Contractual Obligations Following is a summary of the Company's contractual obligations as of December 31, 2018: The estimated payments for reserves for losses and loss expenses in the above table represent the projected (undiscounted) payments for gross loss and loss expense reserves related to losses incurred as of December 31, 2018. The estimated payments in the above table do not consider payments for losses to be incurred in future periods. These amounts include reserves for reported losses and reserves for incurred but not reported losses. Estimated amounts recoverable from reinsurers are not reflected. The estimated payments by year are based on historical loss payment patterns.The actual payments may differ from the estimated amounts due to changes in ultimate loss reserves and in the timing of the settlement of those reserves. In addition, at December 31, 2018, the Company had commitments to invest up to $270.2 million and $253.4 million in certain investment funds and real estate construction projects, respectively. These amounts are not included in the above table. The Company utilizes letters of credit to back certain reinsurance payments and obligations. Outstanding letters of credit were $3 million as of December 31, 2018. The Company has made certain guarantees to state regulators that the statutory capital of certain subsidiaries will be maintained above certain minimum levels. Off-Balance Sheet Arrangements An off-balance sheet arrangement is any transaction, agreement or other contractual arrangement involving an unconsolidated entity under which a company has (1) made guarantees, (2) a retained or contingent interest in transferred assets, (3) an obligation under derivative instruments classified as equity or (4) any obligation arising out of a material variable interest in an unconsolidated entity that provides financing, liquidity, market risk or credit risk support to the Company, or that engages in leasing, hedging or research and development arrangements with the Company. The Company has no arrangements of these types that management believes may have a material current or future effect on our financial condition, liquidity or results of operations. 53 (In thousands) Estimated Payments By Periods 2019 2020 2021 2022 2023  Thereafter Gross reserves for losses $ 3,137,565 $ 2,154,391 $ 1,609,899 $ 1,161,065 $ 848,536 $ 3,637,803 Operating lease obligations 46,592 43,504 39,061 34,444 30,881 75,740 Purchase obligations 87,976 46,676 39,191 39,200 37,869 — Subordinated debentures — — — — — 935,000 Debt maturities 447,433 315,461 — 426,503 — 701,750 Interest payments 155,391 124,616 108,491 105,163 85,647 2,253,250 Other long-term liabilities 3,617 3,317 2,972 2,627 2,362 26,608 Total $ 3,878,574 $ 2,687,965 $ 1,799,614 $ 1,769,002 $ 1,005,295 $ 7,630,151