Here's one way of doing it:

WITH Dates (StartDate, EndDate) AS (
    SELECT '20131201', '20170110' UNION ALL
    SELECT '20131101', '20161110' UNION ALL
    SELECT '20130101', '20140131'
    )
SELECT
     StartDate
,    EndDate
,    CASE
        WHEN StartDate >= '20131231' THEN 0
        WHEN EndDate < '20130101' THEN 0
        WHEN StartDate >= '20130101' AND EndDate < '20131231' THEN DATEDIFF(day,StartDate,EndDate)
        WHEN StartDate >= '20130101' AND EndDate > '20131231' THEN DATEDIFF(day,StartDate,'20131231')
        WHEN StartDate < '20130101' AND EndDate <= '20131231' THEN DATEDIFF(day,'20130101',EndDate) + 1
        WHEN StartDate < '20130101' AND EndDate > '20131231' THEN DATEDIFF(day,'20130101','20131231') + 1
     END AS Daysin2013
,    CASE
        WHEN StartDate >= '20141231' THEN 0
        WHEN EndDate < '20140101' THEN 0
        WHEN StartDate >= '20140101' AND EndDate <= '20141231' THEN DATEDIFF(day,StartDate,EndDate)
        WHEN StartDate >= '20140101' AND EndDate > '20141231' THEN DATEDIFF(day,StartDate,'20141231')
        WHEN StartDate < '20140101' AND EndDate <= '20141231' THEN DATEDIFF(day,'20140101',EndDate) + 1
        WHEN StartDate < '20140101' AND EndDate > '20141231' THEN DATEDIFF(day,'20140101','20141231') + 1
     END AS Daysin2014
,    CASE
        WHEN StartDate >= '20151231' THEN 0
        WHEN EndDate < '20150101' THEN 0
        WHEN StartDate >= '20150101' AND EndDate <= '20151231' THEN DATEDIFF(day,StartDate,EndDate)
        WHEN StartDate >= '20150101' AND EndDate > '20151231' THEN DATEDIFF(day,StartDate,'20151231')
        WHEN StartDate < '20150101' AND EndDate <= '20151231' THEN DATEDIFF(day,'20150101',EndDate) + 1
        WHEN StartDate < '20150101' AND EndDate > '20151231' THEN DATEDIFF(day,'20150101','20151231') + 1
     END AS Daysin2015
,    CASE
        WHEN StartDate >= '20161231' THEN 0
        WHEN EndDate < '20160101' THEN 0
        WHEN StartDate >= '20160101' AND EndDate <= '20161231' THEN DATEDIFF(day,StartDate,EndDate)
        WHEN StartDate >= '20160101' AND EndDate > '20161231' THEN DATEDIFF(day,StartDate,'20161231')
        WHEN StartDate < '20160101' AND EndDate <= '20161231' THEN DATEDIFF(day,'20160101',EndDate) + 1
        WHEN StartDate < '20160101' AND EndDate > '20161231' THEN DATEDIFF(day,'20160101','20161231') + 1
     END AS Daysin2016
,    CASE
        WHEN StartDate >= '20171231' THEN 0
        WHEN EndDate < '20170101' THEN 0
        WHEN StartDate >= '20170101' AND EndDate <= '20171231' THEN DATEDIFF(day,StartDate,EndDate)
        WHEN StartDate >= '20170101' AND EndDate > '20171231' THEN DATEDIFF(day,StartDate,'20171231')
        WHEN StartDate < '20170101' AND EndDate <= '20171231' THEN DATEDIFF(day,'20170101',EndDate) + 1
        WHEN StartDate < '20170101' AND EndDate > '20171231' THEN DATEDIFF(day,'20170101','20171231') + 1
     END AS Daysin2017
FROM Dates d;

John