Week 7.1 Lagranges Method

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Information about Week 7.1 Lagranges Method

Published on March 6, 2014

Author: anhtuantran509

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AMATH 460: Mathematical Methods for Quantitative Finance 7.1 Lagrange’s Method Kjell Konis Acting Assistant Professor, Applied Mathematics University of Washington Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 1 / 29

Outline 1 Optimal Investment Portfolios 2 Relative Extrema of Functions of Several Variables 3 Lagrange’s Method 4 Example 5 Minimum Variance Portfolio Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 2 / 29

Outline 1 Optimal Investment Portfolios 2 Relative Extrema of Functions of Several Variables 3 Lagrange’s Method 4 Example 5 Minimum Variance Portfolio Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 3 / 29

Investment Portfolios Portfolio of n assets Let wi be the proportion of the portfolio invested in asset i Have constraint n wi = 1 i=1 Can take long and short positions =⇒ no constraints on individual wi Let µi be the expected rate of return on asset i Let σi2 be the risk of asset i Let ρij be the correlation between assets i and j Expected rate of return and risk of the portfolio: n Expected Return = w i µi i=1 n Risk = wi2 σi2 + 2 i=1 Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method wi wj σi σj ρij 1≤i<j≤n 4 / 29

Investment Portfolios: Matrix Notation Let w = (w1 , . . . , wn ) and µ = (µ1 , . . . , µn ) The expected rate of return can be written in matrix notation as n wi µi = w T µ Return = i=1 The risk can be written as Risk = w T Σw Σ is the covariance matrix of the n assets  2 σ1     σ2 σ1 ρ21  Σ= .  . .   σ1 σ2 ρ12 · · · σ1 σn ρ1n  2 σ2 . . . ··· .. . σn σ1 ρn1 σn σ2 ρn2 · · · Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method σ2 σn . . . 2 σn        5 / 29

Optimal Investment Portfolios Given µ, Σ, and investor selected w , can compute portfolio return portfolio risk Two notions of optimality For a target expected return, choose w to minimize portfolio risk For a target level of risk, choose w to maximize expected return Both notions are constrained optimization problems that can be solved using Lagrange multipliers Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 6 / 29

Optimal Investment Portfolios Minimum variance optimization n asset case minimize: subject to: w T Σw eTw = 1 µT w = µ P 2 asset case minimize: subject to: 2 2 σ1 w1 2 2 + 2ρσ1 σ2 w1 w2 + σ2 w2 w1 + w2 = 1 µ1 w1 + µ2 w2 = µP Maximum expected return optimization n asset case maximize: subject to: 2 asset case maximize: µT w subject to: eTw = 1 2 w T Σw = σP Kjell Konis (Copyright © 2013) µ1 w1 + µ2 w2 w1 + w2 = 1 2 2 2 2 2 σ1 w1 + 2ρσ1 σ2 w1 w2 + σ2 w2 = σP 7.1 Lagrange’s Method 7 / 29

Outline 1 Optimal Investment Portfolios 2 Relative Extrema of Functions of Several Variables 3 Lagrange’s Method 4 Example 5 Minimum Variance Portfolio Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 8 / 29

Relative Extrema of Single Variable Functions A local minimum (maximum) of a function f is a point x0 where f (x0 ) ≤ (≥) f (x ) for some ∀x ∈ (x0 − , x0 + ) >0 A local extrema is a point that is a local minimum or maximum If f is twice differentiable and f is continuous Any local extremum is a critical point of f : f (x0 ) = 0 Can classify critical points using second derivative test f (x0 ) < 0 f (x0 ) > 0 f (x0 ) = 0 Kjell Konis (Copyright © 2013) local maximum local minimum anything possible 7.1 Lagrange’s Method 9 / 29

Relative Extrema of Functions of n Variables A local minimum (maximum) of a function f : Rn → R is a point x0 ∈ Rn where f (x0 ) ≤ (≥) f (x ) ∀x : x − x0 < Every local extremum is a critical point: Df (x0 ) = 0 If f is twice differentiable and has continuous second order partial derivatives D 2 f (x0 ) is a symmetric matrix with real eigenvalues Second order conditions All eigenvalues of D 2 f (x0 ) > 0 All eigenvalues of D 2 f (x0 ) < 0 D 2 f (x0 ) has ± eigenvalues D 2 f (x0 ) singular Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method local minimum local maximum saddle point anything can happen 10 / 29

Finding Extrema: Functions of 2 Variables Find the local extrema of f (x , y ) = x 2 + xy + y 2 Df (x , y ) = 2x + y x + 2y Df (0, 0) = 0 0 ⇒ (0, 0) is a critical point D 2 f (x , y ) = 2 1 1 2 Can use R to compute the eigenvalues > A <- matrix(c(2, 1, 1, 2), 2, 2) > eigen(A)$values [1] 3 1 Since both eigenvalues are greater than 0 =⇒ (0, 0) a local minimum Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 11 / 29

Finding Extrema: Functions of 2 Variables (Take 2) Find the local extrema of f (x , y ) = −x 2 − xy − y 2 Df (x , y ) = − 2x − y − x − 2y Df (0, 0) = 0 0 ⇒ (0, 0) is a critical point D 2 f (x , y ) = −2 −1 −1 −2 Can use R to compute the eigenvalues > A <- matrix(-c(2, 1, 1, 2), 2, 2) > eigen(A)$values [1] -1 -3 Since both eigenvalues are less than 0 =⇒ (0, 0) a local maximum Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 12 / 29

Finding Extrema: Functions of 2 Variables Find the local extrema of f (x , y ) = x 2 + 3xy + y 2 Df (x , y ) = 2x + 3y 3x + 2y Df (0, 0) = 0 0 ⇒ (0, 0) is a critical point D 2 f (x , y ) = 2 3 3 2 Can use R to compute the eigenvalues > A <- matrix(c(2, 3, 3, 2), 2, 2) > eigen(A)$values [1] 5 -1 One positive and one negative eigenvalue =⇒ (0, 0) a saddle point Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 13 / 29

Finding Extrema: Functions of 2 Variables 3 Find the local extrema of f (x , y ) = 2xy − (1 − y 2 ) 2 First order condition Df (x , y ) = 2y Df (0, 0) = 0 0 2x + 3y 1 − y2 =⇒ (0, 0) is a critical point Second order condition   0 D 2 f (x , y ) =  2  2 3−6y √ 2   1−y 2  D 2 f (0, 0) =  0 2 2 3   Compute the eigenvalues of the Hessian at the critical point > eigen(matrix(c(0, 3, 3, 2), 2, 2))$values [1] 4.162278 -2.162278 One positive and one negative eigenvalue =⇒ (0, 0) a saddle point Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 14 / 29

Outline 1 Optimal Investment Portfolios 2 Relative Extrema of Functions of Several Variables 3 Lagrange’s Method 4 Example 5 Minimum Variance Portfolio Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 15 / 29

Lagrange’s Method Problem: maximize: subject to: f (x1 , x2 , . . . , xn ) g1 (x1 , x2 , . . . , xn ) = 0 g2 (x1 , x2 , . . . , xn ) = 0 . . . (1) gm (x1 , x2 , . . . , xn ) = 0 18th -century mathematician Joseph Louis Lagrange proposed the following method for the solution Form the function m F (x1 , . . . , xn , λ1 , . . . , λm ) = f (x1 , . . . , xn ) + λi gi (x1 , x2 , . . . , xn ) i=1 Optimal value for problem (1) occurs at one of the critical points of F Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 16 / 29

Lagrange’s Method Terminology: The function F (x1 , . . . , xn , λ1 , . . . , λm ) is called the Lagrangian The column vector λ = (λ1 , . . . , λm ) is called the Lagrange multipliers vector Necessary Condition: Let x = (x1 , x2 , . . . , xn ) Let g(x ) = g1 (x ), g2 (x ), . . . , gm (x ) be a vector-valued function of the constraints The gradient D g(x ) must have full rank at any point where the constraint g(x ) = 0 is satisfied, that is rank Dg(x ) = m Kjell Konis (Copyright © 2013) ∀ x where g(x ) = 0 7.1 Lagrange’s Method 17 / 29

Partial Derivatives of the Lagrangian D F (x , λ) has n + m variables, compute gradient in 2 parts D F (x , λ) = Dx F (x , λ) Dλ F (x , λ) Recall Lagrangian: m F (x , λ) = f (x1 , . . . , xn ) + λi gi (x1 , x2 , . . . , xn ) i=1 The partial derivatives are m ∂F ∂f ∂gi = + λi ∂xj ∂xj i=1 ∂xj Gradient of f : Df (x ) = Kjell Konis (Copyright © 2013) ∂F = gi (x ) ∂λi ∂f ∂f ... ∂x1 ∂xn 7.1 Lagrange’s Method 18 / 29

Partial Derivatives of the Lagrangian Gradient of g(x ):  ∂g1 ∂x1 ∂g2 ∂x1 ∂g1 ∂x2 ∂g2 ∂x2 ··· ∂gm ∂x1 ∂gm ∂x2 ···     Dg(x ) =  .  .  .  . . . ∂g1 ∂xn ∂g2 ∂xn      .  .  .   ··· .. . ∂gm ∂xn Can express sum in second term in matrix notation m λi i=1 ∂gi = λT Dg(x ) ∂xj j It follows that D F (x , λ) = Df (x ) + λT Dg(x ) Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method g(x ) T 19 / 29

Outline 1 Optimal Investment Portfolios 2 Relative Extrema of Functions of Several Variables 3 Lagrange’s Method 4 Example 5 Minimum Variance Portfolio Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 20 / 29

Example Want to max/min: subject to: 4x2 − 2x3 2x1 − x2 − x3 = 0 2 2 x1 + x2 − 13 = 0 Start by writing down the Lagrangian F (x , λ) = f (x ) + λ1 g1 (x ) + λ2 g2 (x ) 2 2 = 4x2 − 2x3 + λ1 (2x1 − x2 − x3 ) + λ2 (x1 + x2 − 13) Check necessary condition: Dg(x ) = Kjell Konis (Copyright © 2013) 2 −1 −1 2x1 2x2 0 7.1 Lagrange’s Method 21 / 29

Derivatives of the Lagrangian The Lagrangian 2 2 F (x , λ) = 4x2 − 2x3 + λ1 (2x1 − x2 − x3 ) + λ2 (x1 + x2 − 13) Gradient of the Lagrangian T  2λ1 + 2λ2 x1  4 − λ + 2λ x  1 2 2    −2 − λ1  D F (x , λ) =     2x1 − x2 − x3  2 2 x1 + x2 − 13 get λ1 = −2 for free Set D F (x , λ) = 0 and solve for x and λ set 2λ1 + 2λ2 x1 = 0 set 4 − λ1 + 2λ2 x2 = 0 set 2x1 − x2 − x3 = 0 set 2 2 x1 + x2 − 13 = 0 Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 22 / 29

Example (continued) A little algebra gives x1 = 2 λ2 x2 = −3 λ2 x3 = 7 λ2 Also know that 2 2 x1 +x2 = 13 =⇒ 2 λ2 2 + −3 λ2 2 = 13 = 13 λ2 2 =⇒ λ2 = ±1 The critical points are λ = (−2, −1), x = (−2, 3, −7), f (x ) = 26 λ = (−2, 1), x = (2, −3, 7), f (x ) = −26 Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 23 / 29

Outline 1 Optimal Investment Portfolios 2 Relative Extrema of Functions of Several Variables 3 Lagrange’s Method 4 Example 5 Minimum Variance Portfolio Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 24 / 29

Minimum Variance Portfolio Recall: minimum variance portfolio optimization minimize: subject to: w T Σw eTw = 1 µT w = µ P Lagrange’s method setup f (w ) = w T Σw g(w ) = g1 (w ) µT w − µP = 0 = g2 (w ) eTw − 1 = 0 First, check necessary condition Dg(x ) = Kjell Konis (Copyright © 2013) µT eT 7.1 Lagrange’s Method 25 / 29

Derivative of a Quadratic Form Let A = a b b c 2 2 Let f (x ) = x T Ax = ax1 + 2bx1 x2 + cx2 Then Df (x ) = 2ax1 + 2bx2 2bx1 + 2cx2 = 2x T A In general, let A be an n × n symmetric matrix The derivative (gradient) of the quadratic form f (x ) = x T Ax is Df (x ) = 2x T A Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 26 / 29

Minimum Variance Portfolio The Lagrangian F (y , λ) = w T Σw + λ1 e T w − 1 + λ2 µT w − µP Gradient of the Lagrangian T Df (w ) + λT Dg(w ) = g(w ) 2w T Σ + λ1 e T + λ2 µT D F (w , λ) = eTw − 1 µT w − µ P Find the critical point by solving the linear system      2Σ e µ w 0  T     0 0   λ1  =  1  e µT 0 0 λ2 µP Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 27 / 29

Minimum Variance Portfolio Further reading: Second order conditions, e.g., Theorem 9.2 and Corollary 9.1 in PFME Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 28 / 29

http://computational-finance.uw.edu Kjell Konis (Copyright © 2013) 7.1 Lagrange’s Method 29 / 29

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