This is, of course, a matter of opinion. I would argue that true courage always does require putting something valuable to you at risk. However, I would emphasize that I do not think that the thing you put at risk must be something physical.
Of course, it takes courage to put one's body or life at risk. However, this sort of physical courage is not the only sort. It also takes, for example, a great deal of courage to go against the desires of one's friends. In this case, you are not putting yourself at risk of physical harm. Instead, you are risking the loss of your status in the eyes of your friends. This is a major risk and it is a form of true courage.
If you are not risking anything of much value to you, you are not showing courage because there is no danger of any sort to you. Courage consists of risking the loss of something valuable in order to try to do what you think is right.
Of course it does require for you to put something valuable to you at risk as then only your true strength will show up to protect the thing/person of value to you. If you do not feel or put your best to save that valuable, then it may not be something of real importance to you because real courage comes from within and in the harsh times, it will always be with you...