To model the term structure CIR use equation (2.16) and make several simplifying assumptions about technological change in their specialised equilibrium economy.
The change in production opportunities over time is determined by a single state variable Y .
The means and variances of the rates of return on the production processes are proportional to Y . Hence the state variable Y determines the rate of evolution of capital and neither the means nor variances will dominate the portfolio decisions for increasing values of Y .
The state variable Y follows the following stochastic process ^{ [8] } :
where ¾ and constants, ‰ 0and ½ a vector of constants.
Incorporating the above assumptions into their economic model, CIR [ 18 ] derive an explicit formula for the equilibrium interest rate in terms of the state variable Y , the parameters of its stochastic process and the means and variances of the rates of return on the production processes in the economy. Calculating the drift and variance of this equilibrium interest rate and defining a new Wiener process z ( t ), such that:
they specify the dynamics of the interest rate as ^{ [9] } :
where , ‰ 0. (2.17) represents a continuous time first-order autoregressive process where the stochastic interest rate is pulled to its long-term mean , with speed . Imposing an additional constraint that 2 ‰ ƒ 2 (see §2.7.6) ensures that the rate of interest cannot become negative. The interest rate structure implied by (2.17) displays the following characteristics:
negative interest rates are prevented,
a zero rate of interest can become positive again,
the level of absolute variance increases with increasing interest rates,
the interest rate displays a steady state distribution.
Within the same economic framework CIR determine the factor risk premium in terms of the above-mentioned economic variables . Together with all the simplifying assumptions, the factor risk premium » , is substituted into (2.16) to give the fundamental bond equation which, in equilibrium, must be satisfied by any zero coupon bond:
with P ( r, T, T ) = 1 as the boundary condition. Here the bond price depends on one underlying stochastic variable, the short-term interest rate r , which represents the uncertainty in the underlying economy. This model proposes that the short-term interest rate is the predominant variable in determining the whole term structure. This can only be true if all the simplifying assumptions are satisfied ^{ [10] } .
^{ [8] } This is a specialisation of (2.2)
^{ [9] } See [ 18 ] for the details of this.
^{ [10] } These assumptions can be summarised as: investors have constant relative risk aversion, uncertainty within the economy is modelled by a single variable, and the interest rate is a monotonic function of this variable.