When trading incurs proportional costs, leverage can scale an asset's return only up to a maximum multiple, which is sensitive to the asset's volatility and liquidity. In a continuous-time model with one safe and one risky asset with constant investment opportunities and proportional transaction costs, we find the efficient portfolios that maximize long term expected returns for given average volatility. As leverage and volatility increase, rising rebalancing costs imply a declining Sharpe ratio. Beyond a critical level, even the expected return declines. For funds that seek to replicate multiples of index returns, such as leveraged ETFs, our efficient portfolios optimally trade off alpha against tracking error.

1. The Limits of Leverage
The Limits of Leverage
Paolo Guasoni1,2
Eberhard Mayerhofer2
Boston University1
Dublin City University2
Mathematical Finance and Partial Differential Equations
May 1st
, 2015

2. 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.

3. The Limits of Leverage
Unlimited Leverage?
“If an investor can borrow or lend as desired, any portfolio can be levered up
or down. A combination with a proportion k invested in a risky portfolio and
1 − k in the riskless asset will have an expected excess return of k and a
standard deviation equal to k times the standard deviation of the risky
portfolio.” — 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.

4. 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
µ
σ2
1/2
ε−1/2
ε bid-ask spread, µ excess return, σ volatility.
• Optimal tradeoff between alpha and tracking error.

5. 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.

6. The Limits of Leverage
Leverage Multiplier
Bid-Ask Spread (ε)
Volatility (σ) 0.01% 0.10% 1.00%
10% 71.85 23.15 7.72
20% 50.88 16.45 5.56
50% 32.30 10.54 3.66
Sharpe ratio µ/σ = 0.5
• Approximate value ≈ 0.3815 µ
σ2
1/2
ε−1/2
• Increases with return, decreases with volatility and spread
• Always lower than solvency level ε−1
. Endogenous limit.

7. 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 = rwt dt + ϕt dSt − εdϕ↓
t
• Solvency constraint wt − ε(ϕt )+
St ≥ 0 a.s. for all t ≥ 0.

8. 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 =
1
T
0≤t≤T
t=k∆t
rt ≈
1
T
T
0
dwt
wt
Frictionless: 1
T
T
0
µπt dt + 1
T
T
0
σπt dWt , with πt portfolio.
• Average volatility on [0, T]
1
T
0≤t≤T
t=k∆t
(rt − ¯rT ∆t)2
1/2
≈
1
T
T
0
d w t
w2
t
1/2
Frictionless: σ2
T
T
0
π2
t dt.
• Tracking error of fund w on multiple Λ of benchmark S on [0, T]:
1
T
0≤t≤T
t=k∆t
(rt − ΛrB
t )2
1/2
≈
1
T
·
0
dw
w
− Λ
dS
S T
1/2
Frictionless: σ2
T
T
0
(πt − Λ)2
dt. Same as average volatility for Λ = 0.

9. The Limits of Leverage
Objective
• Maximize return-volatility tradeoff for large T
1
T
E
T
0
dwt
wt
− Λ
dSt
St
−
γ
2
·
0
(πt − Λ)
dSt
St T
• Equals to
r +
1
T
E
T
0
µ(πt − Λ) −
γσ2
2
(πt − Λ)2
dt − ε
T
0
πt
dϕ↓
t
ϕt
• Λ = 0, ε = 0: Usual mean-variance portfolio πt = µ
γσ2 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.

10. The Limits of Leverage
Efficient Frontier (γ > 0, Λ = 0)
Theorem
Trade to keep portfolio weight πt within boundaries π− (buy) and π+ (sell)
π± = θ∗ ± 3
4γ (θ∗)2
(θ∗ − 1)2
1/3
ε1/3
− (1−γ)θ∗+γΛ
γ
γθ∗(θ∗−1)
6
1/3
ε2/3
+ O(ε)
where θ∗ = Λ + µ/(γσ2
) and ζ± solve the free-boundary problem (W, ζ−, ζ+)
1
2 σ2
ζ2
W (ζ) + (σ2
+ µ)ζW (ζ) + µW(ζ) − 1
(1+ζ)2 µ + γσ2
Λ − γσ2
ζ
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.

11. The Limits of Leverage
Limits of Leverage (γ = Λ = 0)
Theorem
Trade to keep portfolio weight πt within boundaries π− (buy) and π+ (sell)
π± = ζ±
1+ζ±
= B±κ1/2
(µ/σ2
)
1
2 ε−1/2
+ 1 + O(ε
1
2 ),
where B− = (1 − κ), B+ = 1 and κ ≈ 0.5828 is the root of 3
2 κ + log(1 − κ) = 0.
ζ± solve the free-boundary problem (W, ζ−, ζ+)
1
2 σ2
ζ2
W (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.

12. 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 limit
for γ ↓ 0 of the solutions for the same ¯ε.
Lemma (Interior solution)
Under the assumption, the optimal strategy is interior:
π+ <
1
ε
.

13. 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.

14. The Limits of Leverage
Alpha vs. Tracking Error (Λ > 0)
Theorem
For 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 = −
√
3
12
σ2
θ2
∗(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.

15. 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).

16. 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
2
π2
t dt − ε
T
s
πt
dϕ↓
t
ϕt
= V(ζs) + λ(T − s)
• V(ζ) + λ(T − s) +
s
0
µπt − γσ2
2 π2
t dt − ε
s
0
πt
dϕ↓
t
ϕt
supermartingale:
V (ζt )dζt + 1
2 V (ζt )d ζt t − λdt + µπt − γσ2
2 π2
t dt − επt
dϕ↓
t
ϕt
= σ2
2 ζ2
t V (ζt ) + µζt V (ζt ) + µ ζ
1+ζ − γσ2
2
ζ
ζ+1
2
− λ dt + V (ζt )ζt σdBt
+ V (ζt )ζt (1 + ζt )
dϕ↑
t
ϕt
+ ε ζt
1+ζt
− V (ζt )ζt (1 + (1 − ε)ζt )
dϕ↓
t
ϕt
.
• dt term nonpositive, and zero on [ζ−, ζ+]
dϕ↑
t , dϕ↓
t terms nonpositive, and zero at ζ−, ζ+ respectively.

17. The Limits of Leverage
Sketch of Argument (2)
• Hamilton-Jacobi-Bellman equation
σ2
2
ζ2
t V (ζt ) + µζt V (ζt ) + µ
ζ
1 + ζ
−
γσ2
2
ζ
ζ + 1
2
− λ = 0
• Take derivative: second-order equation for W = −V . No λ.
σ2
2
ζ2
W (ζ) + (σ2
+ µ)ζW (ζ) + µW(ζ) −
1
(1 + ζ)2
µ −
γσ2
ζ
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.

18. 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.

19. The Limits of Leverage
Thank You!
Questions?