2.5 A more specialised economy

For the problem of modelling the interest rate term structure, CIR specialise the economy. They restrict the class of utility functions to those having constant relative risk aversion ^[6] . Specifically, the utility function is required to be logarithmic and independent of the state variable Y , hence:

where is a constant discount factor.

CIR [ 18 ] show that for this specialised case the indirect utility function takes the form:

for some functions h ( t ) and g ( Y,t ). By the results of the earlier Lemma

Hence we have:

Substituting into (2.8), the form of the equilibrium interest rate reduces to:

Similarly, by (2.11), the return on any contingent claim simplifies to:

For the purposes of developing a model of the interest rate term structure, CIR assume that the contractual terms of all securities are free of explicit dependence on wealth. This implies that the partial derivatives, with respect to wealth, of all securities, equal zero (i.e. F _W = F _WW = F _WY = 0). As shown above the risk-free rate and factor risk premia ^[7] are also independent of wealth. Due to this additional restriction the valuation equation for contingent claims (2.12) reduces to: 1

^[6] That is neither the interest rate, nor the security risk premia depend on the level of investor wealth.

^[7] In (2.15) the factor risk premia, that is the coefficients of F _{Y _i} , reduce to a * ² GS ² .