Exhibit 99.1

ASPEN INSURANCE HOLDINGS CONFIRMS

UNSOLICITED EXCHANGE OFFER BY ENDURANCE

Aspen Rejected Endurance Proposal with Identical Terms

Shareholders Advised to Take No Action Pending Formal Board

Recommendation on Schedule 14D-9

Hamilton, Bermuda – June 9, 2014 – Aspen Insurance Holdings Limited (“Aspen”) (NYSE:AHL) confirmed today that Endurance Specialty Holdings Ltd. (“Endurance”) (NYSE:ENH) has commenced an unsolicited exchange offer to acquire all outstanding shares of Aspen for a combination of common stock and cash.

On June 2, Aspen announced that its directors had unanimously rejected an unsolicited proposal from Endurance with identical terms, noting that the proposal grossly undervalued Aspen, represented a strategic mismatch and, based on conversations with major clients and brokers, would result in significant dis-synergies. Moreover, the Company noted that the majority of the consideration included in the proposal consisted of Endurance stock, which is highly unappealing.

As demonstrated by Aspen’s strong first quarter results, the Company is delivering on a strategic plan to generate strong growth and shareholder value. Aspen achieved strong results across all parts of its business in the first quarter, with a resulting annualized operating ROE of 14.8%. The Company is well positioned to achieve its 10% operating ROE objective in 2014 and to deliver on its expectation that 2015 operating ROE will increase in the order of 100 basis points from 2014.¹

In light of Endurance’s formal commencement of an exchange offer, and consistent with its fiduciary duties, the Board of Directors of Aspen will carefully review the exchange offer in consultation with its independent financial and legal advisors and determine the course of action that it believes is in the best interests of Aspen and its shareholders. The Board intends to advise shareholders of its recommendation regarding the exchange offer within ten business days by making available to shareholders and filing with the U.S. Securities and Exchange Commission a solicitation/recommendation statement on Schedule 14D-9.

Aspen shareholders are advised to take no action at this time pending the Board’s recommendation in the Schedule 14D-9.

Goldman, Sachs & Co. is acting as financial advisor and Wachtell, Lipton, Rosen & Katz and Willkie Farr & Gallagher LLP are acting as legal advisors to Aspen.

- ENDS –

About Aspen Insurance Holdings Limited

Aspen provides reinsurance and insurance coverage to clients in various domestic and global markets through wholly-owned subsidiaries and offices in Bermuda, France, Germany, Ireland, Singapore, Switzerland, the United Kingdom and the United States. For the year ended December 31, 2013, Aspen reported $10.2 billion in total assets, $4.7 billion in gross reserves, $3.3 billion in shareholders’ equity and $2.6 billion in gross written premiums. Its operating subsidiaries have been assigned a rating of “A” (“Strong”) by Standard & Poor’s, an “A” (“Excellent”) by A.M. Best and an “A2” (“Good”) by Moody’s.

Application of the Safe Harbor of the Private Securities Litigation Reform Act of 1995

This press release contains written, and Aspen may make related oral, “forward-looking statements” within the meaning of the U.S. federal securities laws. These statements are made pursuant to the safe harbor provisions of the Private Securities Litigation Reform Act of 1995. Forward-looking statements include all statements that do not relate solely to historical or current facts, and can be identified by the use of words such as “expect,” “intend,” “plan,” “believe,” “do not believe,” “aim,” “project,” “anticipate,” “seek,” “will,” “likely,” “estimate,” “may,” “continue,” “guidance,” “outlook,” “trends,” “future,” “could,” “target,” and similar expressions of a future or forward-looking nature.

All forward-looking statements rely on a number of assumptions, estimates and data concerning future results and events and are subject to a number of uncertainties and other factors, many of which are outside Aspen’s control that could cause actual results to differ materially from such statements.

Forward-looking statements do not reflect the potential impact of any future collaboration, acquisition, merger, disposition, joint venture or investments that Aspen may enter into or make, and the risks, uncertainties and other factors relating to such statements might also relate to the counterparty in any such transaction if entered into or made by Aspen.

All forward-looking statements address matters that involve risks and uncertainties. Accordingly, there are or will be important factors that could cause actual results to differ materially from those indicated in these statements. Aspen believes these factors include, but are not limited to: our ability to successfully implement steps to further optimize the business portfolio, ensure capital efficiency and enhance investment returns; the possibility of greater frequency or severity of claims and loss activity, including as a result of natural or man-made (including economic and political risks) catastrophic or material loss events, than our underwriting, reserving, reinsurance purchasing or investment practices have anticipated; the assumptions and uncertainties underlying reserve levels that may be impacted by future payments for settlements of claims and expenses or by other factors causing adverse or favorable development; the reliability of, and changes in assumptions to, natural and man-made catastrophe pricing, accumulation and estimated loss models; decreased demand for our insurance or reinsurance products and cyclical changes in the highly competitive insurance and reinsurance industry; increased competition from existing insurers and reinsurers and from alternative capital providers and insurance-linked funds and collateralized special purpose insurers on the basis of pricing, capacity, coverage terms, new capital, binding authorities to brokers or other factors and the related demand and supply dynamics as contracts come up for renewal; changes in general economic conditions, including inflation, deflation, foreign currency exchange rates, interest rates and other factors that could affect our financial results; the risk of a material decline in the value or liquidity of all or parts of our investment portfolio; evolving issues with respect to interpretation of coverage after major loss events; our ability to adequately model and price the effect of climate cycles and climate change; any intervening legislative or governmental action and changing judicial interpretation and judgements on insurers’ liability to various risks; the effectiveness

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of our risk management loss limitation methods, including our reinsurance purchasing; changes in the total industry losses, or our share of total industry losses, resulting from past events and, with respect to such events, our reliance on loss reports received from cedants and loss adjustors, our reliance on industry loss estimates and those generated by modeling techniques, changes in rulings on flood damage or other exclusions as a result of prevailing lawsuits and case law; the impact of one or more large losses from events other than natural catastrophes or by an unexpected accumulation of attritional losses; the impact of acts of terrorism, acts of war and related legislation; any changes in our reinsurers’ credit quality and the amount and timing of reinsurance recoverables; changes in the availability, cost or quality of reinsurance or retrocessional coverage; the continuing and uncertain impact of the current depressed lower growth economic environment in many of the countries in which we operate; the level of inflation in repair costs due to limited availability of labor and materials after catastrophes; a decline in our operating subsidiaries’ ratings with S&P, A.M. Best or Moody’s; the failure of our reinsurers, policyholders, brokers or other intermediaries to honor their payment obligations; our ability to execute our business plan to enter new markets, introduce new products and develop new distribution channels, including their integration into our existing operations; our reliance on the assessment and pricing of individual risks by third parties; our dependence on a few brokers for a large portion of our revenues; the persistence of heightened financial risks, including excess sovereign debt, the banking system and the Eurozone debt crisis; changes in our ability to exercise capital management initiatives (including our share repurchase program) or to arrange banking facilities as a result of prevailing market changes or changes in our financial position; changes in government regulations or tax laws in jurisdictions where we conduct business; changes in accounting principles or policies or in the application of such accounting principles or policies; Aspen or Aspen Bermuda Limited becoming subject to income taxes in the United States or the United Kingdom; loss of one or more of our senior underwriters or key personnel; our reliance on information and technology and third party service providers for our operations and systems; and increased counterparty risk due to the credit impairment of financial institutions. For a more detailed description of these uncertainties and other factors, please see the “Risk Factors” section in Aspen’s Annual Report on Form 10-K as filed with the U.S. Securities and Exchange Commission on February 20, 2014 and in Aspen’s Quarterly Report on Form 10-Q as filed with the U.S. Securities and Exchange Commission on May 1, 2014. Aspen undertakes no obligation to publicly update or revise any forward-looking statements, whether as a result of new information, future events or otherwise. Readers are cautioned not to place undue reliance on these forward-looking statements, which speak only as of the dates on which they are made.

The guidance in this press release relating to 10% Operating ROE in 2014 and with a further 100 basis point increase over 2014 in 2015 was and is made as at April 23, 2014. Such guidance assumes for 2014 a pre-tax catastrophe load of $185 million per annum, normal loss experience and given the current interest rate and insurance pricing environment and for 2015 a pre-tax catastrophe load of $200 million, normal loss experience, our expectations for rising interest rates, and a less favorable insurance pricing environment. Aspen has identified and described in the presentation dated May 2014, in the investor relations section of its website actions and additional underlying assumptions in each of its three operating return on equity levers – optimization of the business portfolio, capital efficiency and enhancing investment returns – to seek to achieve the targeted operating ROE in 2014 and 2015. These forward looking statements are subject to the assumptions, risks and uncertainties, as discussed above and in the following slides, which could cause actual results to differ materially from these statements.

In addition, any estimates relating to loss events involve the exercise of considerable judgment and reflect a combination of ground-up evaluations, information available to date from brokers and cedants, market intelligence, initial tentative loss reports and other sources. The actuarial range of reserves and management’s best estimate represents a distribution from our internal capital model for reserving risk based on our then current state of knowledge and explicit and implicit assumptions relating to the incurred pattern of claims, the expected ultimate settlement amount, inflation and dependencies between lines of business. Due to the complexity of factors contributing to the losses and the preliminary nature of the information used to prepare these estimates, there can be no assurance that Aspen’s ultimate losses will remain within the stated amounts.

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Additional Information

This communication does not constitute an offer to buy or solicitation of an offer to sell any securities or a solicitation of any vote or approval. This communication is for informational purposes only and is not a substitute for any relevant documents that Aspen may file with the U.S. Securities and Exchange Commission (“SEC”).

Endurance Specialty Holdings Ltd. has commenced an exchange offer for the outstanding shares of Aspen (together with associated preferred share purchase rights). Aspen will file a solicitation/recommendation statement on Schedule 14D-9 with the SEC.

INVESTORS AND SECURITY HOLDERS OF ASPEN ARE URGED TO READ THIS AND OTHER DOCUMENTS FILED WITH THE SEC CAREFULLY IN THEIR ENTIRETY AS THEY BECOME AVAILABLE BECAUSE THEY WILL CONTAIN IMPORTANT INFORMATION. Investors and security holders will be able to obtain free copies of these documents (when available) and other documents filed with the SEC by Aspen through the web site maintained by the SEC at http://www.sec.gov. These documents will also be available on Aspen’s website at http://www.aspen.co.

Certain Information Regarding Participants

Aspen and certain of its respective directors and executive officers may be deemed to be participants under the rules of the SEC. Security holders may obtain information regarding the names, affiliations and interests of Aspen’s directors and executive officers in Aspen’s Annual Report on Form 10-K for the year ended December 31, 2013, which was filed with the SEC on February 20, 2014, and its proxy statement for the 2014 Annual Meeting, which was filed with the SEC on March 12, 2014. These documents can be obtained free of charge from the sources indicated above.

For further information:

Please visit www.aspen.co or contact:

Investors

Kerry Calaiaro, Senior Vice President, Investor Relations, Aspen

Kerry.Calaiaro@aspen.co

+1 (646) 502 1076

Kathleen de Guzman, Vice President, Investor Relations, Aspen

kathleen.deguzman@aspen.co

+1 (646) 289 4912

Media

Steve Colton, Head of Communications, Aspen

Steve.Colton@aspen.co

+44 20 7184 8337

North America – Sard Verbinnen & Co

Paul Scarpetta or Jamie Tully

+1 (212) 687 8080

International – Citigate Dewe Rogerson

Patrick Donovan or Caroline Merrell

patrick.donovan@citigatedr.co.uk

caroline.merrell@citigatedr.co.uk

+44 20 7638 9571

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