Robust Portfolio Optimization with Derivative Insurance Guarantees Steve Zymler Berç Rustem Daniel Kuhn Department of Computing Imperial College London
Mean-Variance Portfolio Optimization Optimal Asset Allocation Problem Choose the weights vector w R n to make the portfolio return high, whilst keeping the associated risk ρ(w) low. Mean-Variance Portfolio Optimization problem: maximize w T µ λw T Σw w R n subject to w W. Solves tradeoff between expected return w T µ and risk ρ(w) w T Σw, where λ is the risk aversion parameter.
Introduction to Robust Portfolio Optimization Let r denote the asset returns. Portfolio return is w T r. Ben-Tal and Nemirovski [1], Rustem and Howe [3], suggest investor wants to maximize portfolio return: Assume that r U, where max w W w T r U r = {r (r µ) T Σ 1 (r µ) δ 2 } Robust optimization takes worst-case approach: max min w W r U r w T r max w W w T µ δ Σ 1/2 w 2.
Probabilistic Guarantees Assume we know the means µ and covariance matrix Σ 0 of the returns r, but not the entire distribution. Let P be the set containing all the distributions that have mean µ and covariance matrix Σ. El Ghaoui et al. [2] have shown for any w W δ = p/(1 p) = Let φ = max w W min r Ur inf P{w T r min P P r U r w T r, then w T r} = p w T r φ r U r. The non-inferiority property of robust portfolios can be seen as a form of weak insurance.
Support Information and Coherency Assume we have support information about r: B = {r : l r u} (Always true: B = {r : r 0}) Can add support information to U r : U r = {r B (r µ) T Σ 1 (r µ) δ 2 } By strong conic duality: max min w T r max w W r U r w W,s 0 µt (w s) δ Σ 1/2 (w s). 2 Consider the function ρ: ρ(w) = min s 0 µt (w s) + δ Σ 1/2 (w s). 2 Can show that ρ is a coherent risk-measure. maximize worst-case return minimize coherent risk!
Uncertainty Set: Illustration
Parameter Uncertainty Have to estimate means µ and covariance matrix Σ considerable uncertainty. Portfolio optimization is very sensitive to errors in ˆµ error-maximization effect. When r are i.i.d. then ˆµ is approx. normally distributed: ˆµ N (µ, Λ), Λ = (1/M)Σ Can create uncertainty set for ˆµ U µ = {µ (µ ˆµ) T Λ 1 (µ ˆµ) κ 2, e T (µ ˆµ) = 0}. Obtain uncertainty set for r that takes into account U µ : U r = {r B µ U µ, (r µ) T Σ 1 (r µ) δ 2 }
Insurance through Options Options allow us to limit the worst-case portfolio return. V (S T ) = S T + max{k S T, 0} = max{s T, K } V (S T ) = S T + max{s T K, 0} = max{ S T, K } Only consider European options maturing at investment horizon.
Modelling Option Returns Let w d and r d denote option weights and returns resp. r uniquely determines r d, thus r d f ( r). Assume option j is a call with strike K j and premium C j on underlying stock i with initial price S0 i, then r d j is f j ( r) = max { 0, S 0 r i } i K j C j = max { } 0, a j + b j r i, with aj = K j and b j = Si 0. C j C j Likewise, when option j is a put with premium P j, then r d j f j ( r) = max { 0, a j + b j r i }, with aj = K j P j and b j = Si 0 P j. In compact notation: r d = f ( r) = max {0, a + B r}. is
Incorporating Options within Robust Framework Portfolio return is r p = w T r + (w d ) T r d. We will always set w d 0, and 1 T w + 1 T w d = 1. Robust max-min problem formulation: max (w,w d ) W min w T r + (w d ) T r d r U r, r d =f (r) Equivalent semi-infinite problem formulation: maximize w,w d,φ φ subject to w T r + (w d ) T r d φ r U r, r d = f (r) (w, w d ) W At optimality φ is the worst-case portfolio return when r U r.
Incorporating Options within Robust Framework Portfolio return is r p = w T r + (w d ) T r d. We will always set w d 0, and 1 T w + 1 T w d = 1. Robust max-min problem formulation: max (w,w d ) W min w T r + (w d ) T r d r U r, r d =f (r) Equivalent semi-infinite problem formulation: maximize w,w d,φ,y, s subject to φ µ T (w + B T y s) δ Σ 1/2 (w + B T y s) + a T y φ 2 (w, w d ) W, 0 y w d, s 0 At optimality φ is the worst-case portfolio return when r U r.
Worst-Case Return Behaviour 1 0.8 1 0.98 allocations 0.6 0.4 allocations 0.96 0.94 0.92 0.2 0.9 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 uncertainty p put options call options stocks 0.88 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 uncertainty p put options call options stocks conditional worst-case return (φ 1) 0.3 0.25 0.2 0.15 0.1 0.05 0-0.05-0.1-0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 uncertainty p conditional worst-case return (φ 1) 0.3 0.25 0.2 0.15 0.1 0.05 0-0.05-0.1-0.15 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 uncertainty p
Insured Robust Portfolio Optimization At optimality we obtain the non-inferiority guarantee: w T r + (w d ) T r d φ r U r, r d = f (r) Extreme events can cause r to be realised outside U r no more guarantees! Control deterioration of portfolio return below φ for any realisation of r: w T r + (w d ) T r d θφ r B, r d = f (r), where θ [0, 1]. Insurance guarantee expressed as fraction of φ: Only hedge against extreme scenarios not covered by non-inferiority guarantee. Prevents insurance from being overly expensive.
Guarantee Tradeoff The insured robust portfolio optimization model: maximize w,w d,φ φ subject to w T r + (w d ) T r d φ r U r, r d = f (r) w T r + (w d ) T r d θφ (w, w d ) W. Has a SOCP reformulation tractable. r B, r d = f (r) Model exposes tradeoff between non-inferiority and insurance guarantees: As U r increases, φ decreases. When φ decreases, so does insurance level θφ and associated insurance costs (premium).
Guarantee Tradeoff conditional worst-case return (φ 1) 0.3 0.2 0.1 0-0.1-0.2 0 0.2 0.4 insurance level θ 0.6 0.8 1 1 0.4 0.6 0.8 0 0.2 uncertainty p
Backtest Results We consider the following indices in the portfolio: Ticker XMI SPX MID SML RUT NDX Name AMEX Major Market Index S&P 500 Index S&P Midcap 400 Index S&P Smallcap 600 Index Russell 2000 Index NASDAQ 100 Index Adopt a monthly rebalancing strategy. Each month, we include all the options on the indices maturing in one month (data by Optionmetrics). Calculate out-of-sample returns between 19/06/1997 and 10/10/2008.
Backtest Results 5 4.5 4 3.5 MVO RPO IRPO Wealth 3 2.5 2 1.5 1 98 99 00 01 02 03 04 05 06 07 08 Year
A. Ben-Tal and A. Nemirovski. Robust solutions of uncertain linear programs. Operations Research Letters, 25(1):1 13, 1999. L. El Ghaoui, M. Oks, and F. Outstry. Worst-case value-at-risk and robust portfolio optimization: A conic programming approach. Operations Research, 51(4):543 556, 2003. B. Rustem and M. Howe. Algorithms for Worst-Case Design and Applications to Risk Management. Princeton University Press, Princeton, NJ, 2002.