Why was this law formulated this way?
To be consistent with rules established by Congress and prior law.
What is the purpose of requiring specifically 5 year and 10 year periods for such items?
The first use of a 5 year analysis was at the creation of the Congressional Budget Office in 1974. It is likely that the 10 year analysis was added to better control the the budget in regard to deficits.
The 10 year period will necessarily be outside the current administration. Why 10 years, and not more?
The Senate has a long-term rule for legislation that includes four additional decades; currently, to the budget year 2079. There are long-term budget projections going out as far as 70 years.
Long-Term Budgeting within the Congressional Budget Process: In Brief, June 28, 2019
General Budgetary Projections for the Upcoming 10-Year Period
In addition to the information provided on the 10-year budgetary outlook under current law, CBO provides Congress with cost estimates of certain proposed legislation. The Congressional Budget Act of 1974 (the Budget Act) requires that the CBO provide an estimate for any bill reported from committee. These cost estimates provide information on how the legislation would affect spending, revenues, and the deficit over the next 10 years relative to the baseline.
General Budgetary Projections for the Upcoming Decades
Each year, CBO provides Congress with its Long-Term Budget Outlook, which shows the effects of demographic trends, economic developments, and rising health care costs on federal spending, revenues, and deficits over the next 30 years. The report also shows the long-term budgetary and economic effects of some alternative policies.
In addition, in its cost estimates, CBO is required to note whether the underlying legislation would increase deficits in future decades. To assist the Senate in complying with its “long-term deficit rule” (described below), CBO notes whether the legislation would increase on-budget deficits in any of the four consecutive 10-year periods beginning with 2030.
OMB provides long-term projections in the President’s annual budget request in a section titled, “Long Term Budget Outlook.” These projections recently spanned a 25-year period and include projections under different fiscal scenarios.
The Government Accountability Office also provides information and interactive tools on projected spending, revenue, deficits, and debt over the next 70 years.
Additional Rules and Points of Order
[...], in the Senate, a rule exists that is often referred to as the “long-term deficit point of order.” It prohibits the consideration of legislation that would cause a net increase in deficits of more than $5 billion in any of the four consecutive 10-year periods beginning after the upcoming 10 years.