  Consider a random walk whose (light-tailed) increments have positive mean.
Lower and upper bounds are provided for the expected maximal value of the
random walk until it experiences a given drawdown d. These bounds, related to
the Calmar ratio in Finance, are of the form (exp{alpha d}-1)/alpha and (K
exp{alpha d}-1)/alpha for some K>1, in terms of the adjustment coefficient
alpha (E[exp{-alpha X}]=1) of the insurance risk literature. Its inverse
1/alpha has been recently derived by Aumann and Serrano as an index of
riskiness of the random variable X. This article also complements the Lundberg
exponential stochastic upper bound and the Cramer-Lundberg approximation for
the expected minimum of the random walk, with an exponential stochastic lower
bound. The tail probability bounds are of the form C exp{-alpha x} and
exp{-alpha x} respectively, for some 1/K < C < 1. Our treatment of the problem
involves Skorokhod embeddings of random walks in Martingales, especially via
the Azema-Yor and Dubins stopping times, adapted from standard Brownian Motion
to exponential Martingales.
