The Limits of Leverage
^{0}^{0}0For helpful comments, we thank Jakša Cvitanić, Gur Huberman, Johannes MuhleKarbe, Walter Schachermayer, Ronnie Sircar, Matt Spiegel, René Stulz, Peter Tankov and seminar participants at ETH Zürich, Séminaire Bachelier Paris, Vienna Graduate School of Finance, Quant Europe, Quant USA, Banff Workshop on Arbitrage and Portfolio Optimization, Institute of Finance at USI (Lugano).^{0}^{0}0 Boston University, Department of Mathematics and Statistics, 111 Cummington Mall, Boston, MA 02215, USA, and Dublin City University, School of Mathematical Sciences, Glasnevin, Dublin 9, Ireland, paolo.guasoni@dcu.ie. Partially supported by the ERC (278295), NSF (DMS1412529), SFI (16/IA/4443,16/SPP/3347).^{0}^{0}0 University of Limerick, Department of Mathematics & Statistics, Castletroy, Co. Limerick, Ireland, eberhard.mayerhofer@ul.ie. Partially supported by the ERC (278295) and SFI (08/SRC/FMC1389).Paolo Guasoni
Boston University and Dublin City University
Eberhard Mayerhofer
University of Limerick
When trading incurs proportional costs, leverage can scale an asset’s return only up to a maximum multiple, which is sensitive to its volatility and liquidity. In a model with one safe and one risky asset, with constant investment opportunities and proportional costs, we find strategies that maximize long term returns given average volatility. As leverage increases, rising rebalancing costs imply declining Sharpe ratios. Beyond a critical level, even returns decline. Holding the Sharpe ratio constant, higher asset volatility leads to superior returns through lower costs.
Keywords: leverage, transaction costs, portfolio choice.
Mathematics Subject Classification (2010): 91G10, 91G80
JEL Classification: G11, G12
1. Introduction
If trading is costless, leverage can scale returns without limits. Using the words of Sharpe (2011):
“If an investor can borrow or lend as desired, any portfolio can be leveraged up or down. A combination with a proportion invested in a risky portfolio and in the riskless asset will have an expected excess return of [times the excess return of the risky portfolio] and a standard deviation equal to times the standard deviation of the risky portfolio. Importantly, the Sharpe Ratio of the combination will be the same as that of the risky portfolio.”
In theory, this insight implies that the efficient frontier is linear, that efficient portfolios are identified by their common maximum Sharpe ratio, and that any of them spans all the other ones. Also, if leverage can deliver any expected return, then riskneutral portfolio choice is meaningless, as it leads to infinite leverage.
In practice, hedge funds and highfrequency trading firms employ leverage to obtain high returns from small relative mispricing of assets. A famous example is Long Term Capital Management, which used leverage of up to 40 times to increase returns from convergence trades between ontherun and offtherun treasury bonds (for example, see Edwards (1999)).
This paper shows that trading costs undermine these classical properties of leverage and set sharp theoretical limits to its applications. We start by characterizing the set of portfolios that maximize long term expected returns for given average volatility, extending the familiar efficient frontier to a market with one safe and one risky asset, where both investment opportunities and relative bidask spreads are constant. Figure 1 plots this frontier: expectedly, trading costs decrease returns, with the exception of a fully safe investment (the axes origin) or a fully risky investment (the attachment point with unit coordinates), which lead to static portfolios without trading, and hence earn their frictionless returns.^{1}^{1}1As we focus on long term investments, we neglect the oneoff costs of set up and liquidation, which are negligible over a long holding period.
But trading costs do not merely reduce expected returns below their frictionless benchmarks. Unexpectedly, in the leverage regime (to the right of the fullinvestment point) rebalancing costs rise so quickly with volatility that returns cannot increase beyond a critical factor, the leverage multiplier. This multiplier depends on the relative bidask spread , the expected excess return and volatility , and approximately equals
(1.1) 
Table 1 shows that even a modest bidask spread of 0.10% implies a multiplier of 23 for an asset with 10% volatility and 5% expected return (similar to a longterm bond), while the multiplier declines to 10 for an asset with equal Sharpe ratio, but volatility of 50% (similar to an individual stock). Leverage opportunities are much more limited for more illiquid assets with a spread of 1%: the multiplier declines from less than 8 for 10% volatility to less than 4 for 50% volatility. Importantly, these limits on leverage hold even allowing for continuous trading, infinite market depth (any quantity trades at the bid or ask price), and zero capital requirements.
BidAsk Spread ()  

Volatility ()  0.01%  0.10%  1.00% 
10%  71.85 (71.22)  23.15 (22.58)  7.72 (7.12) 
20%  50.88 (50.36)  16.45 (15.92)  5.56 (5.04) 
50%  32.30 (31.85)  10.54 (10.07)  3.66 (3.18) 
Our results have two broad implications. First, with a positive bidask spread even a riskneutral investor who seeks to maximize expected longrun returns takes finite leverage, and in fact a rather low leverage ratio in an illiquid market – riskneutral portfolio choice is meaningful. The resulting multiplier sets an endogenous level of risk that the investor chooses not to exceed regardless of risk aversion, simply to avoid reducing returns with trading costs. In this context, margin requirements based on volatility (such as value at risk and its variations) are binding only when they reduce leverage below the multiplier, and are otherwise redundant. In addition, the multiplier shows that an exogenous increase in trading costs, such as a proportional Tobin tax on financial transactions, implicitly reduces the maximum leverage that any investor who seeks return is willing to take, regardless of risk attitudes.
Second, two assets with the same Sharpe ratio do not generate the same efficient frontier with trading costs, and more volatility leads to a superior frontier. For example (Table 1) with a 1% spread, the maximum leveraged return on an asset with 10% volatility and 5% return is . By contrast, an asset with 50% volatility and 25% return (equivalent to the previous one from a classical viewpoint, as it has the same Sharpe ratio 0.5), leads to a maximum leveraged return of . The reason is that a more volatile asset requires a lower leverage ratio (hence lower rebalancing costs) to reach a certain return. Thus, an asset with higher volatility spans an efficient frontier that achieves higher returns through lower costs.
This paper bears on the established literature on portfolio choice with frictions. The effect of transaction costs on portfolio choice is first studied by Magill and Constantinides (1976), Constantinides (1986), and Davis and Norman (1990), who identify a wide notrade region, and derive the optimal trading boundaries through numerical procedures. While these papers focus on the maximization of expected utility from intertemporal consumption on an infinite horizon, Taksar, Klass and Assaf (1988), and Dumas and Luciano (1991) show that similar strategies are obtained in a model with terminal wealth and a long horizon – time preference has negligible effects on trading policies. This paper adopts the same approach of a long horizon, both for the sake of tractability, and because it focuses on the tradeoff between return, risk, and costs, rather than consumption.
Our asymptotic results for positive risk aversion are similar in spirit to the ones derived by Shreve and Soner (1994), Rogers (2004), Gerhold, Guasoni, MuhleKarbe and Schachermayer (2014), and Kallsen and MuhleKarbe (2015), whereby transaction costs imply a notrade region with width of order and welfare costs of order . We also find that the trading boundaries obtained from a local meanvariance criterion are equivalent at the first order to the ones obtained from power utility.
The riskneutral expansions and the limits of leverage of order are new, and are qualitatively different from the riskaverse case. These results are not regular perturbations of a frictionless analogue, which is illposed. They are rather singular perturbations, which display the speed at which the frictionless problem becomes illposed as the crucial friction parameter vanishes.
Finally, this paper connects to the recent work of Frazzini and Pedersen (2012) on embedded leverage. If different investors face different leverage constraints, they find that in equilibrium assets with higher factor exposures trade at a premium, thereby earning a lower return. Frazzini and Pedersen (2014) confirm this prediction across a range of markets and asset classes, and Asness et al. (2012) use it to explain the performance riskparity strategies. With exogenous asset prices, we find that assets with higher volatility generate a superior efficient frontier by requiring lower rebalancing costs for the same return. This observation suggests that the embedded leverage premium may be induced by rebalancing costs in addition to leverage constraints, and should be higher for more illiquid assets.
The paper is organized as follows: section 2 introduces the model and the optimization problem. Section 3 contains the main results, which characterize the efficient frontier in the riskaverse (Theorem 3.1) and riskneutral (Theorem 3.2) cases. Section 4 discusses the implications of these results for the efficient frontier, the trading boundaries of optimal policies and the embedded leverage effect. The section includes two supporting results, which show that the riskneutral solutions arise as limits of their riskaverse counterparts for low riskaversion (Theorem 4.1), and that the riskneutral solutions are not constrained by the solvency condition (section 4.2). Section 5 offers a derivation of the main freeboundary problems from heuristic control arguments, and concluding remarks are in section 6. All proofs are in the appendix.
2. Model
The market includes one safe asset earning a constant interest rate of and a risky asset with ask (buying) price that follows
where is a standard Brownian motion. The risky asset’s bid (selling) price is , which implies a constant relative bidask spread of , or, equivalently, constant proportional transaction costs.
We investigate the tradeoff between a portfolio’s average return against its realized variance. Denoting by the portfolio value at time , for an investor who observes returns with frequency in the timeinterval , the average return and its continuoustime approximation are^{2}^{2}2All discrete statistics on this section converge in probability to their continuoustime counterparts. The budget equation and the definition of admissible strategies are in appendix A below.
In the familiar setting of no trading costs, , where is the portfolio weight of the risky asset, hence the average return equals the average risky exposure times its excess return, plus the safe rate.
Likewise, the average squared volatility on is obtained by the usual variance estimator applied to returns, and has the continuoustime approximation
reducing to in the absence of trading costs.
With these definitions, the meanvariance tradeoff is captured by maximizing
(2.1) 
where the parameter is interpreted as a proxy for riskaversion.
This objective nests several familiar problems. Without trading costs it reduces to
(2.2) 
which is maximized by the optimal constantproportion portfolio dating back to Markowitz and Merton, and confirms that in a geometric Brownian motion market with costless trading, the objective considered here is equivalent to utilitymaximization with constant relative risk aversion. With or without transaction costs, the riskneutral objective boils down to the average annualized return over a long horizon, while reduces to logarithmic utility.
Trading costs make (2.1) lower than (2.2), as they hinder continuous portfolio rebalancing and make constantproportion strategies unfeasible. The reason is that it is costly to keep the exposure to the risky asset high enough to achieve the desired return, and low enough to limit the level of risk – trading costs reduce returns and increase risk.
To neglect the spurious, nonrecurring effects of portfolio setup and liquidation, we focus on the Equivalent Safe Rate^{3}^{3}3In this equation the is used merely to guarantee a good definition a priori. A posteriori, optimal strategies exist in which the limit superior is a limit, hence the similar problem defined in terms of leads to the same solution.
(2.3) 
which is akin to the one used by Dumas and Luciano (1991) in the context of utility maximization.
3. Main Results
3.1. Risk aversion and efficient frontier
The first result characterizes the optimal solution to the main objective in (2.3) in the usual case of a positive aversion to risk (). In this setting, the next theorem shows that trading costs create a notrade region around the frictionless portfolio , and states the asymptotic expansions of the resulting average return and standard deviation^{4}^{4}4The exact formulae for average return, standard deviation, and average trading costs are in Appendix C., thereby extending the familiar efficient frontier to account for trading costs.
Theorem 3.1.
Let .

For any there exists such that for all , there is a unique solution , with , for the free boundary problem
(3.1) (3.2) (3.3) (3.4) (3.5) 
The trading strategy that buys at and sells at as little as to keep the risky weight within the interval is optimal.

The maximum performance is
(3.6) where is the set of admissible strategies in Definition A.1 below, is the number of shares held at time , and is the cumulative number of shares sold up to time .

The trading boundaries and have the asymptotic expansions
(3.7) The longrun mean (), standard deviation (), Sharpe ratio (), average trading costs () and equivalent safe rate () have expansions^{5}^{5}5We are using the convention for any and odd integer , and .
(3.8) (3.9) SR (3.10) (3.11) (3.12)
3.2. Risk neutrality and limits of leverage
In contrast to the riskaverse objective considered above, the riskneutral objective leads to a solution which does not have a frictionless analogue: for small trading costs, both the optimal policy and its performance become unbounded as the optimal leverage increases arbitrarily. The next result describes the solution to the riskneutral problem, identifying the approximate dependence of the leverage multiplier and its performance on the asset’s risk, return and liquidity.
Theorem 3.2.
Let .

The trading strategy that buys at and sells at as little as to keep the risky weight within the interval is optimal.

The maximum expected return is
(3.13) 
The trading boundaries have the series expansions
(3.14) (3.15) where is the unique solution to
(3.16)
Proof.
See Appendix D below. ∎
The next section discusses how these results modify the familiar intuition about risk, return, and performance evaluation in the context of trading costs.
4. Implications and Applications
4.1. Efficient frontier
Theorem 3.1 extends the familiar efficient frontier to account for trading costs. Compared to the linear frictionless frontier, average returns decline because of rebalancing losses. Average volatility increases because more risk becomes necessary to obtain a given return net of trading costs.
To better understand the effect of trading costs on return and volatility, consider the dynamics of the portfolio weight in the absence of trading, which is
(4.1) 
The central quantity here is the portfolio weight volatility , which vanishes for the singleasset portfolios or , remains bounded above by in the longonly case , and rises quickly with leverage (). This quantity is important because it measures the extent to which a portfolio, left to itself, strays from its initial composition in response to market shocks and, by reflection, the quantity of trading that is necessary to keep it within some region. In the longonly case, the portfolio weight volatility decreases as the notrade region widens to span , which means that a portfolio tends to spend more time near the boundaries. By contrast, with leverage portfolio weight volatility increases, which means that a wider boundary does not necessarily mitigate trading costs.
Consistent with this intuition, equation (3.8) shows that the impact of trading costs is small on longonly portfolios, but rises quickly with leverage, reducing returns at the order of for . Of course, this expansion is valid for small while holding the value of fixed. As declines to zero, both the expected return and volatility diverge, but so does the impact of trading costs, making the asymptotics for uninformative for the riskneutral limit .
The performance (3.12) coincides at the first order with the equivalent safe rate from utility maximization with constant relative risk aversion (Gerhold et al., 2014, Equation (2.4)), supporting the interpretation of as a riskaversion parameter, and confirming that, for asymptotically small costs, the efficient frontier captures the riskreturn tradeoff faced by a utility maximizer.
Figure 2 displays the effect of trading costs on the efficient frontier. As the bidask spread declines, the frontier increases to the linear frictionless frontier, and the asymptotic results in the theorem become more accurate. However, if the spread is held constant as leverage (hence volatility) increases, the asymptotic expansions become inaccurate, and in fact the efficient frontier ceases to increase at all after the leverage multiplier is reached.
4.2. Trading boundaries
Each point in the efficient frontier corresponds to a rebalancing strategy that is optimal for some value of the riskaversion parameter . For small trading costs, equation (3.7) implies that the trading boundaries corresponding to the efficient frontier depart from the ones arising in utility maximization, which are (Gerhold et al., 2014)
(4.2) 
The term of order vanishes for because this case coincides with the maximization of logarithmic utility. For high levels of leverage ( and ), this term implies that the trading boundaries that generate the efficient frontier are lower than the trading boundaries that maximize utility. In Figure 3, corresponds to the safe portfolio in the origin (0,0), while to the risky investment (1,1), which has by definition the same volatility and return as the risky asset. As declines to zero, the trading boundaries converge to the right endpoints, which correspond to the strategy that maximizes average return with no regard for risk, thereby achieving the multiplier.
As leverage increases, the sell boundary rises more quickly than the buy boundary (Figure 3). For example, the riskneutral portfolio tolerates leverage fluctuations from approximately to . The locations of these boundaries trade off the need to keep exposure to the risky asset high to maximize return while also keeping rebalancing costs low. Risk aversion makes boundaries closer to each other by penalizing the high realized variance generated by the wide riskneutral boundaries.
Importantly, these boundaries remain finite even as the frictionless Merton portfolio diverges to infinity with declining to zero. Thus the notrade region is not symmetric around the frictionless portfolio, in contrast to the boundaries arising from utility maximization (Gerhold et al., 2014), which are always symmetric, and hence diverge when is low. The difference is that here the riskneutral objective is to maximize the expected return of the portfolio, while a riskneutral utility maximizer focuses on expected wealth. In a frictionless setting this distinction is irrelevant, and an investor can use a returnmaximizing policy to maximize wealth instead. But trading costs drive a wedge between these two ostensibly equivalent riskneutral criteria – maximizing expected return is not the same as maximizing expected wealth.
In the riskneutral case (Theorem 3.2 (iv)) the optimal trading boundaries satisfy the approximate relation
(4.3) 
which is universal in that it holds for any asset, regardless of risk, return and liquidity. This relation means that an optimal riskneutral rebalancing strategy should always tolerate wide variations in leverage over time, and that the maximum allowed leverage should be approximately 2.5 times the minimum. More frequent rebalancing cannot achieve the maximum return: it can be explained either by risk aversion or by elements that lie outside the model, such as price jumps.
Finally, note that the solvency constraint that wealth remain positive at all times implies that^{6}^{6}6 Any potentially optimal strategy has positive exposure (), as the asset price has a positive risk premium (Remark A.4 below). Denoting by the safe position at time , where is total portfolio wealth, the liquidation value is , which implies and thus the claim. for every admissible trading strategy. As for the optimal trading strategy in Theorem 3.1 and Theorem 3.2, the upper bound is never binding for realistic bidaskspreads.
4.3. Embedded leverage
In frictionless markets, two perfectly correlated assets with equal Sharpe ratio generate the same efficient frontier, and in fact the same payoff space. This equivalence fails in the presence of trading costs: as the more volatile asset has a proportionally higher return, it can be traded to generate higher returns with lower leverage ratios, resulting in an efficient frontier that dominates (for high returns) the one generated by the less volatile asset. Figure 4 (top of the three curves) displays this phenomenon: for example, a portfolio with an average return of 50% net of trading costs is obtained from an asset with 25% return and 50% volatility at a small cost, as an average leverage factor of 2 entails moderate rebalancing.
Achieving the same 50% return from an asset with 20% volatility (and 10% return) is more onerous: trading costs require leverage higher than 5, which in turn increases trading costs. Overall, the resulting portfolio needs about 120% rather than 100% volatility to achieve the desired 50% average return (middle curve in Figure 4).
From an asset with 10% volatility (and 5% return), obtaining a 50% return net of trading costs is impossible (bottom curve in Figure 4), because the leverage multiplier is less than 8 (Table 1, top right), and therefore the return can be scaled to less than 40%. The intuition is clear: increasing leverage also increases trading costs, which in turn call for more leverage to increase return, but also increase costs. At some point, the marginal net return from more leverage becomes zero, and further increases are detrimental.
Because an asset with higher volatility is superior to another one, with equal Sharpe ratio and perfectly correlated, but with lower volatility, the model suggests that in equilibrium they cannot coexist, and that the asset with lower volatility should offer a higher return to be held by investors. Indeed, Frazzini and Pedersen (2012, 2014) document significant negative excess returns in assets with embedded leverage (higher volatility), and offer a theoretical explanation based on heterogeneous leverage constraints, which lead more constrained investors to bid up prices (and hence lower returns) of more volatile assets. Our results hint that the same phenomenon may arise even in the absence of constraints, as a result of rebalancing costs. In contrast to constraintsbased explanations, our model suggests that the premium for embedded leverage should be higher for more illiquid assets.
4.4. From risk aversion to risk neutrality
Theorems 3.1 and 3.2 are qualitatively different: while Theorem 3.1 with positive risk aversion leads to a regular perturbation of the MarkowitzMerton solution, Theorem 3.2 with riskneutrality leads to a novel result with no meaningful analogue in the frictionless setting – a singular perturbation. Furthermore, a close reading of the statement of Theorem 3.1 shows that the existence of a solution to the freeboundary problem, and the asymptotic expansions, hold for less than some threshold that depends on the risk aversion . In particular, if approaches zero while is held constant, Theorem 3.1 does not offer any conclusions on the convergence of the riskaverse to the riskneutral solution. Still, if the riskneutral result is to be accepted as a genuine phenomenon rather than an artifact, it should be clarified whether the risk averse trading policy and its performance converge to their risk neutral counterparts as risk aversion vanishes. The next result resolves this point under some parametric restrictions. Denote by
and associate to any solution of the free boundary problem (3.1) the function
which naturally extends to the left and right of the freeboundaries.
Theorem 4.1.
Let , , and , and assume that for any the free boundary problem (3.1) has a unique solution satisfying and that the function satisfies, for each , the HJB equation
(4.4) 
Then, (4.4) is satisfied also for , and for each , the trading strategy that buys at and sells at to keep the risky weight within the interval is optimal. Furthermore, and as , each .
5. Heuristic Solution
This section offers a heuristic derivation of the HJB equation. Let and denote the cumulative number of shares bought and sold, respectively. The finitehorizon objective (2.1) reduces to the expression (compare eq. (A.8) in Lemma A.2 below)
(5.1) 
From the outset, this objective is scaleinvariant: doubling the initial number of risky shares and safe units, and also doubling the number of shares held at time results in doubling also the number of safe units at time (through the selffinancing condition), thereby leaving , , and hence the objective, unchanged. Thus, we conjecture that the residual value function depends on the calendar time and on the variable , which denotes the number of shares held for each unit of the safe asset. In terms of this variable, the conditional value of the above objective at time becomes:
(5.2) 
By Itô’s formula, the dynamics of is (henceforth the arguments of are omitted for brevity)
where the subscripts of denote partial derivatives. Recall now the selffinancing condition for the safe position and the risky position :
which implies the dynamics for the riskysafe ratio
whence the dynamics of simplifies to
(5.3)  
(5.4) 
Now, by the martingale principle of optimal control (Davis and Varaiya, 1973) the process above needs to be a supermartingale for any trading policy and a martingale for the optimal policy. As and are increasing processes, the supermartingale condition implies^{7}^{7}7In particular, the coefficients of and need to be negative. As short positions are never optimal (cf. Remark A.3 and footnote 6), it follows that , whence only two cases arise: (a) , or (b) . In both cases and , whence (5.5) follows. the inequalities
(5.5) 
and the martingale condition prescribes that the left (respectively, right) inequality becomes an equality at the points of increase of (resp. ). Likewise, it follows that
with the inequality holding as an equality whenever both inequalities in (5.5) are strict. To achieve a stationary (that is, timehomogeneous) system, suppose that the residual value function is of the form for some to be determined, which represents the average optimal performance over a long period of time. Replacing this parametric form of the solution, the above inequalities become
(5.6)  
(5.7) 
Assuming further that the first inequality holds over some interval , with each inequality reducing to an equality at the respective endpoint, the optimality conditions become
(5.8)  
(5.9) 
which lead to a family of candidate value functions, each of them corresponding to a pair or boundaries (). The optimal boundaries are identified by the smoothpasting conditions, formally derived by differentiating eqs. (5.9) with respect to their boundaries
(5.10) 
These conditions identify the value function. The four unknowns are the free parameter in the general solution to the ordinary differential equation (5.8), the free boundaries and , and the optimal rate . These quantities are identified by the boundary and smoothpasting conditions (5.9)–(5.10).
6. Conclusion
The costs of rebalancing a leveraged portfolio are substantial, and detract from its ostensible frictionless return. As leverage increases, such costs rise faster than the return, making it impossible for an investor to lever an asset’s return beyond a certain multiple, net of trading costs.
In contrast to the frictionless theory, trading costs make the riskreturn tradeoff nonlinear. An investor who seeks high return prefers an asset with high volatility to another one with equal Sharpe ratio but lower volatility, because higher volatility makes leverage cheaper to realize. A riskneutral, returnmaximizing investor does not take infinite leverage, but rather keeps it within a band that balances high exposure with low rebalancing costs.
Appendix A Admissible Strategies
In view of transaction costs, only finitevariation trading strategies are consistent with solvency. Denote by and the wealth in the safe and risky positions respectively, and by and the cumulative number of shares bought and sold, respectively. The selffinancing condition prescribes that satisfy the dynamics
(A.1) 
A strategy is admissible if it is nonanticipative and solvent, up to a small increase in the spread:
Definition A.1.
Let (the initial capital) and let and be continuous, increasing processes, adapted to the augmented natural filtration of . Then is an admissible trading strategy if

its liquidation value is strictly positive at all times: There exists such that the discounted asset satisfies
(A.2) 
The following integrability conditions hold^{8}^{8}8 Note that , therefore on the set the quantity is welldefined.
(A.3) where denotes the total variation of on .
The family of admissible trading strategies is denoted by .
The following lemma describes the dynamics of the wealth process , the risky weight , and the riskysafe ratio .
Lemma A.2.
For any admissible trading strategy , ^{9}^{9}9The notation means , hence the SDEs are well defined even for null .
(A.4)  
(A.5)  
(A.6) 
For any such strategy, the functional
(A.7) 
equals to
(A.8) 
Proof.
The selffinancing conditions (A.1) imply that
(A.9)  
(A.10)  
(A.11) 
Equation (A.4) follows from the last equation, and (A.5) holds in view of equations (A.9) and (A.10). Equation (A.6) follows from the identity and (A.4). The expression in (A.8) for the objective functional follows from equation (A.5). ∎
The following lemma shows that, without loss of generality, it is enough to consider trading strategies which do not take short positions in the risky asset.
Lemma A.3.
If is optimal for (2.3), then also the strategy is optimal.
Proof.
Remark A.4.
In view of this Lemma and admissibility, it suffices to consider trading strategies which satisfy , or, in terms of the riskysafe ratio, or .
Appendix B Risk Aversion and Efficient Frontier
This section contains a series of propositions that lead to the proof of Theorem 3.1 (i)–(iii). Part (iv) of the theorem is postponed to Appendix C. Set
(B.1) 
Defining , the free boundary problem (3.1)–(3.5) reduces to
(B.2)  
(B.3)  
(B.4)  
(B.5)  
(B.6) 
Proposition B.1.
Proof of Proposition b.1.
Note that (B.2) is equivalent to the ODE
and thus, the initial conditions (B.3), (B.4) imply that satisfies
By the variation of constants method, and as , any solution of the initial value problem (B.2)–(B.4) is thus of the form
(B.8) 
Suppose is a solution of (B.2)–(B.6). In view of (B.8), . Let
(B.9) 
By the terminal conditions (B.5)–(B.6) at , and setting , satisfy the following system of algebraic equations,
(B.10)  