Pension fund ALM with Multivariate Second order Stochastic - PowerPoint PPT Presentation

Pension fund ALM with Multivariate Second order Stochastic Dominance constraints Sebastiano Vitali, Vittorio Moriggia, Miloš Kopa University of Bergamo Charles University Chemnitz, CMS2019

2 Purpose of the work To model and implement an Asset Liability Management problem of a Pension Fund in a Defined Benefit framework having: - a short-term profitability target, - a medium-term insurance risk-adjusted return - a long-term strategic objective Definition of multivariate second order stochastic dominance between the wealth of the Pension Fund and a benchmark wealth The multivariate Second order Stochastic Dominance (SSD) is formulated with three alternatives which we investigate

3 The multivariate SSD

4 Univariate SSD Let (Ω, 𝐺, 𝑄) denote a probability space and let 𝑌 and 𝑍 be two random variables having as cumulative distribution functions 𝐺 𝑌 and 𝑍 . 𝐺 Let’s define the twice cumulative distribution function as 𝜃 2 (𝜃) = න 𝐺 𝐺 𝑌 𝛽 d𝛽 𝑌 −∞ We say that 𝑌 dominates 𝑍 in the Second order Stochastic Dominance (SSD) sense, 𝑌 ≻ 𝑇𝑇𝐸 𝑍 , if 2 𝜃 ≤ 𝐺 2 𝜃 , 𝐺 ∀𝜃 ∈ 𝑆 𝑌 𝑍 If the random variables are discrete and then represented by random vectors X and Y , and if the realizations are equiprobable , then the SSD is equivalent to X ≤ WY where W is a double stochastic matrix.

5 Multivariate SSD When we consider multivariate random variables, we need to re-think the SSD relation. Assume that a multivariate random variable 𝒀 has 𝑈 dimensions and then we observe 𝑌 𝑢 , 𝑢 = 1, … , 𝑈 univariate random variables. If the random variable is discrete , each univariate random variable 𝑌 𝑢 can be represented with with a vector X 𝑢 , then 𝒀 can be represented with a matrix having 𝑈 columns, one for each dimension: … X 1 X 𝑈 𝑦 1,1 … 𝑦 1,𝑈 … … … = 𝑦 𝑇,1 … 𝑦 𝑇,𝑈 The meaning of 𝒀 ≻ 𝑇𝑇𝐸 𝒁 is not unique and can be declined in various ways. We analyze three of them.

6 Component-wise Multivariate SSD (C-MSSD) 𝑌 ≻ 𝑇𝑇𝐸 𝑍 iff 𝑌 𝑢 ≻ 𝑇𝑇𝐸 𝑢 𝑍 ∀𝑢 (disjointly) 𝑢 ≻ 𝑇𝑇𝐸 1 ≻ 𝑇𝑇𝐸 2 ≻ 𝑇𝑇𝐸 3

7 Linear Multivariate SSD (Lin-MSSD) 𝑈 𝑈 ത lin 𝑍 iff σ 𝑢=1 𝑌 = ෍ 𝑌 𝑢 ⋅ 𝑑 𝑢 , ∀𝑑 𝑢 ≥ 0, ෍ 𝑑 𝑢 = 1 𝑈 𝑈 𝑈 𝑌 ≻ 𝑇𝑇𝐸 𝑑 𝑢 𝑌 𝑢 ≻ 𝑇𝑇𝐸 σ 𝑢=1 𝑑 𝑢 𝑍 𝑢 , ∀𝑑 𝑢 ≥ 0| σ 𝑢=1 𝑑 𝑢 = 1 𝑢=1 𝑢=1 ≻ 𝑇𝑇𝐸 ∀𝑑 𝑢

8 MultiDimension Multivariate SSD (MD-MSSD) 𝑌 ≻ 𝑇𝑇𝐸 𝑍 iff 𝑌 𝑢 ≻ 𝑇𝑇𝐸 𝑍 ∀𝑢 (jointly) 𝑢 ≻ 𝑇𝑇𝐸

9 Multivariate SSD The three possible definitions: • The Component-wise Multivariate SSD (C-MSSD): 𝑌 ≻ 𝑇𝑇𝐸 𝑍 iff 𝑌 𝑢 ≻ 𝑇𝑇𝐸 𝑢 𝑍 ∀𝑢 (disjointly) 𝑢 X 𝑢 ≤ W 𝑢 Y 𝑢 , ∀𝑢 • The Linear Multivariate SSD (Lin-MSSD): Dencheva and Ruszczynski (2009), Dentcheva and Wolfhagen (2015, 2016) lin 𝑍 iff σ 𝑢=1 𝑈 𝑈 𝑈 𝑑 𝑢 𝑌 𝑢 ≻ 𝑇𝑇𝐸 σ 𝑢=1 𝑢 , ∀𝑑 𝑢 ≥ 0| σ 𝑢=1 𝑌 ≻ 𝑇𝑇𝐸 𝑑 𝑢 𝑍 𝑑 𝑢 = 1 𝐝 ⊤ X ≤ W (𝐝) 𝐝 ⊤ Y , 𝑈 ∀𝐝 ≥ 0, Σ 𝑢=1 𝑑 𝑢 = 1 • The MultiDimension Multivariate SSD (MD-MSSD): 𝑌 ≻ 𝑇𝑇𝐸 𝑍 iff 𝑌 𝑢 ≻ 𝑇𝑇𝐸 𝑍 ∀𝑢 (jointly) 𝑢 X 𝑢 ≤ WY 𝑢 , a ∀𝑢

10 Multivariate SSD The three possible definitions: • The Component-wise Multivariate SSD (C-MSSD): C-MSSD 𝑌 ≻ 𝑇𝑇𝐸 𝑍 iff 𝑌 𝑢 ≻ 𝑇𝑇𝐸 𝑢 𝑍 ∀𝑢 (disjointly) 𝑢 X 𝑢 ≤ W 𝑢 Y 𝑢 , ∀𝑢 • The Linear Multivariate SSD (Lin-MSSD): Lin-MSSD Dencheva and Ruszczynski (2009), Dentcheva and Wolfhagen (2015, 2016) lin 𝑍 iff σ 𝑢=1 𝑈 𝑈 𝑈 𝑑 𝑢 𝑌 𝑢 ≻ 𝑇𝑇𝐸 σ 𝑢=1 𝑢 , ∀𝑑 𝑢 ≥ 0| σ 𝑢=1 𝑌 ≻ 𝑇𝑇𝐸 𝑑 𝑢 𝑍 𝑑 𝑢 = 1 𝐝 ⊤ X ≤ W (𝐝) 𝐝 ⊤ Y , 𝑈 ∀𝐝 ≥ 0, Σ 𝑢=1 𝑑 𝑢 = 1 • The MultiDimension Multivariate SSD (MD-MSSD): MD-MSSD 𝑌 ≻ 𝑇𝑇𝐸 𝑍 iff 𝑌 𝑢 ≻ 𝑇𝑇𝐸 𝑍 ∀𝑢 (jointly) 𝑢 X 𝑢 ≤ WY 𝑢 , a ∀𝑢

11 The ALM model

12 Approach structure • Portfolio Universe Financial Datafeed • Risk factors • Econometric model definition • Econometric model estimation Simulation • Population model setting input • Stochastic tree structure • Nodal financial coefficient generation Monte Carlo • Monte Carlo scenario generation simulator • Population actuarial simulation • Dynamic portfolio model Stochastic Programming • Stochastic program solution Solution analysis

13 Extended Asset Universe Asset Class Asset List Lower & Upper Cash 30% Cash 0% Floaters Treasury 1-3y Treasury 3-5y Treasuries 0% 100% Treasury 5-7y Treasury 7-10y Treasury 10+y Securitized 0% 100% Securitized Corporate Inv Grade Corporate 0% 100% Corporate High Yield 0% 50% Public Equity Public Equity Real Estate 0% 20% Real Estate

14 Notation for sets Time partition from time 0 to year 20 T = {𝑢 0 = 0,1,2, … , 𝐼} Set of decision times T 𝑒 = {𝑢 0 = 0,1,2,3, 5, 10, 𝐼 } Set of intermediate stages T 𝑗𝑜𝑢 = T\{T d } = {4,6,7,8, 9, 11, … , 19 ൟ Set of scenarios } 𝑡 = {1,2, … , 𝑇 Financial assets 𝑗 ∈ {1,2, … , 14} Bond asset classes 𝐽 1 : 𝑗 = 1,2, 3, 4, 5 , 𝐽 2 : 𝑗 = {6,7,8} Public Equity, Real Estate and Derivatives 𝐽 3 : 𝑗 = 9 , 𝐽 4 : 𝑗 = {10} , 𝐽 5 : 𝑗 = {11,12,13,14} Asset total set 𝐽 = ራ 𝐽 𝑙 𝑙=1,…,5

15 Notation for investment variables Buying decision in stage t , scenario s , of asset i + 𝑦 𝑗,𝑢,𝑡 Selling in stage t , scenario s , of asset i that was − 𝑦 𝑗,ℎ,𝑢,𝑡 bought in h Expiry of a fixed-income asset in stage t , 𝑓𝑦𝑞 𝑦 𝑗,ℎ,𝑢,𝑡 scenario s , of asset i that was bought in h Holding in stage t , scenario s , of asset i that 𝑦 𝑗,ℎ,𝑢,𝑡 was bought in h + − 𝑨 𝑢,𝑡 − Cash account in stage t , scenario s 𝑨 𝑢,𝑡 = 𝑨 𝑢,𝑡 𝑙 Sponsors ’ unexpected contributions Φ 𝑢,𝑡

16 Variable definitions 𝑂𝐹𝑈 𝑗𝑜𝑞𝑣𝑢 Net pension payments 𝑀 𝑢,𝑡 Defined benefit obligation (DBO) Λ 𝑢,𝑡 𝑗𝑜𝑞𝑣𝑢 𝑌 𝑗,𝑢 𝑘 ,𝑡 = ෍ 𝑦 𝑗,ℎ,𝑢 𝑘 ,𝑡 Asset value 𝑌 𝑗,𝑢,𝑡 ℎ<𝑢 𝑘 Asset portfolio value 𝐷𝑌 𝑢 𝑘 ,𝑡 + 𝐷𝑌 𝑢 𝑘 ,𝑡 = ෍ 𝑌 𝑗,𝑢 𝑘 ,𝑡 + 𝑨 𝑢 𝑘 ,𝑡 𝑗∈𝐽 B 𝑢 𝑘 ,𝑡 B 𝑢 𝑘 ,𝑡 = Λ 𝑢 𝑘 ,𝑡 − 𝐷𝑌 𝑢 𝑘 ,𝑡 Net Defined benefit obligation 𝑎 𝑀 𝑢 𝑘 ,𝑡 = ෍ 𝑦 𝑗,ℎ,𝑢 𝑘 −1 ∙ 𝜊 𝑗,𝑢 𝑘 ,𝑡 + ℎ<𝑢 𝑘 ,ℎ𝜗𝑈 𝑎 Intermediate net payments 𝑀 𝑢 𝑘 ,𝑡 𝑓𝑦𝑞 𝑂𝐹𝑈 ෍ 𝑦 𝑗,ℎ,𝑢 𝑘 ,𝑡 −𝑀 𝑢 𝑘 ,𝑡 ℎ<𝑢 𝑘 ,𝑢 𝑘 −ℎ≥𝑈 𝑗

17 Variable definitions 1 Liquidity gap plus ALM risk 𝛺 𝑢 𝑘 ,𝑡 𝛺 𝑢 𝑘 ,𝑡 = 𝛻 𝑢 𝑘 ,𝑡 + 𝐿 𝑢 𝑘 ,𝑡 + 𝛺 𝑢 𝑘−1 ,𝑡 , Ψ 𝑢 0,𝑡 = 0 𝑂𝐹𝑈 − 𝑨 𝛻 𝑢 𝑘 ,𝑡 = L 𝑢 𝑘 ,𝑡 ෍ 𝑀 ℎ,𝑡 1 + 𝜂 𝑢,𝑡 𝛻 𝑢 𝑘 ,𝑡 𝑢 𝑘−1<ℎ<𝑢𝑘 Liquidity gap 1,𝐽𝑂𝑊 − 𝑓𝑦𝑞 −Π 𝑢 𝑘 ,𝑡 ෍ 𝑦 𝑗,ℎ,𝑢 𝑘 ,𝑡 ℎ<𝑢 𝑘 ,𝑢 𝑘 −ℎ=𝑈 𝑗 1 K 𝑢 𝑘 ,𝑡 + = 𝑒𝑠 + ∙ t j − t j−1 ⋅ Δ 𝑦 𝑢 𝑘 ,𝑡 − Δ Λ 𝑢 𝑘 ,𝑡 1 K 𝑢 𝑘 ,𝑡 ALM risk − − 𝑒𝑠 − ∙ t j − t j−1 ⋅ Δ 𝑦𝑢 𝑘 ,𝑡 − Δ Λ𝑢 𝑘 ,𝑡

18 Variable definitions 1,𝐽𝑂𝑊 + 𝐻 𝑢 𝑘 ,𝑡 𝐽𝑂𝑊 = Π 𝑢 𝑘 ,𝑡 𝐽𝑂𝑊 Realized portfolio return Π 𝑢 𝑘 ,𝑡 Π 𝑢 𝑘 ,𝑡 1,𝐽𝑂𝑊 Coupon return … Π 𝑢 𝑘 ,𝑡 … Capital gain return 𝐻 𝑢 𝑘 ,𝑡 𝐽𝑂𝑊 + 𝑉𝐻𝑀 𝑢 𝑘 ,𝑡 − 𝑉𝐻𝑀 𝑢 0 ,𝑡 Π 𝑢 𝑘 ,𝑡 = Π 𝑢 𝑘 ,𝑡 Total portfolio return Π 𝑢 𝑘 ,𝑡 Unrealized gain and losses 𝑉𝐻𝑀 𝑢 𝑘 ,𝑡 𝑉𝐻𝑀 𝑢 𝑘 ,𝑡 = ෍ ෍ 𝑦 𝑗,ℎ,𝑢 𝑘 ,𝑡 ∙ 𝜓 𝑗,ℎ,𝑢 𝑘 ,𝑡 𝑗∈𝐽 ℎ<𝑢 𝑘 ,ℎ𝜗 ෠ 𝑈 Cumulated 𝐽𝑂𝑊,𝑑𝑣𝑛 𝐽𝑂𝑊,𝑑𝑣𝑛 = ෍ Π 𝑢,𝑡 𝐽𝑂𝑊 Π 𝑢,𝑡 Π 𝑢 𝑙 ,𝑡 realized portfolio return 𝑢 𝑙 ≤𝑢 𝑘 Cumulated 𝐽𝑂𝑊,𝑑𝑣𝑛 + 𝑉𝐻𝑀 𝑢 𝑘 ,𝑡 − 𝑉𝐻𝑀 𝑢 0 ,𝑡 𝑑𝑣𝑛 = Π 𝑢,𝑡 𝑑𝑣𝑛 Π 𝑢 𝑘 ,𝑡 Π 𝑢 𝑘 ,𝑡 total portfolio return