leveraged bets on the S&P 500 index. We put $b:=2$ and $c:=1$ (e.g. buy-and-hold), with $\sigma:=0.15$. We get $C_0=\$1.29$ and $\log(C_0)/T=1\%$, so the replicating strategy is guaranteed to achieve a final compound-growth rate that is $1\%$ lower than the best of $\{b,c\}$ in hindsight. If $b=2$ happens to outperform the index by more than $1\%$ per year, then the trader will beat the market over $t\in[0,25]$. If $b=2$ underperforms the index (or outperforms by less than $1\%$ a year), then the trader’s compound-growth rate will have lagged the market by at most $1\%$ a year.

Note that the corresponding horizon-free strategy (that replicates the perpetual version of the option) can only guarantee to get within $\log(2)/T=2.8\%$ of the hindsight-optmized growth rate at $T=25$

We construct a robust $T:=25$ year scheme for long-run stock market investment that guarantees preservation of capital. We put $b:=1$ ($100\%$ stocks) and $c:=0$ (all cash). Assuming that $\sigma:=0.15$, the practitioner can rest easy, safe in the knowledge that his foray into risk assets will ultimately not cause him to lag the risk-free rate by more than $1\%$ a year. If $r>0.01$, then he is guaranteed not to lose money if he sticks to the Plan for 