Advances in Risk Management

176 OPTIMAL INVESTMENT WITH INFLATION-LINKED PRODUCTS

where the case ofγ=0 corresponds to the choice of the logarithmic utility
function. If in particular the stock price is independent of the inflation index
(for example, we haveσ 2 =0) then we obtain the special form of

π∗(t)=

1

1 −γ

((

b−rN

)

/σ 12

(λσI−rR)/σI^2

)

(9.18)

So in both cases, the optimal fractions of wealth invested in the risky
assets are functions of the excess return suitably weighted by their volatili-
ties. Note especially that the subjective excess returnλσIhas to be bigger than
the real rate – an assumption that rarely seems to hold. So in this – hypothet-
ical – portfolio problem a risk averse investor typically sells inflation when
behaving optimally!

Step 2: Optimal portfolios with inflation linked products.
Problem 1: Inflation-linked bond and non-inflation linked bond.
In this first setting we assume that the investor has access to a market con-
sisting of the riskless bond with priceP 0 (t) and an inflation-linked coupon
bond with coupon payments as described in relation (9.5) and a final pay-
ment as in (9.6), for example, we look at the following special case of the
portfolio problem (OP):

max

φ(.)∈B(x)

E(U(Xφ(T))) (OP1)

with

Xφ(t)=φ 0 (t)P 0 (t)+φ 1 (t)f(1)(t,I(t)) (9.19)

and where the functionf(1)coincides withBIL(t,I(t)) of Proposition 9.2. As
shown in Korn and Trautmann (1999) we only have to use the replication
strategy in the riskless bond and the inflation index for this inflation-linked
bond to determine the optimal trading strategy for the corresponding port-
folio problem (OP), the dynamics of its price processBIL(t,I(t)) play no
explicit role. However, as can be directly verified, the number of the shares
in the inflation index of this replication strategy has the following form:

ψ 1 (t)=

∑

i:ti>t

Ci

I(t 0 )

exp(−rR(ti−t))+

Fexp(−rR(T−t))N(d(t))

I(t 0 )

(9.20)

Combining this with the results of Step 1 and Theorem 1 leads to the optimal
trading strategy in the inflation-linked bond of

φ 1 (t)=

ξ 1 (t)

ψ 1 (t)

=

λσI−rR(t)

(1−γ)σI^2

·

X(t)

ψ 1 (t)I(t)

(9.21)

withX(t) being the optimal wealth process of the basic portfolio problem
(P) (which coincides with the optimal wealth process of (OP1) by Korn and