# Project cash flow distribution by 3iZUqoP

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```									Another Reason for Using Warrants: Risk Averse-Entrepreneurs

Risk-averse people care not only about mean or expected cash flow => They care about

variance (which is a measure of uncertainty) of cash flow as well.

One example of risk-averse utility:

U(w) = w

Numerical Examples

Project cash flow distribution:

Probability                       Cash                        Scenario

1/3                               100                         High

1/3                               50                          Medium

1/3                               25                          Low

     Investment required to implement = \$10

     To be obtained from V-C

     Question: Should the V-C be given equity or warrants?

Numerical Example – I

Assumption: Both V-C and entrepreneur are risk-neutral (care about expected value of cash

flows only).

Then, expected cash flow to V-C should allow him to break-even (set =0 for V-C).
All-Equity Case

10 = x [100/3 + 50/3 + 25/3] => x = 0.17 or 17%

Entrepreneur gets

100-17 = 83% of Equity

Expected Cash Flow to Entrepreneur = 0.83* [100/3 + 50/3 + 25/3]

=\$ 48.41

Warrants, exercisable in (only) high-scenario

V-C share of equity, post exercise = y

10 = (1/3)*y*(100) for V-C breakeven

 y = 0.3 or 30%

Entrepreneur’s share of equity => Depends on scenario now.

Entrepreneur now has:

70% = 1-y      in High

100%           in Medium

100%           in Low

Expected Cash Flow = 1/3 * (25) + 1/3 *50 + 1/3 * (0.7) * (100)

= \$48.41

So entrepreneur is indifferent between giving the V-C equity or warrants.

Numerical Example - 2

Assumption: Risk-Neutral V-C

Risk-Averse Entrepreneur
All Equity Case

V-C’s equity share = 17%

Entrepreneur’s share = 83%

Entrepreneur’s utility function: U(w) = w

Expected Utility

Wealth, w              w                       Probability

H                      0.83*(100)             9.11                     1/3

M                      0.83*(50)              6.44                     1/3

L                      0.83*(25)              4.55                     1/3

Expected Utility = 1/3 * (9.11) + 1/3 * (6.44) + 1/3 * (4.55) = 6.70

All Warrants Case

V-C’s share = 30% in high scenario

0% in others

Entrepreneur’s share = 100% in Low + Medium

70% in High Scenario

Wealth, w              w                       Probability

H                      0.7*(100)              8.36                     1/3

M                      50                     7.07                     1/3

L                      25                     5                        1/3

Expected Utility of Entrepreneur = 1/3 * (8.36) + 1/3* (5) = 6.81

Now the entrepreneur strictly prefers to give warrants to V-C! Why?

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