értekezés - Budapesti Corvinus Egyetem

4. függelék - A Smith-Stulz [1985] modell
Smith and Stulz [1985] show that if a manager can alter the riskiness of his wealth only by
changing the riskiness of firm value 168 , then the incentive to do so depends on the shape of
his utility as a direct function of firm value. They point out that if the manager’s end-ofperiod
wealth is a concave function of the end-of-period firm value, the optimal hedging
strategy is to hedge the firm completely, if feasible. This comes from the fact, also
observed in the cases of convex corporate tax and the existence of financial distress costs,
that the expected value of a concave function of a random variable is smaller then the value
of the function evaluated at the expected value of the random variable (Jensen’s
Inequality). Since the manager’s expected income is maximized when the firm is
completely hedged, the manager will choose to bear no risk.
If the manager’s end-of-period wealth is, however, a convex function of the end-of-period
firm value, but the manager’s expected utility is still a concave function of the end-ofperiod
value of the firm, the optimal strategy will be to eliminate some, but not all,
uncertainty through hedging. There is a trade-off between expected income (positively
related to firm value volatility) and expected utility of income as a function of engaged risk
(negatively related to firm value volatility). The manager will choose the level of risk
where his utility from increased expected wealth is just offset by his disutility due to risk
aversion. 169
On the other hand, a sufficiently convex wealth function will cause the manager’s utility to
be convex in firm value, and the manager will behave like a risk-seeker even though his
expected utility function remains concave in his end-of-period wealth.
Since Smith and Stulz [1985] provide little evidence on the above mechanism, I give an
illustration of the phenomenon of this trade-off below. I show how option-like
compensation features in a firm’s compensation plan change the optimal degree of
hedging.
168 The combination of transaction costs, economies of scale and the large number of managers within any
firm make the firm’s comparative advantage to hedge likely. Note that the size of most future contracts is too
large to make them useful to hedge a manager’s income. (Smith/Stulz, 1985)
169 Guay [1999]
169