chris bemis Whitebox Advisors A Study in Joint Density Modeling in CVaR Optimization

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1 A Study in Joint Density Modeling in CVaR Optimization chris bemis Whitebox Advisors January 7, 2010

2 The ultimate goal of a positive science is the development of a theory or hypothesis that yields valid and meaningful (i.e., not truistic) predictions about phenomena not yet observed. Milton Friedman

3 Many are familiar with the following optimization problem, minimize subject to w Σw µ w α 1 w = 1 w 0, suggested by Markowitz in 1952.

4 Many are familiar with the following optimization problem, minimize subject to w Σw µ w α 1 w = 1 w 0, suggested by Markowitz in 1952.

5 Financial data are notoriously nonstationary, though:

6 Financial data are notoriously nonstationary, though:

7 Financial data are notoriously nonstationary, though: It is clear why this is troublesome.

8 Markowitz formulation for optimal portfolios also presupposes Every investor has the same utility over a fixed horizon That utility is quadratic in risk; viz., variance This necessitates (or is justified by) a geometric brownian motion for the underlying assets Serial independence is assumed for returns at all time levels in the GBM case

9 Markowitz formulation for optimal portfolios also presupposes Every investor has the same utility over a fixed horizon That utility is quadratic in risk; viz., variance This necessitates (or is justified by) a geometric brownian motion for the underlying assets Serial independence is assumed for returns at all time levels in the GBM case

10 Markowitz formulation for optimal portfolios also presupposes Every investor has the same utility over a fixed horizon That utility is quadratic in risk; viz., variance This necessitates (or is justified by) a geometric brownian motion for the underlying assets Serial independence is assumed for returns at all time levels in the GBM case

11 Markowitz formulation for optimal portfolios also presupposes Every investor has the same utility over a fixed horizon That utility is quadratic in risk; viz., variance This necessitates (or is justified by) a geometric brownian motion for the underlying assets Serial independence is assumed for returns at all time levels in the GBM case

12 Markowitz formulation for optimal portfolios also presupposes Every investor has the same utility over a fixed horizon That utility is quadratic in risk; viz., variance This necessitates (or is justified by) a geometric brownian motion for the underlying assets Serial independence is assumed for returns at all time levels in the GBM case

13 Lehmann (1990) provides evidence that returns exhibit negative serial autocorrelation. Lo and MacKinlay (1990) provide more color and suggest positive autocorrelation between assets is exhibited and that this explains Lehmann s results. Shah (2008) shows that daily volatility, when annualized, exceeds annualized monthly volatility, and suggests negative serial autocorrelation as an explanation.

14 Lehmann (1990) provides evidence that returns exhibit negative serial autocorrelation. Lo and MacKinlay (1990) provide more color and suggest positive autocorrelation between assets is exhibited and that this explains Lehmann s results. Shah (2008) shows that daily volatility, when annualized, exceeds annualized monthly volatility, and suggests negative serial autocorrelation as an explanation.

15 Lehmann (1990) provides evidence that returns exhibit negative serial autocorrelation. Lo and MacKinlay (1990) provide more color and suggest positive autocorrelation between assets is exhibited and that this explains Lehmann s results. Shah (2008) shows that daily volatility, when annualized, exceeds annualized monthly volatility, and suggests negative serial autocorrelation as an explanation.

16 What is important to note from the above is that further (e.g., post 1970) studies into the dynamics of returns suggest a modification to the underlying assumption of a GBM dynamic. These new features are not compatible with, and cannot be directly or cogently incorporated into, the above optimization problem. Promising suggestions which maintain Markowitz framework include Goldfarb and Iyengar s (2003) robust portfolio optimization method. We will pursue another avenue...

17 What is important to note from the above is that further (e.g., post 1970) studies into the dynamics of returns suggest a modification to the underlying assumption of a GBM dynamic. These new features are not compatible with, and cannot be directly or cogently incorporated into, the above optimization problem. Promising suggestions which maintain Markowitz framework include Goldfarb and Iyengar s (2003) robust portfolio optimization method. We will pursue another avenue...

18 What is important to note from the above is that further (e.g., post 1970) studies into the dynamics of returns suggest a modification to the underlying assumption of a GBM dynamic. These new features are not compatible with, and cannot be directly or cogently incorporated into, the above optimization problem. Promising suggestions which maintain Markowitz framework include Goldfarb and Iyengar s (2003) robust portfolio optimization method. We will pursue another avenue...

19 What is important to note from the above is that further (e.g., post 1970) studies into the dynamics of returns suggest a modification to the underlying assumption of a GBM dynamic. These new features are not compatible with, and cannot be directly or cogently incorporated into, the above optimization problem. Promising suggestions which maintain Markowitz framework include Goldfarb and Iyengar s (2003) robust portfolio optimization method. We will pursue another avenue...

20 For a vector of portfolio weights, w, and a scenario, y, define the function f, f (w, y) : R n R m R to be the loss of the portfolio allocated according to w under scenario y. We will call a positive value from f a loss.

21 For a vector of portfolio weights, w, and a scenario, y, define the function f, f (w, y) : R n R m R to be the loss of the portfolio allocated according to w under scenario y. We will call a positive value from f a loss.

22 Assuming that the scenarios have probability density function p, the cumulative distribution function of losses, given portfolio weights w, is Ψ(x, γ) = p(y)dy f (x,y)<γ Notice, our framework is about as general as possible. This is intentional

23 Assuming that the scenarios have probability density function p, the cumulative distribution function of losses, given portfolio weights w, is Ψ(x, γ) = p(y)dy f (x,y)<γ Notice, our framework is about as general as possible. This is intentional

24 Assuming that the scenarios have probability density function p, the cumulative distribution function of losses, given portfolio weights w, is Ψ(x, γ) = p(y)dy f (x,y)<γ Notice, our framework is about as general as possible. This is intentional

25 We next define the value at risk for a given threshold, α: VaR α (w) = min{γ R Ψ(w, γ) α} We have that VaR α (w) is the smallest amount of loss that we can expect with probability 1 α And while this particular risk measure has gained traction, we prefer a more robust measure - CVaR

26 We next define the value at risk for a given threshold, α: VaR α (w) = min{γ R Ψ(w, γ) α} We have that VaR α (w) is the smallest amount of loss that we can expect with probability 1 α And while this particular risk measure has gained traction, we prefer a more robust measure - CVaR

27 We next define the value at risk for a given threshold, α: VaR α (w) = min{γ R Ψ(w, γ) α} We have that VaR α (w) is the smallest amount of loss that we can expect with probability 1 α And while this particular risk measure has gained traction, we prefer a more robust measure - CVaR

28 We next define the value at risk for a given threshold, α: VaR α (w) = min{γ R Ψ(w, γ) α} We have that VaR α (w) is the smallest amount of loss that we can expect with probability 1 α And while this particular risk measure has gained traction, we prefer a more robust measure - CVaR

29 The VaR construction ignores tail behavior. Conditional value at risk, or CVaR, incorporates the tail past the VaR value; viz., CVaR α (w) = 1 f (w, y)p(y)dy 1 α f (w,y) VaR α (w) We can discretize this in a natural way by sampling our scenarios discretely according to p

30 The VaR construction ignores tail behavior. Conditional value at risk, or CVaR, incorporates the tail past the VaR value; viz., CVaR α (w) = 1 f (w, y)p(y)dy 1 α f (w,y) VaR α (w) We can discretize this in a natural way by sampling our scenarios discretely according to p

31 The VaR construction ignores tail behavior. Conditional value at risk, or CVaR, incorporates the tail past the VaR value; viz., CVaR α (w) = 1 f (w, y)p(y)dy 1 α f (w,y) VaR α (w) We can discretize this in a natural way by sampling our scenarios discretely according to p

32 Assuming we can do what was just suggested (we can, see Rockafeller (1999)), we may write another optimization problem: min w W CVaR α(w), A linear programming problem. We see now the primacy of correct scenario generation. Our project will simulate asset returns for each scenario, necessitating a structure for the joint density.

33 Assuming we can do what was just suggested (we can, see Rockafeller (1999)), we may write another optimization problem: min w W CVaR α(w), A linear programming problem. We see now the primacy of correct scenario generation. Our project will simulate asset returns for each scenario, necessitating a structure for the joint density.

34 Assuming we can do what was just suggested (we can, see Rockafeller (1999)), we may write another optimization problem: min w W CVaR α(w), A linear programming problem. We see now the primacy of correct scenario generation. Our project will simulate asset returns for each scenario, necessitating a structure for the joint density.

35 Assuming we can do what was just suggested (we can, see Rockafeller (1999)), we may write another optimization problem: min w W CVaR α(w), A linear programming problem. We see now the primacy of correct scenario generation. Our project will simulate asset returns for each scenario, necessitating a structure for the joint density.

36 Assuming we can do what was just suggested (we can, see Rockafeller (1999)), we may write another optimization problem: min w W CVaR α(w), A linear programming problem. We see now the primacy of correct scenario generation. Our project will simulate asset returns for each scenario, necessitating a structure for the joint density.

37 In the tradition of Friedman s quote above, we will examine the out of sample performance of CVaR optimized portfolios under various densities. We will of course examine a multivariate normal assumption. We will also look at using a multivariate Student t distribution as well as a mixed multivariate Student t. Other ideas may be entertained/entertaining.

38 In the tradition of Friedman s quote above, we will examine the out of sample performance of CVaR optimized portfolios under various densities. We will of course examine a multivariate normal assumption. We will also look at using a multivariate Student t distribution as well as a mixed multivariate Student t. Other ideas may be entertained/entertaining.

39 In the tradition of Friedman s quote above, we will examine the out of sample performance of CVaR optimized portfolios under various densities. We will of course examine a multivariate normal assumption. We will also look at using a multivariate Student t distribution as well as a mixed multivariate Student t. Other ideas may be entertained/entertaining.

40 In the tradition of Friedman s quote above, we will examine the out of sample performance of CVaR optimized portfolios under various densities. We will of course examine a multivariate normal assumption. We will also look at using a multivariate Student t distribution as well as a mixed multivariate Student t. Other ideas may be entertained/entertaining.

41 fin.