Long Short Portfolios Formulas Daniel Herlemont - email:[email protected] Copyright, YATS Finances & Technologies

December 28, 2008

Contents 1 The 1.1 1.2 1.3

1.4 1.5

1.6

models Markowitz efficient frontier . . . . . Including the risk free asset . . . . ”Neutral” strategies . . . . . . . . . 1.3.1 Market neutral . . . . . . . 1.3.2 Dollar neutral . . . . . . . . Including more general constraint . One factor model . . . . . . . . . . 1.5.1 Growth Optimal portfolio in 1.5.2 beta neutral portfolio . . . . Two risky assets model . . . . . . . 1.6.1 Without risk free . . . . . . 1.6.2 With risk free . . . . . . . . 1.6.3 One factor model . . . . . .

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2 References

1 1.1

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1 1 2 4 6 7 7 8 9 10 11 11 12 14 14

The models Markowitz efficient frontier min 12 π T V π πT µ = r πT 1 = 1 1

1 THE MODELS Defining A B C D

µT V −1 µ µT V −1 1 = 1T V −1 µ 1T V −1 1 AC − B 2

= = = =

We obtain the efficient frontier σ2 =

Cr2 − 2Br + A D

and

Cr − B −1 A − Br −1 V µ+ V 1 D D That can be written as a linear combination of the maximum slope portfolio (πslope ) and the minimum variance portfolio (πmin ): π=

1 1 −1 V µ = T −1 V −1 µ B 1 V µ 1 −1 1 = V 1 = T −1 V −1 1 C 1 V 1

πslope = πmin

The square Sharpe can be written as: Sharpe2 =

r2 D = 2 σ C − 2B/r + A/r2

is maximal for r = A/B, that is the portfolio πslope S 2 = A = µT V −1 µ When assets are uncorrelated the squared ”Sharpe” are additive: X S2 = Sharpe2i

1.2

Including the risk free asset

Objective is to maximize the growth of the portfolio 1 max gπ = π T µ + π0 µ0 − π T V π π 2 Copyright, Daniel Herlemont, email:[email protected] YATS, All rights reserved,

2

1 THE MODELS under the constraint X

πi + π 0 = 1

i=1,n

where π = (π1 , ..., πn ) are the weights of risky assets, and π0 the holding in the risk free asset µ0 . We can remove the constraint on the weights by considering the risky assets: 1 max gπ = π T (µ − µ0 ) − π T V π π 2 We obtain π ∗ = V −1 (µ − µ0 ) defining µ∗ the rate of return and σ ∗2 the variance of the the optimal portfolio µ∗ − µ0 = (µ − µ0 )T π ∗ = (µ − µ0 )T V −1 (µ − µ0 ) σ ∗2 = π ∗T V π ∗ = (µ − µ0 )T V −1 (µ − µ0 ) It follows µ∗ − µ0 = σ ∗2 Sharpe∗ = σ ∗ Sharpe∗2 = µ∗ − µ0 1 Sharpe∗2 g∗ = 2 The Sharpe ratio is also equal to the volatility σ ∗ . For example, with a Sharpe ratio of 2 the volatility is 200% !!! Clearly, Growth Optitmal Portfolio with high sharpe ratios are not sustainable. A risk aversion parameter can be used to scale down the weights on risky assets. This scaling does not change the Sharpe ratio. Let πγ , µγ and σγ2 being weights in the risky assets, the rate of return and the variance of the optimal portfolio for a γ risk aversion πγ =

1 −1 V (µ − µ0 ) γ (1)

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1 THE MODELS The Sharpe ratio is unchanged µγ − µ0 σγ = Sharpe∗ = σ ∗ = γσγ (2) Example: if a fund reports a 3 Sharpe ratio and 10% volatility, the implicit risk aversion parameter is about Sharpe γ= = 3/0.1 = 30 σ high performance funds corresponds to high aversion parameters. Under a risk constraint (on VaR for example), the objective is to maximize rate of return max gπ = (µ − µ0 )T π π

under the constraint π T V π = σ02 the solution is π=p

σ0 (µ − µ0

)T V −1 (µ

− µ0 )

V −1 (µ − µ0 )

That corresponds to an implicit risk aversion p (µ − µ0 )T V −1 (µ − µ0 ) σ∗ Sharpe γ= = = σ0 σ0 σ0 For example, σ0 can be choose so that the VaR at 95% for 20 p days (one month) shall not exceed 4% of the wealth. The implicit σ0 is defined as 1.65 ∗ 10/252σ0 = 0.04, that is 12.2%. With a 2 Sharpe stratgy, the risk aversion parameter is 2/0.122 = 16.4. The covariance of the Growth Optimal Portfolio with the different assets have the following simple form: σπ,i = µi − µ0 This relation derive directly from the definition of V π = µ − µ0 where the rows of the left hand side are the covariance of the portfolio with the different assets. The correlations are (µi − µ0 ) Sharpei ρπ,i = = σπ σi Sharpeπ Copyright, Daniel Herlemont, email:[email protected] YATS, All rights reserved,

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1 THE MODELS With a γ risk aversion σπ,i =

1 (µi − µ0 ) γ

With every high γ, the covariance with assets are lowered. However, the correlations are unchanged Sharpei ρπ,i = Sharpeπ The correlations with the different are are smaller as the Sharpe ratio become larger.

1.3

”Neutral” strategies

We can impose constraints on the weights to find market neutral or dollar neutral strategies. Suppose that the nth asset represents the market, then the market neutral strategies can be defined as: 1 max(µ − µ0 )T π − γπ T V π π 2 Under the constraint: σπ,n = VnT π = 0 where Vi denote the ith column of the covariance matrix. Vi = V ei with ei is the unitary vector ei (j) = 0 for j 6= i ei (i) = 1. For dollar neutral strategies, the problem is 1 max(µ − µ0 )T π − γπ T V π π 2 Under the constraint: 1T π = 0 Both problems can be put under the same following form. 1 max(µ − µ0 )T π − γπ T V π π 2 Under the constraint: KT π = 0 With K = Vn for market neutral or K = 1 for dollar neutral strategies. The Lagragian is 1 L = (µ − µ0 )T π − γπ T V π − λK T π 2 Copyright, Daniel Herlemont, email:[email protected] YATS, All rights reserved,

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1 THE MODELS deriving with respect to π (µ − µ0 ) − γV π − λK = 0

(3)

Multiplying equation 3 by π, we have the relation between the expected return µπ − µ0 = (µ − µ0 )T π and the variance σπ2 = π T V π µπ − µ0 = γσπ2 This implies that such portfolio have positive excess returns. The Sharpe ratio is Sharpe2π =

(µπ − µ0 )2 = γ(µπ − µ0 ) = γ 2 σπ2 2 σπ

The solution is 1 −1 V (µ − µ0 ) − λK) γ B with λ = C and B = K T V −1 (µ − µ0 ) = (µ − µ0 )T V −1 K and C = K T V −1 K > 0 π =

and µπ − µ0 Sharpe2π 1.3.1

  B2 T −1 (µ − µ0 ) V (µ − µ0 ) − C 2 B = Sharpe∗2 − C 1 = γ

Market neutral

For market neutral, the solution is π=

µn − µ0 1 −1 en V (µ − µ0 ) − γ γσn2

and λ = (µn − µ0 )/σn2 , B = µn − µ0 C = σn2 . This portfolio is very closed to the unconstrained solution, except for the ”market” exposure where the term (µn − µ0 )/(γσn2 ) on asset n has been removed. This term is also the optimal exposure of the single asset problem. Copyright, Daniel Herlemont, email:[email protected] YATS, All rights reserved,

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1 THE MODELS The covariance with the different assets are Vπ =

1 (µ − µ0 − (µn − µ0 )en ) γ

that is σπ,i = µi − µ0 for i = 1, n − 1 and σπ,n = 0 as expected. One can easily find that µπ − µ0 = (µ − µ0 )π
1 = (µ − µ0 )T (V −1 (µ − µ0 ) − λen γ
1 Sharpe∗2 − Sharpe2n = γ and σπ2 =

1 µπ − µ0 = 2 (Sharpe∗2 − Sharpe2n ) γ γ

(Note that Sharpe∗2 is also the excess return of the Growth Optimal Portfolio µ∗ − µ0 ) So that the Sharpe of the strategy is Sharpe2π = Sharpe∗2 − Sharpe2n That is less than the maximum Sharpe strategy without constraint. That can be interpreted as the cost of being market neutral !!! 1.3.2

Dollar neutral

The solution is π=

1 −1 V (µ − µ0 − λ1) γ

with λ = B/C, B = 1T V −1 (µ − µ0 ) = (µ − µ0 )T V −1 1 C = 1T V −1 1. That can be written as 1 π = (π ∗ − Bπmin ) γ where πmin is the minimum variance portfolio. The excess return is
1 ∗ µπ − µ0 = µ − µ0 − B 2 /C γ We have still the relation µπ − µ0 = γσπ2 Copyright, Daniel Herlemont, email:[email protected] YATS, All rights reserved,

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1 THE MODELS So that Sharpe2π = Sharpe∗2 − B 2 /C That could penalize heavily such strategies.

1.4

Including more general constraint

Maximize utility 1 U = π t µ − γπ T V π 2 subject to Aπ = b where A denotes a matrix with k columns equals to the number of assets and m lines equals to the number of equality constraints. Forming the Lagrangian 1 L = π t µ − γπ T V π − λ(Aπ − b) 2 where λ is the m × 1 vector of Lagrangian multipliers. dL/dπ = µ − γV π − λT A = 0 π = 1/γV −1 (µ − λT A) λ=

AV −1 µ b − γ AV −1 AT AV −1 AT

1 π = V −1 AT (AV −1 AT )b + V −1 (µ − AT (AV −1 AT )AV −1 µ) γ The optimal solution is split into a (constrained) minimum variance portfolio and a speculative. The first term depends neither on the expected returns nor on risk tolerance (this the portfolio for infinite risk tolerance), whereas the second term is sensitive to both inputs (see Scherer 2007 [?]).

1.5

One factor model

We consider n + 1 risky assets including the single factor asset n + 1. The asset 0 is the risk free asset with constant rate of returns µ0 . For example, the n risky assets could be the 30 stocks of the DJI index, and the n + 1 asset the DJI itself. The model is the following:

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1 THE MODELS

ri,t − µ0 E[ri,t ] E[i,t ] cov(rn+1,t , i,t ) cov(i,t , j,t ) It follows

1.5.1

= = = = =

αi + βi (rn+1,t − µ0 ) + i,t µi 0 0 0

µi − µ0 = αi + βi (µn+1 − µ0 ) 2 2 + σ,i = βi2 σn+1 σi2 2 σi,j = cov(ri,t , rj,t ) = βi βj σn+1

Growth Optimal portfolio in the one factor model

The objective is to build the growth optimal portfolio with α. In case of αi all null, it is well known that the optimal portfolio is the n + 1 index. the optimal growth portfolio is V −1 (µ − µ0 ) where V is the covariance matrix 2 V = ββ T σn+1 + diag(σ , 0)

The inverse of the covariance matrix U = V −1 has the following structure:  1  0 − σβ21 2 σ1 1   .. ..  . U = V −1 =  0 .   2 P β n 1 i − σβ21 · · · + 2 2 i=1 σ σ 1

n+1

n+1

1 f or i = 1, n σi2 = 0 f or i 6= j i = 1, n and j = 1, n βi = − 2 f or i = 1, n σi n X β2 1 i = + 2 2 σ σn+1 i=1 n+1

uii = uij ui,n+1 = un+1,i un+1,n+1

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1 THE MODELS The optimal weights are αi f or i = 1, n σi2 n X µn+1 − µ0 αi = − βi 2 + 2 σi σn+1 i=1

πi = πn+1 The square Sharpe ratio is T

(µ − µ0 ) V

−1

n X αi2 (µn+1 − µ0 )2 (µ − µ0 ) = + 2 σ2 σn+1 i=1 i

High Sharpe ratios can be obtain with small specific risks and/or high alpha. Therefore, we obtain a long short portfolio long in αi > 0 and short in αi < 0. The position in the index depends on the sign of πn+1 . Note that small specific risk means also high correlation with the index since 1 − ρ2 = σ2 /σ 2 . Note that the Growth Optimal portfolios are not market neutral (beta neutral), since βπ =

n X

βi πi + πn+1 =

i=1

µn+1 − µ0 2 σn+1

The risky weights can be scaled by a risk aversion factor γ to match the risk management constraints (VaR objectives, for example), the expected excess rates of return, the volatility and the β of the portfolio are scaled by the same factor γ with changing the Sharpe ratio that remains the same as the Growth Optimal Portfolio. 1.5.2

beta neutral portfolio

The optimal beta neutral portfolio is defined as: maxπ (µ − µ0 )T π − 12 γπ T V π under the constraint β π = 0 The solution is πγ =

1 −1 V (µ − µ0 − (µn+1 − µ0 )β) γ

that is: 1 αi f or i = 1, n γ σi2 n 1 X αi = − βi γ i=1 σi2

πi = πn+1

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1 THE MODELS This is the same as the growth optimal portfolio, except for the index exposure πn+1 where 2 the term (µn+1 − µ0 )/σn+1 is removed. This term was also the optimal holding in case of index only, it is also the beta of the growth optimal portfolio. The beta neutral portfolio does not depend on the expected return of the index. Therefore there is no need to anticipate the market direction (see also Merton [?], that is consistent with a market neutral strategies. Most importantly, such strategies require estimates of the αi or at least the signs of the alpha. µγ − µ0 = (µ − µ0 )T πγ = γπγT V πγ n 1 X αi2 = γ i=1 σi2 The excess return of the portfolio is γ the variance of the portfolio. The squared Sharpe ratio of the beta neutral is S 2 = γ 2 σγ2 = γ(µγ − µ0 ) n X αi2 = σ2 i=1 i This is smaller than the maximal Sharpe portfolio, the squared Sharpe of the index is removed.

1.6 1.6.1

Two risky assets model Without risk free

Consider the simple case of two risky assets in the one period Markowitz model ! ρ 1 − 2 1 σ1 σ2 σ1 V −1 = 1 − σ1ρσ2 1 − ρ2 σ2 2

the maximum Sharpe is: S2 =

S12 + S22 − 2ρS1 S2 1 − ρ2

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1 THE MODELS

S1 − ρS2 1 σ1 (S1 − ρS2 )/σ1 + (S2 − ρS1 )/σ2 S2 − ρS1 1 = σ2 (S1 − ρS2 )/σ1 + (S2 − ρS1 )/σ2

π1∗ = π2∗ If S1 = S2 ,

S=

2S1 1+ρ

If S1 > S2 , the maximum Sharpe attains a minimal value for ρ = S2 /S1 This value corresponds to π2∗ = 0 and the portfolio is fully invested in the highest Sharpe asset. As a consequence, whatever the parameters, optimal weights lead to a higher Sharpe than the best Sharpe. It is important to note that the Sharpe can attain higher values with highly correlated assets, as soon as ρ > S2 /S1 The portfolio becomes long/short, the structure depends on the sign of (S1 − ρS2 )/σ1 + (S2 − ρS1 )/σ2 . For example, if ρ ≈ 1, this sign mainly depends on σ2 − σ1 . If σ2 >> σ1 , the portfolio is long on the asset 1 and short on the asset 2. 1.6.2

With risk free S1 − ρS2 σ1 (1 − ρ2 ) S2 − ρS1 = σ2 (1 − ρ2 )

π1∗ = π2∗

Considering again S1 > S2 , π1∗ is alway positive. For ρ > S2 /S1 , the portfolio becomes long/short, long in the highest Sharpe asset, and short in the lowest Sharpe asset. The Sharpe ratio can become arbitrarily high as the correlation becomes close to 1. However, the weights becomes higher and higher, that may become a high risk portfolio in case of estimation errors of the time varying correlations and Sharpe ratios ... The following figure displays the Sharpe as a function of the correlation, with S1 = 0.5 S2 = 2.

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1 THE MODELS

40 0

20

optimal growth rate

60

80

Growth rate function of correlation (S1=0.5, S2=1)

−1.0

−0.5

0.0

0.5

1.0

rho

The correlation of the Growth Optimal Portfolio and the assets are S1 S S2 = S

ρπ,1 = ρπ,2

(4) The optimal portfolio is positively correlated to the assets.

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13

1 THE MODELS 1.6.3

One factor model

Now, considering the case of 2 assets in the one factor model, a risky asset and the index. µi are the expected returns in excess of the risk free. The weights of the optimal Sharpe portfolio are π1 =

α1 2 σ1

π2 = −β1

α1 µ2 + 2 2 σ1 σ2 µ

S ∗2

2 π2∗ σ2 = −β + 1 α1 π1∗ 2 σ1  2  2 α1 µ2 = + σ1 σ2

(5)

(6)

α1 σ1

is the Information Ratio. In case α/σ2 >> µ2 /σ22 the optimal holding is also very closed to a beta neutral position ∗ π2 /π1∗ = −β. The sharpe of the beta neutral portfolio is very closed to the maximal sharpe ratio. In this case the advantages are The beta neutral portfolio proportions does not depend on alpha (that is much more difficult to estimate than the beta). In case of β1 close to one, we obtain both a beta and dollar neutral and optimal growth portfolio. However, to get the Growth Optimal Portfolio one a has to take possibly too leveraged positions in risky assets. One can scale the positions to comply risk management constraints without modifying the Sharpe ratio. Then, the success of this L/S portfolio is mostly related the predictability of the sign of α. The optimal beta neutral portfolio is: π1 =

α1 2 σ1

π2 = −β1 π2 = −β1 π1

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α1 2 σ1

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2

References

[1] BRUCE I. JACOBS, K. N. L & STARER, D. ”On the Optimality of Long-Short Strategies” . Financial Analysts Journal, March/April 1998.(3). ... [2] GRINOLD, R. C & KAHN, R. N. ”Active Portfolio Management” . McGraw-Hill, 1999. [3] MERTON, R. ”OPTIMAL USE OF SECURITY ANALYSIS AND INVESTMENT MANAGEMENT”. ... [4] MUNK, C. ”Dynamic Asset Allocation” , 2004.

...

lecture notes.

[5] SCHERER, B. ”Portfolio Construction and Risk Budgeting” . Risk Books, Feb 2007.

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