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Risk Valuation First, we need to identify the risks Second, we need to value the risks To do that, we need to KNOW the losses AND the probabilities with which they will occur Without that, we cannot value risks

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Choice 1 Choice 2 Rectangle Decision

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Decisions Big Factory-600 Small Factory-100 Year 0

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Lottery Event 1 Event 2 Circle Prob 1 Prob 2 Lottery notation (event 1, event 2; prob1, prob2)

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Car stolen p=.1 Car not stolenp=.9

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Stolen p=.1 Not stolen p=.9 Car found p=.2 Car not found p=.8

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Stolen =.1 Not stolen p=.9 Found p=.2 Not found p=.8 Company pays p=.9 Does not pay p=.1 Damaged p=.7 Not damaged p=.3

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Robbed p=.1 Not robbed p=.9 Robbed p=.1 Not robbed p=.9 Insured Not insured

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Example Project: Construction of houses Investment $10 m. Calculates that: If high demand for houses (prob = 0.70) will sell for $15 m.

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Example If low demand (prob = 0.20) only sell for $11 m. Worst scenario (prob =0.10) when there is an economic crisis, will sell only for $7 m. The alternative that the investor has is to invest $10 m in CETES and get a CERTAIN rate of return worth $1 m.

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Example a) ¿What is the average gain from investing in the houses? b) If the utility function is es U (w) = w and it is measured in millions…. ¿Which one will the investor prefer?

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Example c) What if the utility function is U (w) = −e −w/100, ¿In that case, what is the preferred option? d) Suppose now the houses catch fire (prob 0.10) and destoyed completely before they are sold. With the same utility as in c), ¿Which investment will he prefer?

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w+5 P=0.7 P=0.2 P=0.1 w+1 w-3

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Example Average gain 15 (.7) + 11 (.2) + 7 (.1) = 13.4 Net gain13.4 - 10 = 3.4 b) Project: EU (w - 10 + g) =.7 (w + 5) +.2 (w + 1) +.1 (w - 3) = w + 3.4 CETEs U (w - 10 + 11) = w + 1

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With U (w) = −e −w/100 c) Project: EU (w − 10 + g) =.7 −e −(w+5)/100 +.2 −e −(w+1)/100 +.1 −e −(w−1)/100 = −.9669e −w/100 And CETES: U (w − 10 + 11) = −e −(w+1)/100 = −.99e −w/100 Project is better for all w > 10.

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Possibility of fire d) Project: 0.9 x 0.9669e -w/100 + 0.1e -(w-10)/100 = 0.9807e -w/100 Implies that the project is still better for all w > 10.

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