We use a Monte Carlo radiative transfer model (MCRTM) to simulate the UBVRI light curves, images and linear polarization of a light echo from supernova SN$~$1987A in the Large Magellanic Cloud (LMC) using various dust cloud shapes, sizes, and optical properties. We compare the theoretical simulations to the observations of AT2019xis, a light echo detected at a large angular distance (4.05$^{'}$) from SN$~$1987A. We estimate the size and optical thickness of the dust cloud based on the simulation results and the observations of Optical Gravitational Lensing Experiment (OGLE-IV) Transient Detection System (OTDS) I-band light curve. The mass of the dust cloud is calculated using the estimated size, optical thickness and extinction coefficient. If the dust cloud is assumed to correspond to a gas-to-dust ratio of 300, the total mass of the dust cloud is approximately 7.8-9.3 $M_{\odot}$. Based on these theoretical models, we show that the morphological shapes of the light echoes in the wavelength range in or shorter than the U-band to be very different from those in the longer wavelength bands, and the difference carries important information on the early UV radiation of SN$~$1987A., Comment: 21 pages, 20 figures, accepted to ApJ. Corrected several typos