In general, assets are risky and cannot be valued using contractual future payoffs as if they were certain because they are contingent. Stock payoffs are capital gains and dividends which are not contractual. Credit risky bonds pay the contractual promised payment only if they do not default; otherwise the investors get the recovery value of the bond. Option payoffs are contingent upon the underlying asset price. An option on a stock pays off something if the stock price is above the strike price before or at maturity; otherwise, the option does not pay anything. For addressing risky payoffs, whatever the source of risk, which is market risk for stocks and credit risk for risky debts, other valuation principles apply. Discounting future payoffs transform them into their economic value as of today.

If investors were indifferent to risk, they would not make any difference between risky payoffs and risk-free payoffs. The standard example illustrating risk aversion is that of a single uncertain payoff, one period ahead, of which values can be 50 or 150 with equal probabilities. The expected payoff is 100. Under risk neutrality, investors would be indifferent between a certain payoff of 100 and the expected random payoff of 100 (the average of the two possible payoffs). In practice, investors dislike risk and would always prefer the certain payoff of 100. They value the expected payoff at a lower value than the certain payoff of 100. The difference of value between the expected payoff under real world probabilities and the certain payoff of 100 is due to risk aversion, and it measures risk aversion. Assume that the so-called "certainty-equivalent" of the random payoff is 90. The certainty equivalent is the value of the random outcome for investors. The value of risk aversion is the difference between 100 and 90 or 10.

The next chapter reminds the principles for valuation of risky payoffs, given that the risk aversion and the risk premium that it commands over risk-free rate are generally unknown.

APPENDIX:THE TAYLOR EXPANSION FORMULA

Any function value fix) has a value fid) that can be approximated by a Taylor expansion, provided that it is infinitely differentiable in the neighborhood of a, a real number:

4 It is easy to check that, discounting all contractual flows at this rate, we actually find the 975.82 value. The rate j' is the internal rate of return of the stream of flows when using 975.82 as initial value.

If we take a — 1:

lifix) — ln(x), and a= 1, then ln(l) = 0 and d
(x)ldx = l/a = 1, so that: ln(x) = (x- 1). Replacing x = 1 + u, ln(l + u) ~ u when w is small. Similarly, is can be shown that exp(w) ~ 1 - u when u is small.

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