Risk Plus Value Equals Zero
risk (noun) (Exposure to) the possibility of loss, injury, or other adverse or unwelcome circumstance; a chance or situation involving such a possibility.
value (noun) Worth or quality as measured by a standard of equivalence.
This post discusses the idea that the negative of a monetary valuation functional is a monetary risk measure. Symmetrically, the negative of a monetary risk measure functional is a valuation. Valuation functionals are denoted \(V\) and risk measures \(\rho\) because that’s standard in their respective literatures.
Throughout, we operate on real-valued random variables, whose outcome values are interpreted as cash flows. If I own a contingent contract granting me ownership of the cash flows defined by a random variable \(X\), there are two ways to interpret the outcome \(X=+1\). In the payoff convention, it means I am paid (receive) \(1\) and am better off. In the loss convention, it means I must pay (disburse) \(1\) and am worse off. The payoff convention is common in the asset valuation literature; the loss convention in the risk and insurance pricing literature. The quantities specified by the values of \(X\) are vectors in a one-dimensional vector space: they have a magnitude, called amount, and direction, which determines whether \(+1\) is a payment or receipt.
If \(X\) uses the payoff convention then \(-X\) uses the loss convention, and vice versa.
A cash flow of \(-m\), for \(m>0\), means that I pay the amount \(m\) in the payoff convention, and that I receive it in the loss convention. Negation swaps receivable and payable and leaves the magnitude (amount) unchanged.
We talk of functionals, rather than functions, to stress that the argument is itself a function, a random variable.
The monetary in monetary valuation or risk measure means that the range has monetary units: the answer is an amount of money and not an ordinal scale. Measuring in money terms is critical. It means we can combine measures and adjust them by certain cash amounts using addition and subtraction.
A valuation functional is an individual’s assessment of the worth of cash flows specified by \(X\). It can be thought of as the most an individual will pay to acquire \(X\). A risk functional is an individual’s assessment of the cost of risk of cash flows specified by \(X\). It can be thought of as the most an individual will pay to remove \(X\).
Under the payoff convention, if I add \(1\) to each outcome \(X\) to form \(X+1\), then I am better off, value has increased by \(1\). Thus \(V(X+1)=V(X)+1\). Likewise, if I subtract \(1\) I am worse off and \(V(X-1)=V(X)-1\). Objectively, this suggests \[ V(X+m) = V(X) + m \] for all \(m\in\mathbb R\). Utility theorists would argue that marginal value decreases with wealth, subjectively. That leads to a different theory. To require that \(V(X+m)=V(X)+m\) is an assumption called cash additivity. It is assumed as an axiom for monetary valuation functionals.
If \(X\le Y\), meaning that \(X\) pays less than \(Y\) in every state of the world, then the value of \(X\) must be less than that of \(Y\) \[ X \le Y \implies V(X) \le V(Y). \] This is called monotonicity or the monotone condition. Notice that if \(X\le Y\le 0\), then I owe less under \(Y\) than \(X\) and so \(Y\) still has greater value, albeit negative value.
Under the loss convention adding \(1\) increases my loss, makes me worse off, and increases my risk by \(1\). For risk measures, cash additivity is usually called cash invariance, and it requires \[ \rho(X+m) = \rho(X) + m. \] And since a larger loss is worse the monotone condition looks the same: \[ X \le Y \implies \rho(X) \le \rho(Y). \]
So far, so good. But beware, there is a tricky extra shift between our discussion of \(V\) and \(\rho\). We viewed \(V\) under the payoff convention natural for assets, and \(\rho\) under the loss convention natural for losses.
Let’s look at the risk measure \(\rho\) under the payoff convention. Now \(X+1\) is a smaller loss in each state and is less risky. Thus, under the payoff convention, \(\rho(X+1)=\rho(X)-1\), and in general \[ \rho(X+m) = \rho(X) - m,\quad\text{for all }m\in \mathbb R. \] If \(X\le Y\), then \(X\) is a bigger loss than \(Y\) in every state and \[ X \le Y \implies \rho(X) \ge \rho(Y). \] In both cases there is a swap. It is easy to see analogous swaps occur for a valuation functional under the loss convention.
Here’s the point of this post: given a valuation functional \(V\) the functional \[ \rho(X):=-V(X) \] acts like a risk measure functional under the same convention. For example, under the payoff convention, \[ V(X+m) = V(X) + m \iff -\rho(X+m) = -\rho(X) + m \iff \rho(X+m) = \rho(X) - m \] and if \(X \le Y\) then \[ V(X) \le V(Y) \iff -\rho(X)\le -\rho(Y) \iff \rho(Y) \le \rho(X), \] and we recover the expected relationships. Obviously, the same correspondence applies to create a valuation function from the negative of a risk measure functional.
If the sign convention swaps too, and we use the natural payoff convention for assets and loss convention for losses, then we have a double switch. Now, for \(X\) interpreted under the loss convention, \[ \rho(X) = -V(-X). \] The inner \(-X\) converts sign convention to the one \(V\) uses in its domain and range, and the outer one converts the range back to \(\rho\)’s convention. Now we have \[ \rho(X+m) = -V(-X-m) = -(V(-X)-m) = \rho(X) + m \] just as we saw above.
In order for these symmetries to hold we must measure in terms of money. In money units we know how to add cash flows, and understand that the negative of a cash flow swaps payer and payee. In addition to measuring in money units, we need a sign convention to interpret the values of \(X\). Although important, the sign convention is ultimately cosmetic: we can work with payoffs or with losses provided we declare the convention. Declaring “more is good” (payoff) or “more is bad” (loss) suffices, and either choice is acceptable as long as we adjust monotonicity and cash invariance accordingly.
Why would we ever not measure in monetary terms? Partly, because great minds of the past didn’t. Blame von Neumann. Utility theory is ordinal and ranks positions. Non-trivial utility is non-monetary: diminishing marginal utility of wealth. It is monotonic but not cash invariant. (Strictly a utility function is used to represent a monotonic preference.) It can be translated into monetary terms through a certainty equivalent, but in a way that depends on wealth, introducing a nuisance variable that may not make sense in every application. What is the wealth of a unit in a firm?
TL;DR: Measure risk and value in monetary terms. Specify the sign convention. Require cash invariance. Enjoy a simpler analytic life.
See Contracts with Contingent Cash Flows for a longer discussion.