Question

Ms. Blavatsky is proposing to form a new start-up firm. She can invest her money in one of two projects:

(i) a relatively safe project that offers a 40% chance of a $12.5 million payoff and a 60% chance of an $8 million payoff; and

(ii) a risky project that offers a 40% chance of a $20 million payoff and a 60% chance of a $5 million payoff. Ms. Blavatsky initially proposes to finance the firm by an issue of straight debt with a promised payoff of $7 million. Ms. Blavatsky will receive any remaining payoff.

(a) Show the possible payoffs to the lender and to Ms. Blavatsky if (i) she chooses the safe project and (ii) she chooses the risky project. Which project is Ms. Blavatsky likely to choose? Which will the lender want her to choose?

(b) Suppose now that Ms. Blavatsky offers to make debt convertible into 50% of the value of the firm. Show that in this case the lender receives the same expected payoff from the two projects.

Answer 1

a-i). Safe project: In this case, the payoff is greater than $7 million in both situations, so the lender is always assured of a payoff of $7 million.

Probability weighted pay-off = (probability1*payoff1) + (probability2*payoff2)

Probability	Total ($mn)	Lender ($mn)	Ms.Blavatsky (Total - lender) ($mn)
0.4	12.5	7	5.5
0.6	8.00	7	1.00
Probability weighted pay-off		7.00	2.80

Thus, for the safe project, the lender will get a payoff of $7 million while Ms.Blavatsky will get a payoff of $2.8 million.

a-ii). Risky project:

Probability	Total ($mn)	Lender ($mn)	Ms.Blavatsky (Total - lender) ($mn)
0.4	20.00	7.00	13.00
0.6	5.00	5.00	0
Probability weighted pay-off		5.80	5.20

Thus, for the risky project, the lender will get a payoff of $5.8 million while Ms.Blavatsky will get a payoff of $5.2 million.

b). In case of the safe project: The lender has a 0.4 probability of receiving $7 million or receiving (12.5/2) = $6.25 million. He will choose the former with the higher payoff.

Similarly, the lender has a 0.6 probability of receiving $7 million or (8/2) = $4 million. He will choose the former with the higher payoff.

Thus, his overall probability = (0.4*7) + (0.6*7) = $7 million.

In case of risky project:

The lender has a 0.4 probability of receiving $7 million or receiving (20/2) = $10 million. He will choose the latter with the higher payoff.

Similarly, the lender has a 0.6 probability of receiving $5 million or (5/2) = $2.5 million. He will choose the former with the higher payoff.

Thus, his overall probability = (0.4*10) + (0.6*5) = $7 million.

In both cases, the lender will receive a $7 million payoff.