Salutary skewness: risk financing of the third kind?

Citation

Powers, M.R. (2005), "Salutary skewness: risk financing of the third kind?", Journal of Risk Finance, Vol. 6 No. 5. https://doi.org/10.1108/jrf.2005.29406eaa.001

Salutary skewness: risk financing of the third kind?

Salutary skewness: risk financing of the third kind?

Given the complexities of modern risk finance, it is remarkable that virtually all financial risk management products are based upon only two underlying tools:

1. 1.

   diversification; and

2. 2.

   hedging.

As I have noted elsewhere (see Powers, 1999), these two methods match precisely the two earliest forms of “insurance” commonly attributed to the ancients: the division and swapping of cargoes among ships (diversification through risk pooling), and the use of bottomry contracts[1] (hedging through risk transfer). Thus, not only are there only two fundamental tools of risk finance, but there has been no significant innovation at the foundations of the field since the dawn of commerce[2].

The predominance of these two underlying risk-financing techniques raises a couple of interesting questions: Are diversification and hedging the only fundamental techniques possible? And if not, then why don’t we commonly see any third kind of technique operating in practice? To address these questions, I will explore the motivations for diversification and hedging, i.e. how both techniques derive value from reducing the statistical variance (or standard deviation) of financial results.

Consider the variance of a portfolio of two random returns, X and Y[3]:where corr[X,Y]∈[−1,+1] denotes the correlation between X and Y. Clearly, for fixed values of var[X] and var[Y], the variance of the portfolio decreases as the correlation goes from +1 (perfect positive correlation) to −1 (perfect negative correlation).

Now consider a firm with a portfolio consisting only of the random return R. Hedging makes use of equation (1) directly by identifying R with X, and then selecting a new random return, Y, that is sufficiently negatively correlated with R so that:This is true as long as:Diversification makes use of equation (1) more subtly, identifying R with X+Y by subdividing it into two (perfectly positively correlated) pieces, X=aR and Y=(1−a)R , for some a∈(0,1). Then the Y piece is exchanged for a new random return, Y′=(1−a)Z , that is sufficiently less than perfectly positively correlated with X so that:This is true as long as:Since diversification and hedging identify R with X+Y and X, respectively, there is only one additional means of interpreting equation (1): identifying R with Y. However, this third approach merely swaps the roles of X and Y in equation (1), and so necessarily leads back to the concept of hedging. Essentially, diversification and hedging exhaust all potential for variance-minimization. Therefore, to discover a truly novel third method of risk finance, we must look elsewhere.

At this point, it is instructive to reflect on why one might want to minimize variance in the first place. The most straightforward explanation is the mean-variance principle[4], which can be justified formally by considering the Taylor series expansion of the expected utility of a given outcome. Letting u( · ) denote the decision maker’s (infinitely differentiable) utility function, W denote his/her initial wealth, and R denote a random return, we may write:For a risk-averse and prudent decision maker[5] whose utility function can be represented by a cubic (or quadratic) polynomial[6], expected utility is increasing in the mean and decreasing in the variance[7]. To go beyond mean and variance, the most natural thing to do is to consider the third term of the Taylor series, which captures the skewness (asymmetry) of the distribution through the third central moment (TCM)[8]. Then, for a risk-averse and prudent decision maker with a cubic utility function, expected utility is increasing in the TCM[9].

This suggests that we look for new tools of risk finance among methods designed to maximize the TCM. In a manner analogous to equation (1), consider the TCM of a portfolio of two random returns, X and Y:where x=X−E[X] and y=Y−E[Y] . In this case, we see that for fixed values of the various variances, the TCM of the portfolio decreases as either corr[x²,y] or corr[x,y²] goes from +1 to −1, and increases as either TCM[X] or TCM[Y] increases.

If we mimic hedging, by identifying R with X (and r with x), then there are two ways to increase TCM[X+Y] :

1. 1.

   by selecting Y so that corr[r²,y] and corr[r,y²] are as large as possible; and

2. 2.

   by selecting Y so that TCM[Y] is as large as possible.

If we mimic diversification, by identifying R with X+Y (and r with x+y ), then there are two corresponding ways to increase TCM[X+Y] :

(1')by selecting Y'=(1 - a)Z so that corr[a²r²,(1 - a)z]and corr[ar,(1 - a)²z²] are as large as possible; and(2')by selecting Y' = (1 - a)Z so that TCM[1 - a)Z] is as large as possible.

Unfortunately, both methods (1) and (1′) are based upon subtle correlative effects between linear and quadratic forms of r and y that would be extremely difficult to detect in practice. On the other hand, both methods (2) and (2′) are quite straightforward. Noting that (2) is clearly superior to (2′) – because the absolute value of TCM[(1−a)Z] is diminished by the presence of the factor (1−a) – we may take method (2) as the best candidate for a new risk-financing technique.

This leaves us with one concern: will the large TCM associated with Y tend to bring with it a large variance? After all, for the random return Y to “have room” to be skewed, it also must have some degree of spread or dispersion.

To tackle this problem head-on, we can simplify matters by allowing the random returns R and Y to be statistically independent, so that equations (1) and (3) become:andrespectively. Then, to make sure that the contribution of the TCM term in expression (2) does not offset the variance term, we can impose the condition:which simplifies to:or equivalently:where Sk[Y] denotes the coefficient of skewness[10]. One might say that any return satisfying condition (4) possesses salutary skewness.

Assuming that u( · ) can be approximated by exponential utility over the relevant domain, we rewrite condition (4) as:wheredenotes the risk-aversion coefficient. Assuming further that Y is, roughly speaking, “significantly different from zero” (i.e. E[Y]>2 var[Y] ), we obtain the weaker (i.e. necessary) condition:Given that the value of 1/c has been estimated to be approximately 0.16(Firm Equity) in practice[11], it follows that salutary skewness can be achieved only if:where the right-hand side is generally a very large positive number.

To get a concrete idea of how difficult it is to satisfy condition (5), assume that the random return Y represents the return on a venture-capital investment in a high-risk, high-potential start-up company for which the investor receives either zero or a large positive payback, B. Then Y may be written as Y=IB , where I is a Bernoulli (p) random variable with p<1/2 , and condition (5) simplifies to:Noting that the right-hand side of condition (6) reaches a minimum value of about 3.2(Firm Equity) at p0.191 , we see that, even under the most favorable conditions, the Bernoulli model requires an upside return of more than three times the firm’s current equity to provide salutary skewness.

From the above discussion, I would conclude that beyond diversification and hedging, no third kind of risk-financing mechanism is likely to be encountered in practice. While this negative result cannot be proved rigorously, the above heuristic arguments do show convincingly that:

• •

  salutary skewness is the most reasonable candidate for a third type of risk financing; and

• •

  this method is highly unlikely to succeed in the real world.

All things considered, it actually seems much more likely to find a “sophisticated” firm attempting to implement salutary skewness, but failing to satisfy condition (4), than to find a firm successfully employing this technique.

Under a bottomry contract, a land or marine trader would take out a loan of merchandise or money from a merchant, agreeing to a high rate of interest. If all went well, then the principal and interest would be paid at the end of the trading venture; if the merchandise were lost or stolen, then the principal and interest would be forgiven.

Note that modern “insurance” products are simply perfect (or near-perfect) hedging instruments.

The variance of a random variable X is equal to the second central moment, E[(X−E[X])²] , where E[X] denotes the mean or expected value.

The mean-variance principle states that for two random returns, X₁ and X₂, a decision maker will prefer X₁ to X₂ if and only if one of the following two conditions holds: (1) E[X₁]>E[X₂] and var[X₁]≤\kern-2var[X₂] ; or (2) E[X₁]≥E[X₂] and var[X₁]<var[X₂] .

A risk-averse decision maker is characterized by u′( · )>0 and u″( · )<0 , and a prudent decision maker by u( · )>0 . Risk aversion and prudence are common assumptions in the economics of uncertainty.

A cubic polynomial is selected because it can reflect both risk aversion and prudence, without having to make further assumptions about u⁽⁴⁾( · ) and higher-order derivatives.

To be more precise, the partial derivative of expected utility with respect to the mean is positive, and the partial derivative of expected utility with respect to the variance is negative. If there is a functional relationship between the mean and the variance (e.g. if the variance is the square of the mean, as in the exponential distribution), then clearly the two partial derivatives will be related and dependent.

The third central moment of a random variable X is E[(X−E[X])³] .

The finance literature provides convincing evidence that investors are willing to pay a risk premium for the positive skewness of financial returns. See, for example, Harvey and Siddique (2000).

The coefficient of skewness of a random variable X is defined as TCM[X]/(var[X])^3/2.

See Howard (1988) and McNamee and Celona (1990). The former author finds that 1/c falls within a range of 0.150 to 0.167 of firm equity for companies in the oil and chemicals industry. The latter authors suggest that this range translates reasonably well across firms from different industries.

Abbreviations: (1′); by selecting Y′\kern-2=(1−a)Z so that corr[a²r²,(1−a)z] and corr[ar,(1−a)²z²] are as large as possible; and; (2′)by selecting Y′=(1−a)Z so that TCM[(1−a)Z] is as large as possible.

Michael R. Powers