 Finite-horizon drawdown of a two-sign-memory process

For increments $X_t=a_{t-2}{a}_{t-1}+a_t$ with freely admissible starting sign history, the worst absolute financial drawdown over $N$ steps is the displayed value. A four-state recursion $d_{n+1}(b,c)={\min }_{a=\pm 1}[d_n(a,b)+ab+c]$ computes the minimum cumulative increment, with zero initial costs. The minimum is attained by nonpositive increments: $(-2,0{)}^k,-2$ for odd $N=2k+1$, and $(-2,0{)}^{k-1},-2,-2$ for even $N=2k$. The horizon-free bound is infinite. At most two strictly negative increments can be consecutive, but confusing that losing-run bound with drawdown would be incorrect.