BIS Working Papers. Liquidity risk in markets with trading frictions: What can swing pricing achieve? No 663. Monetary and Economic Department

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1 BIS Working Paers No 663 Liquidity risk in markets with trading frictions: What can swing ricing achieve? by Ulf Lewrick and Jochen Schanz Monetary and Economic Deartment October 207 JEL classification: G0, G23, G28, C72 Keywords: Financial stability, mutual funds, regulation, liquidity insurance, trading frictions

2 BIS Working Paers are written by members of the Monetary and Economic Deartment of the Bank for International Settlements, and from time to time by other economists, and are ublished by the Bank. The aers are on subjects of toical interest and are technical in character. The views exressed in them are those of their authors and not necessarily the views of the BIS. This ublication is available on the BIS website ( Bank for International Settlements 207. All rights reserved. Brief excerts may be reroduced or translated rovided the source is stated. ISSN (rint) ISSN (online)

3 Liquidity risk in markets with trading frictions: What can swing ricing achieve? Ulf Lewrick a,, Jochen Schanz a,2 a Bank for International Settlements, Centralbahnlatz 2, CH-4002 Basel, Switzerland Abstract Oen-end mutual funds exose themselves to liquidity risk by granting their investors the right to daily redemtions at the fund's net asset value. We assess how swing ricing can damen such risks by allowing the fund to settle investor orders at a rice below the fund's net asset value. This reduces investors' incentive to redeem shares and mitigates the risk of large destabilising outflows. Otimal swing ricing balances this risk with the benefit of roviding liquidity to cash-constrained investors. We derive bounds, deending on trading costs and the share of liquidity-constrained investors, within which a fund chooses to swing the settlement rice. We also show how the otimal settlement rice resonds to unanticiated shocks. Finally, we discuss whether swing ricing can hel mitigate the risk of self-fulfilling runs on funds. Keywords: Financial stability, mutual funds, regulation, liquidity insurance, trading frictions JEL: G0, G23, G28, C72 We thank Stijn Claessens, Ingo Fender, Leslie Kain, Kostas Tsatsaronis as well as seminar articiants at the Bank for International Settlements and the Bank of Jaan for helful comments. Disclaimer: The views exressed in this article are those of the authors and do not necessarily reflect those of the Bank for International Settlements. ulf.lewrick@bis.org. Phone: Corresonding author. jochen.schanz@bis.org. Phone:

4 Introduction A key role of nancial intermediaries is to rovide liquidity essentially, on-demand access to cash to their investors. Tyically, nancial intermediaries that rovide liquidity also engage in maturity transformation. For examle, banks issue long-term loans but grant their deositors the right to withdraw their funds on demand. Similarly, oen-end mutual funds ("funds") that invest in comaratively illiquid securities, such as cororate bonds, give their investors the otion of redeeming their shares in cash every day. Daily redemtions allow fund investors to insure against their liquidity needs while articiating in the higher return their fund earns on less liquid assets. At the same time, funds need to adequately insure the residual liquidity risk that they incur. amlify nancial crises. Insu ciently insured liquidity risk can trigger and In this aer, we assess the e ect of swing ricing a tool for managing liquidity risk in funds, which several tyes of US funds will be able to use from November 208 onwards. 2 Swing ricing ermits a fund to ay out less than net asset value (NAV 3 ) er share when net redemtions are large. It thereby alleviates a situation in which the fund, in the absence of swing ricing, would have to sell assets at a large discount to generate su cient cash to ay out its redeeming shareholders. Symmetrically, swing ricing allows the fund to raise the rice er share above the NAV er share when the fund exeriences large net in ows. It can thus hel to ensure that the costs associated with urchasing additional assets are borne by incoming investors. If investors anticiate that the fund will settle share transactions above the NAV when net demand for its shares is high, and below the NAV when net demand is negative, swing ricing can hel reduce the volatility of ows into and out of the fund. Our model builds on Diamond and Dybvig (983) and Jacklin (997). We derive an otimal rule for adjusting the single rice at which a fund settles both share redemtions and subscritions (the "settlement rice"). The settlement rice in our model summarises a range of contract features that in ractice the fund can o er to its investors: for examle, a higher settlement rice could also corresond to faster settlement of redemtion requests or more generous maximum daily redemtion amounts. In our model, we assume that investors can either urchase assets or invest indirectly in those assets by urchasing fund shares. There is a large number of small funds whose managers maximise their investors utility. At the time the fund is set u, fund managers commit to a rule whereby they set the settlement rice. (In ractice, the fund discloses the general terms of its swing ricing olicy in its rosectus.) Fund managers take into account that investors might become cash constrained, romting them to redeem their shares earlier than they exected and indeendently of the settlement rice. The higher the settlement rice, the more investors obtain if they turn out to be cash constrained, at the exense of those who stay with the fund. In this context, three arameters determine the otimal settlement rice: the cost of trading in the asset market; the likelihood with which investors become cash constrained; and the degree of investor risk aversion. Their imact is best understood against the background of the otimal investment contract. This contract entails investors receiving less when they need to redeem their shares early than if they hold onto their shares until the fund s assets mature. Thus, even if the return of the fund s assets was certain, the return of an investment in fund shares is risky because the investor is uncertain whether he will have to redeem his shares early. isk-averse investors dislike this uncertainty. The bene t of raising the settlement rice for early redemtions is that investors ayo s become less volatile. The fund manager balances these bene ts with the larger costs he incurs when selling assets to serve redemtion requests. See Financial Stability Board (207) for olicy recommendations to address liquidity risk in asset management activites. 2 Securities and Exchange Commission (206). 3 The net asset value is the market value of the fund s assets net of the fund s liabilities.

5 Against this background, consider rst the imact of trading frictions on the otimal settlement rice. 4 The settlement rice is lower when trading frictions rise because the fund manager aims to reduce the costs associated with asset sales. Higher trading costs raise the marginal cost of raising the settlement rice for early redemtions without altering the bene ts. However, the fund does not ass on the entire marginal costs of selling assets, thereby raising the return of cash-constrained investors and reducing the variability of investors returns. Second, the otimal settlement rice falls the more likely investors are to become cash constrained: ceteris aribus, more cash-constrained investors imly greater redemtions, so the marginal cost of raising the settlement rice increases. Finally, the settlement rice increases if investors become more risk averse. This follows because the more risk-averse investors, the more they dislike variations in the return of their investment. The marginal bene t of raising the settlement rice increases, without altering the costs. We also show that the otimal settlement rice lies within a bound that is determined by the trading frictions. If investors can anticiate the fund s settlement rice, they may move to arbitrage any di erence between the settlement rice and the market value of the fund s shares: for examle, by redeeming fund shares and urchasing directly the assets the fund holds. If a fund manager settled net redemtion requests too far below the market value of the shares, investors would start rushing into the fund, diluting the value of the fund s assets er share. Corresondingly, if the settlement rice was too far above the market value of the fund s shares, its existing investors would redeem their shares and cause the fund to be liquidated early. A fund manager wants to avoid both situations because they would reduce the ayo for the fund s existing shareholders. He therefore only varies the settlement rice to the extent that it does not generate arbitrage-driven ows in and out of the fund and holds su cient liquid assets to ay out cash-constrained investors. Finally, we discuss whether swing ricing might hel mitigate the risk of self-ful lling runs on the fund. Within our theoretical framework, we show that swing ricing can revent self-ful lling runs: the fund manager can commit to setting a su ciently low settlement rice to discourage redemtions by investors that are not cash constrained. In ractice, however, swing ricing might be less e ective, rimarily because of uncertainty about how low the settlement rice would need to be set without unduly enalising cash-constrained investors that need to redeem shares. The remainder of the aer is structured as follows. Section 2 discusses how the aer relates to the literature. Section 3 describes the modeling framework. To rovide a benchmark for the fund s choice of settlement rices, Section 4 derives the welfare-otimal solution from the view of a lanner who is able to directly allocate consumtion goods to households. Section 5 comares the lanner s solution with the decentralised one and derives the rule for setting the settlement rice and the fund s otimal liquidity bu er. Section 6 discusses comarative static roerties of otimal settlement rices and asks whether swing ricing might hel mitigate the risks of self-ful lling runs on funds. Section 7 discusses the assumtions made and their in uence on the results. Section 8 concludes. 2 elated literature The aer is related to the literature that studies liquidity insurance rovided by nancial intermediaries and markets. 5 A key question is whether the existence of nancial intermediaries can be exlained by their ability to insure risk-averse households against idiosyncratic liquidity shocks. Diamond and Dybvig (983) show that a well-designed deosit contract can rovide households with the welfare-otimal degree 4 In ractice, a multitude of market frictions create costs for trading in asset markets. See Madhavan (2000) for a discussion. 5 More broadly, insights from the literature on otimal taxation can also be relevant for swing ricing in that the trading costs caused by share redemtions or subscritions imose a negative externality on existing shareholders. For a discussion of how taxation can be used to internalise externalities, see eg Sandmo (975). 2

6 of insurance: an outcome that households would be unable to achieve if they invested directly in nancial markets. The deosit ayout corresonds to the settlement rice of fund shares in our model. The otimal deosit ayout itself deends on arameters such as households risk aversion and the likelihood of them requiring immediate access to their deosits. The same factors in uence the otimal settlement rice in our model. The key di erence between the two contracts is that the settlement rice is allowed to deend on net redemtion requests. The fund manager can commit to setting the settlement rice su ciently low to discourage redemtions even if each fund investor believes that a large share of other fund investors will redeem their shares. In contrast, romised deosit ayouts are indeendent of the amount of deosits withdrawn, making a bank vulnerable to self-ful lling runs. Jacklin (987) showed that equity contracts can achieve the same liquidity risk sharing as deosit contracts while being immune to runs. Sales of equity shares corresond to redemtions in our model, and the dividend ayout lays a similar role to the settlement rice. Similar to deosit contracts, equity contracts romise a dividend ayout that is indeendent of aggregate liquidity needs. However, the rice at which shareholders can sell their shares, and hence their level of consumtion, deends on the aggregate amount of dividend ayouts relative to the aggregate amount of shares sold. The more shares that are sold (the larger redemtions, in our model), the lower the rice, and the lower the incentive to sell equity shares (to redeem fund shares, in our model). These welfare-otimal deosit and equity contracts are only feasible if households cannot invest directly in the assets the intermediary holds. Otherwise, as eg Jacklin (987) shows, nancial intermediaries cannot o er contracts that rovide their investors with a higher level of utility than if households invested directly in nancial markets. The reason is that any di erence in ayo s would be exloited by arbitrage trades. Corresondingly, we show that if there are no trading frictions in the asset market, the fund otimally sets the settlement rice equal to the market value of its assets er share (the fund s NAV er share). Von Thadden (998, 999) derives an otimal deosit contract in the resence of investment frictions. The in uence of such frictions is also the main interest of this aer. Von Thadden s context is di erent: in his model, households cannot trade assets but can liquidate them, against a loss, and then start a new investment roject. Desite this di erence, we nd that the same factors that matter for the otimal deosit contract also determine otimal settlement rices and the otimal size of the fund s liquidity bu er in our model: the size of the frictions, the likelihood with which investors become cash constrained, and the degree of investor risk aversion. Lewrick and Schanz (207) emirically investigate the imact of swing ricing on the ows, the liquidity bu er, and the ro tability of oen-end mutual funds. In that aer, we also rovide a stylised artial-equilibrium model to motivate our estimation hyotheses. Other than that, we are not aware of any other aers that model swing ricing. However, the risk of runs on mutual funds, which swing ricing might mitigate, has been investigated in the recent theoretical and emirical literature. Chen et al (200), Goldstein et al (forthcoming) and Zheng (206) show how an incomlete allocation of liquidation costs to those investors redeeming their shares can give rise to a run on an oen-end investment fund. Malik and Lindner (207) discuss whether swing ricing might reduce systemic risk. They suggest measuring the e ect of swing ricing by its ability to damen the imact of large out ows on the fund s NAV. Seci cally, they comare changes in the NAV with out ows during normal and stressed eriods of funds that imlemented swing ricing with those that did not. They nd some suggestive evidence for swing ricing to be e ective in a small samle of funds. In Lewrick and Schanz (207), we emloy a related method to comare the erformance of funds that were allowed to use swing ricing with that of funds not ermitted to swing settlement rices. We show that swing ricing damens out ows in resonse to weak fund erformance, but has a limited e ect during stress eisodes. Furthermore, swing ricing suorts fund returns while raising the volatility of fund share rices, and may incentivise funds to hold less cash. We comare some of these ndings with our redictions in section 6. 3

7 3 Framework Our modeling framework is based on Diamond and Dybvig (983), extended by an asset market, in which households can directly trade the assets the fund invests in. Our economy has three eriods, t = 0; ; 2 and two tyes of agents: households and oen-end investment funds. At date 0, each household is endowed with one unit of a hysical good. There are no other goods or endowments. The good, which serves as the numeraire, can be stored for one eriod in both eriods 0 and, yielding a gross return of after one eriod. In this sense, it can be thought of as cash. But it is a real good that also serves as an inut to a long-term investment oortunity at date 0. This investment has constant returns to scale, is arbitrarily divisible and yields a certain gross return of at date 2. It cannot be liquidated early, but agents can issue and trade claims (equity contracts) on their investment s ayo at date in a cometitive "asset" market at rice. These claims are risk-free and ay er unit of investment at date 2. There are initially two tyes of households: a fraction of households is risk neutral; the remainder is risk averse. We will focus on an equilibrium in which at date 0 only risk-averse households invest in funds: these households value the fund s smoothing of investment returns. isk-neutral households are otential trading artners for the funds and may decide to subscribe to fund shares at date. They have identical references over future consumtion of the good given by u N = c 2. isk-averse households references, u A, are given by ( u (c ) with robability u A (c ; c 2 ) = () u (c 2 ) with robability where the utility function u is twice continuously di erentiable in consumtion, increasing, strictly concave, and satis es Inada conditions. reresents the robability of being subject to a liquidity shock. This shock determines whether the household is imatient and needs to consume at date, or atient and consumes at date 2. The households individual shocks materialise at date, are identically distributed, and satisfy the Law of Large Numbers. Hence, there is no uncertainty about aggregate consumtion needs nor about rices. As standard in this literature (eg von Thadden (999)), the utility function reresents the simlifying case where agents consume only once in their lives. Individual consumtion needs are rivate information. Therefore, if agents interact, tye-deendent consumtion allocations must be incentive comatible. Households decide in eriod zero how to invest their endowment (storage or the long-term technology) and, at date, whether to issue claims on the long-term technology or subscribe to or redeem investment fund shares. For simlicity, we allow households to only hold either fund shares or invest in the long-term technology but not both. Investment funds do not have own endowments but issue shares to households. They are small and take the market rice as given when considering whether to trade in the asset market. For simlicity, we do not allow them also to issue debt. Each fund manager aims to maximise the aggregate utility of those households that invest in fund shares at date 0: that is, we abstract from agency con icts between the fund manager and the fund s shareholders. This simli es the comarison of our results with those of Diamond and Dybvig (983) and Jacklin (987); their justi cation - that cometing fund managers can attract investors only if they best serve investors interests - also alies here. Each fund manager chooses an investment contract (the fund s "investment rosectus"). This contract seci es the fraction! 0 of the roceeds from share issuance, S 0, that the fund invests at date 0 in storage rather than in the long-term technology (the fund s "liquidity bu er"). It also describes how s, the rice at which the fund settles requests for share issuance and redemtions at date, and s 2 the ayout er share to investors at date 2, deend on the number of fund shares that households wish to 4

8 redeem or acquire at date. Because the entire wealth of the fund is distributed to its investors at the end of eriod 2, s 2 is equal to the fund s assets er share at that time and determined by the fund s choice of s. Our focus is therefore on the fund s choice of s, which we refer to as the fund s settlement rice. In ractice, a fund s rosectus would exress s as a "swing factor" in relation to the mid-market rice of the fund s assets, : =! 0 + (! 0 ) (2) s Accordingly, a ositive swing factor corresonds to a settlement rice below the unswung net asset value. Both exressions are equivalent in our model because there is no aggregate uncertainty, so the market rice is known. We normalise the number of shares the fund issues at date 0 to one er share. Thus, s is equal to the consumtion of a household that urchases a fund share at date 0 and redeems it at date, while s 2 is its consumtion if it redeems its share at date 2. At date, the fund urchases (or sells) claims on the long-term technology to invest roceeds from net share issuance (or to obtain su cient amounts of the good to settle redemtion requests). Trading claims is costly. These costs are reresented by a bid-ask sread [( ) ; = ( )] around the mid-market rice. That is, a seller of one claim receives ( ), whereas a buyer of a claim ays = ( ). Figure summarises the timeline of the model. Figure : Timeline 4 First best As a benchmark for the decentralised solution, we resent the solution of a social lanner s roblem in this section. We assume that the lanner can distinguish atient from imatient households and is able to allocate the consumtion good directly to households without having to trade. The lanner s objective is to maximise the weighted welfare of the two tyes of households, W = c N + ( ) E [u (c A )] (3) = c N + ( ) u c I A + ( ) u c P A where is the share of risk-neutral households, c N their (eriod-2) consumtion, the share of risk-averse households, c I A the (eriod-) consumtion of risk-averse imatient households, and cp A the (eriod-2) consumtion of risk-averse atient households. 5

9 Lemma (First best) If atient and imatient households are identi able, and if consumtion can be allocated directly to households, the welfare otimal solution is u 0 u 0 c I A c P A = (4) = (5) c N = ( ) ca I ( ) ( ) c P A (6) Proof. To fund consumtion of imatient households, the lanner needs to store! 0 = ( ) c I A (7) units of the endowment good and invests the remainder in the long-term technology. (! 0 ) to be consumed by risk-neutral and atient risk-averse households, that is, This leaves c N + ( ) ( ) c P A = ( ) c I A (8) Using the resource constraint (8) to relace the consumtion levels of risk-averse households in the welfare function (3) yields!! W = c N + ( ) u c I ( ) c I A + ( ) u A cn (9) ( ) ( ) The lanner s roblem can now be written as max cn ;c I W. The rst-order constraints are ( ) c I = + ( ) ( N ( ) ( ) u0 A cn ( ) ( )!! = u 0 c I A cn = 0 ( ) I = ( ) u 0 c I ( ) ( ) c I A + ( ) A ( ) ( ) u0 A ( ) ( )!! = ( ) u 0 c I A u 0 c I A cn = 0 ( ) ( )!! cn (0) () Entering the rst into the second yields the equilibrium allocation (4) - (6). In the rst best, households consume less when imatient than when atient. This is a variant of the standard result that in the rst best, liquidity insurance is incomlete (see eg Freixas and ochet (2008), chater 2.2). The resence of risk-neutral households means that the marginal utility of risk-averse households when atient, u 0 ca P, is equal to that of (atient) risk-neutral households (= ). 5 Decentralised solution We start by resenting the decentralised solution in Section 5. and comare it with the rst best described in the revious section. Section 5.2 then derives the otimal settlement rice in this equilibrium and characterises the roofs for the otimal liquidity bu er and investment decisions at date 0. 6

10 5. Overview and comarison with the lanner s solution There are two frictions that cause the decentralised solution to di er from the welfare otimum: rst, markets are incomlete, because individual liquidity shocks are not ublicly observable and securities contingent on these shocks cannot be traded; and second, because consumtion needs to be allocated via trading in asset markets and trading is costly. De nition A decentralised (Nash) equilibrium consists of the following: for each household j, investment choices that maximise households utility (u N and u A, resectively) for each investment fund k, investment and contract choices! 0;k ; s ;k ; s 2;k that maximise the exected aggregate utility of the households that urchased the fund s shares A market clearing condition for traded claims on the long-term technology. We show existence of an equilibrium in which, at date 0, risk-neutral households invest in the longterm technology whereas risk-averse households invest in fund shares; in which all agents of the same tye make the same choices; in which all funds have the same size; and in which there is no trade in the asset market at date. Proosition (Decentralised equilibrium) There is a decentralised equilibrium with the following roerties:. isk-neutral households invest their endowment in the long-term technology. isk-averse households invest their endowment in a fund s shares. If imatient, they redeem their shares at date and consume. If atient, they hold onto their shares. 2. Each fund sets the settlement rice at date equal to 8 >< s H if ^s s H s = ^s if ^s 2 (s L ; s H ) (2) >: s L if ^s s L where and the bounds s L, s H are given by ^s = s : u 0 (s ) = s ( ) u0 (3) s H = s L = ( ) ( ) + ( ) = ( ) + (4) (5) 3. Each fund share ays out, at date 2, s 2 = s (6) 4. Each fund s liquidity bu er is equal to the ayouts to imatient investors at date (! 0 = s ). Notice that there is no trade in the asset market in this equilibrium. Nevertheless, the market rice at which investors, and the fund, believe they would be able to trade needs to meet a certain condition for this equilibrium to exist. This is stated in Lemma 2. If the market rice was lower than, 7

11 all households would refer to store their endowment at date 0 to urchase claims on the long-term technology at date in the asset market (a contradiction, because there would not be any investment in the long-term technology), desite the trading cost of. Corresondingly, if the market rice exceeded = ( ), all households would refer to invest at date 0 in the long-term technology; if imatient, they would sell claims at date to obtain more than (a contradiction, because none would have any endowment to urchase those claims). Lemma 2 In equilibrium, the mid-market rice ful ls 2 ( ; = ( )) (7) To ease the comarison with the rst best described in Lemma, the following corollary to roosition translates the equilibrium investment contract terms (s ; s 2) into consumtion levels for the various tyes of households. Corollary In equilibrium, households consume c N = (8) c I A = s (9) c P A = s 2 = s (20) Proosition illustrates how trading frictions a ect the fund manager s choice for the settlement rice. If the otimal settlement rice is su ciently close to the fund s NAV, trading costs deter investors from exloiting the di erence between returns on the fund s share and on a direct investment in the claims on the long-term technology. The fund manager then settles net redemtion requests at the unconstrained otimal solution, ^s. It di ers from the rst best only because the fund manager would incur trading costs if he sold or urchased claims on the long-term technology in the asset market. To see this, notice that h i (3) can be written as u 0 ca I = = (( ) ) u 0 c P A. Lemma 2 imlies that ( ) 2 ( ) 2 ;. Di erences between rst best consumtion, determined by u 0 ca I = u 0 ca P, and consumtion in the decentralised economy when ^s is otimal would disaear if the trading cost was zero. If the fund settled redemtions above s H, the fund s shares would be so exensive that atient households with fund shares would refer to sell those shares and to invest the roceeds in claims on the long-term technology. Similarly, if the fund settled net redemtions below s L, the fund s shares would be so chea that households that invested in the long-term technology would sell claims on their investment in order to subscribe to the fund s shares. The fund manager dislikes both situations: the former, because atient households redeeming fund shares unnecessarily incur trading costs; and the latter, because of the dilution of the fund s value induced by additional shares that are issued too chealy. Lemma 6, below, shows that these ows reduce the ayo s of the fund s investors both if atient and if imatient. On balance, the fund settles shares at the unconstrained otimal settlement rice if this is su ciently close to the fund s NAV such that trading costs deter arbitrage ows. Otherwise, it chooses the settlement rice closest to the unconstrained otimal rice that just kees those arbitrage-driven ows at bay. Trading frictions have both good and bad e ects on the welfare of risk-averse households. They oen an interval, [s L ; s H ], in which the fund manager can set the settlement rice with the aim of maximising his investors utility without triggering arbitrage trades. At the same time, trading costs raise the marginal cost of raising the settlement rice. Larger trading frictions therefore lower the unconstrained otimal settlement rice, ^s. The ga widens between risk-averse households consumtion when atient and imatient. The smaller the trading costs, the narrower the interval [s L ; s H ]. It shrinks to zero in the absence of trading costs. In this case, the fund manager settles shares at a rice that ensures that his 8

12 fund, and claims on the long-term technology, o er the same return (s L = s H = = ). Then c I A =, c P A =. This is not generally equal to the rst best allocation, which, as shown in Lemma, deends on households utility function. 5.2 Derivation of the otimal settlement rice for a given level of a fund s cash bu er Swing ricing enables the fund manager to set the settlement rice as a function of net redemtions. We therefore de ne net redemtions before deriving the otimal settlement rice. A fund s aggregate redemtion requests are the sum of requests of imatient and atient investors. Given that the likelihood of becoming imatient is, and the size of the fund (the number of shares it issued at date 0, each against one unit of the endowment) is S 0;k, its imatient investors withdraw S 0;k shares in equilibrium. We denote atient investors net redemtions of fund k s shares by k. If k < 0, the fund exeriences net subscritions by atient investors; if k > S 0;k, these net subscritions more than o set redemtions by its imatient investors. We further denote the number of shares redeemed at which the fund s liquidity bu er (! 0;k ) just su ces to ay out redeeming shareholders by ^ k, ie ^ k =! 0;k S 0;k =s ;k S 0;k (2) In Section 5.2., we derive the "unconstrained" solution, ^s k, which is otimal if only imatient households adjust their ortfolios at date : that is, the otimal settlement rice conditional on k = 0. In Section 5.2.2, we derive the "no-arbitrage bounds", s L;k and s H;k. These incentive constraints ensure that atient investors refer not to alter their ortfolio at date ; hence by construction, k = 0 for any s ;k 2 (s L;k ; s H;k ). Lemma 6 argues that the fund manager would not choose a settlement rice outside [s L;k ; s H;k ]. This lies behind the result (2) in roosition. As mentioned above, we focus on a situation in which only risk-averse households invest in the fund. We derive conditions for the existence of the equilibrium in the annex (these are the date-0 articiation constraints for households that make their equilibrium investment decisions otimal) and rovide an examle of the equilibrium for a CA utility function. We omit fund subscrits (k) in the following for ease of exosition Unconstrained solution (^s) ^s is the solution to max s U s.t. = 0, where U is the fund s objective function. In an equilibrium in which only risk-averse investors invest in the fund at date 0, U = E [~u A (c ; c 2 )]. The fund manager takes the size S 0 of his fund as given when maximising U. Given that each risk-averse household urchased one share at date 0, ayouts to investors at date 2, s 2, are equal to the value of the fund s assets er share at date 2. These deend on net redemtions at date. If > ^, then net redemtion requests, S 0 +, exceed the fund s cash bu er,! 0 S 0, leaving it short s (S 0 + )! 0 S 0 < 0 units of the endowment good. Because of trading costs, the fund receives not units of the endowment good er asset sold but only ( are, at date 2, ) units. Assets er share s 2j>^ = (! 0) S 0 (s (S 0 + )! 0 S 0 ) = (( ) ) (22) ( ) S 0 The numerator is the fund s remaining assets after having met net redemtion requests: (! 0 ) S 0 from its initial investment into the long-term technology, minus date- sales. The denominator is the remaining number of shares in issuance after redemtion requests have been settled: the number issued at date 0, S 0, minus shares held by its rst-eriod investors who turn out to be imatient, S 0, minus net redemtions by atient investors,. 9

13 If, in contrast, < ^, net redemtions are less than the fund s cash bu er. The fund invests sare cash (a total of s (S 0 + )! 0 S 0 > 0), receiving claims on the long-term technology for each units of the endowment good sent. Assets er share are, at date 2, s 2j<^ = (! 0) S 0 + ( ) (! 0 S 0 s (S 0 + )) = (23) ( ) S 0 Notice that assets er share, and hence date-2 ayouts to investors, are strictly decreasing in the settlement rice if net redemtions are ositive (ie if + S 0 > 0): the more aid out er share at date, the fewer assets are left to back shares that are redeemed only at date 2. Corresondingly, assets er share are is strictly increasing in the settlement rice if net redemtions are negative ( + S 0 < 0): the more new subscribers have to ay er share, the larger the fund s resulting assets er share. Lemma 3 derives an exression for the "unconstrained" otimal settlement rice, ^s, ie the settlement rice that is otimal conditionally on = 0. Lemma 3 Conditionally on = 0, the otimal settlement rice, ^s, ful ls ( ^s = s : u0 (s ) u 0 (s 2 ) = Proof. If = 0, the fund manager s objective function can be written as = (( ) ) if! 0 s ( ) = if! 0 > s (24) U = u (c ) + ( ) u (c 2 ) (25) = u (s ) + ( ) u (s 2 ) (26) The solution to the fund manager s roblem is given by the = u 0 (s ) + ( u 0 (s 2 ) = 0 (27) where the resonse of the fund s date-2 ayout (its assets er share at date 2) to a marginal increase in its date- ayout (the settlement rice) = Substituting (28) into (27) yields (24). ( = (( ) ) if! 0 s ( ) = if! 0 > s (28) No-arbitrage bounds (s L, s H ) Lemma 4 describes an interval for the return of an investment in fund k s shares at date within which atient investors would not alter their ortfolio at date. Deriving the interval in terms of gross returns of such an investment, s 2 =s, is be simler than deriving the interval directly for the settlement rice, s. The two exressions are equivalent because s 2 is a function of s. Lemma 4 If all funds set the same settlement rice, then = 0 if s 2 2 ( ) s ; (29) Proof. If all funds set the same settlement rice, households that at date 0 subscribed to fund shares have no incentive to switch funds. The ayo of a atient household that at date 0 subscribed to fund 0

14 k s share and does not change its ortfolio is s 2. If it redeemed its share and invested the roceeds in the asset market, it would earn s ( ) =. Thus, for the investor to remain with the fund, s 2 s ( ) (30) The ayo of a atient household that at date 0 invested in the long-term technology and does not change its ortfolio is. If the household instead issued claims on the investment and used the roceeds, ( ), to subscribe to the shares of a fund, it would earn ( ) s 2 =s. Thus, for the investor to remain invested in the long-term technology, s 2 s (3) Lemma 5 translates this return interval into an interval for the settlement rice for the fund s shares. Lemma 5 If the fund sells assets at date (if ^), (29) holds if and only if 2 3! 0 + (! 0 ) ( ) s j>^ 2 4! 0 + (! 0 ) ( ) ; + ( ) ( ) 2 + ( ) 2 5 (32) =S 0 If the fund urchases assets at date (if < ^), (29) holds if and only if 2 3 s j<^ 2 4! 0 + (! 0 ) = ( ) + ( ) = ( ) 2 + = ( ) 2 ;! 0 + (! 0 ) 5 (33) =S 0 The roof is in the annex; it uses the fact that the fund s assets er share, s 2, are a strictly monotone function of its settlement rice, and solves for the settlement rices that aly at the bounds of (29). Just as for assets er share (see (22) and (23)), a ositive bid-ask sread imlies a discontinuity at ^, such that the interval is de ned searately for the case in which the fund sells or urchases assets in equilibrium. Notice that s j>^ exceeds the market value of a share net of transaction costs,! 0 + (! 0 ) ( ). When the fund is not constrained in its choice of the otimal settlement rice by arbitrage trades (ie if s = ^s), then it does not charge the entire cost of liquidation to investors redeeming their shares at date. Lemma 6 states that the otimal settlement rice falls within that interval. Lemma 6 In an equilibrium in which funds are symmetric, the settlement rice, s, ful ls s 2 s 2 ( ) ; ( ) (34) Proof. The roof is by contradiction. Suose that s 2 =s < ( ) = (imlying s is above the uer bound of (32)). Then the fund s atient investors would strictly refer to redeem their shares because the return on fund k s shares would be less than the return on investments in the market. The fund would liquidate its assets, retrieving ( ) er unit sold. Each investor would obtain a ayout of! 0 + (! 0 ) ( ) er share. As a result, each household with fund shares would consume, if imatient, c I A =! 0 + (! 0 ) ( ) (35)

15 and c P A = c I A ( ) = (36) if atient, having reinvested its ayout in the market. Because s reaches a minimum at! 0 +(! 0 ) ( ) when the fund sells assets (see (32)), these consumtion levels are lower than what atient and imatient households, resectively, would obtain if s 2 =s ( ) =. Suose instead that s 2 =s > = (imlying s is below the lower bound of (33)). Then households that invested at date 0 in the long-term technology would strictly refer to sell claims on their investment and use the roceeds to subscribe to fund shares. As a result,! (the fund is small relative to the share of households that would ow in). Because s is below the lower bound of (33), c I A < ci A. In addition, atient households would also obtain less (ie cp A < cp A ) because of dilution: the fund s assets er share would fall in resonse to the in ows to a lower level than that imlied had the settlement rice been set to the lower bound of (33). To see this, notice that for all s j<^ in the interval (33), lim s 2j<^ = lim!! (! 0 ) S 0 + ( ) (! 0 S 0 s (S 0 + )) = (37) ( ) S 0 (! 0 ) S 0 + ( )! 0 S 0 s j<^ (S 0 + ) = < lim (38)! ( ) S 0 = s j<^ ( ) = (39) That is, lim! s 2j<^ =s j<^ is less than the lower bound of the range of returns set in equilibrium; see (34). By construction, = 0 if s is in the interior of the no-arbitrage interval, [s L ; s H ]. In contrast, if s is equal to one of the bounds of [s L ; s H ], is undetermined because atient investors are, by construction, indi erent. We focus on an equilibrium in which = 0 for all s 2 [s L ; s H ]. The rule for the otimal settlement rice in (2) then follows because the fund manager s objective function is concave in s 2 U (@s ) 2 = u 2 2 s (s ) + ( ) (@s ) 2 u0 (s 2 ) + u 00 (s 2 2 = u (s ) + ( ) u 00 (s 2 ) < 5.3 Equilibrium cash bu er and rst-eriod choices In this section, we rovide the intuition for our results for the equilibrium cash bu er and households rst-eriod choices. The roofs are in the annex. Consider rst the otimal liquidity bu er. In order to settle net redemtions at date, the fund can either kee a cash bu er at date 0 or sell assets at date. Because the date- market rice ful ls 2 ( ; = ( )) (see Lemma 2), the oortunity costs of investing in the cash bu er are lower than the trading costs the fund would incur if it had to sell assets at date ( = = ( )). The fund chooses the cash bu er with a view to avoiding transaction costs and holds just enough cash in order to settle date- net redemtion requests. Now consider households rst-eriod choices. We are interested in an equilibrium in which riskneutral households invest in the long-term technology at date 0 and risk-averse households invest in fund shares. Two conditions need to be met for this equilibrium to exist. The rst is that the exected return of investing in the long-term technology from date 0 to date 2 is larger than that of holding fund shares over the same horizon. This ensures that risk-neutral households, all of which are atient, invest in the long-term technology. The condition holds if the fund o ers investors that redeem early a return on its 2

16 shares larger than that of storage, ie if s >. This corresondingly reduces the ayout of investors staying with the fund until date 2, discouraging risk-neutral households from investing in the fund. 6 The second condition is that the return on investments in fund shares has su ciently lower volatility than that of investing in the long-term technology. If so, risk-averse households refer investing in fund shares, desite their lower exected returns. In the annex, we show that these conditions are met for lausible arameterisations of the model. 6 Proerties of the otimal settlement rice In this section, we show how the equilibrium described in roosition is a ected if the share of liquidityconstrained investors,, and the trading costs,, rise. In Section 6., we assume that the new values of and are known at date 0, when the investment contract is written and the fund chooses the otimal liquidity bu er: ie we derive the comarative static roerties of the equilibrium. In Section 6.2, we assume instead that the share of liquidity-constrained investors or of trading costs unanticiatedly increases at date, when it is too late for the fund to build its cash bu er to meet redemtions. This scenario might hel understand funds resonses to sudden stress in nancial markets. Finally, we ask whether swing ricing can mitigate the risk of self-ful lling runs. 6. Comarative static roerties We exress the roerties of the otimal settlement rice also in terms of the swing factor, which is given by the relative di erence between the fund s net asset value and the settlement rice. Entering the equilibrium values for the liquidity bu er and the settlement rice into (2) yields the otimal swing factor. It is strictly declining in the otimal settlement rice: =! 0 + (! 0) s = s + ( s ) s = + (40) s We de ne ^ as the value of (40) when the no-arbitrage bounds do not constrain the fund manager, ie if s = ^s (see (3)). Lemma 7 summarises key roerties of the otimal swing factor. The roofs are in the annex (Section 0.5). Lemma 7 (Proerties of the otimal swing factor and the otimal settlement rice).. The otimal swing factor ful ls 2 ( ) ; ( ) (4) 2. Larger trading costs,, widen the interval (4) and raise ^ (reduce ^s), and lower the liquidity bu er,! A greater likelihood of becoming imatient,, shrinks the interval (4) and raises ^ (reduces ^s). The rst result in Lemma 7 is the no-arbitrage interval which contains the otimal swing factor, (4). It re ects the frictions induced by trading costs. But the interval is also shaed by the share of 6 A similar condition alies in Diamond and Dybvig (983): deosits need to o er households that withdraw early a higher return than storage. This condition ensures that households utility in a world in which banks o er deosit contracts is higher than in a situation in which banks do not exist, forcing households to invest directly in the asset market. 3

17 imatient investors. The larger that share, the smaller the interval. This is because, in equilibrium, only imatient investors redeem their shares. The larger their share,, the more resonsive the fund s return to the settlement rice. To see this, notice from (6) that that the fund s ayout at date 2, s 2, is strictly declining in. This means that when is large, a small change in the settlement rice leads to a substantial change in the return of fund shares from date to date 2. The no-arbitrage interval (4) shrinks, re ecting that the fund manager has less scoe to vary the settlement rice without triggering ows of atient investors. In ractice, funds often commit to a maximum swing factor. 7 The second result is that trading frictions widen the no-arbitrage interval. If they were zero, the interval would collase to zero. Trading frictions also raise the marginal cost to raising the settlement rice. To see this, notice that a larger makes the rst derivative of the fund s assets er share at date 2, given by (22), more negative. This translates into a stronger decline of ayouts at date 2, s 2, in resonse to an increase in s. As a result, larger trading frictions induce the fund manager to lower the settlement rice. He thereby reduces the degree of liquidity insurance the fund rovides to its investors. Finally, notice that the otimal liquidity bu er inherits the roerties of the otimal settlement rice because the fund manager chooses to hold a su ciently large bu er to avoid trading in the market, instead funding redemtions from the bu er. 6.2 Otimal settlement rice following unanticiated shocks Some events in nancial markets can best be characterised as unanticiated, in the sense that market articiants have assigned only a very small, or indeed no, robability to their occurrence. For examle, few investors would have thought ossible the raid increase in long-term bond yields during the 203 "taer tantrum", and the associated large out ows from bond funds. 8 We therefore brie y consider the fund s resonse to unancitiated increases in the share of cash-constrained investors,, and an unanticiated increase in the trading cost,, at date, for a given value of the fund s cash bu er. Lemma 8 The otimal settlement rice is declining following an unanticiated increase in the share of cash-constrained investors,, and an unanticiated increase in the trading cost,, at date. An unanticiated increase in leads to an increase in redemtion requests. At the equilibrium settlement rice, the fund would not have enough cash to accommodate the redemtion requests. It cannot sell assets to raise cash because the increase in a ects all funds in the same way, and imatient investors consume the endowment good received rather than reinvest it in the asset market. The fund therefore distributes the available cash among redeeming investors. The settlement rice falls, raising the date-2 fund s assets er share. As a result, only imatient investors redeem their shares. The resonse to an unanticiated increase in follows because the fund manager trades o the marginal costs and bene ts of raising the settlement rice. The marginal costs have increased following the increase in, while the marginal bene ts (an increase in the consumtion of investors that stay with the fund) has remained unchanged. This induces the fund manager to lower the settlement rice. The roof is in the annex (Section 0.6). In Lewrick and Schanz (207), we nd evidence that is consistent with these results. We comare the erformance of funds that are ermitted to use swing ricing with that of funds that are not, immediately following the taer tantrum. Arguably, the taer tantrum led to both an unanticiated increase in cashconstrained investors and an increase in the costs of trading. We nd that the return of funds that are ermitted to swing their settlement rice on average falls less in resonse to net redemtions than that of funds constrained to set their settlement rice equal to their net asset value. 7 See ALFI (205). 8 In Lewrick and Schanz (forthcoming), we assess the e ect of swing ricing on fund ro tability and the extent of their out ows during this eisode from an emirical ersective. 4

18 6.3 Can swing ricing mitigate the risk of self-ful lling runs? The model for which Diamond and Dybvig (983) derived the otimal deosit contract has two equilibria: the e cient one, in which only imatient investors withdraw their funds at date, and an ine cient one, in which all investors withdraw and the bank s assets are liquidated at a loss in an attemt to fund those withdrawals. As mentioned in Section 2, a key di erence between a deosit contract and the contract o ered by an oen-end investment fund that uses swing ricing is that the fund manager can x the settlement rice after he has collected redemtion and subscrition requests. In articular, the fund manager can commit to setting a su ciently low settlement rice following large net redemtions, aiming to induce atient investors not to redeem their shares early. Within the context of our model, this can make a fund that uses swing ricing immune to the risk of self-ful lling runs. Lemma 9 There are (su ciently low) settlement rices such that no atient fund investor redeems its shares at date, even if it believes that other atient fund investors will redeem their shares at date. Proof. Suose each atient fund investor believes that a share of =S 0 other atient fund investors will redeem their shares, and that the fund manager sets a settlement rice of _s should net redemtions be equal to S 0 +. Then each atient investor anticiates that if he redeems his share, he obtains a ayo of _s ( roceeds in the market ) = after reinvesting the redemtion if he stays with the fund, he obtains a ayo given by (22), s 2 ( _s ) = (! 0) S 0 ( _s (S 0 + )! 0 S 0 ) = (( ) ) ( ) S 0 If _s < s 2, each atient fund investor strictly refers to remain invested with the fund. Positive values that satisfy _s < s 2 exist because s 2 (0) > 0, 2 =@s < 0. In ractice, the scoe for swing ricing to revent self-ful lling runs is more limited than Lemma 9 suggests. For one, the fund manager would need to be able to correctly assess the share of investors that are liquidity constrained (). In our framework, the fund manager knows, so he can commit to lowering the settlement rice whenever net redemtions exceed without any negative imact on investors utility. In ractice, however, is stochastic. In this case, any commitment to lower the settlement rice when net redemtions are unusually large comes at a cost: cash-constrained fund investors would su er from receiving only a low settlement rice if is higher than what the fund manager believes it to be. 7 Discussion In this section, we assess how relaxing some of the main assumtions of the model might a ect our results. First, we assume that there is no aggregate uncertainty in the model. All rices, and the share of cash-constrained investors, are known. This imlies that fund managers can erfectly insure against redemtions by holding a liquidity bu er that is equal to the amount aid out to redeeming investors. Nevertheless, we can rovide an interesting characterisation of otimal swing ricing olicies. The reason is that the marginal cost and bene ts of varying the settlement rice are in uenced by the size of redemtions and trading frictions, even if, in equilibrium, there is no need to trade. That said, our framework may underestimate the leeway that funds have in swinging the settlement rice (ie the width of the no-arbitrage interval, (34)). If the fund manager had rivate information about the realisation of a shock to the (aggregate) redemtion requests his fund exeriences, he would only ensure that no arbitrage oortunities arise in exectation over those redemtion requests other agents believe ossible. 5