I don't know that it's faster, but it's certainly simpler.

;
WITH Dates (ID, StartDate, EndDate) AS (
  SELECT 1, '20131201', '20170110' UNION ALL
  SELECT 2, '20131101', '20161110' UNION ALL
  SELECT 3, '20130101', '20140131'
  )
SELECT
    ID
,    MAX(StartDate) AS StartDate
,    MAX(EndDate) AS EndDate
,    MAX(CASE WHEN n = 5 THEN DaysInPeriod END) AS DaysInYear1
,    MAX(CASE WHEN n = 4 THEN DaysInPeriod END) AS DaysInYear2
,    MAX(CASE WHEN n = 3 THEN DaysInPeriod END) AS DaysInYear3
,    MAX(CASE WHEN n = 2 THEN DaysInPeriod END) AS DaysInYear4
,    MAX(CASE WHEN n = 1 THEN DaysInPeriod END) AS DaysInYear5
FROM Dates
CROSS APPLY
(
    SELECT n, PeriodDate, DATEDIFF(DAY, LAG(PeriodDate, 1, StartDate) OVER(PARTITION BY ID ORDER BY PeriodDate), PeriodDate) AS DaysInPeriod

    FROM
    (
        SELECT n, DATEADD(YEAR, DATEDIFF(YEAR, -1, GETDATE()) - n, -1) AS PeriodDate
        FROM ( VALUES(1), (2), (3), (4), (5) ) t(n)
        UNION
        SELECT DATEDIFF(YEAR, EndDate, GETDATE()), EndDate
    ) v
    WHERE PeriodDate > StartDate
        AND PeriodDate <= EndDate
) v
GROUP BY ID
;