Exhibit (a)(1)(F)

This announcement is neither an offer to purchase nor a solicitation of an offer to sell shares. The Offer (as defined below) is made solely by the Offer to Purchase, dated December 8, 2015, and the related Letter of Transmittal, as they may be amended or supplemented from time to time. The information contained therein is incorporated herein by reference. The Offer is not being made to, nor will tenders be accepted from or on behalf of, holders of shares in any jurisdiction in which the making or acceptance of offers to sell shares would not be in compliance with the laws of that jurisdiction. If, after a good faith effort, the Company (as defined below) cannot comply with applicable law, the Offer will not be made to (nor will tenders be accepted from or on behalf of) the stockholders residing in such jurisdiction. In any jurisdiction where the securities, blue sky or other laws require the Offer to be made by a licensed broker or dealer, the Offer shall be deemed to be made on behalf of the Company by the Dealer Manager (as defined below), or by one or more registered brokers or dealers licensed under the laws of that jurisdiction.

Notice of Offer to Purchase by

Insperity, Inc.

of

Shares of Its Common Stock for an Aggregate Purchase Price

of Not More Than $125 Million

at a Per Share Purchase Price Not Less Than $43.50 Per Share

Nor Greater Than $50.00 Per Share

Insperity, Inc., a Delaware corporation (the “Company”), is offering to purchase for cash shares of its common stock, par value $0.01 per share, pursuant to (i) auction tenders at prices specified by the tendering stockholders of not less than $43.50 nor greater than $50.00 per share (“Auction Tenders”) or (ii) purchase price tenders (“Purchase Price Tenders”), in either case upon the terms and subject to the conditions described in the Offer to Purchase, dated December 8, 2015 (the “Offer to Purchase”), and in the related Letter of Transmittal (the “Letter of Transmittal” and together with the Offer to Purchase, as they may be amended or supplemented from time to time, constitute the “Offer”).

THE OFFER AND WITHDRAWAL RIGHTS WILL EXPIRE AT 5:00 P.M., NEW YORK CITY TIME, ON JANUARY 7, 2016, UNLESS THE OFFER IS EXTENDED OR TERMINATED (SUCH TIME, AS IT MAY BE EXTENDED, THE “EXPIRATION TIME”).

The Offer is not conditioned upon any minimum number of shares being tendered. The Offer is, however, subject to a number of other terms and conditions as specified in the Offer to Purchase.

The Company’s Board of Directors has approved the Offer. However, none of the Company, the Company’s Board of Directors, the Dealer Manager, the Depositary (as defined below) or the Information Agent (as defined below) makes any recommendation to the Company’s stockholders as to whether to tender or refrain from tendering their shares or as to the price or prices at which stockholders may choose to tender their shares. The Company has not authorized any person to make any such recommendation. Stockholders must make their own decision as to whether to tender their shares and, if so, how many shares to tender and the price or prices at which their shares should be tendered. In doing so, stockholders should read carefully the information in, or incorporated by reference in, the Offer to Purchase and in the related Letter of Transmittal, including the purposes and effects of the Offer. Stockholders are urged to discuss their decision with their tax advisers, financial advisers and/or brokers.

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The Company’s Board of Directors, with the assistance of management, has decided to conduct the Offer after reviewing a variety of alternatives for using the Company’s available financial resources. The Board considered the Company’s existing and anticipated capitalization and financial position, including its outstanding common stock, liquidity, cash position and working capital, lack of long-term debt, and its operations, strategy and expectations for the future.

The Board believes that the Offer represents a prudent use of the Company’s financial resources in light of its business profile, assets, current indebtedness and debt capacity. At September 30, 2015, the Company had working capital of $76.4 million, no long-term debt and $124.4 million of available borrowing capacity under its existing revolving credit facility. The primary purpose of the Offer is to return cash to the Company’s stockholders by providing them with the opportunity to tender all or a portion of their shares and, thereby, receive a return of some or all of their investment if they so elect. In addition, if the Company completes the Offer, stockholders who do not participate in the Offer will automatically increase their relative percentage ownership interest in the Company and its future operations.

In accordance with the instructions to the Letter of Transmittal, stockholders desiring to tender shares must specify (1) whether shares are tendered pursuant to an Auction Tender or a Purchase Price Tender and (2) if an Auction Tender is made, the price, not less than $43.50 nor greater than $50.00 per share (in increments of $0.25), at which they are willing to tender their shares to the Company in the Offer. After the Expiration Time, the Company will, upon the terms and subject to the conditions of the Offer, determine a single price per share (the “Purchase Price”), which will be not less than $43.50 and not more than $50.00 per share, that it will pay for shares of the Company’s common stock validly tendered in the Offer and not validly withdrawn, taking into account the number of shares tendered pursuant to Auction Tenders and Purchase Price Tenders and the prices specified by stockholders tendering shares pursuant to Auction Tenders. Shares tendered pursuant to Purchase Price Tenders will be deemed to have been tendered at a price of $43.50 per share (which is the minimum price per share under the Offer) for purposes of determining the Purchase Price. Accordingly, elections to tender shares pursuant to Purchase Price Tenders may cause the Purchase Price to be lower and could result in the tendered shares being purchased at the minimum price of $43.50 per share. The Purchase Price will be the lowest price per share of not less than $43.50 and not more than $50.00 per share, at which shares have been tendered or have been deemed to be tendered in the Offer, that will enable the Company to purchase the maximum number of shares validly tendered pursuant to the Offer and not withdrawn having an aggregate purchase price not exceeding $125 million. Shares validly tendered pursuant to an Auction Tender will not be purchased if the Purchase Price determined by the Company for the shares is less than the price selected by the stockholder. All shares purchased in the Offer will be purchased at the same Purchase Price regardless of whether the stockholder tendered at a lower price.

As of December 1, 2015, there were 24,277,515 shares issued and outstanding. Assuming that the Offer is fully subscribed, if the Purchase Price is determined to be $43.50 per share, the minimum Purchase Price under the Offer, the approximate number of shares that would be purchased under the Offer is 2,873,563, or 11.8% of the shares outstanding on December 1, 2015. Assuming that the Offer is fully subscribed, if the Purchase Price is determined to be $50.00 per share, the maximum Purchase Price under the Offer, the approximate number of shares that would be purchased under the Offer is 2,500,000, or 10.3% of the shares outstanding on December 1, 2015.

Upon the terms and subject to the conditions of the Offer, if the number of shares of common stock validly tendered at or below the Purchase Price and not validly withdrawn would result in an aggregate purchase price of more than $125 million, the Company will purchase shares:

• First, from all holders of “odd lots” of less than 100 shares who validly tender all of their shares at or below the Purchase Price, and do not validly withdraw them (tenders of less than all of the shares owned, beneficially or of record, by such “odd lot” shareholders will not qualify for this preference);

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• Second, from all other stockholders who validly tender shares at or below the Purchase Price (and do not validly withdraw such shares), on a pro rata basis, with appropriate adjustments to avoid the purchase of fractional shares (except for stockholders who tendered shares conditionally for which the condition was not satisfied), until the Company has purchased shares resulting in an aggregate purchase price of $125 million; and

• Third, only if necessary to permit the Company to purchase shares resulting in an aggregate purchase price of $125 million, from holders who validly tender (and do not validly withdraw) shares at or below the Purchase Price conditionally (for which the condition was not initially satisfied) by random lot, to the extent feasible. To be eligible for purchase by random lot, stockholders whose shares are conditionally tendered must have validly tendered and not validly withdrawn all of their shares prior to the Expiration Time.

If any tendered shares are not purchased, or if less than all shares evidenced by a stockholder’s certificates are tendered, a Direct Registration System Book-Entry Statement for unpurchased shares will be returned promptly after the expiration or termination of the Offer or the valid withdrawal of the shares, or, in the case of shares tendered by book-entry transfer at DTC (as defined in the Offer to Purchase), the shares will be credited to the appropriate account maintained by the tendering stockholder at DTC, in each case at the Company’s expense.

Stockholders wishing to tender their shares must follow the procedures set forth in Section 3 of the Offer to Purchase and in the Letter of Transmittal. Stockholders wishing to tender their shares but who are unable to deliver them physically or by book-entry transfer prior to the Expiration Time, or who are unable to make delivery of all required documents to the Depositary prior to the Expiration Time, may tender their shares by complying with the procedures set forth in Section 3 of the Offer to Purchase for tendering by Notice of Guaranteed Delivery.

For purposes of the Offer, the Company will be deemed to have accepted for payment, subject to the “odd lot” priority, proration and conditional tender provisions of the Offer, shares that are validly tendered at or below the Purchase Price and not validly withdrawn, only when, as and if the Company gives oral or written notice to Computershare Trust Company, N.A. (the “Depositary”) of its acceptance of the shares for payment pursuant to the Offer.

Upon the terms and subject to the conditions of the Offer, the Company will accept for payment and pay the Purchase Price per share for all of the shares accepted for payment pursuant to the Offer promptly after the Expiration Time. In all cases, payment for shares tendered and accepted for payment pursuant to the Offer will be made promptly, taking into account any time necessary to determine any proration, but only after timely receipt by the Depositary of (1) certificates for shares, or a timely book-entry confirmation of the deposit of shares into the Depositary’s account at DTC, (2) a validly completed and duly executed Letter of Transmittal including any required signature guarantees, or, in the case of a book-entry transfer, an Agent’s Message, and (3) any other required documents.

The Company expressly reserves the right, in its sole discretion and subject to applicable law, at any time and from time to time, and regardless of whether or not any of the conditions to the Offer set forth in Section 7 of the Offer to Purchase have occurred or be deemed by the Company to have occurred, to extend the period of time the Offer is open and delay acceptance for payment of, and payment for, any shares by giving oral or written notice of such extension to the Depositary and making a public announcement of such extension no later than 9:00 a.m., New York City time, on the next business day after the last previously scheduled or announced Expiration Time. In the event of an extension, the term “Expiration Time” will refer to the latest time and date at which the Offer, as extended by the Company, will expire. During any such extension, all shares previously tendered and not validly withdrawn will remain subject to the Offer and to the right of a tendering stockholder to withdraw such stockholder’s shares.

The Company also expressly reserves the right, in its sole discretion, to terminate the Offer and reject for payment and not pay for any shares not theretofore accepted for payment or paid for, subject to applicable law, or to postpone payment for shares, upon the occurrence of any of the conditions to the Offer specified in Section 7 of the Offer to Purchase, by giving oral or written notice of such termination or postponement to the Depositary and making a public announcement of such termination or postponement. The Company’s reservation of the right to delay payment for shares that it has accepted for payment is limited by Rule 13e-4(f)(5) and Rule 14e-1 under the Securities Exchange Act of 1934, as amended (the “Exchange Act”), which requires that the Company must pay the consideration offered or return the shares tendered promptly after termination or withdrawal of the Offer.

Shares tendered in the Offer may be withdrawn at any time prior to the Expiration Time. In addition, unless the Company has already accepted such tendered shares for payment, stockholders may withdraw their tendered shares at any time after 11:59 p.m., New York City time, on February 9, 2016. Except as otherwise provided in the Offer to Purchase, tenders of shares pursuant to the Offer are irrevocable. For a withdrawal to be effective, a written or facsimile notice of withdrawal must be received in a timely manner by the Depositary at one of its addresses set forth on the back cover page of the Offer to Purchase, and any notice of withdrawal must specify the name of the tendering stockholder, the number of shares to be withdrawn, the price at which such shares

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were tendered, if an Auction Tender is being withdrawn, and the name of the registered holder of the shares to be withdrawn, if different from the person who tendered the shares. A stockholder who has tendered shares at more than one price must complete a

separate notice of withdrawal for shares tendered at each price. If the certificates for shares to be withdrawn have been delivered or otherwise identified to the Depositary, then, before the release of those certificates, the tendering stockholder also must submit the serial numbers shown on those particular certificates for shares to be withdrawn and, unless an Eligible Institution (as defined in the Offer to Purchase) has tendered those shares, the signature(s) on the notice of withdrawal must be guaranteed by an Eligible Institution. If shares have been tendered pursuant to the procedure for book-entry transfer described in Section 3 of the Offer to Purchase, the notice of withdrawal also must specify the name and the number of the account at DTC to be credited with the withdrawn shares and must otherwise comply with DTC’s procedures.

All questions as to the form and validity, including the time of receipt, of any notice of withdrawal will be determined by the Company, in its sole discretion and will be final and binding on all parties absent a finding to the contrary by a court of competent jurisdiction. The Company reserves the absolute right to waive any defect or irregularity in the notice of withdrawal or method of withdrawal of shares by any stockholder, whether or not the Company waives similar defects or irregularities in the case of any other stockholder. None of the Company, the Dealer Manager, the Depositary, the Information Agent or any other person will be obligated to give notice of any defects or irregularities in any notice of withdrawal, nor will any of them incur liability for failure to give any such notice.

Generally, U.S. stockholders will be subject to U.S. federal income taxation when they receive cash from the Company in exchange for the shares they tender. Their receipt of cash for tendered shares will generally be treated as either (1) consideration received in a sale or exchange or (2) a dividend. All stockholders should read carefully the Offer to Purchase for additional information regarding certain tax issues and should consult their own tax adviser regarding the tax consequences of the Offer.

The Offer to Purchase and the Letter of Transmittal contain important information that stockholders should read carefully before they make any decision with respect to the Offer.

The information required to be disclosed by Rule 13e-4(d)(1) under the Exchange Act, is contained in the Offer to Purchase and is incorporated herein by reference.

Please direct any questions or requests for assistance to Innisfree M&A Incorporated (the “Information Agent”) or Credit Suisse Securities (USA) LLC (the “Dealer Manager”) at their respective telephone numbers and addresses set forth below. Please direct requests for copies of the Offer to Purchase, the Letter of Transmittal or the Notice of Guaranteed Delivery to the Information Agent at the telephone numbers and address set forth below. The Information Agent will promptly furnish to stockholders additional copies of these materials at the Company’s expense. Stockholders may also contact their broker, dealer, commercial bank, trust company or other nominee for assistance concerning the Offer.

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The Information Agent for the Offer is:

501 Madison Avenue, 20th Floor

New York, New York 10022

Banks and Brokers Call: (212) 750-5833

All Others Call Toll-Free: (888) 750-5834

The Dealer Manager for the Offer is:

Credit Suisse Securities (USA) LLC

Eleven Madison Avenue

New York, New York 10010

Call Toll-Free (800) 318-8219

December 8, 2015

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