1,153.8	8.4	5.5		-66.1	2,548.3	224
	2019	1,165.4	998.2	c	60.1	107.1		1,257.2	1,242.6	8.9	5.7		-91.8	2,456.5	203
	2020	1,222.0	1,050.8	c	66.1	105.1		1,353.8	1,338.5	9.3	5.9		-131.7	2,324.8	181
	2021	1,284.2	1,112.4	c	72.6	99.2		1,456.4	1,440.9	9.8	5.7		-172.2	2,152.5	160

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a

Includes reimbursements from the General Fund of the Treasury to the OASI and DI Trust Funds for: (1) the cost of noncontributory wage credits for military service before 1957; (2) the cost of benefits to certain uninsured persons who attained age 72 before 1968; (3) the cost of payroll tax credits provided to employees in 1984 and self-employed persons in 1984-89 by Public Law 98-21; (4) the cost in 2009-17 of excluding certain self-employment earnings from SECA taxes under Public Law 110-246; and (5) payroll tax revenue forgone under the provisions of Public Laws 111-147, 111-312, and 112-96.

b

The “Trust fund ratio” column represents assets at the beginning of a year (which are identical to assets at the end of the prior year shown in the “Amount at end of year” column) as a percentage of cost for the year.

c

Between -$50 million and $50 million.

Note: Totals do not necessarily equal the sums of rounded components.