  We perform a detailed theoretical study of the value of a class of
participating policies with four key features: $(i)$ the policyholder is
guaranteed a minimum interest rate on the policy reserve; $(ii)$ the contract
can be terminated by the holder at any time until maturity (surrender option);
$(iii)$ at the maturity (or upon surrender) a bonus can be credited to the
holder if the portfolio backing the policy outperforms the current policy
reserve; $(iv)$ due to solvency requirements the contract ends if the value of
the underlying portfolio of assets falls below the policy reserve.
  Our analysis is probabilistic and it relies on optimal stopping and free
boundary theory. We find a structure of the optimal surrender strategy which
was undetected by previous (mostly numerical) studies on the same topic.
Optimal surrender of the contract is triggered by two `stop-loss' boundaries
and by a `too-good-to-persist' boundary (in the language of \cite{EV20}).
Financial implications of this strategy are discussed in detail and supported
by extensive numerical experiments.
