  In this paper we consider backward stochastic differential equations with
time-delayed generators of a moving average type. The classical framework with
linear generators depending on $(Y(t),Z(t))$ is extended and we investigate
linear generators depending on $(\frac{1}{t}\int_0^tY(s)ds,
\frac{1}{t}\int_0^tZ(s)ds)$. We derive explicit solutions to the corresponding
time-delayed BSDEs and we investigate in detail main properties of the
solutions. An economic motivation for dealing with the BSDEs with the
time-delayed generators of the moving average type is given. We argue that such
equations may arise when we face the problem of dynamic modelling of
non-monotone preferences. We model a disappointment effect under which the
present pay-off is compared with the past expectations and a volatility
aversion which causes the present pay-off to be penalized by the past exposures
to the volatility risk.
