(5) A rent exemption order shall be issued to each tenant who applies to the department of finance in the case of a tenant who is sixty-two years of age or older or to such agency as the mayor shall designate in the case of a tenant who is a person with a disability, in accordance with such department's or agency's applicable regulations and who is found to be eligible under this subdivision. Such order shall take effect on the first day of the first month after receipt of such application by the department of finance or such agency as the mayor shall designate, except that where there is any other increase in the legal regulated rent within ninety days of the issuance of the order increasing the tenant's maximum rent which a tenant is not exempted from paying, the rent exemption order shall without further order of the department of finance or such agency as the mayor shall designate take effect as of the effective date of said order increasing the tenant's rent including any retroactive increments collectible pursuant to such order.