BERND PAPE Asset Allocation, Multivariate Position Based Trading ...

86 ACTA WASAENSIASwitches between chartists and fundamentalists are driven by expected or realizedexcess profits above the real rate of the economy R, whichisassumedtoequalthereturn of the risky asset in the case of a constant trading price p at fundamental valuep f . That is R = r/p f ,wherer denotes the dividend of the stock. In that case theexpected excess returns by fundamentalists are given by s|(p f − p)/p|, wheresisadiscount factor, since fundamentalists profits will first occur in the future when thetrading price will have returned to its fundamental value.Bullish chartists, who invest into the risky security, receive its nominal dividend r andthe price change ṗ/v 2 , but forego the average rate of return of the economy R, suchthattheir excess return is (r +ṗ/v 2 )/p −R. The utilities of moving from the fundamentalistto the optimist subgroup U 2,+ and vice versa −U 2,+ arethereforegivenby r + ṗ/v2U 2,+ = α 3 − R − sp f − pp p , (4.47a)where α 3 measures the sensitivity of traders to differences in profits. Bearish chartistson the other hand, who short the risky asset in order to invest into the overall economyreceive R−(r+ṗ/v 2 )/p. The utilities of moving from the fundamentalist to the pessimistsubgroup U 2,− and to move from the pessimist to the fundamentalist subgroup −U 2,−are thereforeU 2,− = α 3 R − r + ṗ/v 2p− sp f − pp . (4.47b)Taking into account the probability of interaction of the relevant trader subgroups asmeasured by their relative frequency yields for the transition rates from fundamentaliststo the two kind of chartists p f+/− and vice versa p +/−f in analogy to (4.46):n +p f+ = v 2N exp(U 2,+),n −p f− = v 2N exp(U 2,−),n fp +f = v 2N exp(−U 2,+),n fp −f = v 2N exp(−U 2,−).(4.48a)(4.48b)The quasi-meanvalue equations for n + and n − read now 126ṅ + = n − p −+ + n f p f+ − n + (p +− + p +f ),ṅ − = n + p +− + n f p f− − n − (p −+ + p −f ),(4.49a)(4.49b)126 This is the case of L = 3 investment styles in equation (A7.14).