Help me

I am trying to find a shoujo/romance manga where female mc (princess) have an arranged marriage with a rumored cold prince (kill ppl) which he actually not. The mc has a bad reputation being cold hearted too. After marriage, both of them realize they are not what the rumor said.
MC has a lil sister who has a bright personality and care about her ppl. The arranged marriage was for lil sister but she didn't want to get married with a rumored cold-hearted prince. Lil sister misunderstood mc thinking she is cold hearted and spend money on parties.... but she (mc) has reasons why she does those. MC takes cares of all the business of her household. MC is very independent, caring and smart. But no one in her country knows her very well. SO when she left to marriage, the town ppl was happy that she left. The prince lives in a winter country. There was never a problem between them. They fell in love. Both respect each other