With the advent of Recommendation ITU-R BT.2020 (Rec. 2020) color space for ultra-high definition television (UHDTV), the development of a gamut mapping algorithm for wide to standard gamuts is greatly needed. In wide-gamut UHDTV production, particularly under the assumption of simultaneous live UHDTV and high-definition television (HDTV) broadcast scenarios, real-time gamut mapping from UHDTV to HDTV is vital. Designing such an algorithm requires a careful consideration of the shape and size of significantly different gamuts. In this paper, we set some requirements for gamut mapping from UHDTV to HDTV, and propose a hue-preserved gamut mapping algorithm. The mapping paths are designed in the CIE 1976 L*a*b* (CIELAB) color space through modifications of the hue prediction based on our subjective experimental results on the constant perceived hue. To prevent an excessive chroma reduction, the lightness of the out-of-gamut colors is changed. The design process and mathematical expressions of our proposed gamut mapping algorithm are provided. In addition, we demonstrate the improvements in image quality.