I'm not fond of the static years in this code.  Perhaps it could be converted to a dynamic pivot
declare @t table (    
    startdate date,
    enddate date
)
declare @n table (
    Nbr tinyint
)
;
insert @n values(0),(1),(2),(3),(4)
;
insert @t values
    ('12/1/2013',    '1/10/2017'),
    ('11/1/2013',    '11/10/2016'),                    
    ('1/1/2013',    '1/31/2014')
;
select StartDate, Enddate, [2013] as [2013], [2014] as [2014], [2015] as [2015], [2016] as [2016], [2017] as [2017]
from (
    select
        StartDate,
        EndDate,
        DATEPART(YEAR, (DATEADD(YEAR, n.Nbr, t.StartDate))) AS [Year],
        DATEDIFF(
            DAY,
            case DATEPART(YEAR, (DATEADD(YEAR, n.Nbr, t.StartDate)))
                when datepart(year,startdate) then t.startdate
                else CAST(DATEPART(YEAR, (DATEADD(YEAR, n.Nbr, t.StartDate))) AS CHAR(4)) + '-01-01'
            end,
            case DATEPART(YEAR, (DATEADD(YEAR, n.Nbr, t.StartDate)))
                when datepart(year,enddate) then t.enddate
                else CAST(DATEPART(YEAR, (DATEADD(YEAR, n.Nbr, t.StartDate))) AS CHAR(4)) + '-12-31'
            end
        ) AS DayCnt
    from @t t
    inner join @n n on n.Nbr < datepart(year,t.enddate) - datepart(year,t.startdate) + 1
) p
pivot (
    min(DayCnt)
    for [Year] in ([2013], [2014], [2015], [2016], [2017])
) as pvt
order by startdate, enddate