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BUY AND HOLD STRATEGIES INOPTIMAL PORTFOLIO SELECTION PROBLEMS:
COMONOTONIC APPROXIMATIONS
J. Marın-Solano (UB), M. Bosch-Prıncep (UB), J. Dhaene (KUL),C. Ribas (UB), O. Roch (UB), S. Vanduffel (KUL)
Buy & Hold Strategies Comonotonic Approximations
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MULTIPERIOD OPTIMAL PORTFOLIO SELECTIONPROBLEM
1. Investment strategies.
2. Buy and hold strategy. Terminal wealth.
3. Upper and lower bounds for the terminal wealth.
4. Optimal portfolio.
Buy & Hold Strategies Comonotonic Approximations
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1. INVESTMENT STRATEGIES
There are (m+1) securities. One of them is riskfree:dP 0(t)P 0(t)
= rdt.
There are m risky assets:dP i(t)P i(t)
= µidt +d∑
j=1
σijdW j(t).
By defining Bi(t) =1σi
d∑
j=1
σijWj(t),
dP i(t)P i(t)
= µidt + σidBi(t) , i = 1, . . . , m .
Buy & Hold Strategies Comonotonic Approximations
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From the solution to this equation,
P i(t) = pi exp[(
µi − 12σ2
i
)t + σiB
i(t)],
we obtain that the random yearly returns of asset i in year k, Y ik ,
are independent and have identical normal distributions with
E [Y ik ] = µi − 1
2σ2
i ,
Var [Y ik ] = σ2
i , and
Cov [Y ik , Y j
l ] =
0 if k 6= l,
σij if k = l .
Buy & Hold Strategies Comonotonic Approximations
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Let Π(t) = (Π0(t), Π1(t), . . . , Πm(t)) denote the vector describingthe proportions of wealth invested in each asset at time t.
In general, a vector Π(t) will define an investment strategy.
If one unit of a security is constructed according to the investmentstrategy Π(t), let P (t) be the price of that unit at time t. Then,
dP (t)P (t)
=m∑
i=0
Πi(t)dP i(t)P i(t)
=
[m∑
i=1
Πi(t)(µi − r) + r
]dt+
m∑
i=1
Πi(t)σidBi(t) .
If Π(t) is prefixed, P (t) can be obtained by solving the stochasticdifferential equation above (constantly rebalanced portfolio).
Buy & Hold Strategies Comonotonic Approximations
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2. BUY AND HOLD STRATEGY. TERMINAL WEALTH
• The new amounts of money α(t) are invested at timet = 0, 1, . . . , n− 1 in some prefixed proportionsπ(t) = (π0(t), π1(t), . . . , πm(t)).
• Fractions πi(t) are always the same. Denoting πi(0) = πi, then(π0(t), . . . , πm(t)) = (π0, . . . , πm), for every t = 0, 1, . . . , n− 1.
• New quantities are invested once in a period of time (typically,once in a year), i.e.,
α(t) =
αi if t = i , for i = 0, 1, . . . , n− 1 ,
0 otherwise .
• The decision maker follows a buy and hold strategy, i.e., nosecurities are sold.
Buy & Hold Strategies Comonotonic Approximations
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Objective: To compute the terminal wealth Wn(π) for a givenbuy and hold strategy π = (π0, π1, . . . , πm).
Let Zij be the sum of returns of 1 unit of capital invested at time
t = j of asset i from time t = j to the final time t = n,
Zij =
n∑
k=j+1
Y ik .
The terminal wealth invested in asset i is
W in(π) =
n−1∑
j=0
πiαjeZij ,
whereas the terminal wealth will be given by
W (π) =m∑
i=0
W i(π) =m∑
i=0
n−1∑
j=0
πiαjeZij .
Buy & Hold Strategies Comonotonic Approximations
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3. UPPER AND LOWER BOUNDS FOR THE TERMINALWEALTH
Let X = (X1, X2, . . . , Xn) and let S = X1 + X2 + · · ·+ Xn.
It can be shown that
Sl ≤cx
n∑
i=1
Xi ≤cx Sc,
where Sc =n∑
i=1
F−1Xi
(U) and Sl =n∑
i=1
E [Xi | Λ].
Buy & Hold Strategies Comonotonic Approximations
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If S =n∑
i=1
αi eZi with αi ≥ 0,
Sc =n∑
i=1
F−1
αi eZi(U) =
n∑
i=1
αi eE [Zi]+σZiΦ−1(U) .
For a given Λ =n∑
j=1
γjZj ,
Sl =n∑
i=1
αi E [eZi | Λ] =n∑
i=1
αi eE [Zi]+
12 (1−r2
i )σ2Zi
+riσZiΦ−1(U)
.
We need values of γj that minimize of the “distance” between S
and Sl.
Buy & Hold Strategies Comonotonic Approximations
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‘Maximal Variance’ lower bound approach. As we have thatVar[S] = Var[Sl] + E[Var[S | Λ]], it seems reasonable to choose thecoefficients γj such that the variance of Sl is maximized:
γk = αk eE [Zk]+ 1
2 σ2Zk .
‘Taylor-based’ lower bound approach. Λ is a lineartransformation of a first order approximation to S:
γk = αk eE [Zk].
Buy & Hold Strategies Comonotonic Approximations
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Comonotonic Upper Bound B&H strategy:
W c(π) =m∑
i=0
n−1∑
j=0
πiαj e(n−j)(µi− 12 σ2
i )+√
n−jσiΦ−1(U) .
Note that W c(π) is a linear combination of fractions πi,i = 0, . . . , m.
Comonotonic Lower Bound B&H strategy:
W l(π) =m∑
i=0
n−1∑
j=0
πiαj e(n−j)(µi− 12 r2
ijσ2i )+rij
√n−jσiΦ
−1(U)
where the correlation coefficients rij are given by
rMVij =
∑mk=0
∑n−1l=0 πkαl(n−max(j, l))σike(n−l)µk
σi
[(n− j)
∑ms,k=0
∑n−1t,l=0 πsπkαtαl(n−max(t, l))σske(n−t)µs+(n−l)µk
]1/2.
Buy & Hold Strategies Comonotonic Approximations
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and
rTij =
∑mk=0
∑n−1l=0 πkαl(n−max(j, l))σik e(n−l)[µk− 1
2 σ2k]
σi(n− j)1/2·
·
m∑
s,k=0
n−1∑
t,l=0
πsπkαtαl(n−max(t, l))σsk e(n−t)[µs− 12 σ2
s]+(n−l)[µk− 12 σ2
k]
−1/2
Buy & Hold Strategies Comonotonic Approximations
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Numerical illustration: 2 risky, 1 risk-free. µ1 = 0.06, µ2 = 0.1,σ1 = 0.1, σ2 = 0.2, Pearson’s correlation: 0.5, r = 0.03.
Every period αi = 1, invested in proportions: 19% risk-free asset,45% first risky asset, 36% in the second risky asset. This amount isinvested for i = 0, ..., 19, whereas in i = 20 the invested amount isα20 = 0. The simulated results were obtained with 500,000 randompaths.
Buy & Hold Strategies Comonotonic Approximations
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p MC LBMV LBT UB
0.01 20.018 +2.39% +1.46% -20.01%
0.025 23.072 +1.49% +0.79% -18.54%
0.05 25.056 +1.07% +0.57% -16.80%
0.25 33,488 +0.00% -0.00% -10.14%
0.5 42.307 -0.11% +0.09% -3.92%
0.75 55.161 +0.12% -0.34% +3.32%
0.95 86.258 +0.24% +0.10% +14.49%
0.975 101.5 +0.11% -0.27% +18.07%
0.99 123.99 -0.23% -0.93% +22.08%
Buy & Hold Strategies Comonotonic Approximations
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4. OPTIMAL PORTFOLIO
Possible criteria: maximizing an expected utility, Yaari’s dualtheory of choice under risk:
maxπ
ρf [Wn(π)] = maxπ
∫ ∞
0
f(Pr(Wn(π) > x))dx ,
risk measures (some of them correspond to distorted expectationsρf [Wn(π)] for appropriate choices of the distorsion function f).
Buy & Hold Strategies Comonotonic Approximations
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Value at Risk at level p:Qp[X] = F−1
X (p) = inf{x ∈ R | FX(x) ≥ p}. If FX is an strictlyincreasing function, then it coincides with the related risk measureQ+
p [X] = sup{x ∈ R | FX(x) ≤ p} , p ∈ (0, 1).
Additive for sums of comonotonic risks.
Conditional Left Tail Expectation at level p (CLTEp[X]):
CLTEp[X] = E[X | X < Q+
p [X]]
, p ∈ (0, 1) .
Buy & Hold Strategies Comonotonic Approximations
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For the upper and lower bounds in B&H strategy:
Qp[W c(π)] =m∑
i=0
n−1∑
j=0
πiαj e(n−j)(µi− 12 σ2
i )+√
n−jσiΦ−1(p) ,
Qp[W l(π)] =m∑
i=0
n−1∑
j=0
πiαj e(n−j)(µi− 12 r2
ijσ2i )+rij
√n−jσiΦ
−1(p) ,
CLTEp[W c(π)] =m∑
i=0
n−1∑
j=0
πiαjeµi(n−j) 1− Φ(
√n− jσi − Φ−1(p))
p,
CLTEp[W l(π)] =m∑
i=0
n−1∑
j=0
πiαjeµi(n−j) 1− Φ(
√n− jrijσi − Φ−1(p))
p.
Buy & Hold Strategies Comonotonic Approximations
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Maximizing the Value at Risk: for a given probability p and a giveninvestment strategy, let Kp(π) be the p-target capital defined as the(1− p)-th order ‘+’-quantile of terminal wealth,Kp(π) = Q+
1−p[W (π)].
For the optimal case:
K∗p = max
πQ+
1−p[W (π)] .
Alternatives:
Kc∗p = maxπ Q+
1−p[Wc(π)] or Kl∗
p = maxπ Q+1−p[W
l(π)].
Buy & Hold Strategies Comonotonic Approximations
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≥ r MC LBMV LBT UB
π0 100% 100% 100% 100%
π1 0% 0% 0% 0%
π2 0% 0% 0% 0%
K∗ 27.817 27.817 27.817 27.817
≥ 6% MC LBMV LBT UB
π0 33.24% 33.83% 34.36% 57.14%
π1 41.82% 40.79% 39.87% 0%
π2 24.94% 25.38% 25.77% 42.86%
K∗ 25.718 25.914 25.805 22.78
Buy & Hold Strategies Comonotonic Approximations
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Maximizing the CLTE: maxπ CLTE1−p[W (π)]. This optimizationproblem describes decisions of risk averse investors.
The CLTE1−p has the following nice property (lacking with theVaR):
CLTE1−p[W c(π)] ≤ CLTE1−p[W (π)] ≤ CLTE1−p[W l(π)] .
Alternative: maxπ CLTE1−p[W l(π)].
Buy & Hold Strategies Comonotonic Approximations
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≥ r MC LBMV LBT UB
π0 100% 100% 100% 100%
π1 0% 0% 0% 0%
π2 0% 0% 0% 0%
K∗ 27.817 27.817 27.817 27.817
≥ 6% MC LBMV LBT UB
π0 37.70% 37.36% 38.44% 57.14%
π1 34.12% 34.62% 32.74% 0%
π2 28.18% 28.02% 28.82% 42.86%
K∗ 23.431 24.089 23.962 21.163
Buy & Hold Strategies Comonotonic Approximations