Working Paper 149 Appendix

Theorem I: If p ( t, s ) m ( t, s ) = a ( t, 0 ) under the Market Risk Proposition and
( 0 < a ( t, 0 < 1 ) in APT, this implies:

IA. ( 0 < p ( t, s ) < a ( t, s ) < 1 ) for s < 0 Depressions

IB. [ 0 < p ( t, s ) < 1 ] for s < 0 Booms

IC. If p ( t, s ) > 0, then p ( t, s ) > a ( t, 0 ) for s > 0 Booms
Proof: From Equations ( 1 ), ( 2 ), and ( 3 ) and the p ( t, s ) m ( t, s ) = a ( t, 0 ) condition we derive

p ( t, s ) = a ( t, 0 ) / ( 1 - s ) for s < 0.

p ( t, s ) = a ( t, 0 ) for s = 0.

p ( t, s ) = a ( t, 0 ) ( 1 + s ) for s > 0.

Depression and boom second derivatives are

 2p / s = a ( t, 0 ) ( 1 - s ) -3 > 0 for s < 0.

 2p / s = 0 for s > 0.

Hence, p ( t, s ) is nonlinear and strictly convex for x < 0 depressions, asymptotically approaches zero ( as s ® - ¥ ) and a ( t, 0 ) as s ® 0 .

Across s > 0 boom states, p ( t, s ) is linear and reaches the upper limit of a probability measure of p ( t, s' ) = 1 when s' = 1 / a ( t, 0 ). Since a ( t, 0 ) < 1, there exists a set of infeasible boom states having p ( t, s ) = 0 for ( s' < s < + ¥ ). The feasible boom states are ( 0 < s < s' ) for which the probability bounds are ( a ( t, 0 ) < p ( t, 0 ) < 1 ).
Corollary: Under Theorem I, the m ( t, s ) risk preferences are bounded as follows:

[ 1 < m ( t, s ) < + ¥ ] for s < 0 Depressions

[ a ( t, 0 ) < m ( t, s ) < 1 ] for s > 0 Booms
Proof: Since p ( t, s ) = a ( t, 0 ) / m ( t, s ), the second derivative for all boom and depression states is

2p / m = a ( t, 0 ) ( m ( t, s ) ) -3 > 0 for m ( t, s ) > 0.

Hence p ( t, s ) on m ( t, s ) is nonlinear and strictly convex for all m ( t, s ) on the real line. However, from Theorem I, boom states ( s' < s < + ¥ ) are infeasible since p ( t, s ) = 0. Thus, from Equation ( 2 ) the s' = 1 / a ( t, 0 ) implies m ( t, s' ) = a ( t, 0 ) is a lower bound on m ( t, s ) such that intense booms ( 0 < m ( t, s ) < a ( t, 0 ) ) are infeasible because p ( t, s ) = 0. For s < 0 depression states in Theorem I, p ( t, s ) > 0 for all s < 0. Hence from Equation ( 1 ) it follows that all depression states have feasible m ( t, s ) depression intensities whose probabilities are greater than zero.