=E
№+/
Ds,
ST.ds
dSs Ds
h+ + J Ss-ds

τ
I dμv ( Ss ) - E
t
τ
У dμv (Ss)
t


By equation (5), μv(St) = ηt,DηS,DσD + rf and ΣS(St) = ηS,DσD. Since
Vt = αtSt , we obtain ηtS,D = ηtV,D and ΣV (St) = ΣS(St) = ηtV,DσD . Hence
we can rewrite the covariance as
τ
f J ηV,DσDdWs
t
VD dηs,D , Φ,D dηV,D 1 3 DdW
ηs ∂Ds+ ηs ∂Ds σDD Ds s
j E ( ʃ vγrD∂ dηs,D + „ФD dηV,D 1 σ4 D ry,D
J ɪs ∂Ds + ηs ∂Ds jDDDsηs
Dt ds,
since by Ito’s Lemma the stochastic part of dμv(St) is given by
Dt
Φ,D V,D 2
d ( ηt η, σD + rf ) σD Dt dWt.
∂Dt
Theelasticities ηV,D and ηs,d are positive. Hence, Cov (CERt,τ,μv (Vτ) — rf ∣ Dt) <
[>] 0 if aggregate RRA is declining [increasing] and ηsV,D is non-increasing
[non-declining] in Ds . The latter condition is equivalent to the condition that
the instantaneous volatility of the return index, ΣV (Ss), is not increasing
[not declining] because ΣV(Ss) = ηV,DσD. ■
6.5 Proof of Lemma 2
Let X denote the aggregate supply of claims. Then equation (9) yields for
ηi(xi)=γi,i = 1...n
1 = αi ( X )
ηM ( X ) γi ’
Differentiating with respect to X yields
ηM( X ) = ^ αi( X )
[пм(X)]2 = V Yi '
23
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