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Optimal portfolio composition for sovereign wealth funds
1. Optimal Portfolio Composition for Sovereign
Wealth Funds
Khouzeima Moutanabbir* Diaa Noureldin**
*Department of Mathematics **Department of Economics
American University in Cairo
ERF and World Bank Workshop on Sovereign Wealth Funds: Stabilization,
Investment Strategies and Lessons for the Arab Countries
September 9-10, 2016
The World Bank, Washington, DC
2. Outline
I Introductory remarks.
I Paper’s objective.
I Overview of sovereign wealth funds (SWFs).
I Modelling framework.
I Model estimation.
I Analysis of the optimal allocation.
I Concluding remarks.
3. Introductory remarks
I Signi…cance of SWFs in international …nancial markets.
I US$ 7.2 trillion in total assets.
I 80% of funds (by assets) in Middle East and Asia.
I 56% of total assets are in oil-based funds.
I Signi…cance of SWFs for their respective economies.
I Large assets relative to GDP and annual oil income.
I Small ratio of assets to oil reserves (except Norway).
I The investment strategies of large oil-based SWFs:
I Gradual increase of equity share to roughly 60%.
I Rest is divided between bonds and alternative investments.
I Transparency and governance issues.
I Agency problems.
I Home/regional bias in investment strategies.
I Fiscal rules (or lackthereof).
5. Introductory remarks: SWFs asset allocations in 2014
0
1 0
2 0
3 0
4 0
5 0
6 0
7 0
N o r w a y U A E - A D I A S a u d i A r a b i a K u w a i t Q a t a r
Equity Bonds O ther
(%)
6. Paper’s objective
Our objective is to determine the optimal asset allocation for a SWF
based on income from oil. The main issues are:
I Investment horizon.
I Utility function (risk aversion and time preference).
I Nature of the available investment opportunity.
I Oil income subject to random shocks and correlated with risky asset.
The questions of interest:
I Should …nancial assets (e.g. equity or bonds) be used to hedge
against oil shocks?
I Should oil be considered as an asset in the optimal allocation
problem? In other words, should we consider the optimal tradeo¤
between above ground and underground wealth?
I What happens if risk aversion and the elasticity of intertemporal
substitution change?
I What happens if the return dynamics for existing …nancial assets
change?
7. Related literature
Our paper draws on the following strands of the literature:
I Optimal asset allocation for a SWF:
I Gintschel & Scherer (2008); Scherer (2011); van den Bremer, van
der Ploeg & Wills (2016).
I Asset allocation given a stochastic stream of income:
I Bodie, Merton & Samuelson (1992); Koo (1995, 1998): Veceira
(2001).
I Asset allocation in the presence of state variables:
I Campbell and Veceira (1999); Campbell, Chan and Veceira (2003).
8. Model 1 setup: Exogenous oil income
Main equations
I We assume
Ft+1 = (Ft + Yt Ct ) RF ,t+1
where Ft is the value of the fund at time t, i.e. at the beginning of
the period [t, t + 1[.
I Yt is the income from oil allocated to the fund at time t. It follows
the dynamic:
Yt+1 = Yt exp g + ξt+1 .
I Ct is the consumption out of the fund over the interval [t, t + 1[
evaluated at time t.
I Utility of consumption is given by the Epstein-Zin (1989) stochastic
di¤erential utility:
Ut = (1 δ)C
1 γ
θ
t + δ
h
Et U
1 γ
t+1
i1
θ
θ
1 γ
,
where γ > 0 is the RRA coe¢ cient, ψ > 0 is the EIS, 0 < δ < 1 is
the time discount factor, and θ = 1 γ
1 ψ 1 .
9. Model 1 setup: Exogenous oil income
Financial Market Assumptions
I RF ,t = π0R0,t + π
0
Rt is the continuously compounded return on
the fund.
I Time-varying investment opportunities: zt+1 = Φ0 + Φ1zt + νt+1,
where
zt+1 =
0
@
r0,t+1
xt+1
st+1
1
A , xt+1 =
0
B
B
B
@
r1,t+1 r0,t+1
r2,t+1 r0,t+1
...
rn,t+1 r0,t+1
1
C
C
C
A
.
I The shocks νt+1 and ξt+1 are correlated through the vector β:
ξt+1 = β
0
νt+1 + σ(o)ξ
(o)
t+1,
and ξ
(o)
t+1 is the own oil shock.
10. Model 1 setup: Exogenous oil income
The objective of the fund manager is to maximize the expected present
value of future consumption at discount rate δ:
maxfCt ,πt gE
"
∞
∑
t=0
δt
U(Ct )
#
,
subject to the intertemporal budget constraint:
Ft+1 = (Ft + Yt Ct ) RF ,t .
11. Model 1 setup: Exogenous oil income
Under the log-linear approximation, the optimal log consumption and
portfolio composition are (lower case variables are in logs):
ct yt = a + b(ft yt ) + B
0
1zt + z
0
t B2zt ,
πt = A0 + A1zt .
While the optimal allocation is linear in the state vector, the optimal
consumption path is quadratic.
Focusing on the optimal allocation coe¢ cients, we have:
A0 =
1
1 θ + bθ
ψ
Σ 1
xx Hx Φ0 +
1
2
σ2
x + σ0x
θ(1 b)
ψ
Hx β
0
Σν
θ
ψ
Λ0 σ0x ,
A1 =
1
1 θ + bθ
ψ
Σ 1
xx Hx Φ1
θ
ψ
Λ1 .
12. Model 1 setup: Exogenous oil income
Let κ = 1
1 θ+ bθ
ψ
Σ 1
xx , the optimal allocation coe¢ cients can be
decomposed as follows:
A0
|{z}
Total demand
= κ Hx Φ0 +
1
2
σ2
x + σ0x σ0x
| {z }
Speculative demand
+ κ
θ
ψ
Λ0
| {z }
Hedging demand
+κ
θ(1 b)
ψ
Hx β
0
Σν
| {z }
Hedging oil shocks
,
A1
|{z}
Total demand
= κ [Hx Φ1]
| {z }
Speculative demand
+ κ
θ
ψ
Λ1
| {z }
Hedging demand
13. Model 2 setup: Oil as an asset in optimal allocation
Main equations
I We assume total wealth Wt = W o
t + W f
t , where W o
t is oil wealth
and W f
t represent …nancial assets.
I Ct is the consumption out of the fund over the interval [t, t + 1[
evaluated at time t.
I Total wealth evolves according to
Wt+1 = (Wt Ct ) RW ,t+1.
I Utility of consumption is given by the Epstein-Zin (1989) stochastic
di¤erential utility:
Ut = (1 δ)C
1 γ
θ
t + δ
h
Et U
1 γ
t+1
i1
θ
θ
1 γ
.
14. Model 2 setup: Oil as an asset in optimal allocation
Financial Market Assumptions
I De…ne ηt = W o
t
Wt
as the relative weight of oil wealth in total wealth.
I Let RW ,t+1 be the return on total wealth over the period [t, t + 1[
given by
RW ,t+1 = ηt
∆W o
t+1
W o
t+1
!
+ (1 ηt )
∆W f
t+1
W f
t+1
!
= Ro,t+1 +
n
∑
i=1
(1 ηt )αi,t (Ri,t+1 Ro,t+1) .
I Time-varying investment opportunities: zt+1 = Φ0 + Φ1zt + νt+1,
where
zt+1 =
0
@
ro,t+1
xt+1
st+1
1
A , xt+1 =
0
B
B
B
@
r1,t+1 ro,t+1
r2,t+1 ro,t+1
...
rn,t+1 ro,t+1
1
C
C
C
A
.
15. Model 2 setup: Oil as an asset in optimal allocation
Under the log-linear approximation, the optimal log consumption and
portfolio composition are (lower case variables are in logs):
ct wt = b0 + B
0
1zt + z
0
t B2zt ,
πt = A0 + A1zt .
Considering the parameters of the optimal allocation, we have
A0 =
1
γ
Σ 1
xx Hx Φ0 +
1
2
σ2
x + (1 γ)σox + 1
1
γ
Σ 1
xx
Λ0
1 ψ
,
A1 =
1
γ
Σ 1
xx Hx Φ1 + 1
1
γ
Σ 1
xx
Λ1
1 ψ
,
16. Model 2 setup: Oil as an asset in optimal allocation
I The total value of accumulated wealth and the optimal consumption
at time t is given by
Ct = Wt exp b0 + B
0
1zt 1 + z
0
t 1B2zt 1 ,
and the total wealth is updated using the budget constraint
Wt+1 = (Wt Ct ) RW ,t+1.
I The optimal solution provides the values of πt and the weight of the
optimal oil wealth is given by
1 ηt =
n
∑
i=1
πi,t .
I Then we obtain the optimal oil wealth at time t + 1
W o
t+1 = ηt Wt+1.
I We can also …nd optimal oil reserves if we divide W o
t+1 by the
expected price of oil at time t + 1.
17. Data for empirical analysis
Variable Mean St. Dev. Min Max ρ (1)
Return on oil 0.05900 0.29630 -0.66100 1.121000 0.06451
Return on risk-free asset 0.04977 0.03307 0.00030 0.143000 0.86470
Dividend yield 0.03026 0.01305 0.01137 0.055700 0.89281
Excess returns over RF asset
Excess returns on equity 0.06219 0.17309 -0.38100 0.32100 0.00680
Excess returns on bonds 0.02504 0.09683 -0.14200 0.21800 -0.19418
Excess returns over oil returns
Excess returns on cash -0.00928 0.29160 -1.04300 0.72200 0.05969
Excess returns on equity 0.05298 0.38576 -1.38000 0.84600 0.03692
Excess returns on bonds 0.01583 0.32524 -1.10100 0.90400 0.12028
18. Model 1: VAR speci…cation
The VAR(1) model is
zt+1 = Φ0 + Φ1zt + νt+1, νt+1jFt N (0, Σν)
where
zt =
0
B
B
@
rrf ,t
ereq,t
erbn,t
dt
1
C
C
A =
0
B
B
@
rrf ,t
req,t rrf ,t
rbn,t rrf ,t
dt
1
C
C
A .
20. Model 1: Optimal allocation
1 2 3 4 5 6 7 8 9 10 11
-100
0
100
200
300
Impact of changing γ on equityallocation
1 2 3 4 5 6 7 8 9 10 11
-30
-20
-10
0
10
Impact of changing γ on equityallocation
0.5 0.6 0.7 0.8 0.9 1
60
65
70
75
80
Impact of changing δ on equityallocation
1 2 3 4 5 6 7 8 9 10 11
-100
0
100
200
300
Impact of changing γ on bond allocation
1 2 3 4 5 6 7 8 9 10 11
-40
-20
0
20
Impact of changing γ on bond allocation
0.5 0.6 0.7 0.8 0.9 1
65
70
75
80
Impact of changing δ on bond allocation
Speculative (%)
Total (%)
Hedging (%)
Hedging oil (%)
Total (%)
Speculative (%)
Total (%)
Hedging (%)
Hedging oil (%)
Total (%)
21. Model 2: VAR speci…cation
The VAR(1) model is
zt+1 = Φ0 + Φ1zt + νt+1, νt+1jFt N (0, Σν)
where
zt =
0
B
B
B
B
@
roil,t
breq,t
brbn,t
brrf ,t
dt
1
C
C
C
C
A
=
0
B
B
B
B
@
roil,t
req,t roil,t
rbn,t roil,t
rrf ,t roil,t
dt
1
C
C
C
C
A
.
26. Model 2: Long-term projections for consumption, fund
value and oil extraction
2015 2020 2025 2030 2035 2040 2045 2050
0
500
1000
1500
Evolution of the total wealth
2015 2020 2025 2030 2035 2040 2045 2050
0
50
100
150
200
250
300
Evolution of oil wealth
2015 2020 2025 2030 2035 2040 2045 2050
0
1
2
3
4
Evolution of the oil reserves
2015 2020 2025 2030 2035 2040 2045 2050
0
20
40
60
80
100
120
Evolution of the consumption
27. Concluding remarks
I The objective of the paper is to study the optimal asset allocation
for an oil-based SWF:
I First: The case where oil revenue is given as an exogenous stochastic
stream of income.
I Second: The case where oil is considered as an asset in the optimal
allocation problem.
I The …rst model allows us to decompose the demand components as
speculative demand, hedging demand, and hedging-against-oil
demand.
I The second model given myopic and hedging demand components.
I The historical allocation from both models provides surprisingly
similar results, especially with regard to allocation tradeo¤ between
equity and bonds.
I The model’s medium-term projections indicate a need to reduce
exposure to equity in favor of bonds.
