#### PRIMARY CARE INVESTORS, SEVEN, INC., Bert Schweizer, III, as Trustee for Primary Care Investors Nineteen, Inc., Donna J. Hill, as Trustee for Primary Care Investors, Thirty-Two, Inc., William Londoff, as Trustee for Primary Care Investors, Thirty-Two, Inc., Primary Care Investors, Thirty-Five, Inc., Primary Care Investors, Forty-Two, Inc., Primary Care Investors, Forty-Five, Inc., Dagmar S. Graham, as Trustee for Primary Care Investors Forty-Seven, Inc., Robert E. Moldthan, as Trustee for Primary Care Investors Forty-Seven, Inc., Steven Bernstein, as Trustee for Primary Care Investors Fifty, Inc., Gwen Bernstein, as Trustee for Primary Care Investors Fifty, Inc., Jacqueline F. Mulwee, as Trustee for Primary Care Investors Thirty, Inc., Jon G. Carlson, as Trustee for Primary Care Investors Thirty-Six, Inc., Primary Care Investors, Forty-Four, Inc., Appellants, v. PHP HEALTHCARE CORP., formerly known as PHP Corporation, a Delaware corporation, Primary Care Corp., a Missouri corporation, Charles H. Robbins, Michael D. Starr; George E. Schaffer, M.D., Jack M. Mazur, Stephen I. Frates; Ronald J. Raben, David L. Jones, as Trustee for Primary Care Investors Nineteen, Inc., Appellees.

No. 92-2055.

United States Court of Appeals, Eighth Circuit.

Submitted Dec. 16, 1992.

Decided Feb. 25, 1993.

*1209Counsel who presented argument on behalf of the appellant Roger B. Harris, of Chicago, IL, argued (Samuel J. Betar, C. Vincent Maloney, Chicago, IL, and Eugene Portman, St. Louis, MO, on brief), for appellants.

Alan Popkin, St. Louis, MO, argued (Mark Arnold, Daniel Bloom, Eugene Buckley, and Thomas Fiala, St. Louis, MO, on brief), for appellees.

Before WOLLMAN, Circuit Judge, MORRIS SHEPPARD ARNOLD, Circuit Judge, and BATTEY,* District Judge.

*

 The HONORABLE RICHARD H. BATTEY, United States District Judge for the District of South Dakota, sitting by designation.

MORRIS SHEPPARD ARNOLD, Circuit Judge.

This appeal arises from summary judgment dismissal entered by the district court1 of the plaintiffs’ section 10(b)/Rule 10b-5 claim under the Securities Exchange Act of 1934 and of their claim of a pattern of racketeering activity in violation of the RICO statute.

Plaintiffs are corporations created for the purpose of forming a joint venture partnership with Primary Care Corporation (Primary). The joint venture was named Primary Care Associates (PCA). PCA was organized for the purpose of developing and operating medical convenience centers (PrimeCare Centers) that would offer medical treatment on convenient terms. Plaintiffs and other investors were to finance the joint venture, and Primary was to develop and operate the medical centers, while also managing the affairs of the joint venture, PCA.

Defendant Primary was a wholly-owned subsidiary of PHP Healthcare Corporation (PHP). PHP apparently formed Primary for the purpose of acting as managing partner in PCA. Defendant PHP was engaged in the business of providing health care services and health care personnel to governmental agencies and administering various health care facilities or departments in hospitals and institutions. Defendants Robbins, Starr, Schaffer, Frates, Mazur, and Raben were the directors and offi*1210cers of PHP and held (directly and through their trusts and corporations) about 75 percent of PHP’s stock. In addition, Mazur and Raben were lawyers for the principals of seven of the 11 plaintiff corporations.

PCA was formed in February of 1984 as a joint venture with outside investment of $1.4 million. The Joint Venture Agreement (Agreement) was drafted by Mazur and Raben. The provision of the agreement from which this action primarily arises addressed conversion of PCA unit interests into PHP stock:

> In the event that at any time during the term of the Joint Venture PHP ... determines to make a public offering of its common stock, the Managing Participant [Primary] *may*call a meeting of the Participants pursuant to Section 5.2 hereof for the purpose of voting on the issue of whether the General Participants [investors including Plaintiffs] wish to convert their ... Units into publicly registered stock of PHP at a specified conversion rate____ If the holders of at least 28 of the [55 total] Votes allocated to the General Participants vote in favor of such conversion, all General Participants shall be required to exchange their ... Units for PHP common stock at the specified conversion rate.

Agreement, § 11.1 (emphasis added). According to this provision, only general participants could vote on whether to convert their partnership interests into shares of PHP stock. The offering circular for the joint venture characterized section 11.1 as a “Risk Factor” of the investment.

Regarding meetings at which votes could be taken, section 5.2 of the Agreement provided that Primary, as managing participant, “may” call a meeting of participants at any time. It further provided that Primary “shall" call a meeting in response to a written request of general participants who hold at least ten votes of those allocated to general participants.

The lawyer defendants solicited their clients and friends to invest in the PCA venture Plaintiffs assert that the lawyer defendants touted the conversion feature in their sales pitches to prospective investors, and they testified in depositions that they invested in PCA in anticipation that their interests would be converted to public stock in PHP.

By October of 1985, the officers, directors, and attorneys for PHP had begun considering a public offering of PHP’s stock. They met with officials at Stifel, Nicolaus & Co. (Stifel), a securities broker-dealer, and in June, 1986, Stifel’s commitment committee authorized the broker to undertake a public offering of PHP stock. PHP’s shareholders and board of directors then formally adopted a resolution approving Stifel’s management of the offering, and a formal letter of intent was signed by Stifel and PHP. The public offering was to commence on October 1, 1986; however, because of filing delays, it commenced November 20, 1986.

At about the same time that PHP began to plan its public offering, it also began to buy the PCA unit interests of outside investors, along with those investors’ PHP shares. Plaintiffs assert that defendants had decided to eliminate PCA’s outside investors and that defendants had two motives for doing so: (1) the asserted contingent right of plaintiffs and other outside investors to convert their PCA unit interests into PHP public stock; and (2) the existence of actual and potential conflicts of interest between PHP and PCA because both were engaged in the general medical service field. Regarding the latter issue, the PCA Agreement provided:

> Prior to any investment *or involvement*by [Primary] or its Affiliates [presumably including PHP] with Medical Convenience Centers other than the Centers [under the PCA Partnership] ... [Primary] shall call a meeting of the Participants for the purpose of voting on the issue of whether the General Participants wish to retain an amount equal to at least 50% of Cash Flow for the purpose of funding the development and equipping of Medical Convenience Centers in addition to the [two] Centers already funded by the Participants’ Capital Contributions.

*1211Agreement, § 6.2 (emphasis added). If the participants voted in favor of retention of cash flow for development of additional PrimeCare Centers, the Agreement further prohibited Primary and its affiliates from “investing in or developing Medical Convenience Centers other than those developed by the Joint Venture____” A caveat to that provision, however, allowed Primary to manage other medical convenience centers, so long as it had no equity interest in them.

In the spring of 1984, Defendant Robbins, president of both Primary and PHP, approached U.S. military officials about the use of off-base medical convenience centers for military personnel and their families. Primary stated in its February, 1985, Annual Report to PCA investors that it expected to submit a contractual proposal on behalf of PCA for providing these health care services and that the project held “great potential.”

In May, 1985, Primary submitted a proposal for these military-related clinics; it did so, however, on behalf of PHP rather than on behalf of PCA. In its initial proposal to military officials, Primary stated that it had acquired knowledge and experience needed to make the proposal, and to develop and operate the proposed PRIMUS centers, from its development and operation of the PCA-owned PrimeCare medical centers.2 PHP was awarded the contract for what became known as PRIMUS centers. No meeting was called in order to allow PCA general participants to vote on retention of cash flow or on the possibility of pursuing the PRIMUS centers through PCA, pursuant to section 6.2.

The initial PRIMUS center was similar to PCA’s PrimeCare centers, and it was very successful, very quickly. By August of 1986, the government had awarded PHP four separate contracts for additional PRIMUS centers. PHP’s development and operation of PRIMUS Centers was projected to grow to well over $5 million annually in revenue by the summer of 1987.

Plaintiffs assert that, as PCA investors, they had at this point two potential claims against Primary: one for breach of section 6.2 of the agreement and the other for breach of fiduciary duty for misappropriation of the PRIMUS opportunity. Stifel, as underwriter, instructed PHP that, in order to proceed with the public offering, PCA’s outside investors should be bought out in order to obviate prospective “lawsuits.” Records of a PHP board meeting indicate that the company’s directors also were concerned to “eliminate lawsuits” and thought buying out the outside investors would accomplish that goal. Defendants maintain that these comments refer to suits that might have arisen because PCA was losing money and appeared to have been a bad investment.

In furtherance of its goal of eliminating the outside investors, defendants prepared and circulated to all such investors a “Purchase Offer” dated July 29, 1986. The offer was accompanied by a “Purchase Memorandum” explaining that PHP was offering to purchase the investors’ units in PCA because, contrary to original projections, PCA had incurred losses since its formation. Each investor was offered the cash price paid for his or her interest in PCA, plus 12 percent interest per annum for the duration of ownership. Part of the payment was to be made with cash, part with a promissory note. The memorandum did not disclose that PHP expected PCA to show a profit of $120,000 for 1986, a projection disclosed in a May, 1986, letter to Stifel. Nor did the memorandum disclose that PHP was planning a public offering in three months. The memorandum did state that payment on the promissory note element of the purchase price would be accelerated “in the event PHP ... offers its capital stock in a public offering----”

Plaintiffs agreed to sell in August, 1986, and received the cash portion of the purchase price the following month. PHP’s public offering subsequently became effective, and trading of its stock began in late *1212November of 1986. The following month, PHP wrote to the plaintiffs, notifying them of the public offering and tendering accelerated payment on the promissory note portion of the purchase price. Plaintiffs were paid a total of approximately $468,101 for their unit interests, which represented 31.4 percent of PCA.

Plaintiffs filed this action in federal district court, alleging that defendants violated the Securities Exchange Act of 1934, section 10(b), as well as Rule 10b-5 promulgated thereunder, by failing to disclose material facts, including the planned public offering of PHP stock. Based upon that omission and on other allegedly fraudulent activities continuing over the course of 21 months, plaintiffs also alleged that defendants engaged in a “pattern of racketeering activity” in violation of the RICO statute, 18 U.S.C. § 1965.

The district court granted 11 motions *in limine*filed by the defendants. These evidentiary rulings excluded a great deal of plaintiffs’ proffered evidence and largely formed the basis for granting defendants’ summary judgment motion, dismissing both plaintiffs’ section 10(b) and RICO claims. Rulings granting defendants’ motions *in limine*excluded evidence related to, among other things, the alleged misappropriation of the PRIMUS opportunity, other litigation and settlements, financial projections, and certain expert testimony. The rulings also made the introduction of certain evidence contingent on presentation of other evidence. For example, evidence of alleged fraud in connection with the 1985 buy-out of PCA investors and PHP stockholders was excluded until plaintiffs were able to state a prima facie fraud case related to the 1986 buy-out. Evidence relating to the PHP public offering was excluded until plaintiffs were able to state a prima facie case of a right to convert their unit interests into publicly held PHP stock.

On appeal, plaintiffs assert that "the district judge erred in granting eleven of defendants' motions *in limine,*which rulings formed the basis for the summary judgment." Plaintiffs' brief fails, however, to go beyond this cursory and summary statement. This error is compounded because the plaintiffs in effect assert 11 different errors with this single sentence while providing no hint as to the nature of the asserted error on any of the 11 rulings. Rule 28(a)(5) of the Federal Rules of Appellate Procedure provides that an appellant's brief shall include an argument containing "contentions . . . with respect to the issues presented, and the reasons therefor, with citations, statutes and parts of the record relied on." F.R.A.P. 28(a)(5); *see also Turner Co. S.D. v. Miller,*170 F.2d 820, 828 (8th Cir.1948) ("[a]n unargued assertion of error is no more helpful to an appellate court than is an unsupported allegation of fact to a trial court"; errors assigned but not argued are waived) (preceding the Federal Rules of Appellate Procedure and quoting *Kimball Laundry Co. v. United States,*166 F.2d 856, 859 (8th Cir.1948)); *see generally Wright, Miller, Cooper & Gressman, Federal Practice and Procedure:*Jurisdiction § 3974. By their failure to specify errors and to provide citations to authorities that would lead us to resolve any questions, plaintiffs have waived the right to have us consider whether each ruling was correct. Seeing no plain error in any of the 11 rulings and noting the district court's broad discretion in ruling on evidentiary and order-of-proof issues, *see, e.g., Warner v. Transamerica Ins. Co.,*739 F.2d 1347, 1350 (8th Cir.1984), we affirm these rulings.

Plaintiffs raise five additional issues on appeal. Four of the issues go to the correctness of rulings which formed the basis of the district court’s summary judgment on the section 10(b) claim: (1) whether the district court correctly ruled that defendants’ failure to disclose PHP’s plan to make a public offering was not a material fact requiring disclosure when PHP purchased the PCA units; (2) whether the district court correctly concluded that plaintiffs had no contract right to convert their partnership units into publicly traded PHP stock; (3) whether the district court correctly accepted defendants’ contention that they had no fiduciary duty to offer plaintiffs and other investors the opportunity to vote to convert their partnership units into *1213PHP stock; and (4) whether the district court correctly inferred that PHP would not have voted to convert plaintiffs’ and other investors partnership units into publicly traded PHP stock had PHP failed to purchase plaintiffs’ convertible partnership interests. Plaintiffs raise a final issue related to their RICO theory: whether the district court correctly granted defendants’ summary judgment motion based on its determination that no genuine issue of fact existed regarding the alleged pattern of racketeering activity.

For the reasons that follow, we affirm the district court in all respects.

I.

Rule 10b-5, promulgated under section 10(b) of the Securities Exchange Act of 1934, 15 U.S.C. § 78j, is the basis of a cause of action for securities fraud when any person making a transaction in securities and using instrumentalities of interstate commerce, "omit[s] to state a material fact necessary in order to make the statements made, in light of the circumstances under which they were made, not misleading." 17 C.F.R. § 240.10b-5. A fact is material if "it is substantially likely that a reasonable investor would consider the matter important in making an investment decision." *Harris v. Union Elec. Co.,*787 F.2d 355, 366 (8th Cir.), *cert. denied,*479 U.S. 823, 107 S.Ct. 94, 93 L.Ed.2d 45 (1986). A plaintiff in a Rule 10b-5 case must show actual damages in order to make out a prima facie case. *See, e.g., Davis v. Merrill Lynch, Pierce, Fenner & Smith,*906 F.2d 1206, 1217-19 (8th Cir. 1990).

Plaintiffs’ stated theories of both actual damages and materiality3 assume that they had a right to vote on conversion of their PCA unit interests into PHP stock in the event that PHP became a publicly traded company. That is, plaintiffs alleged that PHP’s failure to disclose that it was planning a public offering was a material omission because if plaintiffs had known of the impending offering, they would have retained their PCA unit interests in anticipation of converting them into PHP public stock. Similarly, plaintiffs’ theory of damages is that they were effectively cheated out of valuable PHP public stock when they were fraudulently induced to sell their PCA unit interests. Because plaintiffs strictly link both elements of their Rule 10b-5 action to the right to convert their PCA interests into PHP stock, we first address the two theories on which plaintiffs base their asserted right of conversion.

A.

Plaintiffs contend that they, in conjunction with other investors, had a right under the PCA Joint Venture Agreement to convert their PCA unit interests into PHP stock in the event that PHP made a public stock offering. Section 11.1 of the Agreement provides that if PHP “determines to make a public offering of its common stock, Primary may call” a participants’ meeting for the purpose of voting on the stock conversion option. Plaintiffs argue that PHP “determine[d]” to make a public offering, triggering the outside investors’ right to vote on conversion. The district court found that section 11.1 of the Agreement grants Primary, as managing participant, discretion to call a meeting for the purpose of voting on conversion. The district court wrote, moreover, “[i]t does not require such a meeting, nor does [it] confer on any participant the right to have such a meeting called.” We agree. The auxiliary verb used in section 11.1 is the permissive “may”, which stands in contrast to the use of the mandatory “shall” elsewhere in the same section and elsewhere in the Agreement.

Appellants contend that section 5.2 of the Agreement lends additional support to their theory of a contractual right to conversion. That section provides that Pri*1214mary “shall” call a participants meeting in response to a written request by participants holding at least ten votes. The district judge ruled that section 5.2 “does not confer on any participant the right to have a section 11.1 meeting.” We agree. This section simply states the method for calling a general meeting, and the specific provisions of section 11.1 override this general provision. *See generally Phillips v. Authorized Investors Group, Inc.,*625 S.W.2d 917, 921 (Mo.App.1981). Additionally, we note that section 3.3(c) of the Agreement provides that “[n]o participant shall have the right to demand or receive any property other than cash for his Capital Contribution.” Finally, the fact that the offering circular for the joint venture identified section 11.1 as a “participant risk” supports the finding that the conversion option belonged to Primary rather than to PCA investors. We affirm the district court’s conclusion that plaintiffs enjoyed no contractual right to convert their PCA unit interests into PHP stock.

B.

Plaintiffs assert that Primary’s fiduciary duty to the PCA investors as their partner required Primary to exercise its discretion in favor of calling a meeting under the circumstances in this case.4 We disagree. We look to state law to determine whether a fiduciary relationship exists. *Davis v. Merrill Lynch, Pierce, Fenner & Smith,*906 F.2d 1206, 1215 (8th Cir.1990). Missouri law indicates that, as to matters specifically addressed in it, an agreement between parties to such a business arrangement determines the extent of the fiduciary duty owed by the parties to that agreement. In *Unigroup, Inc. v. O’Rourke Storage & Transfer Co.,*980 F.2d 1217, 1218-1219 (8th Cir.1992), this court considered a corporate by-law that gave the board of directors discretion to pay more than book value for re-purchased shares. The by-law stated that the re-purchase price “shall be such price as the Board ... may from time to time determine but in no event less than the greater of (1) the book value ... or (ii) the price paid to the corporation” when the shares were issued. Construing Missouri law, we held that the directors had no fiduciary duty to exercise their discretion to pay more than the greatest of these alternative measures for re-purchased shares. Similarly, Primary had no fiduciary duty to exercise the discretion accordéd by section 11.1 of the Agreement to call a meeting for the purpose of voting on conversion.

Because plaintiffs had no right to convert,5 nor even any control over whether a vote on conversion would be called, knowledge of the impending public offering would have been meaningless to them. Plaintiffs have failed, therefore, to state a material omission and cannot make out a prima facie case under Rule 10b-5. Accordingly, we need not consider the issue of damages. We affirm the district court’s grant of summary judgment, dismissing plaintiffs’ Rule 10b-5 action.

II.

Based on defendants’ alleged mail fraud and securities fraud, plaintiffs sued under the RICO statute, alleging a “pattern of racketeering activity” in violation of 18 U.S.C. § 1962(a)-(d). Plaintiffs assert that the fraudulent acts alleged as the predicates spanned a period of 21 months and that the purpose of the overall scheme was to acquire PCA unit interests and PHP shares held by outside minority investors, all for an inadequate price. In March, 1985, PHP sent a mailing to its shareholders and to PCA participants. In the letter, defendant Robbins gave the impression that PHP was on the verge of financial *1215collapse, warning that it was “necessary” to secure six-month options to purchase the shares and interests of respective investors. Plaintiffs claim that this was not true but that it enabled defendants to acquire, at bargain prices, shares of minority holders in PHP and also to acquire and control a majority of unit interests in PCA. Also allegedly in furtherance of and concluding the overall scheme or pattern, PHP sent a December, 1986, group mailing to those former shareholders, including plaintiffs, who had sold their PCA unit interests in August, 1986. Pursuant to the purchase offer and memorandum, the letter informed the sellers of PHP’s public offering and paid each the balance due for the purchase.

Assuming but not finding that a common scheme existed, the district court concluded that the initial predicate act of any alleged fraud occurred on October 31, 1985, when PHP purchased some investors’ interests in both PHP and PCA. The court further found that the alleged scheme was concluded in August, 1986, when plaintiffs accepted PHP’s offer to purchase their unit interests. Accordingly, the court reasoned that any alleged scheme endured only ten or eleven months and failed to span the requisite “substantial period” required for establishing a pattern of racketeering activity under RICO.

We agree. We consider first the predicate acts purportedly marking the beginning and end of the alleged scheme. The March, 1985, letter to investors evinces no fraud. It warned of a dire financial situation and, indeed, in October, 1985, PHP did default on its loans. As for the pay-off in December of 1986, the alleged final act, we find that the ministerial act of tendering payment for securities that plaintiffs had already sold cannot be a predicate act. *See Rodriguez v. Banco Central,*777 F.Supp. 1043, 1063 (D.P.R.1991) (acceptance of payment on a note is not a separate predicate act) (citing *Roberts v. Peat, Marwick, Mitchell & Co.,*857 F.2d 646 (9th Cir. 1988)); *accord Lange v. Hocker,*940 F.2d 359, 362 (8th Cir.1992) (ratifying a predicate act is not a predicate act). We agree with the district court that the acts marking the beginning and ending of the alleged scheme lasted, at most, ten or eleven months, and next consider whether this is legally sufficient to constitute the requisite pattern of racketeering activity.

The Supreme Court in *H.J., Inc. v. Northwestern Bell Tel. Co.,*492 U.S. 229, 109 S.Ct. 2893, 106 L.Ed.2d 195 (1989), held that in order to prove a "pattern of racketeering activity", a plaintiff must show both "relationship" and "continuity" — that the racketeering predicates are related and that they either constitute or threaten longterm criminal activity. *Id.*at 237-39, 109 S.Ct. at 2899-2901. Continuity over a closed period — that is, not threatening ongoing activity — is proved by showing a series of related predicate acts extending over a "substantial period of time." *Id.*at 242, 109 S.Ct. at 2902. No Eighth Circuit case has set a minimum period of time over which the predicate acts must extend in order to be "substantial." Other Circuits have consistently held that the requirement of continuity over a closed period is not met when the predicate acts extend less than a year. *See, e.g., Uni*Quality, Inc. v. Infotronx, Inc.,*974 F.2d 918, 922 (7th Cir.1992) (seven to eight months insufficient); *Aldridge v. Lily-Tulip, Inc. Salary Retirement Plan Benefits Committee,*953 F.2d 587, 593 (11th Cir.) (six months to a year insufficient), *rehearing denied,*961 F.2d 224 (11th Cir.1992); *Hughes v. Consol-Pennsylvania Coal Co.,*945 F.2d 594, 609-11 (3rd Cir.1991) ("twelve months is not a substantial period of time"), *cert. denied,*— U.S. —, 112 S.Ct. 2300, 119 L.Ed.2d 224 (1992); *American Eagle Credit Corp. v. Gaskins,*920 F.2d 352, 354-55 (6th Cir.1990) (six months insufficient). Many cases in which courts have found a "substantial period of time" have involved schemes extending for a number of years. *See, e.g., Dana Corp. v. Blue Cross & Blue Shield Mut. of Ohio,*900 F.2d 882, 887 (6th Cir.1990) (17 years); *Fleet Credit Corp. v. Sion,*893 F.2d 441, 447 (1st Cir.1990) (four and a half years); *Walk v. Baltimore and Ohio R.R.,*890 F.2d 688, 690 (4th Cir.1989) (ten years). In this case, the activity lasted between ten and eleven months and, in *1216light of the growing body of case law that we have just reviewed, we deem this period insubstantial. Accordingly, plaintiffs did not prove the continuity requirement necessary for their RICO claim to survive summary judgment.

III.

For the foregoing reasons, we affirm.

1

. The Honorable Jean C. Hamilton, United States District Judge for the Eastern District of Missouri.

2

. Consistent with this posture, PHP board meeting notes from March, 1988, stated that the PCA medical convenience centers had been established "to lead us into the PRIMUS opportunity."

3

. Plaintiffs appear also to allege the omission of an additional material fact: In spite of losses in 1984, 1985, and in early 1986, PHP expected PCA to show a profit of $120,000 for the 1986 fiscal year. We do not, however, consider this omission because it was not pleaded in the complaint and was raised for the first time on appeal.

4

. Plaintiffs also maintain that PHP owed them a fiduciary duty because PHP, by purchasing unit interests in PCA in 1985, had become their partner. The specific duty that plaintiffs claim arose from PHP’s fiduciary relationship to the PCA investors was that, if a vote had been taken, PHP would have been compelled by its duty to vote in favor of conversion. Because we do not reach the issue of the outcome of such a vote, we need not consider PHP’s fiduciary duties.

5

. Based on this conclusion, we need not address the question of whether a majority of general participants would have favored conversion.