Дилемма инвестора и управляющего хедж-фонда
Assets under management A0Management fee: mfA0 (0<mf<1)
Performance fee: pfA1-A0=pfA0r (0<pf<1)
r – annual return for investments
W0 – amount of withdrawal
Model 1: Equilibrium for a managing director
act aggressively, while investors withdraw W0payoff:
mfA0-W0+pfA0-W0raact conservative and investors do not withdraw money
mfA0+pfA0rcIf the utility from both scenarios is the same, then
W0pfra+mf=pfA0(ra-rc)Introduce ω, where ω=W0A0 - fraction of assets withdrawn by investors
ω=pf(ra-rc)mf+pfraWe have a relationship between risky and conservative strategies linked to the proportion of withdrawal.
Model 2: equilibrium for an investor
withdraw W0, while MD acts aggressively
-mfA0-W0+(1-pf)A0-W0rado not withdraw money, while MD act conservative
-mfA0+(1-pf)A0rcSo, in the equilibrium:ω=(1-pf)(ra-rc)1-pfra-mfSince ω>0, 1-pfra>mf which logically constitute to the fact that return from the investments should outweigh the management costs.
In equilibrium investors accept lower return for their investments than MD, for every level of withdrawal.