Roy’s Safety-First Criterion in Portfolio Optimization Problem

The main objective of portfolio selection is the construction of a portfolio that maximises expected return given a certain tolerance for risk. There is an intuitive problem with the portfolio variance as a measure of risk. Using the variance in the portfolio optimization context, it means that outcomes that are above the expected portfolio returned are deemed as risky as outcomes that are below the expected portfolio return. The use of portfolio variance as a risk measure has been a subject to many discussions. There’s a thin line we step onto.

Since the class of risk for financial portfolios has been dividing into two sub-branches, namely, dispersion and downside risk measures, the Roy’s safety-first criterion (after the paper of Arthur Roy, Safety First and the Holding of Assets, Econometrica 1952 July, 431–450) refers more to the investor’s control over investment in the way such the investor makes sure that a certain amount of the invested principal is preserved, so the investor tries to minimise the probability that the return earned is less than or equal to the threshold $t$. Therefore, the optimization problem for portfolio with $N$ assets and associated weights of $w$ is:
$$
\min_{w} Pr(r_p\le t)
$$ or
$$
\min_{w} Pr\left( \sum_{i=1}^{N} w_ir_i \le t \right)
$$ where, by a clever observation one can notice that Roy’s safety-first criterion is a special case of the lower-partial moment risk measure formula,
$$
\left(E[\min\{r_p-t,0\}^q]\right)^{1/q} \equiv \int_{-\infty}^{t} (t-r)^k f(r) dr
$$ when $k=0$. In financial risk management, Roy’s safety-first criterion is referred to as shortfall risk, and additionally, for $t=0$, the shortfall risk is called the risk of loss.