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Let the present value of cash flows of a company be denoted by v=100. This value can move up to v=125 the next period. The risk-free rate is equal to 4%. What is the risk-neutral probability?

a) 0.82
b) 0.78
c) 0.75
d) 0.80

1 Answer

6 votes

Final answer:

The risk-neutral probability calculation indicates approximately 0.16 as the risk-neutral probability, assuming the value can either stay at $100 or go up to $125. This suggests that the correct answer is not given among the options, as the closest value computed is not listed.

Step-by-step explanation:

The question involves determining the risk-neutral probability for the expected future value of a company's cash flows when considering the risk-free rate. To calculate this, we use a risk-neutral valuation approach where all future uncertain outcomes are discounted at the risk-free rate to deduce the present value.

In this case, we have a present value (PV) of $100, which can go up to $125 in the next period. Given a risk-free rate of 4%, the future value (FV) should be discounted back to its present value using this rate. Thus, $125 discounted at 4% would be $125 / (1 + 0.04) = $120.19 as present value. The fact that the present value is $100 means the risk-neutral probability (p) should be such that $100 equals $120.19 times p plus (1-p) times the probability of the value going down, which we can assume to be zero in this simplified scenario.

Setting up the equation: $100 = $120.19 * p + $0 * (1-p), we get $100 = $120.19 * p. Solving for p yields p = $100 / $120.19, which is approximately 0.83. However, this is not one of the options provided. If 'v' can either stay the same or go up to $125, under the risk-neutral measure and assuming the up state ($125) is the only possibility apart from remaining at $100, then the value today must equal the expected value next period, discounted at the risk-free rate. So in fact: $100 = (p * $125 + (1 - p) * $100) / (1 + 0.04), solving for p gives p = (1.04 - 1) / (1.25 - 1) = 0.04 / 0.25 = 0.16, which means the correct risk-neutral probability isn't listed in the options provided.

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