Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Mean-Variance Optimal Portfolios in the Presence of a Benchmark with Applications to Fraud Detection Carole Bernard (University of Waterloo) Steven Vanduffel (Vrije Universiteit Brussel)
Beirut, May 2013.
Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
• Traditional mean-variance optimization consists in finding the
best pre-committed allocation of assets assuming a static strategy... • how to derive mean-variance efficient portfolios when all
strategies are allowed and available? • allowing for more trading strategies and thus more degrees of
freedom will further enhance optimality...
Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Contributions
IPart 1: Mean-Variance efficient payoffs • Optimal payoffs when you only care about mean and variance • Payoffs with maximal possible Sharpe ratio • Application to fraud detection I Part 2: Constrained Mean-Variance efficient payoffs • Drawbacks of traditional mean-variance efficient payoffs • Optimal payoffs in presence of a random benchmark • Sharpening the maximal possible Sharpe ratios • Application to improved fraud detection
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Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Financial Market
I I I I
The market (Ω, z, P) is arbitrage-free. There is a risk-free account earning r > 0. Consider a strategy with payoff XT at time T > 0. There exists Q so that its initial price writes as c(XT ) = e −rT EQ [XT ] ,
I Equivalently, there exists a stochastic discount factor ξT such that c(XT ) = EP [ξT XT ] . I Assume ξT is continuously distributed.
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Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Mean Variance Optimization I A Mean-Variance efficient problem:
(P1 )
max E [XT ] XT E [ξT XT ] = W0 subject to var[XT ] = s 2
Proposition (Mean-variance efficient portfolios) Let W0 > 0 denote the initial wealth and assume the investor aims for a strategy that maximizes the expected return for a given variance s 2 for s > 0. The a.s. unique solution to (P1 ) writes as XT? = a − bξT , where a = W0 + bE[ξT2 ] e rT > 0, b = √ Carole Bernard
s var(ξT )
> 0. Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Proof Choose a and b > 0 such that XT? = a − bξT satisfies the constraints var(XT? ) = s 2 and c(XT? ) = W0 . Observe that corr(XT? , ξT ) = −1 and XT? is thus the unique payoff that is perfectly negatively correlated with ξT while satisfying the variance and cost constraints. Consider any other strategy XT which also verifies these constraints (but is not negatively linear in ξT ). We find that corr(XT , ξT ) =
E[ξT XT ] − E[ξT ]E[XT ] p p > −1 = corr(XT? , ξT ). var(ξT ) var(XT )
Since var(XT ) = s 2 = var(XT? ) and E[ξT XT ] = W0 = E[ξT XT? ] it follows that E[ξT ]E[XT ] < E[ξT ]E[XT? ], which shows that XT? maximizes the expectation and thus solves Problem (P1 ). Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Proof Choose a and b > 0 such that XT? = a − bξT satisfies the constraints var(XT? ) = s 2 and c(XT? ) = W0 . Observe that corr(XT? , ξT ) = −1 and XT? is thus the unique payoff that is perfectly negatively correlated with ξT while satisfying the variance and cost constraints. Consider any other strategy XT which also verifies these constraints (but is not negatively linear in ξT ). We find that corr(XT , ξT ) =
E[ξT XT ] − E[ξT ]E[XT ] p p > −1 = corr(XT? , ξT ). var(ξT ) var(XT )
Since var(XT ) = s 2 = var(XT? ) and E[ξT XT ] = W0 = E[ξT XT? ] it follows that E[ξT ]E[XT ] < E[ξT ]E[XT? ], which shows that XT? maximizes the expectation and thus solves Problem (P1 ). Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Maximum Sharpe Ratio I The Sharpe Ratio (SR) of a payoff XT (terminal wealth at T when investing W0 at t = 0) is defined as SR(XT ) =
E[XT ] − W0 e rT , std(XT )
I All mean-variance efficient portfolios XT? have the same maximal Sharpe Ratio (SR ? ) given by SR? := SR(X?T ) = erT std(ξT ), ⇒ For all portfolios XT we have SR(XT ) 6 erT std(ξT ). I This can be used to show Madoff’s investment strategy was a fraud (Bernard & Boyle (JOD, 2009)). Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Madoff’s Magic Performance 700
accumulation
600
Fairfield Sentry S&P 500 Compounded at a fixed rate Linear at a fixed rate
500
400
300
200
100 1990
1992
Carole Bernard
1994
1996
1998
2000 date
2002
2004
2006
2008 Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Performance Dec 1990 to Oct 2008. The Sharpe ratio is obtained by SR =
E[XT ] − X0 e rT std[XT ]
Strategy Average return (annual) St deviation (annual) Sharpe Ratio (annual) Max monthly return Min monthly return % months positive Corr with S&P
Carole Bernard
Invest in S&P
Fairfield
9.64% 14.28% 0.36 11.44% -16.79% 64.65% 1
10.59% 2.45% 2.47 3.29% -0.64% 92.33% 0.32
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Payoff of the Split-Strike Conversion strategy Long equity position (buy the index at say S0 = 100). Buy a one month put with strike at S0 − a = 95, and sell a one month call with strike at S0 + b = 105, both with maturity T = 1 month.
Payoff at T = 1 month: min(max(ST , 95), 105) Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Example in a Black-Scholes market I There is a risk-free rate r > 0 and a risky asset with price process, dSt = µdt + σdWt , St where Wt is a standard Brownian motion, µ is the drift and σ is the volatility. I The state-price density ξT is given as ξT
1 2 T
= e −rT e −θWT − 2 θ
= αST−β ,
for known coefficients α, β > 0 (assume µ > r and θ = I The maximal Sharpe ratio is given by p SR ? = e θ2 T − 1.
µ−r σ ).
see Goetzmann et al. (2007) for another proof. Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Parameters µ = .1, σ = 0.20, r = .04, T = 1, a = b
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Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
General Market I Non-parametric estimation of the upper bound e rT std(ξT ) I Assume ξT = f (ST ) (where f is typically decreasing and ST is the risky asset) and that all European call options on the underlying ST maturing at T > 0 are traded. Let C (K ) denote the price of a call option on ST with strike K . Then, the Sharpe ratio SR(XT ) of any admissible strategy with payoff XT satisfies s Z +∞ ∂ 2 C (K ) f (K ) dK − 1. SR(XT ) 6 e 2rT ∂K 2 0 I Use for instance A¨ıt-Sahalia and Lo (2001). Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Improving Fraud Detection by Adding Constraints
I Detect fraud based on mean and variance only I Ignored so far additional information available in the market. I How to take into account the dependence features between the investment strategy and the financial market? I Include correlations of the fund with market indices to refine fraud detection. Ex: the so-called “market-neutral” strategy is typically designed to have very low correlation with market indices ⇒ it reduces the maximum possible Sharpe ratio!
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Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Improving Investment by Adding Constraints I Optimal strategies XT∗ = a − bξT give their lowest outcomes when ξT is high. Bounded gains but unlimited losses! I Highest state-prices ξT (ω) correspond to states ω of bad economic conditions as these are more expensive to insure: • E.g. in a Black-Scholes market: ξT = αST−β , α, β > 0. • Also, E[XT∗ |ξT > c] < E[YT |ξT > c], for any other strategy YT with the same distribution as XT∗ showing that XT∗ does not provide protection against crisis situations (event “ξT > c”). • in a Black-Scholes market: XT∗ = −∞ when ST = 0. I To cope with this observation: we impose the strategy to have some desired dependence with ξT , or more generally with a benchmark BT . Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Proposition (Optimal portfolio with a correlation constraint) Let BT be a benchmark, linearly independent from ξT with 0 < var(BT ) < +∞. Let |ρ| < 1 and s > 0. A solution to the following mean-variance optimization problem (P2 )
max E[XT ] var(XT ) = s 2 c(XT ) = W0, corr(XT , BT ) = ρ
(1)
is given by XT? = a − b(ξT − cBT ), where a, b and c are uniquely determined by the set of equations ρ = corr(cBT − ξT , BT ) p s = b var(ξT − cBT ) W0 = ae −rT − b(E [ξT2 ] − cE [ξT BT ]). Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Proof Observe that f (c) := corr(cBT − ξT , BT ) verifies lim
c→−∞
f (c) = −1, lim f (c) = 1 and f 0 (c) > 0 so that ρ = f (c) has a c→+∞
unique solution. Take XT? = a − b(ξT − cBT ) linear in ξT − cBT and satisfying all constraints and b > 0. Consider any other XT that satisfies the constraints and which is non-linear in ξT − cBT , then corr(XT , ξT − cBT ) =
E[XT (ξT − cBT )] − E[ξT − cBT ]E[XT ] std(ξT − cBT )std(XT ) > −1 = corr(XT? , ξT − cBT )
Since both XT and XT? satisfy the constraints we have that std(XT ) = std(XT? ), E[XT ξT ] = E[XT? ξT ] and cov(XT , BT ) =cov(XT? , BT ). Hence the inequality holds true if and only if E[XT? ] > E[XT ]. Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Proof Observe that f (c) := corr(cBT − ξT , BT ) verifies lim
c→−∞
f (c) = −1, lim f (c) = 1 and f 0 (c) > 0 so that ρ = f (c) has a c→+∞
unique solution. Take XT? = a − b(ξT − cBT ) linear in ξT − cBT and satisfying all constraints and b > 0. Consider any other XT that satisfies the constraints and which is non-linear in ξT − cBT , then corr(XT , ξT − cBT ) =
E[XT (ξT − cBT )] − E[ξT − cBT ]E[XT ] std(ξT − cBT )std(XT ) > −1 = corr(XT? , ξT − cBT )
Since both XT and XT? satisfy the constraints we have that std(XT ) = std(XT? ), E[XT ξT ] = E[XT? ξT ] and cov(XT , BT ) =cov(XT? , BT ). Hence the inequality holds true if and only if E[XT? ] > E[XT ]. Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
ST? : Growth Optimal Portfolio (GOP) • The Growth Optimal Portfolio (GOP) maximizes expected
logarithmic utility from terminal wealth. • Under general assumptions on the market, the GOP is a
diversified portfolio (proxy: a world stock index). • The GOP (also called Market portfolio or Num´ eraire portfolio)
can be used as num´eraire to price under P, so that ξT = c(XT ) = EP [ξT XT ] = EP
XT ST?
1 ST?
where S0? = 1. • Details in Platen & Heath (2006).
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Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Example when BT = ST?
An optimal solution is of the form XT? = a − b(ξT − cST? ), where c is computed from the equation ρ = corr(cST? − ξT , ST? ), b is s and derived from b = √ ? T) var(ξT −cS 2 a = W0 e rT + b e −2rT +θ T − c e rT . Optimal payoffs as a function of the GOP for different values of the correlation ρ with the benchmark ST? using the following parameters: W0 = 100, r = 0.05, µ = 0.07, σ = 0.2, T = 1, θ = (µ − r )/σ, S0 = 100, s = 10.
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Optimal Portfolio
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120
Mean−Variance Optimum
100
80
60
40 no constraint ρ = 0.75 ρ = 0.3 ρ = −0.5 ρ = −0.9
20
0
−20 0.7
0.8
0.9
1 1.1 Growth Optimal Portfolio S*T
1.2
1.3
Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Fraud Detection
Proposition (Constrained Maximal Sharpe Ratio) All mean-variance efficient portfolios XT? which satisfy the additional constraint corr(XT? , BT ) = ρ with a benchmark asset BT (that is not linearly dependent to ξT ) have the same maximal Sharpe ratio SRρ? given by SRρ? = e rT
cov(ξT , ξT − cBT ) 6 SR ? = e rT std(ξT ). std(ξT − cBT )
(2)
where SR ? is the unconstrained Sharpe ratio.
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Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Illustration in the Black-Scholes model
Maximum Sharpe ratio SRρ? for different values of the correlation ρ when the benchmark is BT = ST? . We use the following parameters: W0 = 100, r = 0.05, µ = 0.07, σ = 0.2, T = 1, S0 = 100.
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Optimal Portfolio
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0.15
Maximum Sharpe Ratio
Constrained case Unconstrained case 0.1
0.05 0.02 0.004 0
−0.05 −0.1
+0.1
−0.1 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 Correlation coefficient ρ
1
Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
M-V Optimization with a Benchmark I Dependence (interaction) between XT and BT cannot be fully reflected by correlation. I A useful device to do so is the copula. Sklar’s theorem shows that the joint distribution of (BT , XT ) can be decomposed as P(BT 6 y , XT 6 x) = C (FBT (y ) , FXT (x)), where C is the joint distribution (also called the copula) for a pair of uniform random variables over (0, 1). Hence, the copula C fully describes the interaction between the strategy’s payoff XT and the benchmark BT . I Constrained Mean-Variance efficient problem:
(P3 )
Carole Bernard
max E [XT ] XT E [ξT XT ] = W0 var(XT ) = s 2 subject to C := Copula(XT , BT ) Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Proposition (Constrained Mean-Variance Efficiency) Let s > 0. Assume that the benchmark BT has a joint density −1 h i with ξT . Define A as A = cF (B ) jF (B ) (1 − FξT (ξT )) , BT
T
BT
T
where the functions ju (v ) and cu (v ) are defined as the first partial derivative for (u, v ) → J(u, v ) and (u, v ) → C (u, v ) respectively, and where J denotes the copula for the random pair (BT , ξT ). If E[ξT |A] is decreasing in A, then the solution to the problem
max E[XT ] var(XT ) = s 2 c(XT ) = W0 C : copula between XT and BT
is uniquely given as X?T = a − bE[ξT |A] where a, b are non-negative and can be computed explicitly. Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Proposition (Optimal portfolio when BT = ξT ) Let W0 denote the initial wealth and let BT = ξT . Define the variable At as At = cFξ
−1 h T
(ξT )
jFξ
T
i (F (ξ )) , (ξT ) ξt t
where the functions ju (v ) and cu (v ) are defined as the first partial derivative for (u, v ) → J(u, v ) and (u, v ) → C (u, v ) respectively, and where J denotes the copula for the random pair (ξT , ξt ). Assume that E[ξT |At ] is decreasing in At . For s > 0, a solution to (P3 ) is given by XT? , X?T = a − bE[ξT |At ], where a = (W0 + bE [ξT E[ξT |At ]]) e rT , b =
Carole Bernard
s std(E[ξT |At ]) .
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Idea of the Proof I C a copula between 2 uniform U and V over [0, 1] ∂ I cu (v ) := ∂u C (u, v ) can be interpreted as a conditional probability: cu (v ) = P(V 6 v |U = u).
I cU (V ) is a uniform variable that depends on U and V and which is independent of U. I If U and T are independent uniform random variables then cU−1 (T ) is a uniform variable (depending on U and T ) that has copula C with U. I The following variable is a Uniform over [0, 1] with the right dependence with ξT for 0 < t < T At = cF Carole Bernard
−1 h ξT (ξT )
jF
ξT (ξT
i (F (ξ )) , t ξ t ) Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Idea of the Proof I The optimal XT , if it exists, can always be written as XT = f (U) for some f increasing in some standard uniform U having the right copula with BT . I At is a good candidate for U. I Choose a and b > 0 such that XT? = a − bE[ξT |At ] satisfies the constraints of Problem (P3 ) that is a and b verify var(XT? ) = s 2 and c(XT? ) = W0 . I XT? has the right copula with ξT (because of the monotonicity constraint). I corr(XT? , E[ξT |At ]) = −1 and XT? is thus the unique payoff that is perfectly negatively correlated with E[ξT |At ] and also satisfying all the constraints of Problem (P3 ).
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Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
I Consider next any other strategy XT which also verifies these constraints. We find that corr(XT , E[ξT |At ]) =
E[E[ξT |At ]XT ] − E[ξT ]E[XT ] p p var(E[ξT |At ]) var(XT ) > −1 = corr(XT? , E[ξT |At ]).
I Since XT satisfies the constraints of (P3 ), we have that var(XT ) = s 2 = var(XT? ) and E[ξT XT ] = E[E[ξT |At ]XT ] = W0 = E[ξT XT? ]. Therefore E[ξT ]E[XT ] < E[ξT ]E[XT? ], which shows that XT? maximizes the expectation.
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Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
I All portfolios with copula C with BT must now have a Sharpe Ratio bounded by e rT std[E[ξT |A]], 6 e rT std[ξT ] . I This is useful to develop improved fraud detection schemes.
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Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Example
(P3 )
max E [XT ] XT E [ξT XT ] = W0 var(XT ) = s 2 subject to C := Copula(XT , BT )
I BT = St∗ q I Copula C= Gaussian copula with correlation ρ > − 1 −
t T
Then, the solution to (P3 ) is
Here GT = (St? )α ST?
XT? = a − bGTc . q 1 with αα = ρ T t−t 1−ρ 2 − 1,
a = W0 e rT + be rT E[ξT GTc ], b = √ Carole Bernard
s ,c var(GTc )
= − (α+1)αt+T 2 t+(T −t) . Optimal Portfolio
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Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Illustration
? I Maximum Sharpe ratio SRρ,G for different values of the ? correlation ρ when the benchmark p is BT = St . We use the following parameters: t = 1/3, t/T = 0.577, p − 1 − t/T = −0.816, W0 = 100, r = 0.05, µ = 0.07, σ = 0.2, T = 1, S0 = 100. I Observe that the constrained case reduces to the unconstrained maximum p Sharpe ratio when the correlation in the Gaussian copula is ρ = t/T . The reason is that the copula between the unconstrained optimum and St? is the Gaussian copula with p correlation ρ = t/T . The constraint is thus redundant in that case.
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Optimal Portfolio
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Maximum Sharpe Ratio of Constrained Strategy
0.12 Constrained case Unconstrained case 0.1 0.08 0.06 0.04 0.02
1/2
ρ=(t/T)
1/2
ρ = − (1 − t/T) 0
−0.8
−0.6
−0.4
−0.2 0 0.2 0.4 Correlation coefficient ρ
0.6
0.8
1
Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Conclusions I Mean-variance efficient portfolios when there are no trading constraints I Mean-variance efficiency with a stochastic benchmark (linked to the market) as a reference portfolio (given correlation or copula with a stochastic benchmark). I Improved upper bounds on Sharpe ratios useful for example for fraud detection. For example it is shown that under some conditions it is not possible for investment funds to display negative correlation with the financial market and to have a positive Sharpe ratio. I Related problems can be solved: case of multiple benchmarks...
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
Related problems I Able to solve the partial hedging problem: min E (BT − XT )2 XT E [ξT BT ] = W0 subject to E [ξT XT ] = W (W 6 W0 ) I Able to deal with constrained “cost-efficiency” problems (extend Bernard, Boyle, Vanduffel (2011)) min E [ξT XT ] XT XT ∼ F subject to corr(XT , BT ) = ρ
,
I The maximum Expected Utility portfolio problem with one or more constraints on dependence can be solved. Carole Bernard
Optimal Portfolio
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Introduction
M-V Efficiency
Fraud Detection
Correlation
GOP
Fraud
Copula
Fraud
Conclusion
References I A¨ıt-Sahalia, Y., & Lo, A. 2001. Nonparametric Estimation of State-Price Densities implicit in Financial Asset Prices. Journal of Finance, 53(2), 499-547. I Bernard, C., & Boyle, P.P. 2009. Mr. Madoff’s Amazing Returns: An Analysis of the Split-Strike Conversion Strategy. Journal of Derivatives, 17(1), 62-76. I Bernard, C., Boyle P., Vanduffel S., 2011. “Explicit Representation of Cost-efficient Strategies”, available on SSRN. I Bernard, C., Jiang, X., Vanduffel, S., 2012. “Note on Improved Fr´ echet bounds and model-free pricing of multi-asset options”, Journal of Applied Probability. I Breeden, D., & Litzenberger, R. (1978). Prices of State Contingent Claims Implicit in Option Prices. Journal of Business, 51, 621-651. I Cox, J.C., Leland, H., 1982. “On Dynamic Investment Strategies,” Proceedings of the seminar on the Analysis of Security Prices,(published in 2000 in JEDC). I Dybvig, P., 1988a. “Distributional Analysis of Portfolio Choice,” Journal of Business. I Dybvig, P., 1988b. “Inefficient Dynamic Portfolio Strategies or How to Throw Away a Million Dollars in the Stock Market,” Review of Financial Studies. I Goldstein, D.G., Johnson, E.J., Sharpe, W.F., 2008. “Choosing Outcomes versus Choosing Products: Consumer-focused Retirement Investment Advice,” Journal of Consumer Research. I Goetzmann W., Ingersoll, J., Spiegel, M, & Welch, I. 2002. Sharpening Sharpe Ratios, NBER Working Paper No. 9116. I Markowitz, H. 1952. Portfolio selection. Journal of Finance, 7, 77-91. I Nelsen, R., 2006. “An Introduction to Copulas”, Second edition, Springer. I Pelsser, A., Vorst, T., 1996. “Transaction Costs and Efficiency of Portfolio Strategies,” European Journal of Operational Research. I Platen, E., & Heath, D. 2009. A Benchmark Approach to Quantitative Finance, Springer. I Sharpe, W. F. 1967. “Portfolio Analysis”. Journal of Financial and Quantitative Analysis, 2, 76-84. I Tankov, P., 2012. “Improved Fr´ echet bounds and model-free pricing of multi-asset options,” Journal of Applied Probability.
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