Sul tasso implicito di operazioni finanziarie aleatorie

Abstract

Si danno limitazioni inferiori e superiori per il valor medio del tasso implicito di operazioni finanziarie aleatorie di una classe determinata. Per l’ottenimento di alcune disuguaglianze si fa uso di una versione generalizzata della ben nota disuguaglianza di Čebyšev tra medie con diverse ponderazioni che viene dimostrata in appendice e che vale anche per distribuzioni di probabilitá con componenti discreta e continua congiuntamente.

Abstract

We consider random investment projects, which are described through their cumulative cash-flow functionS(t, ω)=cash fund of the project at the epocht, whereω is a point in a probability space. The discounted cash-flow of the random project is:

$$G(\delta , \omega ) = S(0, \omega ) + \int_0^{ + \infty } { e^{ - \delta t} } dS(t,\omega )$$

δ being the discount rate. We assume thatG(δ, ω) is for every fixedω a decreasing and convex function ofδ with a unique real rootδ*(ω), i.e.δ*(ω) is the unique real number such thatG(δ*(ω),ω)=0. The numberδ*(ω) is therandom internal rate of return of the project. Under suitable conditionsδ*(ω) is a real random variable with mean$\bar \delta ^ *$. The aim of the paper is to give bounds for$\bar \delta ^ *$. Theorem 1 provides a lower bounda for$\bar \delta ^ *$ considering the straight line tangent to the graph ofG whereδ=0. Theorem 2 provides an upper boundb for$\bar \delta ^ *$, whereb is the unique real root of the equation:M(G(δ, ω))=0.

The proof of Theorem 2 requires a general version of the well known Čebyšev inequality between means with different weights. In the Appendix we give the proof of such general version that holds for distributions (possibly) having both a continuous and a discrete component.