Special Report
HOW TO (STATISTICALLY) AVOID GAMBLERS RUIN IN HIGH-RISK SITUATIONS*
or
Interrelations of probability of success, percent of capital that can be risked, and return on investment.
By Michael Holmes
Denver, Colorado
Editor's Note:
This article, first published by Doctor Holmes in June, 1980, is equally valuable today. LGM
Part One: Utility or Preference Functions
INTRODUCTION
Of major concern to the investor in high-risk situations (such as the drilling of oi1 and gas wells) is how to avoid a "run of bad luck" with consequent capital exhaustion. A variety of empirical, game-plans may be invoked, including diversification of the (risky) investment opportunities, rules of not investing more than some minimum value in any one project, and requiring at least some minimum potential return on investment before the gamble is made. For example, an oil and gas operator may diversify in exploration projects in a variety of different areas, or drill a mixed program of exploration and development wells. He may further invest no more than ten percent of his total capital in any one well, and require at least a 3:l ultimate cash return on each dollar invested, A more sophisticated investor may determine his own individual "utility" (or preference) function - a mathematically defined aversion to risk - and judge whether or not to risk his capital in any specific venture. In either approach, the chances of success wi11 enter the decision a project with fifty percent probability of success will always be chosen over one with a thirty percent probability of success, presuming that other parameters of both projects are identical.
This paper combines considerations of capital availability with three other parameters: (l) potential loss, (2) potential profit, (3) probability of success, to determine the percentage of capital that can be risked in any one venture. The analyses are based on the assumption that an infinite number of similar opportunities are available. A surprising conclusion reached is that the empirical rules, followed by many in the oil and gas business, are leaving the investor wide open to financial ruin, because of over- investment in any one opportunity. A corollary is that most operators have an (often) much too limited number of prospects in any one program. The only savior from financial ruin is luck early in the venture.
UTILITY OR PREFERENCE FUNCTIONS
A utility function is defined as:
u = Profit
Profit + Capital
For example, an individual with a capital available of 5000 units, with the opportunity to invest in a project with the potential of realizing 20,000 units, has a utility function of:
u = 20,000 = 0.8
20,000 + 5,000
Let us consider the following example under consideration by two investors, one with a capital of 5,000 units (Investor I), and one with a capital of 20,000 units (Investor II):
Investment Opportunity A
50% probability of realizing 50,000 units profit
50% probability of losing 2,000 units
Investment Opportunity B
70% probability of realizing 10,000 units
30% probability of losing 2,000 units
Expected values of utility functions can now be determined for each investor, and for each opportunity:
Investor I
Opportunity A
us = 50,000= 0.909
50,000 + 5,000
uf =-2000= 0.667
-2,000 + 5,000
0.909 x 0.5 + (-0.667) x 0.5 = 0.121
Opportunity B
us = 10,000= 0.667
10,000 + 5,000
-2,000
uf =-2000= 0.667
-2,000 + 5,000
0.667 x 0.7 + (-0.667) x 0.3 = 0.267
Investor II
Opportunity A
us = 50,000= 0.714
50,000 + 20,000
uf =-2,000= -0.111
-2,000 + 20,000
0.714 x 0.5 + (-0.111) x 0.5 = 0.301
Opportunity B
us = 10,000= 0.333
10,000 + 20,000
-2,000
uf =-2,000= 0.111
-2,000 + 20,000
0.333 x 0.7 + (-0.111) x 0.3 = 0.200
From the relative magnitude of the expected values of the utility functions, the conclusion is reached that, because of capital restrictions, Investor I should choose the less profitable (but less risky) venture A, whereas Investor II can afford to gamble on the more risky, (but potentially more rewarding) venture B.
Coming Next Month: Application of Utility Functions to Limiting Cases of Capital Restrictions
*REFERENCES:
1. Arps, J.J. and Arps, J.L. "Prudent Risk-Taking," J.P.T, July, 1974
2. Newendorp, Paul. "Decision Analysis for Petroleum Exploration," Petroleum Publishing Company, l975
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