Return To Risk Ratio 22

1. Core Concept Overview & Intuition

The return-to-risk ratio is a decision criterion used when a manager must choose among risky alternatives whose payoffs are random. Two quantities summarize each alternative: its expected monetary value, which measures the average return the alternative is expected to generate, and its standard deviation, which measures the dispersion of the payoffs about that average and therefore serves as the measure of risk. The return-to-risk ratio combines these two summaries into a single index by dividing the expected return by the risk.

The intuition is that of reward per unit of risk borne. An alternative that promises a large expected payoff but whose outcomes swing wildly is not automatically preferable to an alternative with a smaller expected payoff that is highly stable. By expressing expected return relative to the spread of outcomes, the ratio places alternatives on a common footing: the alternative with the larger ratio delivers more expected return for each unit of variability accepted. A ratio of $2$ means that the expected return is twice the size of the standard deviation of the payoffs; a ratio of $0.5$ means the expected return is only half the standard deviation, so the risk is large relative to the reward.

The criterion is applied comparatively. The ratio is computed separately for each alternative course of action, and the alternative with the highest ratio is selected. This makes the criterion a ranking device rather than an absolute test: a ratio has no universal pass mark, and its meaning derives entirely from comparison with the ratios of the competing alternatives. The criterion is therefore a member of the family of decision rules for decision making under risk, in which probabilities of the states of nature are known and the decision maker uses expected values together with a measure of variability.

Two structural features of the ratio deserve emphasis at the outset. First, because the standard deviation is non-negative, the sign of the ratio is governed by the sign of the expected value. An alternative with a negative expected value yields a negative ratio and is dominated in ratio terms by any alternative with a positive expected value, however small. Second, the ratio is undefined when the standard deviation is zero, that is, when an alternative has a certain payoff. Such a riskless alternative must be handled separately rather than forced through the formula.

The ratio is closely related to the earlier criteria of the course. The expected monetary value criterion ranks alternatives by expected return alone and ignores risk; the return-to-risk ratio refines that ranking by penalizing variability. The expected opportunity loss criterion and the expected value of perfect information quantify the cost of uncertainty in money terms; the return-to-risk ratio instead normalizes expected return by the scale of uncertainty. Together these tools allow a decision maker to move from a pure expected-value ranking to a risk-adjusted ranking without abandoning the expected-value framework.

2. Mathematical Formalization & Core Definitions

> Definition (Expected Monetary Value). Let an alternative course of action $a$ have payoffs $x_1, x_2, \ldots, x_n$ under states of nature $s_1, s_2, \ldots, s_n$ with prior probabilities $p_1, p_2, \ldots, p_n$ satisfying $p_i \ge 0$ and $\sum_{i=1}^{n} p_i = 1$. The expected monetary value of $a$ is

$$
\mathrm{EMV}(a) = \sum_{i=1}^{n} p_i x_i.
$$

> The expected monetary value is the probability-weighted average payoff of the alternative and is measured in the same monetary units as the payoffs.

> Definition (Risk of an Alternative). The risk of an alternative is measured by the standard deviation of its payoffs about the expected monetary value,

$$
\sigma(a) = \sqrt{\sum_{i=1}^{n} p_i \left( x_i - \mathrm{EMV}(a) \right)^2 }.
$$

> The standard deviation is measured in the same monetary units as the payoffs and is non-negative. A larger standard deviation indicates payoffs that are more dispersed about their mean and hence a riskier alternative.

> Definition (Return-to-Risk Ratio). For an alternative $a$ with expected monetary value $\mathrm{EMV}(a)$ and standard deviation $\sigma(a) > 0$, the return-to-risk ratio is

$$
R(a) = \frac{\mathrm{EMV}(a)}{\sigma(a)}.
$$

> The ratio is dimensionless: it expresses expected return in units of risk. It is defined only when $\sigma(a) > 0$.

> Decision Rule (Return-to-Risk Criterion). Among a set of alternatives, compute $R(a)$ for each alternative with positive standard deviation and select the alternative whose ratio is largest. Equivalently, the chosen alternative $a^{*}$ satisfies

$$
R(a^{*}) = \max_{a} R(a).
$$

The following properties follow directly from the definition and govern how the ratio behaves in practice.

• Sign behaviour. Since $\sigma(a) > 0$, the sign of $R(a)$ equals the sign of $\mathrm{EMV}(a)$. A negative expected value produces a negative ratio.
• Scale behaviour. If every payoff of an alternative is multiplied by a positive constant $k$, then both $\mathrm{EMV}(a)$ and $\sigma(a)$ are multiplied by $k$, so $R(a)$ is unchanged. The ratio is invariant to the monetary unit in which payoffs are expressed.
• Shift behaviour. If a constant $c$ is added to every payoff of an alternative, the expected value increases by $c$ while the standard deviation is unchanged, so the ratio changes by $c/\sigma(a)$. The ratio is therefore not invariant to additive shifts, which is why alternatives must be compared on a common payoff basis.
• Riskless alternatives. If $\sigma(a) = 0$ the ratio is undefined. A certain payoff is not ranked by the ratio; it is treated as a separate benchmark against which risky alternatives are judged.

The computation of the ratio for a decision problem proceeds in a fixed sequence. First, the payoff table is assembled with alternatives as rows, states of nature as columns, and prior probabilities attached to the states. Second, the expected monetary value of each alternative is computed as the probability-weighted sum of its row. Third, the deviation of each payoff from its row mean is squared, weighted by the state probability, summed, and square-rooted to obtain the standard deviation. Fourth, the expected value is divided by the standard deviation. Fifth, the ratios are compared and the largest is selected.