<PAGE>   1
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                UNITED STATES SECURITIES AND EXCHANGE COMMISSION
                             WASHINGTON, D.C. 20549

                                    FORM 10-Q

  [X]    QUARTERLY REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE 
         SECURITIES EXCHANGE ACT OF 1934

FOR THE QUARTERLY PERIOD ENDED:  JUNE 30, 1998

                                       OR

  [ ]    TRANSITION REPORT PURSUANT TO SECTION 13 OR 15(d) OF THE 
         SECURITIES EXCHANGE ACT OF 1934

FOR THE TRANSITION PERIOD FROM ____________ TO ______________

COMMISSION FILE NUMBER:  33-47913

                         CMC SECURITIES CORPORATION III
             (Exact name of Registrant as specified in its Charter)

                DELAWARE                              75-2431913
     (State or other jurisdiction of               (I.R.S. Employer
     incorporation or organization)                Identification No.)

    2711 NORTH HASKELL, DALLAS, TEXAS                    75204
(Address of principal executive offices)               (Zip Code)

        Registrant's telephone number, including area code (214) 874-2323

The Registrant meets the conditions set forth in General Instruction H(1)(a) and
(b) for Form 10-Q and is therefore filing this Form under the reduced disclosure
format.

Indicate by check mark whether the Registrant (1) has filed all documents and
reports required to be filed by Section 13 or 15(d) of the Securities Exchange
Act of 1934 during the preceding 12 months (or for such shorter period that the
Registrant was required to file such reports), and (2) has been subject to such
filing requirements for the past 90 days. YES  X    NO
                                              ---      ---

                      APPLICABLE ONLY TO CORPORATE ISSUERS:

Indicate the number of shares outstanding of each of the issuer's classes of
common stock, as of the last practicable date.

Common Stock ($1.00 par value)                      1,000 as of August 12, 1998

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<PAGE>   2

                         CMC SECURITIES CORPORATION III
                                    FORM 10-Q
                       FOR THE QUARTER ENDED JUNE 30, 1998


                                      INDEX



                        PART I. -- FINANCIAL INFORMATION

<TABLE>
<CAPTION>
                                                                                                            PAGE
                                                                                                            ----

<S>           <C>                                                                                          <C>
ITEM 1.       Financial Statements

   Balance Sheet -- June 30, 1998 and December 31, 1997...................................................    1

   Statement of Operations -- Quarter and Six Months Ended
     June 30, 1998 and 1997...............................................................................    2

   Statement of Cash Flows -- Six Months Ended June 30, 1998 and 1997.....................................    3

   Notes to Financial Statements..........................................................................    4

ITEM 2.       Management's Discussion and Analysis of
              Financial Condition and Results of Operations...............................................    7


                                           PART II. -- OTHER INFORMATION

ITEM 6.       Exhibits and Reports on Form 8-K............................................................    7

SIGNATURES................................................................................................    8
</TABLE>


<PAGE>   3
                        PART I. -- FINANCIAL INFORMATION
                         CMC SECURITIES CORPORATION III
                                 BALANCE SHEET
                     (IN THOUSANDS, EXCEPT PER SHARE DATA)



ITEM 1.       FINANCIAL STATEMENTS


<TABLE>
<CAPTION>
                                            JUNE 30, 1998    DECEMBER 31, 1997
                                            -------------    -----------------
                                             (UNAUDITED)

<S>                                          <C>                 <C>        
ASSETS
   Mortgage securities collateral            $ 2,103,407         $ 1,630,600
   Cash and other equivalents                       --                     1
                                             -----------         -----------

                                             $ 2,103,407         $ 1,630,601
                                             ===========         ===========

LIABILITIES
   Collateralized mortgage securities        $ 2,103,090         $ 1,630,220
   Payable to Parent                                  45                  40
                                             -----------         -----------
                                               2,103,135           1,630,260
                                             -----------         -----------

STOCKHOLDER'S EQUITY
   Common stock - $1.00 par value,
     1 shares authorized, issued
     and outstanding                                   1                   1
   Paid-in capital                                   835                 824
   Accumulated deficit                              (564)               (484)
                                             -----------         -----------
                                                     272                 341
                                             -----------         -----------

                                             $ 2,103,407         $ 1,630,601
                                             ===========         ===========
</TABLE>







See accompanying notes to financial statements.




                                      -1-
<PAGE>   4

                         CMC SECURITIES CORPORATION III
                             STATEMENT OF OPERATIONS
                                   (UNAUDITED)
                                 (IN THOUSANDS)


<TABLE>
<CAPTION>
                                                  QUARTER ENDED                    SIX MONTHS ENDED
                                                     JUNE 30                            JUNE 30
                                             -------------------------         -------------------------
                                               1998             1997             1998             1997
                                             --------         --------         --------         --------

<S>                                          <C>              <C>              <C>              <C>     
Interest income on mortgage
   securities collateral                     $ 37,251         $ 30,122         $ 64,889         $ 60,893

Interest expense:
   Collateralized mortgage securities          37,283           30,153           64,952           60,956
   Payable to Parent                             --                  9             --                 17
                                             --------         --------         --------         --------
     Total interest expense                    37,283           30,162           64,952           60,973
                                             --------         --------         --------         --------

       Net interest expense                       (32)             (40)             (63)             (80)
                                             --------         --------         --------         --------

Other expenses:
   Management fees                                  2                2                5                5
   Professional fees and other                      3                3               12               11
                                             --------         --------         --------         --------
     Total other expenses                           5                5               17               16
                                             --------         --------         --------         --------

Net loss                                     $    (37)        $    (45)        $    (80)        $    (96)
                                             ========         ========         ========         ========
</TABLE>










See accompanying notes to financial statements.



                                      -2-
<PAGE>   5


                         CMC SECURITIES CORPORATION III
                             STATEMENT OF CASH FLOWS
                                   (UNAUDITED)
                                 (IN THOUSANDS)


<TABLE>
<CAPTION>
                                                       SIX MONTHS ENDED JUNE 30
                                                      ---------------------------
                                                        1998               1997
                                                      --------           --------

<S>                                                   <C>               <C>       
OPERATING ACTIVITIES:
   Net loss                                           $     (80)        $     (96)
   Noncash item - amortization of
     discount and premium                                    58                31
                                                      ---------         ---------
         Net cash used by
           operating activities                             (22)              (65)
                                                      ---------         ---------

INVESTING ACTIVITIES:
   Mortgage securities collateral:
     Purchases of collateral                           (594,390)             --
     Principal collections on collateral                125,723            78,151
     Decrease (increase) in accrued interest
       receivable                                        (2,825)              438
                                                      ---------         ---------
         Net cash provided (used) by
           investing activities                        (471,492)           78,589
                                                      ---------         ---------

FINANCING ACTIVITIES:
   Collateralized mortgage securities:
     Issuance of securities                             594,390              --
     Principal payments on securities                  (125,723)          (78,151)
     Increase (decrease) in accrued
       interest payable                                   2,830              (406)
   Increase (decrease) in payable to Parent                   5              (686)
   Capital contribution                                      11               719
                                                      ---------         ---------
         Net cash provided (used) by financing
           activities                                   471,513           (78,524)
                                                      ---------         ---------

Net change in cash and cash equivalents                      (1)             --

Cash and cash equivalents at beginning
   of period                                                  1                 1
                                                      ---------         ---------

Cash and cash equivalents at end of
   period                                             $    --           $       1
                                                      =========         =========
</TABLE>









See accompanying notes to financial statements.



                                      -3-
<PAGE>   6

                         CMC SECURITIES CORPORATION III
                          NOTES TO FINANCIAL STATEMENTS
                                  JUNE 30, 1998
                                   (UNAUDITED)


NOTE A -- BASIS OF PRESENTATION

The accompanying unaudited financial statements have been prepared in accordance
with generally accepted accounting principles for interim financial information
and with the instructions to Form 10-Q and Rule 10-01 of Regulation S-X.
Accordingly, it does not include all of the information and footnotes required
by generally accepted accounting principles for complete financial statements.
In the opinion of management, all adjustments (consisting of normal recurring
accruals) considered necessary for a fair presentation have been included.
Operating results for the quarter and six months ended June 30, 1998 are not
necessarily indicative of the results that may be expected for the calendar year
ending December 31, 1998. For further information refer to the financial
statements and footnotes thereto included in the CMC Securities Corporation III
annual report on Form 10-K for the year ended December 31, 1997.

NOTE B -- DISCLOSURES REGARDING FAIR VALUES OF MORTGAGE SECURITIES COLLATERAL

The estimated fair values of mortgage securities collateral have been determined
by using available market information and appropriate valuation methodologies;
however, considerable judgment is required in interpreting market data to
develop these estimates. In addition, fair values fluctuate on a daily basis.
Accordingly, estimates presented herein are not necessarily indicative of the
amounts that could be realized in a current market exchange. The use of
different market assumptions and/or estimation methodologies may have a material
effect on estimated fair value amounts.

The fair values of mortgage securities collateral were estimated using quoted
market prices, when available, including quotes made by Capstead Mortgage
Corporation's lenders in connection with designating collateral for repurchase
arrangements.

The following table summarizes the fair values of mortgage securities collateral
(in thousands):

<TABLE>
<CAPTION>
                                              JUNE 30, 1998     DECEMBER 31, 1997
                                              -------------     -----------------

<S>                                            <C>                 <C>        
         Carrying amount                       $ 2,103,407         $ 1,630,600
         Unrealized gains                           22,872              14,768
         Unrealized losses                          (1,568)             (2,389)
                                               -----------         -----------

           Fair value                          $ 2,124,711         $ 1,642,979
                                               ===========         ===========
</TABLE>

All mortgage securities collateral is held-to-maturity. The maturity of mortgage
securities collateral is directly affected by the rate of principal prepayments
by mortgagors. In addition, upon the redemption of remaining bonds outstanding
pursuant to clean-up calls, released collateral may be sold. Such sales are
deemed maturities under the provisions of Statement of Financial Accounting
Standards No. 115. No such redemptions occurred during the six months ended June
30, 1998 or during 1997.



                                      -4-
<PAGE>   7

NOTE C -- MORTGAGE SECURITIES COLLATERAL

During March 1998 the Company acquired from an affiliate conventional mortgage
loans with unpaid principal balances of $594 million as collateral for the
issuance of CMO Series 1998-I. These loans were acquired at an amount equal to
the net proceeds of the issuance.

NOTE D -- COLLATERALIZED MORTGAGE SECURITIES

During March 1998 the Company issued CMO Series 1998-I with a 30-year stated
maturity and a total obligation (including accrued interest and premium) of $598
million. The Company retained no beneficial interest in this CMO and as such, no
economic benefit will be received and no related net income or loss will be
recognized other than the amortization of unreimbursed shelf issuance costs.
This issuance was accounted for as a financing.

NOTE E -- NET INTEREST INCOME ANALYSIS

The following tables summarize the amount of interest income and interest
expense and the average effective interest rates for mortgage securities
collateral and collateralized mortgage securities (dollars in thousands):

<TABLE>
<CAPTION>
                                                             QUARTER ENDED JUNE 30
                                               ---------------------------------------------------
                                                       1998                           1997
                                               --------------------         ----------------------
                                                            AVERAGE                        AVERAGE
                                               AMOUNT         RATE          AMOUNT           RATE
                                               ------       -------         ------         -------

<S>                                            <C>         <C>             <C>            <C>  
Interest income on mortgage
   securities collateral                       $ 37,251       7.50%        $ 30,122         6.92%
Interest expense on
   collateralized mortgage securities            37,283       7.50           30,153         6.92
                                               --------                    --------

Net interest expense                           $    (32)                   $    (31)
                                               ========                    ========
</TABLE>


<TABLE>
<CAPTION>
                                                             SIX MONTHS ENDED JUNE 30
                                               ---------------------------------------------------
                                                       1998                           1997
                                               --------------------         ----------------------
                                                            AVERAGE                        AVERAGE
                                               AMOUNT         RATE          AMOUNT           RATE
                                               ------       -------         ------         -------

<S>                                            <C>         <C>             <C>            <C>  
Interest income on mortgage
   securities collateral                       $ 64,889       7.12%         $ 60,893         6.92%
Interest expense on
   collateralized mortgage securities            64,952       7.12            60,956         6.92
                                               --------                     --------

Net interest expense                           $    (63)                    $    (63)
                                               ========                     ========
</TABLE>




                                      -5-
<PAGE>   8


The following tables summarize changes in interest income and interest expense
due to changes in interest rates, versus changes in volume for the quarter and
six months ended June 30, 1998, compared to the same periods in 1997 (in
thousands):

<TABLE>
<CAPTION>
                                                          QUARTER ENDED JUNE 30, 1998
                                                ----------------------------------------------------
                                                 RATE*                 VOLUME*                TOTAL
                                                 -----                 -------                -----

<S>                                              <C>                  <C>                    <C>    
Interest income on mortgage
   securities collateral                         $ 2,668              $ 4,461                $ 7,129
Interest expense on
   collateralized mortgage securities              2,664                4,466                  7,130
                                                 -------              -------                -------

                                                 $     4              $    (5)               $    (1)
                                                 =======              =======                =======
</TABLE>

<TABLE>
<CAPTION>
                                                          SIX MONTHS ENDED JUNE 30, 1998
                                                ----------------------------------------------------
                                                 RATE*                 VOLUME*                TOTAL
                                                 -----                 -------                -----

<S>                                              <C>                  <C>                    <C>    
Interest income on mortgage
   securities collateral                         $ 1,762              $ 2,234                $ 3,996
Interest expense on
   collateralized mortgage securities              1,762                2,234                  3,996
                                                 -------              -------                -------

                                                 $     -              $     -                $     -
                                                 =======              =======                =======
</TABLE>


*    THE CHANGES IN INTEREST INCOME AND EXPENSE DUE TO BOTH VOLUME AND RATE HAVE
     BEEN ALLOCATED TO VOLUME AND RATE CHANGES IN PROPORTION TO THE RELATIONSHIP
     OF THE ABSOLUTE DOLLAR AMOUNTS OF THE CHANGE IN EACH.




                                      -6-
<PAGE>   9


ITEM 2.   MANAGEMENT'S DISCUSSION AND ANALYSIS OF FINANCIAL CONDITION AND 
          RESULTS OF OPERATIONS


FINANCIAL CONDITION AND RESULTS OF OPERATIONS

CMC Securities Corporation III (the "Company"), was incorporated in Delaware on
May 6, 1992, as a limited purpose finance corporation and is a wholly-owned
subsidiary of Capstead Mortgage Corporation ("CMC").

During the first quarter of 1998, the Company issued CMO Series 1998-I with
total obligations (including accrued interest and premium) of $597.9 million.
Since inception the Company has issued 8 series of collateralized mortgage
obligations ("CMOs") with an aggregate original principal balance of
$2,748,809,000, $4,306,000 of which was issued through private placements, with
the remainder being issued under the registration statement. These issuances
have been accounted for as financings. At June 30, 1998 the Company had
$1,255,496,000 remaining under the registration statement for future CMO
issuances. Since the Company did not retain any investment in the CMOs issued,
no economic benefit was or will be received, thus no related net income or loss
was or will be recognized other than amortization of unreimbursed shelf issuance
costs.

The Company's net losses are due to operational costs incurred (management and
professional fees).

LIQUIDITY AND CAPITAL RESOURCES

All ongoing CMO expenses of the Company are paid out of the excess cash flows on
the CMOs issued before the residual holders receive their residual interest. The
Company believes that the excess cash flows will be sufficient to pay ongoing
CMO expenses. Cash flow requirements due to ongoing operational costs are 
funded by CMC.

                          PART II. -- OTHER INFORMATION


ITEM 6. EXHIBITS AND REPORTS ON FORM 8-K:

(a)     Exhibits:

        Exhibit 27 Financial Data Schedule (electronic filing only).

(b)     Reports on Form 8-K:  None.




                                      -7-
<PAGE>   10

                                   SIGNATURES


Pursuant to the requirements of the Securities Exchange Act of 1934, the
Registrant has duly caused this report to be signed on its behalf by the
undersigned thereunto duly authorized.


                                 CMC SECURITIES CORPORATION III




Date:  August 12, 1998            By:   /s/ RONN K. LYTLE
                                       --------------------------------------
                                       Ronn K. Lytle
                                       Chairman and Chief Executive Officer




Date:  August 12, 1998            By:   /s/ ANDREW F. JACOBS
                                       --------------------------------------
                                       Andrew F. Jacobs
                                       Executive Vice President - Finance,
                                           Treasurer and Secretary




                                      -8-
<PAGE>   11
                               INDEX TO EXHIBITS


<TABLE>
<CAPTION>
EXHIBIT
NUMBER         DESCRIPTION
-------        -----------
<S>            <C>
  27           Financial Data Schedule
</TABLE>