Assume i with probability 1, for some constant . Via the following steps, show

Assume ε˜i ≥ −γ with probability 1, for some constant γ . Via the following steps, show that

|δi| ≤

αw0πi exp(αγ w0πi)var(ε˜i)

Rf

.

(a) Show that

δi = E[exp(−αw˜ m)ε˜i]

RfE[exp(−αw˜ m)]

.

(b) Show that δi = E[exp(−αw0πiε˜i)ε˜i]
RfE[exp(−αw0πiε˜i)]
.
Hint: Use independence and the fact that end-of-period market wealth is w˜ m = w0
j=i πjR˜j + w0πiR˜i = w0
j=i πjR˜j + w0πiai + w0πib
i F˜ +w0πiε˜i .

(c) Show that E[exp(−αw0πiε˜i)] ≥ 1 .
Hint: Use Jensen’s inequality.

(d) Show that #
#E[exp(−αw0πiε˜i)ε˜i]
#
# ≤ αw0πi exp(αγ w0πi)var(ε˜i).
Hint: Use an exact first-order Taylor series expansion of the exponential function.