  This paper compares the optimal investment problems based on monotone
mean-variance (MMV) and mean-variance (MV) preferences in the L\'{e}vy market
with an untradable stochastic factor. It is an open question proposed by
Trybu{\l}a and Zawisza. Using the dynamic programming and Lagrange multiplier
methods, we get the HJBI and HJB equations corresponding to the two investment
problems. The equations are transformed into a new-type parabolic equation,
from which the optimal strategies under both preferences are derived. We prove
that the two optimal strategies and value functions coincide if and only if an
important market assumption holds. When the assumption violates, MMV investors
act differently from MV investors. Thus, we conclude that the difference
between continuous-time MMV and MV portfolio selections is due to the
discontinuity of the market. In addition, we derive the efficient frontier and
analyze the economic impact of the jump diffusion risky asset. We also provide
empirical evidences to demonstrate the validity of the assumption in real
financial market.
