
630 F.3d 1300 (2011)
Patrick CHAVEZ, on behalf of himself and all other City employees who have been paid overtime that was improperly determined under the Fair Labor Standards Act; Jeannine Chavez; Rudy Campos; Michael Cocchiola; Fortino Ortega, on behalf of themselves and all other City employees who have been paid overtime that was improperly determined under 29 U.S.C.A. 207(a)(1) of the Fair Labor Standards Act; Robert C. Gutierrez, Plaintiffs-Appellants and Cross-Appellees,
v.
CITY OF ALBUQUERQUE, Defendant-Appellee and Cross-Appellant.
Nos. 09-2274, 09-2288.
United States Court of Appeals, Tenth Circuit.
January 12, 2011.
*1303 Paul Livingston, Attorney at Law, Placitas, NM, (Sam Bregman and Eric Loman of Bregman & Loman, P.C., Albuquerque, NM, with him on the briefs), for Plaintiffs-Appellants and Cross-Appellees.
Edward W. Bergmann of Seyfarth Shaw LLP, Chicago, IL, (Robert J. Perry, City Attorney, City of Albuquerque, NM; Michael I. Garcia, City of Albuquerque Legal Department, Albuquerque, NM; Jerry A. Walz of Walz & Associates, Cedar Crest, New Mexico, with him on the briefs), for Defendant-Appellee and Cross-Appellant.
Before BRISCOE, Chief Judge, HOLLOWAY, and O'BRIEN, Circuit Judges.
BRISCOE, Chief Judge.
This is a collective action brought under 29 U.S.C. § 216(b) by current and former employees (collectively, the Employees) of the defendant City of Albuquerque, New Mexico (the City), alleging violations of the Fair Labor Standards Act, 29 U.S.C. § 201 et seq. (FLSA). The FLSA generally requires employers to compensate overtime hours at one and one-half times an employee's "regular rate" of pay. The district *1304 court concluded that the City violated the FLSA by failing to include compensation for unused vacation and sick time (vacation and sick leave buy-backs) in the FLSA regular rate. The district court ruled in the City's favor on all other claims. Both the Employees and the City have appealed from these rulings. Exercising jurisdiction under 28 U.S.C. § 1291, we affirm the district court in all regards, except we reverse and remand its inclusion of vacation buy-backs in the FLSA regular rate.

I
The factual background of this case is best understood in the context of the relevant statute.
The FLSA generally requires covered employers to pay its employees overtime pay for work in excess of forty hours a week.[1] The purpose of FLSA overtime is "to compensate those who labored in excess of the statutory maximum number of hours for the wear and tear of extra work and to spread employment through inducing employers to shorten hours because of the pressure of extra cost." Bay Ridge Operating Co. v. Aaron, 334 U.S. 446, 460, 68 S.Ct. 1186, 92 L.Ed. 1502 (1948). Overtime hours must be compensated "at a rate not less than one and one-half times the regular rate at which [the employee] is employed." 29 U.S.C. § 207(a)(1). The regular rate "shall be deemed to include all remuneration for employment paid to, or on behalf of, the employee," with eight exceptions. 29 U.S.C. § 207(e).[2]
The first step in many FLSA disputes is to determine an employee's regular rate. See Walling v. Youngerman-Reynolds Hardwood Co., 325 U.S. 419, 424, 65 S.Ct. 1242, 89 L.Ed. 1705 (1945) ("The keystone of Section 7(a) is the regular rate of compensation. On that depends the amount of overtime payments which are necessary to effectuate the statutory purposes. The proper determination of that rate is therefore of prime importance."). The regular rate is "the *1305 hourly rate actually paid for the normal, non-overtime workweek." Walling v. Helmerich & Payne, 323 U.S. 37, 40, 65 S.Ct. 11, 89 L.Ed. 29 (1944). It is determined by looking to the employment arrangement. See Bay Ridge, 334 U.S. at 461, 68 S.Ct. 1186 ("Every contract of employment, written or oral, explicitly or implicitly includes a regular rate of pay for the person employed."). The regular rate may include more than just an employee's contractually-designated hourly wage if the employee is, in fact, paid more than that hourly wage. Contractual stipulations as to the regular rate are not controlling, because the regular rate is an "actual fact," rather than "an arbitrary label chosen by the parties." Youngerman-Reynolds, 325 U.S. at 424, 65 S.Ct. 1242. This means that a contract cannot designate an artificially low regular rate in order to reduce the minimum statutory overtime due. See Walling v. Wall Wire Prods. Co., 161 F.2d 470, 473 (6th Cir.1947) (explaining that parties cannot avoid the purposes of the FLSA by designating a fictitious regular rate).

II

The Employment Contracts
The Employees include police officers, firefighters, transit workers, clerical and technical workers, and others. They are members of labor unions that have entered into collective bargaining agreements with the City (each, a CBA, and together, the CBAs).[3] Employees other than firefighters have contractual "normal" workweeks of forty hours. All Employees are paid a base hourly (or straight time) rate set out in the relevant CBA. In addition to straight time pay, certain of the CBAs provide for various types of add-on pay, such as longevity, hazard, shift differential, assignment, education, and firearms qualification pay (collectively, add-on payments). These add-on payments are made as lump sums on a bi-weekly basis.
Generally, the CBAs provide overtime compensation for work beyond an employee's normal daily and weekly schedule, and on holidays. The contractual overtime rate is one and one-half times an employee's straight time rate. The CBAs generally provide for paid sick leave and vacation leave, and count this paid time off as hours worked for overtime purposes. Certain of the CBAs provide that employees who have accumulated a minimum amount of unused vacation or sick leave may sell that leave back to the City.

The City's Wage Calculations
For all employees other than police officers, the City calculates an employee's wage entitlements under the FLSA and under the applicable CBA, and then pays the employee the greater of the two. No FLSA calculation is made for police officers because the City has determined that their contractual entitlements will always exceed their FLSA entitlements. This is because police officers receive overtime under the FLSA after working forty-three hours, whereas they receive overtime under their CBA after working forty hours. City's Opening Br., Attach. 5 at 14.
The City includes straight time and add-on payments in an employee's FLSA regular rate, but does not include vacation or sick leave buy-back. To calculate an employee's *1306 FLSA regular rate for a certain pay period, the City multiplies the total hours worked by the straight time rate and adds the add-on payments. This sum is divided by the total number of hours worked, and the quotient is the hourly regular rate. Total FLSA wages are calculated by multiplying the hourly regular rate by total hours worked (paying the regular rate for each hour worked) and then adding one-half the regular rate multiplied by overtime hours (adding an overtime premium for the extra hours). The City does not count hours of paid leave towards the FLSA overtime threshold. Thus, under the City's calculations, an employee is not entitled to FLSA overtime unless he or she actually works more than forty hours in a week.

Procedural History
The Employees claim that the City violates the FLSA in the following ways: it improperly calculates FLSA and contractual wage entitlements separately; it does not include vacation and sick leave buy-backs in the regular rate; it improperly uses total hours worked rather than the normal workweek as the divisor when calculating an employee's hourly regular rate; it incorrectly multiplies the regular rate by one-half rather than one and one-half; and it improperly takes credits or offsets against its FLSA liability.
The parties filed cross-motions for summary judgment. The district court granted summary judgment to the City on the divisor and multiplier claims. It held a bench trial on the remaining issues. The district court found that the Employees did not show that the City failed to pay any Employee FLSA overtime that he or she was owed. J.A. at 135-38. It compared the FLSA and contractual entitlements of various Employees and found that each Employee's CBA compensation was higher than what was required under the FLSA. Because the Employees failed to show that the City had any FLSA liability to offset, the district court found that the Employees could not show that the City was taking improper credits against FLSA liability. Id. at 143. It also determined that hours paid but not worked did not count towards the FLSA overtime threshold. Id. at 125. Finally, it determined that the City was required to include vacation and sick leave buy-back in its calculation of the regular rate. Id. at 141-42. This was the only claim upon which the Employees prevailed.

III

Jurisdiction
After the district court entered its order on summary judgment and its findings of fact and conclusions of law, the parties entered into a stipulation as to damages and the district court entered a final judgment. Therefore, we have jurisdiction under 28 U.S.C. § 1291.

Standard of Review
The district court made its determinations regarding the divisor, multiplier, and inclusion of bonuses on summary judgment. It made its determinations regarding the buy-backs, threshold hours, and credits after a bench trial.
"We review the grant of summary judgment de novo, applying the same standard as the district court. . . ." Gwinn v. Awmiller, 354 F.3d 1211, 1215 (10th Cir.2004). Summary judgment is appropriate if there is no genuine issue as to any material fact and the movant is entitled to judgment as a matter of law. Fed.R.Civ.P. 56(c). We view the record on summary judgment in the light most favorable to the nonmoving party. Gwinn, 354 F.3d at 1215. "In an appeal from a bench trial, we review the district court's factual findings for clear error and its legal conclusions de novo." Keys Youth Svcs., Inc. v. City of Olathe, Kan., 248 F.3d 1267, 1274 (10th Cir.2001).


*1307 Dual Calculation Method

The City calculates compensation due under the CBAs and under the FLSA and then pays the employee the higher of the two. The Employees argue that the City should calculate what is due under the CBAs and then apply the FLSA on top of that. The Employees misinterpret the FLSA. The FLSA is not a statutory enhancement of negotiated compensation, and the regular rate does not necessarily include all compensation provided for in the CBAs. There is nothing inherently wrong with the City's approach of calculating wage entitlements under the FLSA and under the applicable CBA, and then paying the greater of the two.
The FLSA guarantees minimum overtime compensation. See Barrentine v. Ark.-Best Freight Sys., Inc., 450 U.S. 728, 737, 101 S.Ct. 1437, 67 L.Ed.2d 641 (1981) (describing the FLSA as a "statute designed to provide minimum substantive guarantees to individual workers"). It is not a supplement to contractual compensation. If an employee's contractual pay exceeds what the FLSA would require, an employer has no additional FLSA liability. See 29 U.S.C. § 216(b) (providing that an employer who violates the minimum overtime provisions shall be liable in the amount of unpaid overtime compensation (emphasis added)); cf. Valerio v. Putnam Assocs., Inc., 173 F.3d 35, 40 (1st Cir.1999) (noting that an employer's "obligation under the FLSA was extinguished" where the employee received more overtime payment than she was entitled to receive under the FLSA). Therefore, we reject the Employees' contention that the FLSA requires that they receive compensation on top of whatever they are due under the CBAs. The FLSA guarantees the Employees a minimum of one and one-half times the regular rate for overtime hours. The regular rate does not include all compensation due under the CBAs. It is "remuneration for employment paid to, or on behalf of, the employee," not within an exception. 29 U.S.C. § 207(e). Not all CBA compensation meets this definition.
The Employees cite to the City's Personnel Rule 302.2, which states that "when overtime is required for non-exempt employees, compensation must be in accordance with the [FLSA] and any applicable collective bargaining agreement." Employees' Answer Br. at 2 (emphasis in original). This language means that the City must comply with the FLSA and the CBAs. It does not mean that employees are to receive FLSA wages plus CBA wages. Further, the Employees did not bring a claim for violation of Personnel Rule 302.2, and the rule has no bearing on the City's FLSA liability. We conclude the City's dual calculation method does not violate the FLSA.

Vacation and Sick Leave Buy-Back
The City currently includes add-on payments in the regular rate, but does not include payments for buy-back of sick time and vacation time.[4] The Employees argue that these buy-backs must be included in the regular rate and the district court agreed and included both. We conclude that the district court was correct to include sick leave buy-back in the FLSA regular rate, but the court erred by also including vacation buy-back.
*1308 Compensation for vacation and sick days that are actually taken is not part of the regular rate. See 29 U.S.C. § 207(e)(2). The instant issue arises when an employee works instead of using a vacation or sick day. Under the CBAs, the employee is essentially paid double on such a day: he or she is paid for the day worked, and also paid some portion of a day's worth of vacation or sick pay.[5]
The Department of Labor's (DOL) position is that vacation buy-back is not part of the regular rate, but sick leave buy-back is. DOL issues interpretive bulletins that constitute the agency's official interpretation of the FLSA overtime requirements. 29 C.F.R. § 778.1. The rulings, interpretations, and opinions of DOL, while not controlling, are entitled to the level of deference announced in Skidmore v. Swift & Co., 323 U.S. 134, 65 S.Ct. 161, 89 L.Ed. 124 (1944). They "constitute a body of experience and informed judgment to which courts and litigants may properly resort for guidance." Id. at 140, 65 S.Ct. 161; see also Rodriguez v. Farm Stores Grocery, Inc., 518 F.3d 1259, 1268 n.5 (11th Cir.2008) (noting that 29 C.F.R. § 778.113 is an interpretive bulletin, not a regulation, and that it is entitled to Skidmore deference). One such bulletin states that extra pay for foregone vacation is not included in the regular rate:
[C]ertain payments made to an employee for periods during which he performs no work because of a holiday or vacation are not required to be included in the regular rate because they are not regarded as compensation for working. [Where] an employee who is entitled to. . . a paid vacation foregoes his . . . vacation and performs work for the employer. . . during the vacation period. . . [i]f, under the terms of his employment, he is entitled to a certain sum as. . . vacation pay, whether he works or not, and receives pay at his customary rate (or higher) in addition for each hour that he works on the . . . vacation day, the certain sum allocable to . . . vacation pay is still to be excluded from the regular rate.
29 C.F.R. § 778.219(a). According to DOL, an employee who is entitled to be paid for a week's vacation does not increase his regular rate by working instead of taking vacation, even though he or she is paid for the week of work and for an additional week of vacation time. See 29 C.F.R. § 778.219(a)(1).
Unlike pay for foregone vacation, DOL includes production and attendance bonuses in the regular rate. "Bonuses which are announced to employees to induce them to work more steadily or more rapidly or more efficiently or to remain with the firm are regarded as part of the regular rate of pay. Attendance bonuses. . . are in this category. They must be included in the regular rate of pay." 29 C.F.R. § 778.211(c). Thus, we must determine whether vacation and sick leave buy-backs are more analogous to payments for "periods when no work is performed due to vacation . . . [or] illness," or attendance bonuses. The Wage and Hour Division of DOL (the Wage and Hour Division) takes the position that vacation buy-back is not part of the regular rate, but that sick leave buy-back is part of the regular rate because sick leave buy-back is analogous to an attendance bonus. U.S. Department of Labor, Wage & Hour Opinion Letter FLSA 2009-10 dated Jan. 16, 2009, 2009 WL 649021.[6] We reach the same conclusion.
*1309 The Sixth and Eighth Circuit have considered the buy-back of sick time, but have reached conflicting conclusions. Compare Acton v. City of Columbia, Mo., 436 F.3d 969 (8th Cir.2006) (holding that sick time buy-back must be included in the regular rate) with Featsent v. City of Youngstown, 70 F.3d 900 (6th Cir.1995) (holding that sick time buy-back is not part of the regular rate). In Featsent, the Sixth Circuit held that sick time buy-back did not need to be included in police officers' regular rates because:
These payments are unrelated to the police officers' compensation for services and hours of service. Moreover, awards for nonuse of sick leave are similar to payments made when no work is performed due to illness, which may be excluded from the regular rate. Thus, bonuses for the absence of medical claims and nonuse of sick leave may be excluded from the regular rate.
Id. at 905.
In contrast, the Eighth Circuit has held that sick leave buy-back must be included in the regular rate. The Acton plaintiffs were firefighters who could sell back unused sick days in exchange for seventy-five percent of their straight time pay. 436 F.3d at 976. The Acton court found that "the primary effect of the buy-back program is to encourage firefighters to come to work regularly over a significant period of their employment tenure." Id. at 977. It reasoned that consistent workplace attendance is a general duty of employment and, therefore, "sick leave buy-back monies constitute remuneration for employment." Id. The Acton court rejected Featsent's reasoning, stating that the Sixth Circuit "did [not] recognize and explain how payments awarded to an employee for not using accrued sick leave, which necessarily requires employees to work more days than they are required, is not tantamount to payment for services rendered." Id. at 979.
We agree with the reasoning of the Wage and Hour Division and the Eighth Circuit and hold that sick leave buy-back payments must be included in the regular rate. We further hold that vacation buy back-payments are not part of the regular rate. To be sure, both vacation and sick leave buy-back reward attendance, in some sense, because they reward an employee for not taking days off.[7] The key difference lies in the way each type of day off operates. A sick day is usually unscheduled or unexpected, and is a burden because the employer must find last-minute coverage for the sick employee. In contrast, vacation days are usually scheduled in advance, so their use does not burden the employer in the way that unscheduled absences do. An employee has a duty not to abuse sick days, whereas there is no corresponding duty not to use vacation days. Buying back sick days rewards an employee for consistent and as-scheduled attendance, which are the aspects of good *1310 attendance that provide additional value to an employer. Thus, sick leave buy-backs are compensation for additional service or value received by the employer, and are analogous to attendance bonuses. In contrast, payments for non-use of vacation days are analogous to holiday work premiums or bonuses for working particular undesirable days.

FLSA Overtime Threshold
For employees other than police officers and firefighters, the FLSA sets an overtime threshold of forty hours. This means that employees are entitled to FLSA overtime payments for each hour worked in excess of forty. The Employees argue that the City should count paid time off towards the FLSA overtime hours threshold because the CBAs count paid time off as time worked for the purposes of contractual overtime. We hold that the City is not required to count this time as time worked for FLSA purposes.
The Employees argue that, just as the FLSA regular rate incorporates some CBA compensation provisions, the FLSA should incorporate the CBAs' definitions of hours worked. The First Circuit has rejected this contention, and we do as well. See O'Brien v. Town of Agawam, 350 F.3d 279, 289 (1st Cir.2003) ("The CBAs label such extra pay `overtime,' but that does not control. For purposes of the FLSA, all hours worked under the statutory maximum are non-overtime labor."). Contractual overtime and FLSA overtime are not the same things. The FLSA's policy goals are not implicated when an employee does not actually work over forty hours in a week.
The DOL's position is that FLSA overtime applies only to hours actually worked. 29 C.F.R. § 778.101 states that "[h]ours worked in excess of the statutory maximum in any workweek are overtime hours under the statute; a workweek no longer than the prescribed maximum is a nonovertime workweek under the Act, to which the pay requirements of section 6 (minimum wage and equal pay) but not those of section 7(a) are applicable." Similarly, 29 C.F.R. § 778.102 states that "[i]f no more than the maximum number of hours prescribed in the Act are actually worked in the workweek, overtime compensation pursuant to section 7(a) need not be paid." One of the FLSA's purposes is to compensate employees for the strain of working long hours. This purpose is not implicated if the employee does not actually work the requisite number of hours in a week, regardless of whether the employee is compensated as though he or she did work.[8] It would be illogical to conclude that the FLSA would require overtime payments where, for example, an employee takes a full week's vacation but also does three hours of work. An employee in that situation would not be subject to the stresses of long hours. However, were we to adopt the Employees' interpretation of the FLSA, statutory overtime compensation would be required. We hold that compensated non-work hours do not count towards the overtime threshold.
The Employees argue that the Third Circuit's decision in Wheeler v. Hampton Township, 399 F.3d 238 (3d Cir.2005), should be persuasive here. The CBA at issue in Wheeler provided for paid vacations, holidays, and sick days (described as "non-work pay"), and stated that these payments would be included in the regular rate. It also provided for lump-sum yearly payments for longevity, educational attainment, *1311 and similar add-ons, but did not state that these would be included in the regular rate. The plaintiff police officers argued that the defendant township violated the FLSA because it did not include add-on payments in the regular rate. In response, the township "argued that the Officers traded their right to have [add-on] pay added to their basic annual salary in the CBA's overtime calculation in exchange for the inclusion of non-work pay, which is not required under the FLSA." Id. at 241.
The Wheeler court determined that the township had to include add-on pay and non-work pay in the regular rate. It held that the township could not contract away the officers' right to FLSA overtime on add-on pay. Regarding the non-work pay, the Wheeler court reasoned that, while the FLSA itself did not require non-work pay to be included in the regular rate, the CBA required non-work pay to be included in the regular rate for FLSA purposes. See id. at 244. Relying on the language "shall not be deemed," in 29 U.S.C. § 207(e)(2), the Third Circuit reasoned that "the function of [that subsection] is to forbid this Court from deeming that the CBA include non-work pay." Id. As the CBA already provided that non-work pay would be included in the regular rate, the Third Circuit concluded that there was nothing for the court to "deem." Id.
The Wheeler court, like the Employees, conflated the CBA and the FLSA. In an FLSA case, the issue is the FLSA regular rate, not the contractual overtime rate. Including or not including vacation pay in the FLSA regular rate is not a matter of "deeming that the CBA include non-work pay," id. (emphasis added), because the CBA rate is not affected by the FLSA rate. The proper way to calculate the FLSA regular rate is to look at all the compensation provided for in the CBA and identify what part of that compensation is "remuneration for employment paid to, or on behalf of, the employee," not within an exception. 29 U.S.C. § 207(e).
Further, Wheeler is not apposite to the Employees' hours threshold argument. There is a difference between what compensation is included in the regular rate and which hours count toward the FLSA overtime threshold. While the FLSA regular rate incorporates the negotiated compensation rate, the FLSA overtime threshold is defined in the statute itself. The Wheeler defendant argued that the plaintiffs agreed to take certain compensation out of the FLSA regular rate that, by statute, is part of that regular rate. Here, the City does not argue that the Employees agreed to give up anything to which they would otherwise be entitled under the FLSA. The FLSA does not provide a right to overtime until an employee has first worked forty hours; therefore, the City is not attempting to take away any compensation guaranteed by the FLSA.

The Proper Divisor When Calculating the Hourly Regular Rate
In this case, the first step in calculating the regular rate over a particular week is to total the week's straight time pay and add-ons. The parties disagree as to the second step: whether this total should be divided by the CBA's "normal workweek," or the hours actually worked by the employee. Conceptually, the issue is whether the add-ons are additional compensation for forty hours[9] worth of work, or whether they are additional compensation for all work an employee may do in a given week. If actual hours is used as the *1312 divisor, the value of the add-ons decreases as an employee works more overtime. If the normal workweek is used, an employee would receive additional add-ons for each overtime hour worked. We hold that the proper divisor is the actual hours worked.
Some courts have characterized longevity payments and other add-ons as "reflect[ing] the greater value of an experienced employee's labor." See Scott v. City of New York, 592 F.Supp.2d 475, 488 (S.D.N.Y.2008). If the add-ons are meant to compensate for additional value of each hour of labor, then it makes sense to compensate for this additional value in every overtime hour as well. See id. ("[T]he value of that experience is received by the employer for every hour that the employee works."). The problem with this approach is that the CBAs do not discuss the add-ons in terms of an hourly rate increase; the add-ons are described as biweekly lump-sum payments and are not dependent on hours worked. Sums paid as a reward for service, "the amounts of which are not measured by or dependent on hours worked, production, or efficiency," are not part of the regular rate. 29 U.S.C. § 207(e)(1); see also Zumerling v. Devine, 769 F.2d 745, 751 (Fed.Cir.1985) (holding that the proper divisor is hours worked because lump-sum premium pay is "not received in return for any particular hours of work . . . [rather, it is] a means of generally compensating the employees").
According to DOL, "the regular hourly rate of pay of an employee is determined by dividing his total remuneration for employment (except statutory exclusions) in any workweek by the total number of hours actually worked by him in that workweek for which such compensation was paid." 29 C.F.R. § 778.109. The parties offer differing constructions of this statement. The City argues that it means that compensation is divided by total hours worked. The Employees argue that over-time hours are not hours "for which [add-on] compensation was paid," so overtime hours are not included in the divisor. We believe that 29 C.F.R. § 778.110(b) forecloses the Employees' interpretation. It provides:
If the employee receives, in addition to his earnings at the hourly rate, a production bonus of $9.20, the regular hourly rate of pay is $6.20 an hour (46 hours at $6 yields $276; the addition of the $9.20 bonus makes a total of $285.20; this total divided by 46 hours yields a rate of $6.20). The employee is then entitled to be paid a total wage of $303.80 for 46 hours (46 hours at $6.20 plus 6 hours at $3.10, or 40 hours at $6.20 plus 6 hours at $9.30).
DOL views a lump-sum production bonus as general compensation rather than additional hourly compensation. This interpretation is consistent with the language of 29 U.S.C. § 207(e)(1).
The Employees argue that using an hours worked divisor causes the bonus to diminish as overtime increases. This is the case any time there is a fixed and hourly component to an employee's compensation. Absent evidence of an intent to manipulate compensation to avoid FLSA liability, this result does not violate the FLSA. See Helmerich & Payne, 323 U.S. at 42, 65 S.Ct. 11 (explaining that parties are free to set the regular rate through contract, but that the regular rate may not be calculated "in a wholly unrealistic and artificial manner so as to negate the statutory purposes [of the FLSA]"). It is also the result for which the parties bargained under the CBAs, because CBA overtime is based solely on the straight time rate. Under the CBAs' overtime schemes, the value of the add-on payments decreases if an employee works overtime. Therefore, the Employees cannot claim that this result is unfair or surprising. Cf. Overnight *1313 Motor Transp. Co. v. Missel, 316 U.S. 572, 580, 62 S.Ct. 1216, 86 L.Ed. 1682 (1942) (stating that "the longer the hours the less the rate and the pay per hour" was not a persuasive argument against using hours worked to determine the regular rate of a salaried employee), superseded on other grounds by statute, 29 U.S.C. § 260, as recognized in Trans World Airlines, Inc. v. Thurston, 469 U.S. 111, 128 n. 22, 105 S.Ct. 613, 83 L.Ed.2d 523 (1985). Further, the suggestion that the normal workweek divisor is more appropriate because it would produce a consistent regular rate is unpersuasive. "The regular rate may vary from week to week."[10]Bay Ridge, 334 U.S. at 460, 68 S.Ct. 1186.
DOL's interpretationthat the proper divisor is hours workedis consistent with the statutory language and is not illogical, unreasonable, or unfair. We find no reason to adopt a contrary position. Therefore, we hold that the FLSA hourly regular rate is calculated by dividing the relevant weekly compensation by the actual hours worked, rather than the normal workweek.

Proper Multiplier
The Employees contend that the City's use of a one-half multiplier, rather than one and one-half, violates the FLSA. The City uses a one-half multiplier because it first applies the regular rate to all hours worked. To the extent that the Employees argue that the FLSA requires that they receive the regular rate for all hours worked and the regular rate and one-half for overtime hours, such an argument is without merit. Under that scenario, the Employees would receive two and one-half times the regular rate for each overtime hour worked, which is more than the one and one-half that the FLSA requires.
The same result is achieved if the City pays straight time for all hours and an additional one-half straight time on overtime hours, or if the City pays straight time for non-overtime hours and one and one-half straight time on overtime hours. To illustrate, assume that an employee's straight time pay is $10.00 an hour and he or she receives $55.00 in add-on pay each week. The employee works 44 hours in a particular week. To calculate the hourly regular rate for that week, divide the add-on by the total hours worked ($55.00/44 = $1.25) and add that to the straight time rate ($1.25 + $10.00 = $11.25).[11]
1. ($11.25)(44) + (.5)($11.25)(4)  =
   $495.00  + $22.50 = $517.50
2. ($11.25)(40) + (1.5)($11.25)(4) =
   $450.00  + $67.50 = $517.50
This is because the two equations are mathematically the same.
1. (x)(44) + (.5)(x)(4)  = (x)(44) + (2)(x)
   = (46)(x)
2. (x)(40) + (1.5)(x)(4) = (x)(40) + (6)(x)
   = (46)(x).
The City's use of a one-half multiplier does not violate the FLSA.

Credits
The Employees assert that the City is taking improper credits against its FLSA liability. If an employer actually pays an employee certain compensation *1314 that is not included in the regular rate, the employer may credit those payments towards any FLSA liability. See 29 U.S.C. § 207(h). However, if an employer has no FLSA liability, there is nothing against which to take credits.
The district court ruled in the City's favor on this issue because the Employees failed to show that the City had any FLSA liability at all. Instead of pointing to evidence that the City improperly calculates FLSA credits, the Employees simply assert that this is so. They have not identified a single instance of an improper credit. In fact, they did not identify any actual FLSA underpayment at the summary judgment or trial phases of this matter. The Employees had ample opportunity to support their allegations of improper credits, but they failed to do so. Their credits claim fails.

IV
In summary, we hold the following: the City's dual calculation method does not violate the FLSA; buy-back of unused vacation time is not part of the FLSA regular rate; buy-back of unused sick time is part of the FLSA regular rate; paid time off does not count towards the FLSA overtime threshold; when calculating the hourly regular rate, weekly compensation is to be divided by total hours worked, rather than an employee's regular workweek; the City's use of a one-half multiplier does not violate the FLSA; and the City is not improperly taking credits against its FLSA liability.
Therefore, the district court's decision is affirmed in part and reversed in part. The decision is reversed with respect to the buy-back of vacation time, and the remainder is affirmed. This case is remanded for further proceedings not inconsistent with this opinion.
NOTES
[1]  Police officers and firefighters are subject to slightly different overtime thresholds. See 29 U.S.C. § 207(k).
[2]  The subsection defining the regular rate provides, in relevant part:

As used in this section the "regular rate" at which an employee is employed shall be deemed to include all remuneration for employment paid to, or on behalf of, the employee, but shall not be deemed to include
(1) sums paid as gifts; payments in the nature of gifts made at Christmas time or on other special occasions, as a reward for service, the amounts of which are not measured by or dependent on hours worked, production, or efficiency;
(2) payments made for occasional periods when no work is performed due to vacation, holiday, illness, failure of the employer to provide sufficient work, or other similar cause; reasonable payments for traveling expenses, or other expenses, incurred by an employee in the furtherance of his employer's interests and properly reimbursable by the employer; and other similar payments to an employee which are not made as compensation for his hours of employment;. . .
(5) extra compensation provided by a premium rate paid for certain hours worked by the employee in any day or workweek because such hours are hours worked in excess of eight in a day or in excess of the maximum workweek applicable to such employee under subsection (a) of this section or in excess of the employee's normal working hours or regular working hours, as the case may be;
(6) extra compensation provided by a premium rate paid for work by the employee on Saturdays, Sundays, holidays, or regular days of rest, or on the sixth or seventh day of the workweek, where such premium rate is not less than one and one-half times the rate established in good faith for like work performed in nonovertime hours on other days;
. . . .
29 U.S.C. § 207(e).
[3]  The CBAs at issue here are those between the City and: the Albuquerque Police Officer's Association (APOA); American Federation of State, County, and Municipal Employees (AFSCME) Local 624; the Albuquerque Area Fire Fighters Union (AAFU); AFSCME Local 3022; the Albuquerque Officers' Association (AOA) Local 1888; Albuquerque Clerical and Technical Employees (ACTE); the New Mexico Transportation Union (NMTU); and AFSCME Local 624 Transit Union.
[4]  The APOA, AFSCME Local 624, and AAFU CBAs provide for buy-back of sick time and vacation time. The AFSCME Local 3022, AOA, and ACTE CBAs provide for buy-back of sick time. The AFSCME Local 624 Transit Union CBA provides for buy-back of vacation time. The NMTU CBA does not provide for buy-back of any leave time. See City's Opening Br. Attach. 5, at 5-11.
[5]  At oral argument, counsel explained that the buy-backs occur annually and are pro-rated over the year. Any resulting increase in the regular rate must also be pro-rated over the year.
[6]  "The [Wage and Hour Division] issues opinion letters to explain the requirements of the FLSA and its regulations and how they apply to particular circumstances." In re Wal-Mart Stores, Inc., 395 F.3d 1177, 1184 (10th Cir.2005) (internal quotation omitted). This court gives "great weight" to these letters when they interpret the DOL's own ambiguous regulations. Id. Although the opinion letter referenced herein does not interpret regulations, it "nonetheless provide[s] some persuasive authority, [as it is issued] by the government agency responsible for the enforcement of the federal wage and hour laws. . . ." Myers v. Copper Cellar Corp., 192 F.3d 546, 554 (6th Cir.1999).
[7]  The City's contention that "payments for vacation and sick leave buy-back are intended to provide employees with compensation for days they are not at work," City's Opening Br. at 30 (emphasis in original), is simply untrue. The buy-backs are extra compensation for days that the employees do come to work.
[8]  The interpretive bulletins and cases treating meals or "on call" time as hours worked are not relevant here. In those circumstances, the issue is whether the employee should be considered working while on call. Here, there is no question that employees are not working while on paid leave.
[9]  The firefighters' normal workweek is not forty hours, and is set forth in the AAFU CBA. For brevity's sake, we use forty hours as shorthand for an Employee's normal workweek. This opinion applies with equal force to the firefighters.
[10]  Including sick leave buy-backs in the regular rate will also cause it to change from week to week. The fact that the term "regular rate" includes the word "regular" does not mean that it must be consistent.
[11]  This is the same as multiplying the straight time rate by forty hours, adding the add-ons, and dividing the sum by hours worked. ($10.00)(40) = $400.00. $400.00 + $55.00 = $455.00. $455.00/44 = $11.25.
