CSCI 1951-G Optimization Methods in Finance Part 07: Portfolio - - PowerPoint PPT Presentation

CSCI 1951-G – Optimization Methods in Finance Part 07: Portfolio Optimization

March 9–16, 2018

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The portfolio optimization problem

How to best allocate our money to n risky assets S1, . . . , Sn with random returns?

• µi: expected return of asset i in a time interval;

Stocks Bonds Money Market µi 0.1073 0.0737 0.0627

• Σ: variance-covariance n × n matrix of returns, with:
• σii: variance of the return of asset i;
• σij: covariance of the returns of assets i and j.

Covariance Stocks Bonds MM Stocks 0.02778 0.00387 0.00021 Bonds 0.00387 0.01112

• 0.00020

MM 0.00021

• 0.00020

0.00115

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The portfolio optimization problem

Portfolio x = (x1, . . . , xn), where xi: proportion of money invested in asset i. Expected return: E[x] = µ1x1 + · · · + µnxn = µTx Variance: Var[x] =

i,j σijxixj = xTΣx

Var[x] ≥ 0, so Σ ...is positive semidefinite (we assume positive definite) Feasible portfolios: set X = {x : Ax = b, Cx ≥ d} One constraint is

n

• i=1

xi = 1

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The portfolio optimization problem

Efficient portfolio w.r.t. R > 0: the portfolio with minimum variance among all those with expected return at least R. (variants possible, e.g., ) Markowitz’ mean-variance optimization: find the efficient portfolio: min xTΣx s.t. µTx ≥ R Ax = b Cx ≥ d This optimization problem is ...convex. We assumed Σ ≥ 0, so the optimal solution is ...unique.

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The portfolio optimization problem

Stocks Bonds MM µi 0.1073 0.0737 0.0627 Covariance Stocks Bonds MM Stocks 0.02778 0.00387 0.00021 Bonds 0.00387 0.01112

• 0.00020

MM 0.00021

• 0.00020

0.00115

min 0.02778x2

S + 2 · 0.00387xSxB + 2 · 0.00021xSxM

+ 0.01112x2

B − 2 · 0.00020xBxM + 0.00115x2 M

s.t. 0.1073xS + 0.0737xB + 0.0627xM ≥ R xS + xB + xM = 1 xS ≥ 0, xB ≥ 0, xM ≥ 0

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The portfolio optimization problem

Rate of Return R Variance Stocks Bonds MM 0.065 0.0010 0.03 0.10 0.87 0.070 0.0014 0.13 0.12 0.75 0.075 0.0026 0.24 0.14 0.62 0.080 0.0044 0.35 0.16 0.49 0.085 0.0070 0.45 0.18 0.37 0.090 0.0102 0.56 0.20 0.24 0.095 0.0142 0.67 0.22 0.11 0.100 0.0189 0.78 0.22 0.105 0.0246 0.93 0.07 Table 8.1: Efficient Portfolios

2 4 6 8 10 12 14 16 6.5 7 7.5 8 8.5 9 9.5 10 10.5 Standard Deviation (%) Expected Return (%) 6.5 7 7.5 8 8.5 9 9.5 10 10.5 10 20 30 40 50 60 70 80 90 100 Expected return of efficient portfolios (%) Percent invested in different asset classes Stocks Bonds MM

Figure 8.1: Efficient Frontier and the Composition of Efficient Portfolios

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The efficient frontier

Rmin, Rmax: minimum and maximum expected returns for efficient portfolios. σ(R) : [Rmin, Rmax] → R, σ(R) =

• xT

RΣxR

1/2 where xR is the efficient portfolio w.r.t. R ∈ [Rmin, Rmax]. The efficient frontier is the graph E = {(R, σ(R)) : R ∈ [Rmin, Rmax]}

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Maximizing the Sharpe ratio

Consider a riskless asset with deterministic return rf ≤ Rmin (why does it make sense?) Consider convex combinations between a risky portfolio x with the riskless asset xθ = [(1 − θ)x θ]T As θ varies, (for fixed x) the combinations form a line on the stdev/mean plot:

Mean Variance CAL

rf

Figure 8.4: Capital Allocation Line

For different choices of x, the slope of the line changes.

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Maximizing the Sharpe ratio

Which Capital Allocation Line (CAL) is the best?

Mean Variance CAL

rf

Figure 8.4: Capital Allocation Line

The CAL with the largest slope: the corresponding portfolio will have the lowest stdev for any given value of R ≥ rf.

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Maximizing the Sharpe ratio

To which portfolio x does the optimal CAL corresponds to? The feasible x that maximizes the slope: h(x) = µTx − rf (xTΣx)1/2 The quantity h(x) is known as the Sharpe ratio or the reward-to-volatility ratio.

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Maximizing the Sharpe ratio

To find the optimal risky portfolio we solve max µTx − rf (xTΣx)1/2 s.t. Ax = b Cx ≥ d The feasible region is polyhedral, but the objective function may be non-concave. Let’s build an equivalent convex quadratic program.

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Maximizing the Sharpe ratio

X = {x : Ax = b, Cx ≥ d} (includes full alloc. const., assumes ∃ˆ x ∈ X s.t. µTˆ x > rf) X + = {(x, k) : x ∈ Rn, k ∈ R++, x

k ∈ X} ∪ {(0, 0}

The optimal risky portfolio is x∗ = y∗/k∗ where (y∗, k∗) is the

• ptimal solution of:

min yTΣy s.t. (y, k) ∈ X + (µ − rf 1)Ty = 1 This is a quadratic convex program (why?)

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Returns-based style analysis

• You are a portfolio manager (!) and would like to understand how

your portfolio manager“friend” Sally, the style of her portfolio i.e., the mix of stocks in it;

• Sally is secretive on the mix, but publish the returns of her

portfolio over time;

• You also have access to the returns of index funds tracking different

sectors of the market; Definition (Return-Based Style Analysis (RBSA)) A technique using constrained optimization to determine the style of a portfolio using the return time series of the portfolio and of a number of other asset classes (factors).

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RBSA Mathematical Model

Fundamentally a linear model for regression. Data

• Rt, t = 1, . . . , T: the return of Sally’s portfolio over T fixed time

intervals (e.g., Rt is the monthly return, and T = 12 months);

• Fit, i = 1, . . . , n, t = 1, . . . , T: the returns of factor i over T fixed

time intervals (same intervals as Rt); Model Rt = w1tF1t + w2tF2t + · · · + wntFnt + εt = F T

t wt + εt

• wit: sensitivity of Rt to factor i;
• εt: non-factor return.

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Interpretation

Rt = w1tF1t + w2tF2t + · · · + wntFnt + εt = F T

t wt + εt

Assume the Fit are returns of passive investments (e.g., index funds); Then:

• F T

t wt is the return of a benchmark portfolio of passive

investments;

• εt is the difference between the passive benchmark and the active

strategy followed by Sally. If the passive investments considered together are representative of the market, then εt measures the additional (or negative) return due to Sally’s ability as a portfolio manager.

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Optimization problem

Rt = w1tF1t + w2tF2t + · · · + wntFnt + εt = F T

t wt + εt

Additional assumptions/constraints:

• wit = wi, i.e., the weights do not change over time.
• wi > 0, n

i=1 wt = 1.

Constraints of our optimization problem: min ??? s.t.

n

• i=1

wi = 1 wi ≥ 0, i = 1, . . . , n What about the objective function? Any idea?

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Optimization problem (cont.)

• If εt measures Sally’s ability as a portfolio manager, we can

assume that it is approximately constant over time.

• I.e., we want the plots of the returns of Sally’s portfolios and of the

benchmark portfolios to be curves with approximately constant distance.

• I.e., we want εt to have the smallest possible variance over time.

Formulation min

w∈Rn Var(et) = Var(Rt − F T t w)

s.t.

n

• i=1

wi = 1 wi ≥ 0, i = 1, . . . , n The objective function is convex.

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Objective function

Let R =    R1 . . . RT    , and F =    F T

1

. . . F T

T

   , and e =    1 . . . 1    We have Var(Rt − F T

t w) = 1

T

T

• i=1

(Rt − F T

t w)2 −

T

t=1(Rt − F T t w

T 2 = 1 T R − Fw2 − eT(R − Fw) T 2 = R2 − 2RTFw + wTF TFw T − (eTR)2 − 2eTR − wTF TeeTFw T 2

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Objective function

Var(Rt − F T

t w) = R2 − 2RTFw + wTF TFw

T − (eTR)2 − 2eTR − wTF TeeTFw T 2 Reorganizing the terms as function of w: Var(Rt − F T

t w) =

R2 T − eTR)2 T 2

• − 2

RTF T − eTR T 2 eTF

• w

+ wT 1 T F TF − 1 T F TeeTF

• w

We have 1 T F TF − 1 T F TeeTF = 1 T F T

• I − eeT

T

• F

The matrix M = I − eeT/T is symmetric and positive semidefinite (eigenvalues: 0 and 1), and so is F TMF. Hence the objective function is convex.

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