The Limits of Leverage

The Limits of Leverage

Paolo Guasoni1,2 Eberhard Mayerhofer2

Boston University1

Dublin City University2

Mathematical Finance and Partial Differential EquationsMay 1st , 2015

The Limits of Leverage

Efficient Frontier

0

2

4

6

8

10

12

14

16

0 2 4 6 8 10 12 14 16

µ = 8%, σ = 16%, ε = 1%

• Average Return (y) against volatility (x) as benchmarks’ multiples.• No transaction costs at zero (0,0) or full investment (1,1).• Higher Leverage = Lower Sharpe Ratio.• Maximum return at around 9x leverage. More leverage decreases return.

The Limits of Leverage

Unlimited Leverage?“If an investor can borrow or lend as desired, any portfolio can be levered upor down. A combination with a proportion k invested in a risky portfolio and1 − k in the riskless asset will have an expected excess return of k and astandard deviation equal to k times the standard deviation of the riskyportfolio.” — Sharpe (2011)

• Implications:• Efficient frontier linear. One Sharpe ratio.• Any efficient portfolio spans all the others.• Portfolio choice meaningless for risk-neutral investors.

• Applications:• Levered and inverse ETFs: up to 3x and -3x leverage. A 10x ETF?• Leverage to increase returns from small mispricings.• Capital ratios as regulatory leverage constraints.

• Limitations:• Constant leverage needs constant trading. Rebalancing costs?• Higher beta with lower alpha (Frazzini and Pedersen, 2013).• Levered ETFs on illiquid indexes have substantial tracking error.

The Limits of Leverage

What We Do

• Model• Maximize long-term return given average volatility.• Constant proportional bid-ask spread.• IID returns. Geometric Brownian motion.• Continuous trading allowed. No constraints.

• Results• Sharpe ratio declines as leverage increases.• Limits of leverage.

Beyond a a certain threshold, even expected return declines.• Leverage Multiplier.

Maximum factor by which the asset return can be increased:

0.3815( µσ2

)1/2ε−1/2

ε bid-ask spread, µ excess return, σ volatility.• Optimal tradeoff between alpha and tracking error.

The Limits of Leverage

More Volatility = Cheaper Leverage

0.0

0.5

1.0

1.5

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Constant Sharp Ratio 50%, 100 BPs Transaction costs

Standard Deviation

µ/σ = 0.5, ε = 1%, σ = 10%(bottom), 20%, 50%(top)

• Average Return (y) against volatility (x), annualized.• Frontier of asset with 10% return with 20% volatility superior to that of

asset with 5% return and 10% volatility.• More asset volatility = less rebalancing costs for same portfolio volatility.

The Limits of Leverage

Leverage Multiplier

Bid-Ask Spread (ε)Volatility (σ) 0.01% 0.10% 1.00%

10% 71.85 23.15 7.7220% 50.88 16.45 5.5650% 32.30 10.54 3.66

Sharpe ratio µ/σ = 0.5

• Approximate value ≈ 0.3815(µσ2

)1/2ε−1/2

• Increases with return, decreases with volatility and spread• Always lower than solvency level ε−1. Endogenous limit.

The Limits of Leverage

Market

• Safe rate r . Ask price St geometric Brownian motion, Bid Price = (1− ε)St

dSt

St= (µ+ r)dt + σdBt , S0, σ, µ > 0,

• ϕt = ϕ↑t − ϕ↓t number of shares at time t as purchases minus sales.

• Fund value at ask prices:

dwt = rwtdt + ϕtdSt − εdϕ↓t

• Solvency constraint wt − ε(ϕt )+St ≥ 0 a.s. for all t ≥ 0.

The Limits of Leverage

Return, Volatility, Tracking Error• Usual fund performance statistics in terms of returns rt = wt−wt−∆t

wt−∆t

• Average return on [0,T ]

rT =1T

0≤t≤T∑t=k∆t

rt ≈1T

∫ T

0

dwt

wt

Frictionless: 1T

∫ T0 µπtdt + 1

T

∫ T0 σπtdWt , with πt portfolio.

• Average volatility on [0,T ](1T

0≤t≤T∑t=k∆t

(rt − rT ∆t)2

)1/2

≈

(1T

∫ T

0

d〈w〉tw2

t

)1/2

Frictionless: σ2

T

∫ T0 π2

t dt .• Tracking error of fund w on multiple Λ of benchmark S on [0,T ]:(

1T

0≤t≤T∑t=k∆t

(rt − ΛrBt )2

)1/2

≈(

1T

⟨∫ ·0

(dww− Λ

dSS

)⟩T

)1/2

Frictionless: σ2

T

∫ T0 (πt − Λ)2dt . Same as average volatility for Λ = 0.

The Limits of Leverage

Objective• Maximize return-volatility tradeoff for large T

1T

E

[∫ T

0

(dwt

wt− Λ

dSt

St

)− γ

2

⟨∫ ·0

(πt − Λ)dSt

St

⟩T

]

• Equals to

r +1T

E

[∫ T

0

(µ(πt − Λ)− γσ2

2(πt − Λ)2

)dt − ε

∫ T

0πt

dϕ↓tϕt

]

• Λ = 0, ε = 0: Usual mean-variance portfolio πt = µγσ2 optimal.

• Λ = 0, γ = 0: Maximize return, forget volatility. Ill-posed for ε = 0.• Λ = 0, γ = 1: logarithmic utility. Taksar et al. (1988), Gerhold et al. (2012).• µ = 0: Minimize tracking error, forget return. Leveraged ETF replication.• Tradeoff between high leverage and high trading costs.

Well-posed even without risk.

The Limits of Leverage

Efficient Frontier (γ > 0,Λ = 0)TheoremTrade to keep portfolio weight πt within boundaries π− (buy) and π+ (sell)

π± = θ∗ ±(

34γ (θ∗)

2(θ∗ − 1)2)1/3

ε1/3 −(

(1−γ)θ∗+γΛγ

)(γθ∗(θ∗−1)

6

)1/3ε2/3 + O(ε)

where θ∗ = Λ + µ/(γσ2) and ζ± solve the free-boundary problem (W , ζ−, ζ+)

12σ

2ζ2W ′′(ζ) + (σ2 + µ)ζW ′(ζ) + µW (ζ)− 1(1+ζ)2

(µ+ γσ2Λ− γσ2ζ

1+ζ

)= 0,

W (ζ−) = 0, W (ζ+) = ε(1+ζ+)(1+(1−ε)ζ+) ,

W ′(ζ−) = 0, W ′(ζ+) = ε(ε−2(1−ε)ζ+−2)(1+ζ+)2(1+(1−ε)ζ+)2

• Solution similar to utility maximization. Same first-order approximation.• No-trade region around the frictionless portfolio.• Result valid for ε small enough.

The Limits of Leverage

Limits of Leverage (γ = Λ = 0)TheoremTrade to keep portfolio weight πt within boundaries π− (buy) and π+ (sell)

π± = ζ±1+ζ±

= B±κ1/2(µ/σ2)12 ε−1/2 + 1 + O(ε

12 ),

where B− = (1− κ),B+ = 1 and κ ≈ 0.5828 is the root of 32κ+ log(1− κ) = 0.

ζ± solve the free-boundary problem (W , ζ−, ζ+)

12σ

2ζ2W ′′(ζ) + (σ2 + µ)ζW ′(ζ) + µW (ζ)− µ(1+ζ)2 = 0,

W (ζ−) = 0, W (ζ+) = ε(1+ζ+)(1+(1−ε)ζ+) ,

W ′(ζ−) = 0, W ′(ζ+) = ε(ε−2(1−ε)ζ+−2)(1+ζ+)2(1+(1−ε)ζ+)2

• Frictionless problem meaningless. Infinite leverage.• Pure tradeoff between leverage and rebalancing costs.• π− is the multiplier. Maximum return is µπ−.• Approximate relation π−

π+≈ 0.4172.

The Limits of Leverage

Does it make sense?• As risk-aversion vanishes, do solutions converge to risk-neutral ones?

Yes.• Spread of ε implies maximum leverage of 1/ε. Is this driving the results?

No.

Assumption

For any γ ∈ [0, γ] and ε = ε the free boundary problem has a solution.

Lemma (Convergence)

Under the assumption, the solution for γ = 0 and ε = ε coincides with the limitfor γ ↓ 0 of the solutions for the same ε.

Lemma (Interior solution)

Under the assumption, the optimal strategy is interior:

π+ <1ε.

The Limits of Leverage

Trading Boundaries

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6 7 8 9

µ = 8%, σ = 16%, ε = 1%

• Buy (bottom) and Sell (top) boundaries (y) vs. volatility (x), as multiples.• Trivial at zero (0,0) or full investment (1,1).• Boundaries finite even for γ = 0 or γ < 0.

The Limits of Leverage

Alpha vs. Tracking Error (Λ > 0)

TheoremFor small ε,

α =− 3σ2

γ

(γθ∗(θ∗ − 1)

6

)4/3

ε2/3 + O(ε),

TrE =σ√

3(θ∗(θ∗ − 1)

6√γ

)2/3

ε1/3 + O(ε),

whence

α = −√

312

σ2θ2∗(θ∗ − 1)2 ε

TrE+ O(ε4/3).

• Optimal tradeoff between alpha and tracking error.• With low γ lower costs (higher alpha), but more tracking error.• With high γ low tracking error, but also high trading costs.

The Limits of Leverage

Alpha vs. Tracking Error

σ = 16%, ε = 1%

• Log tracking error (y) against log negative alpha (x).• Approximate log-linear relation for small tracking error (right)• Tracking error levels off as no-trading region widens (left).

The Limits of Leverage

Sketch of Argument (1)• Summarize holdings by risky/safe ratio ζt = πt/(1− πt ).• For some λ, conjecture finite-horizon value of the form

Es

[∫ T

s

(µπt −

γσ2

2π2

t

)dt − ε

∫ T

sπt

dϕ↓tϕt

]= V (ζs) + λ(T − s)

• V (ζ) + λ(T − s) +∫ s

0

(µπt − γσ2

2 π2t

)dt − ε

∫ s0 πt

dϕ↓tϕt

supermartingale:

V ′(ζt )dζt + 12 V ′′(ζt )d〈ζt〉t − λdt +

(µπt − γσ2

2 π2t

)dt − επt

dϕ↓tϕt

=

(σ2

2 ζ2t V ′′(ζt ) + µζtV ′(ζt ) + µ ζ

1+ζ −γσ2

2

(ζζ+1

)2− λ)

dt + V ′(ζt )ζtσdBt

+ V ′(ζt )ζt (1 + ζt )dϕ↑tϕt

+(ε ζt

1+ζt− V ′(ζt )ζt (1 + (1− ε)ζt )

)dϕ↓tϕt.

• dt term nonpositive, and zero on [ζ−, ζ+]

dϕ↑t , dϕ↓t terms nonpositive, and zero at ζ−, ζ+ respectively.

The Limits of Leverage

Sketch of Argument (2)

• Hamilton-Jacobi-Bellman equation

σ2

2ζ2

t V ′′(ζt ) + µζtV ′(ζt ) + µζ

1 + ζ− γσ2

2

(ζ

ζ + 1

)2

− λ = 0

• Take derivative: second-order equation for W = −V ′. No λ.

σ2

2ζ2W ′′(ζ) + (σ2 + µ)ζW ′(ζ) + µW (ζ)− 1

(1 + ζ)2

(µ− γσ2ζ

1 + ζ

)= 0

• Four unknowns (c1, c2, ζ−, ζ+), two boundary conditions.• Smooth pasting conditions at ζ− and ζ+.• Now four equations and four unknowns. One solution.• Recover λ from first equation.

The Limits of Leverage

Conclusion

• Maximize average return for fixed volatility. Without frictions, usual frontier.• Trading costs!• Leverage cannot increase expected returns indefinitely.

Maximum leverage multiplier finite.• Multiplier increases with liquidity and returns. Decreases with volatility.• Between two assets with equal Sharpe ratio, more volatility better.

Superior frontier.• Embedded leverage without constraints, but with trading costs.• Optimal tradeoff between alpha and tracking error.

The Limits of Leverage

Thank You!Questions?