A manager is doing capital investment for a company. His performance depends on

Question

A manager is doing capital investment for a company. His performance depends on his past investment performance. Specifically, if he succeeded in previous one investment, then he would have increasing probability to success for next investment; if he lost in previous one investment, then he would have increasing probability to lose for the next investment. As shown in the diagram, where the number on the edge is the probability the manager succeeds or loses money. Each investment, if succeeded, he gains $100; if not, he loses $100. Thus, the value in the node is the total money the manager has at each stage.

(1) What is the probability for the manager to succeed for the first investment, lose for the second and succeed for the third investment ?

(2) What is the probability for the manager to succeed in his third investment?

(3) What is the expected payoff of the manager after he invests three times?

Explanation / Answer

P( for the manager to succeed for the first investment, lose for the second and succeed for the third investment )

= P( the manager succeed for the first investment) + P( lose the second investment | he succeeds at first) + P( a success at third investment | he succeeds at first and loses at second)

= (1/2)(1/3)(1/4) = (1/24)

2.)P( manager succeed in the third investment)

= (1/2)(2/3)(3/4) + (1/2)(1/3)(1/4) + (1/2)(1/3)(3/4) + (1/2)(2/3)(1/4)

= (1/4) + (1/24) + (1/8) + (1/12)

= 1/2

3.) Expected payoff = (300 × 3/4 ) + (100 × 1/4) + (100 × 1/4) + (-100 × 3/4) + (100 × 3/4) + (-100 × 1/4) + ( -100 × 1/4) + (-300 × 3/4)

= 0

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