buy and hold strategies in optimal portfolio … · 2014. 5. 15. · h (n ¡ j) pm s;k=0 pn ......

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

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

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

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

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

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

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

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

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

‘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

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

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

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

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

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

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

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

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

≥ 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

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

≥ 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