Mean-Variance Optimal Portfolios in the Presence of a Benchmark with Applications to Fraud Detection

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) Fields Institute, September 2012. Carole Bernard Optimal Portfolio 1/33

Contributions Part 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 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 Carole Bernard Optimal Portfolio 2/33

Financial Market The market (Ω, Ϝ, P) is arbitrage-free. There is a risk-free account earning r > 0. Consider a strategy with payoff X T at time T > 0. There exists Q so that its initial price writes as c(x T ) = e rt E Q [X T ], Equivalently, there exists a stochastic discount factor ξ T such that c(x T ) = E P [ξ T X T ]. Assume ξ T is continuously distributed and var(ξ T ) <. Carole Bernard Optimal Portfolio 3/33

Mean Variance Optimization A Mean-Variance efficient problem: (P 1 ) max E [X T ] X T { E [ξt X subject to T ] = W 0 var[x T ] = s 2 Proposition (Mean-variance efficient portfolios) Let W 0 > 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 (P 1 ) writes as X T = a bξ T, where a = ( W 0 + be[ξ 2 T ]) e rt 0, b = s 0. var(ξt ) Carole Bernard Optimal Portfolio 4/33

Proof Choose a and b 0 such that X T = a bξ T satisfies the constraints var(x T ) = s2 and c(x T ) = W 0. Observe that corr(xt, ξ T ) = 1 and XT is thus the unique payoff that is perfectly negatively correlated with ξ T while satisfying the variance and cost constraints. Consider any other strategy X T which also verifies these constraints (but is not negatively linear in ξ T ). We find that corr(x T, ξ T ) = E[ξ T X T ] E[ξ T ]E[X T ] var(ξt ) var(x T ) > 1 = corr(x T, ξ T ). Since var(x T ) = s 2 = var(x T ) and E[ξ T X T ] = W 0 = E[ξ T X T ] it follows that E[ξ T ]E[X T ] < E[ξ T ]E[X T ], which shows that XT maximizes the expectation and thus solves Problem (P 1 ). Carole Bernard Optimal Portfolio 5/33

Proof Choose a and b 0 such that X T = a bξ T satisfies the constraints var(x T ) = s2 and c(x T ) = W 0. Observe that corr(xt, ξ T ) = 1 and XT is thus the unique payoff that is perfectly negatively correlated with ξ T while satisfying the variance and cost constraints. Consider any other strategy X T which also verifies these constraints (but is not negatively linear in ξ T ). We find that corr(x T, ξ T ) = E[ξ T X T ] E[ξ T ]E[X T ] var(ξt ) var(x T ) > 1 = corr(x T, ξ T ). Since var(x T ) = s 2 = var(x T ) and E[ξ T X T ] = W 0 = E[ξ T X T ] it follows that E[ξ T ]E[X T ] < E[ξ T ]E[X T ], which shows that XT maximizes the expectation and thus solves Problem (P 1 ). Carole Bernard Optimal Portfolio 5/33

Maximum Sharpe Ratio The Sharpe Ratio (SR) of a payoff X T (terminal wealth at T when investing W 0 at t = 0) is defined as SR(X T ) = E[X T ] W 0 e rt std(x T ), All mean-variance efficient portfolios XT maximal Sharpe Ratio (SR ) given by have the same For all portfolios X T we have SR := SR(X T ) = ert std(ξ T ), SR(X T ) e rt std(ξ T ), This can be used to show Madoff s investment strategy was a fraud (Bernard & Boyle (2007)). Carole Bernard Optimal Portfolio 6/33

Example in a Black-Scholes market There is a risk-free rate r > 0 and a risky asset with price process, ds t S t = µdt + σdw t, where W t is a standard Brownian motion, µ is the drift and σ is the volatility. The state-price density ξ T is given as ξ T = e rt e θw T 1 2 θ2t = αs β T, for known coefficients α, β > 0 (assume µ > r and θ = µ r σ ). The maximal Sharpe ratio is given by SR = e θ2t 1. see Goetzmann et al. (2007) for another proof. Carole Bernard Optimal Portfolio 7/33

General Market Non-parametric estimation of the upper bound e rt std(ξ T ) Assume ξ T = f (S T ) (where f is typically decreasing and S T is the risky asset) and that all European call options on the underlying S T maturing at T > 0 are traded. Let C(K) denote the price of a call option on S T with strike K. Then, the Sharpe ratio SR(X T ) of any admissible strategy with payoff X T satisfies SR(X T ) e 2rT + use for instance Aït-Sahalia and Lo (2001). 0 f (K) 2 C(K) K 2 dk 1. Carole Bernard Optimal Portfolio 8/33

Improving Fraud Detection by Adding Constraints Detect fraud based on mean and variance only Ignored so far additional information available in the market. How to take into account the dependence features between the investment strategy and the financial market? 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! Carole Bernard Optimal Portfolio 9/33

Improving Investment by Adding Constraints Optimal strategies X T = a bξ T give their lowest outcomes when ξ T is high. Bounded gains but unlimited losses! Highest state-prices ξ T (ω) correspond to states ω of bad economic conditions as these are more expensive to insure: E.g. in a Black-Scholes market: ξ T = αs β T, α, β > 0. Also, E[XT ξ T > c] < E[Y T ξ T > c], for any other strategy Y T with the same distribution as XT showing that X T does not provide protection against crisis situations (event ξ T > c ). in a Black-Scholes market: X T = when S T = 0. To cope with this observation: we impose the strategy to have some desired dependence with ξ T, or more generally with a benchmark B T. Carole Bernard Optimal Portfolio 10/33

Proposition (Optimal portfolio with a correlation constraint) Let B T be a benchmark, linearly independent from ξ T with 0 < var(b T ) < +. Let ρ < 1 and s > 0. A solution to the following mean-variance optimization problem (P 2 ) max var(x T ) = s 2 c(x T ) = W 0, corr(x T, B T ) = ρ E[X T ] (1) is given by X T = a b(ξ T cb T ), where a, b and c are uniquely determined by the set of equations ρ = corr(cb T ξ T, B T ) s = b var(ξ T cb T ) W 0 = ae rt b(e[ξ 2 T ] ce[ξ T B T ]). Carole Bernard Optimal Portfolio 11/33

Proof Observe that f (c) := corr(cb T ξ T, B T ) verifies f (c) = 1, lim c lim f (c) = 1 and f (c) > 0 so that ρ = f (c) has a c + unique solution. Take X T = a b(ξ T cb T ) linear in ξ T cb T and satisfying all constraints and b > 0. Consider any other X T that satisfies the constraints and which is non-linear in ξ T cb T, then corr(x T, ξ T cb T ) = E[X T (ξ T cb T )] E[ξ T cb T ]E[X T ] std(ξ T cb T )std(x T ) > 1 = corr(x T, ξ T cb T ) Since both X T and XT satisfy the constraints we have that std(x T ) = std(xt ), E[X T ξ T ] = E[XT ξ T ] and cov(x T, B T ) =cov(xt, B T ). Hence the inequality holds true if and only if E[XT ] > E[X T ]. Carole Bernard Optimal Portfolio 12/33

Proof Observe that f (c) := corr(cb T ξ T, B T ) verifies f (c) = 1, lim c lim f (c) = 1 and f (c) > 0 so that ρ = f (c) has a c + unique solution. Take X T = a b(ξ T cb T ) linear in ξ T cb T and satisfying all constraints and b > 0. Consider any other X T that satisfies the constraints and which is non-linear in ξ T cb T, then corr(x T, ξ T cb T ) = E[X T (ξ T cb T )] E[ξ T cb T ]E[X T ] std(ξ T cb T )std(x T ) > 1 = corr(x T, ξ T cb T ) Since both X T and XT satisfy the constraints we have that std(x T ) = std(xt ), E[X T ξ T ] = E[XT ξ T ] and cov(x T, B T ) =cov(xt, B T ). Hence the inequality holds true if and only if E[XT ] > E[X T ]. Carole Bernard Optimal Portfolio 12/33

ST : Growth Optimal Portfolio (GOP) The Growth Optimal Portfolio (GOP) maximizes expected logarithmic utility from terminal wealth. It has the property that it almost surely accumulates more wealth than any other strictly positive portfolios after a sufficiently long time. Under general assumptions on the market, the GOP is a diversified portfolio (proxy: a world stock index). The GOP can be used as numéraire to price under P, so that ξ T = 1 ST [ ] XT c(x T ) = E P [ξ T X T ] = E P ST where S 0 = 1. Details in Platen & Heath (2006). Carole Bernard Optimal Portfolio 13/33

Example when B T = S T /2 The optimal solution is of the form XT = a b(ξ T cst /2 ), where c is computed from the equation ρ = corr(cst /2 ξ T, ST s /2), b is derived from b = ) and a = W 0 e rt + b (e 2rT +θ2t ce[ξ T ST /2 ] e rt. var(ξt cs T /2 ) Optimal payoffs as a function of the GOP for a given correlation level ρ = 0.5 with the benchmark ST /2 using the following parameters: W 0 = 100, r = 0.05, µ = 0.07, σ = 0.2, T = 1, S 0 = 100, s = 10. Carole Bernard Optimal Portfolio 14/33

Mean Variance Optimum 140 130 120 110 100 90 80 70 ρ = 0.5, B T =S t *, t=t/2 no constraint 60 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 * Growth Optimal Portfolio S T

Example when B T = S T An optimal solution is of the form XT = a b(ξ T cst ), where c is computed from the equation ρ = corr(cst ξ T, ST ), b is s derived from b = var(ξt cs ( and T ) ) a = W 0 e rt + b e 2rT +θ2t 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: W 0 = 100, r = 0.05, µ = 0.07, σ = 0.2, T = 1, θ = (µ r)/σ, S 0 = 100, s = 10. Carole Bernard Optimal Portfolio 16/33

120 100 Mean Variance Optimum 80 60 40 20 0 no constraint ρ = 0.75 ρ = 0.3 ρ = 0.5 ρ = 0.9 20 0.7 0.8 0.9 1 1.1 1.2 1.3 * Growth Optimal Portfolio S T

Fraud Detection Proposition (Constrained Maximal Sharpe Ratio) All mean-variance efficient portfolios XT which satisfy the additional constraint corr(xt, B T ) = ρ with a benchmark asset B T (that is not linearly dependent to ξ T ) have the same maximal Sharpe ratio SRρ given by SR ρ = e rt cov(ξ T, ξ T cb T ) std(ξ T cb T ) SR = e rt std(ξ T ). (2) where SR is the unconstrained Sharpe ratio. Carole Bernard Optimal Portfolio 18/33

Illustration in the Black-Scholes model Maximum Sharpe ratio SRρ for different values of the correlation ρ when the benchmark is B T = ST. We use the following parameters: W 0 = 100, r = 0.05, µ = 0.07, σ = 0.2, T = 1, S 0 = 100. Carole Bernard Optimal Portfolio 19/33

Maximum Sharpe Ratio 0.15 0.1 0.05 0.02 0.004 0 0.05 0.1 Constrained case Unconstrained case +0.1 0.1 1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Correlation coefficient ρ

M-V Optimization with a Benchmark Dependence (interaction) between X T and B T cannot be fully reflected by correlation. A useful device to do so is the copula. Sklar s theorem shows that the joint distribution of (B T, X T ) can be decomposed as P(B T y, X T x) = C(F BT (y), F XT (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 X T and the benchmark B T. Constrained Mean-Variance efficient problem: (P 3 ) max E [X T ] X T subject to E [ξ T X T ] = W 0 var(x T ) = s 2 C := Copula(X T, B T ) Carole Bernard Optimal Portfolio 21/33

Proposition (Optimal portfolio when B T = ξ T ) Let W 0 denote the initial wealth and let B T = ξ T. Define the variable A t as A t = ( ) 1 [ ] c FξT (ξ T ) j FξT (ξ T )(F ξt (ξ t )), where the functions j u (v) and c u (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 (ξ T, ξ t ). Assume that E[ξ T A t ] is decreasing in A t. For s > 0, a solution to (P 3 ) is given by X T, where a = (W 0 + be [ξ T E[ξ T A t ]]) e rt, b = X T = a be[ξ T A t ], (3) s std(e[ξ T A t]). Carole Bernard Optimal Portfolio 22/33

Idea of the Proof C a copula between 2 uniform U and V over [0, 1] c u (v) := u C(u, v) can be interpreted as a conditional probability: c u (v) = P(V v U = u). (4) c U (V ) is a uniform variable that depends on U and V and which is independent of U. If U and T are independent uniform random variables then c 1 U (T ) is a uniform variable (depending on U and T ) that has copula C with U. The following variable is a Uniform over [0, 1] with the right dependence with ξ T for 0 < t < T A t = ( ) 1 [ ] c FξT j (F (ξ T ) FξT (ξ T ) ξ t (ξ t )), Carole Bernard Optimal Portfolio 23/33

Idea of the Proof The optimal X T, if it exists, can always be written as X T = f (U) for some f increasing in some standard uniform U having the right copula with B T. A t is a good candidate for U. Choose a and b 0 such that XT = a be[ξ T A t ] satisfies the constraints of Problem (P 3 ) that is a and b verify var(xt ) = s2 and c(xt ) = W 0. XT has the right copula with ξ T (because of the monotonicity constraint). corr(xt, E[ξ T A t ]) = 1 and XT is thus the unique payoff that is perfectly negatively correlated with E[ξ T A t ] and also satisfying all the constraints of Problem (P 3 ). Carole Bernard Optimal Portfolio 24/33

Consider next any other strategy X T which also verifies these constraints. We find that corr(x T, E[ξ T A t ]) = E[E[ξ T A t ]X T ] E[ξ T ]E[X T ] var(e[ξt A t ]) var(x T ) > 1 = corr(x T, E[ξ T A t ]). Since X T satisfies the constraints of (P 3 ), we have that var(x T ) = s 2 = var(xt ) and E[ξ T X T ] = E[E[ξ T A t ]X T ] = W 0 = E[ξ T XT ]. Therefore E[ξ T ]E[X T ] < E[ξ T ]E[X T ], which shows that X T maximizes the expectation. Carole Bernard Optimal Portfolio 25/33

Proposition (Constrained Mean-Variance Efficiency) Let s > 0. Assume that the benchmark B T has a joint density with ( ) 1 [ ] ξ T. Define A as A = c FBT j (1 F (B T ) FBT (B T ) ξ T (ξ T )), where the functions j u (v) and c u (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 (B T, ξ T ). If E[ξ T A] is decreasing in A, then the solution to the problem max var(x T ) = s 2 c(x T ) = W 0 C : copula between X T and B T E[X T ] (5) is uniquely given as X T = a be[ξ T A] where a, b are non-negative and can be computed explicitly. Carole Bernard Optimal Portfolio 26/33

In the paper we apply this to Black-Scholes markets. All portfolios with copula C with B T must now have a Sharpe Ratio bounded by e rt std[e[ξ T A]], ( ) e rt std[ξ T ]. In the paper we use these results to develop improved fraud detection schemes. Carole Bernard Optimal Portfolio 27/33

Proposition (Case B T = S t ) Let W 0 denote the initial wealth and let B T = St (0 < t < T ) be the benchmark. Assume that ρ 1 t T. Then, the solution to (P 3 ) when the copula C is the Gaussian copula with correlation ρ, Cρ Gauss is given by XT, XT = a bg T c. (6) Here G T is a weighted average of the benchmark and the GOP. It is given as G T = (St ) α ST with α, α = ρ T t t Furthermore a = W 0 e rt + be rt E[ξ T G c T ], b = 1 1 ρ 2 1. s, c = αt+t var(g c T ) (α+1) 2 t+(t t). Carole Bernard Optimal Portfolio 28/33

Illustration Maximum Sharpe ratio SRρ,G for different values of the correlation ρ when the benchmark is B T = St. We use the following parameters: t = 1/3, t/t = 0.577, 1 t/t = 0.816, W 0 = 100, r = 0.05, µ = 0.07, σ = 0.2, T = 1, S 0 = 100. Observe that the constrained case reduces to the unconstrained maximum 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 correlation ρ = t/t. The constraint is thus redundant in that case. Carole Bernard Optimal Portfolio 29/33

Maximum Sharpe Ratio of Constrained Strategy 0.12 0.1 0.08 0.06 0.04 0.02 Constrained case Unconstrained case ρ = (1 t/t) 1/2 ρ=(t/t) 1/2 0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Correlation coefficient ρ

Conclusions Mean-variance efficient portfolios when there are no trading constraints Mean-variance efficiency with a stochastic benchmark (linked to the market) as a reference portfolio (given correlation or copula with a stochastic benchmark). 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. Related problems can be solved: case of multiple benchmarks... Carole Bernard Optimal Portfolio 31/33

Related problems Able to solve the partial hedging problem: min E [ (B T X T ) 2] X T { E [ξt B subject to T ] = W 0 E [ξ T X T ] = W (W W 0 ) Able to deal with constrained cost-efficiency problems (extend Bernard, Boyle, Vanduffel (2011)) min E [ξ T X T ] X T { XT F subject to corr(x T, B T ) = ρ Multiple constraints can be dealt with., Carole Bernard Optimal Portfolio 32/33

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