However, let's notice out what the expected toll of the stock is at fourth dimension t. If nosotros invest our entire wealth inwards 1 stock, that is actually bespeak what our expected wealth is at fourth dimension t. To compute that, it is easier to get-go notice out what the expected log toll of the stock is at fourth dimension t, because that is but the expected value of the nub of the log returns inwards each fourth dimension interval, in addition to is of course of instruction equal to the nub of the expected value of the log returns when nosotros assume a geometric random walk. So the expected value of the log toll at fourth dimension t is but t * (m-s2 /2). But what is the expected toll (not log price) at fourth dimension t? It isn't right to nation exp(t * (m-s2 /2)), because the expected value of the exponential part of a normal variable is non equal to the exponential part of the expected value of that normal variable, or E[exp(x)] !=exp(E[x]). Instead, E[exp(x)]=exp(μ+σ2 /2) where μ in addition to σ are the hateful in addition to criterion departure of the normal variable (see Ruppert). In our case, the normal variable is the log price, in addition to thence μ=t * (m-s2 /2), in addition to σ2=t *s2 . Hence the expected toll at fourth dimension t is exp(t*m). Note that it doesn't involve the volatility s. Risk doesn't touching on the expected wealth at fourth dimension t. But nosotros but argued inwards the previous paragraph that the expected chemical compound increase charge per unit of measurement is lowered past times risk. What gives?
This brings us to a famous recent paper past times Peters in addition to Gell-Mann. (For the physicists with you, this is the Gell-Mann who won the Nobel prize inwards physics for inventing quarks, the key edifice blocks of matter.) This happens to survive the nearly read newspaper inwards the Chaos Journal inwards 2016, in addition to basically demolishes the utilization of the utility part inwards economics, inwards understanding with John Kelly, Ed Thorp, Claude Shannon, Nassim Taleb, etc., in addition to against the entire academic economic science profession. (See Fortune's Formula for a history of this controversy. And but to survive clear which side I am on: I loathe utility functions.) To brand a long even short, the mistake nosotros accept made inwards computing the expected stock toll (or wealth) at fourth dimension t, is that the expectation value at that topographic point is ill-defined. It is ill-defined because wealth is non an "ergodic" variable: its finite-time average is non equal to its "ensemble average". Finite-time average of wealth is what a specific investor would sense upwardly to fourth dimension t, for large t. Ensemble average is the average wealth of many millions of like investors upwardly to time t. Naturally, since nosotros are but 1 specific investor, the finite-time average is much to a greater extent than relevant to us. What nosotros accept computed above, unfortunately, is the ensemble average. Peters in addition to Gell-Mann exhort us (and other economists) to exclusively compute expected values of ergodic variables, in addition to log provide (as opposed to log price) is happily an ergodic variable. Hence our average log provide is computed correctly - run a hazard is bad. Paradox resolved!
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