1. International Diversification
with and without Stochastic Inflation
Elmar Mertens
University of Basel, WWZ/Department of Finance
elmar.mertens@unibas.ch
October 12, 2002
1
2. Some technical things to remember: We want to . . .
• Add / subtract inflation rates and exchange rate changes from
returns → Use log-changes (Why?)
• Construct portfolios, i.e. aggregate across assets
→ Use simple returns (Why?)
The only way to do both, is to go to continuous time. Right now, we
need this to make our approximation 1+r = 1+R
1+π → r ≈ R−π work as
good as it gets. Please note: We did not say 1+˜r = 1+ ˜R
1+˜π → ˜r ≈ ˜R−˜π
This will be crucial later.
2
3. Solnik-Sercu Setup
N Risky Assets N risky assets with mean returns in vector E ˜R and
covariance matrix Σ. Each risky asset has portfolio weight wi
stacked in vector w. They can be split into two groups: “assets”
(A) and foreign money markets (C)
1 Riskfree Asset with constant return Rf. Has portfolio weight
wf = 1 − 1 w.
No (Stochastic) Inflation It is easiest to derive the Solnik-Sercu
model without any inflation. We will later see that it works also
with deterministic inflation (some constant) or an inflation rate
which is stochastic but uncorrelated with assets
3
4. Solnik-Sercu Solution
max
w
V = w µ
e
+ Rf
µp
−
η
2
w Σw
σp
⇒
∂V
∂w
= µe − ηΣw
!
= 0
⇐⇒ w =
1
η
Σ−1µe
⇐⇒ w =
1
η
Σ−1µe + 1 −
1
η
0
w
wf
=
1
η
Σ−1µe
1 − 1 Σ
−1
µe
“Log-Portfolio”
+ 1 −
1
η
0
1 − 1 Σ
−1
µe
“Hedge”
Even though the role of the “hedge” portfolio is rather void here, it
mirrors the structure of the more general Adler-Dumas result
4
5. Invariance of Log-Portfolio
The log-portfolio Σ−1µe is not only the portfolio held by a log-utility
investor (η = 1). It is also invariant to the reference currency of
the investor. The direct proof of transforming Σ and µe and showing
that w remains the same is cumbersome (Ito Lemma and some matrix
algebra, see Sercu 1980).
But we can appeal to a more illuminating argument: Maximizing
log-wealth in one numeraire (ln W), yields the same portfolio as max-
imizing log-wealth measured in another numeraire ln W
S = ln W − ln S
5
6. Partitioning Σ−1µe
This is actually the international side of Solnik-Sercu
Σ =
ΣA,A ΣA,C
ΣC,A ΣC,C
B = Σ−1
C,CΣC,A
ΣA,A|C = ΣA,A − B ΣC,CB
Σ−1 =
Σ−1
A,A|C
−Σ−1
A,A|C
B
−B Σ−1
A,A|C
Σ−1
C,C + B Σ−1
A,A|C
B
w =
wA
wC
=
Σ−1
A,A|C
µe
A − Σ−1
A,A|C
B µe
C
Σ−1
C,C µe
C − B wA
= Σ−1µe
Note: ΣA,A|C is the covariance matrix of the assets after minimum
variance hedging against currency risks
6
8. Solnik-Sercu: Numerical Example (2 × 2) with Matrix Formula
Same data as before. Now we use the formula for a 2 × 2 inverse:
µe =
0.15
−0.05
Σ =
σ2
A σAB
σAB σ2
B
=
0.252 0.02
0.02 0.202
|Σ| = σ2
Aσ2
B − (σAB)2
Σ−1 =
1
|Σ|
σ2
B −σAB
−σAB σ2
A
Σ−1µe =
1
0.0021
0.202 −0.02
−0.02 0.252
=Σ−1
0.15
−0.05
=µe
=
1
0.0021
0.202 · 0.15 + (−0.02) · (−0.05)
(−0.02) · 0.15 + 0.252 · (−0.05)
≈
3.33
−2.92
wf = 1 − wA − wC ≈ 1 − 3.33 + 2.92 = 0.59
8
9. Solnik-Sercu deviation: Currency Overlay
A currency overlay proceeds sequentially, first optimizing over the un-
hedged assets, then doing some minimum-variance currency hedging
and speculating with currencies – no feedback with asset positions,
though!
wOV =
wA
wC
=
Σ−1
A,A µe
A
Σ−1
C,C µe
C − B wA
This is only optimal when B = 0, then
w
B=0
= wOV
B=0
=
wA
wC
B=0
=
Σ−1
A,A µe
A
Σ−1
C,C µe
C B=0
9
10. Adler-Dumas Setup:
N Risky Assets N risky assets with nominal mean returns in vector
E ˜R and covariance matrix Σ. Each risky asset has portfolio weight
wi stacked in vector w
1 Riskfree Asset is only nominally riskfree. Return is Rf with zero
variance. Note that this is a really risky investment! Has portfolio
weight wf. Note: wf = 1 − 1 w.
Inflation ˜π is stochastic with mean µπ and variance σp. Covariances
with the N risky assets are stacked in vector σAπ. Note that the
covariance with riskfree asset is zero (why?)
10
11. Objective:
Again we want to maximize
V = µp −
η
2
σ2
p
but now we will have stochastic inflation and need to be careful how to
calculate the real returns. It is here that the trickiness of continuous
time sneaks in!!
Steps:
• Figure out how to write real µp and σ2
p as function of asset weights
• Solve FOC with respect to weights to get optimal portfolio
11
12. Moments of real portfolio returns: THE WRONG WAY
µp = w E ˜R + wfRf − µπ
Something missing here . . .
= w µ
e
+ Rf − µπ
Where µe ≡ E ˜R − Rf1
σp = Var w ˜R + wf Rf − ˜π = Var w ˜R − ˜π = w Σw − 2w σAπ + σ2
π
But then we would get an inflation hedge independent from η . . .
Does the log-investor suddenly care about numeraire-risks?
∂V
∂w
= µe − ηΣw + ησAπ
!
= 0 ⇐⇒ w =
1
η
Σ−1µe + Σ−1σAπ
Inflation Hedge
DON’T DO THIS AT HOME!!!
12
13. Step 1: Real Moments
Once we have realized that we need to apply stochastic calculus
in order account for stochastic inflation when computing the real
portfolio returns, we get:
µp = w µ
e
− w σAπ
KEY!
+Rf − µπ + σ2
π
Variance has been right:
σp = w Σw − 2w σAπ + σ2
π
13
14. Step 2: Adler-Dumas FOC
max
w
V = w µ
e
− w σAπ + Rf − µπ + σ2
π −
η
2
w Σw − 2w σAπ + σ2
π
⇒
∂V
∂w
= µe − σAπ − ηΣw + η σAπ
!
= 0
⇐⇒ w =
1
η
Σ−1(µe − σAπ) + Σ−1σAπ
⇐⇒ w =
1
η
Σ−1µe + 1 −
1
η
Σ−1σAπ
With wf = 1 − 1 w we get Adler-Dumas formula (9)
w
wf
=
1
η
Σ−1µe
1 − 1 Σ
−1
µe
“Log-Portfolio”
+ 1 −
1
η
Σ−1σAπ
1 − 1 σAπ
Inflation Hedge
With σAπ = 0, we obtain again the Solnik-Sercu portfolio
14
16. Stochastic processes for real portfolio
N Risky Assets dSi = µi Si dt + σi Si dXi
1 Riskfree Asset dM = Rf M dt
Price Level dP = µπ P dt + σπ P dXP
Where E(dXi) = E(dXP ) = 0, E(dXi)2 = E(dXP )2 = 1, E(dXidXj) =
σij (∀i = j) and E(dXidXP ) = σiπ
16
17. Stochastic calculus for real portfolio
We want to know how the portfolio
W = i ni Si + nf M
W · P
where wi ≡
ni Si
W
·
1
P
and wf ≡
nf M
W
·
1
P
evolves in real terms. Applying Ito’s Lemma yields
dW
W =
N
i=1
wi(µi − Rf − σiπ) + Rf − µπ + σ2
π
µp
dt+
N
i=1
wiσidXi − σπdXP
σp=E(·)2
THIS, YOU MAY DO AT HOME!!!
. . . and you will learn how to do Ito by heart
. .
17
18. Multidimensional Ito
dW =
N
i=1
∂W
∂Si
dSi +
∂W
∂M
dM +
2
2
N
i=1
∂2W
∂Si∂P
σiπ Si P dt +
1
2
∂2W
∂P2
σ2
πP2dt
+
∂W
∂P
dP +
1
2
N
i=1
N
j=1
∂2W
∂Si∂Sj
σij Si Sj dt +
1
2
∂2W
∂M2
σ2
M
=0
M2dt
+
1
2
N
i=1
∂2W
∂Si∂M
σiM
=0
Si M dt +
1
2
∂2W
∂P∂M
σπM
=0
P M dt +
∂W
∂t
dt
∂W
∂Si
=
ni
P
∂2W
∂Si∂P
= −
ni
P2
∂W
∂P
= − i ni Si + nf M
P2
∂W
∂M
=
nf
P
∂2W
∂Si∂Sj
= 0 =
∂W
∂t
∂2W
∂P2
= 2 · i ni Si + nf M
P3
18
