In a continuous-time framework an individual with time-additive expected power utility induces the state-price deflator t =

In a continuous-time framework an individual with time-additive expected power utility induces the state-price deflator

ζt = e

−δt

 ct c0

−γ

, where γ is the constant relative risk aversion, δ is the subjective time preference rate, and c = (ct)t∈[0,T] is the optimal consumption process of the individual. If the dynamics of the optimal consumption process is of the form dct = ct



μct dt + σ

ct dzt

then the dynamics of the state-price deflator is dζt = −ζt

δ + γ μct − 1 2

γ (1 + γ )σ ct2



dt + γ σ

ct dzt



.

(a) State the market price of risk λt in terms of the preference parameters and the expected growth rate and sensitivity of the consumption process.

In many concrete models of the individual consumption and portfolio decisions, the optimal consumption process will be of the form ct = Wtef(Xt,t)

, where Wt is the wealth of the individual at time t, Xt is the time t value of some state variable, and f is some smooth function. Here X can potentially be multidimensional.

(b) Give some examples of variables other than wealth that may affect the optimal consumption of an individual and which may therefore play the role of Xt.

Suppose the state variable X is one-dimensional and write the dynamics of the wealth of the individual and the state variable as dWt = Wt

(μWt − ef(Xt,t)

)

dt + σ

Wt dzt



, dXt = μXt dt + σ

Xt dzt.

(c) Characterize the market price of risk in terms of the preference parameters and the drift and sensitivity terms of Wt and Xt.

Hint: Apply Itô’s Lemma to ct = Wtef(Xt,t) to express the required parts of the consumption process in terms of W and X.

(d) Show that the instantaneous excess expected rate of return on risky asset i can be written as

μit + δit − r f

t = βiW,tηWt + βiX,tηXt, where βiW,t and βiX,t are the instantaneous betas of the asset with respect to wealth and the state variable, respectively. Relate ηWt and ηXt to preference parameters and the drift and sensitivity terms of W and X.