Mark-to-Market Accounting and Cash-in-the-Market Pricing

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1 Mark-to-Market Accounting and Cash-in-the-Market Pricing Franklin Allen Wharton School University of Pennsylvania Elena Carletti Center for Financial Studies University of Frankfurt March 2, 2006 Abstract When liquidity plays an important role as in times of financial crisis, asset prices in some markets may reflect the amount of liquidity available in the market rather than the future earning power of the asset. Mark-to-market accounting is not a desirable way to assess the solvency of a financial institution in such circumstances. An example is given where a shock in the inusurance sector leads the current value of banks assets to be less than the current value of their liabilities so the banks are insolvent. However, if the banks were allowed to continue, which would be the case if historic cost accounting was used, then the banks could meet all their future liabilities. Mark-to-market accounting can thus lead to contagion where none would occur with historic cost accounting. JEL Codes: G21, G22, M41. Keywords: Mark-to-market, historical cost, incomplete markets. Preliminary version. Comments are welcome. We would like to thank Alessio Devincenzo for useful comments and discussions.

2 1 Introduction In recent years there has been a considerable debate on the advantages and disadvantages of moving towards a full mark-to-market accounting system for financial institutions such as banks and insurance companies. This debate has been triggered by the move of the International Accounting Standards Board (IASB) and the US Financial Accounting Standards Board (FASB) to make changes in this direction as part of an attempt to globalize accounting standards (Hansen 2004). There are two sides to the controversy in the debate. Proponents of mark-to-market accounting argue that this accounting method reflects the true (and relevant) value of the balance sheets of financial institutions. This in turn should allow investors and policy makers to better assess their risk profile and undertake more timely market discipline and corrective actions. In contrast, opponents claim that mark-to-market accounting leads to excessive and artificial volatility. As a consequence, the value of the balance sheets of financial institutions would be driven by short-term fluctuations of the market that do not reflect the value of the fundamentals and the value at maturity of assets and liabilities. In this paper we argue that adopting a mark-to-market system to value the assets of financial institutions may not be beneficial when financial markets are incomplete and illiquid. In times of financial crises the interaction of institutions and markets can lead to situations where prices in illiquid markets do not reflect future payoffs but rather reflect the amount of cash available to buyers in the market. The level of liquidity in such markets is endogenously determined and there is cash-in-the-market pricing. When historic cost accounting is in use, this problem does not compromise the solvency of banks as it is does not affect the value of their assets. In contrast, when mark-to-market accounting is used, the volatility of asset prices directly affects the value of banks assets. This leads to contagion and forces banks into insolvency even though they would be fully able to cover their commitments if they were allowed to continue until the assets mature. The potential problems that might have arisen had Long Term Capital Management (LTCM) been allowed to go bankrupt illustrate the issue. The Federal Reserve Bank of New York justified its action of facilitating a private sector bailout of LTCM by arguing that if the fund had been liquidated many prices in illiquid markets would have fallen and this would have caused further liquidations and so on in a downward spiral. The point of our paper is to argue that using mark-to-market accounting significantly exacerbates the 2

3 problem of contagion in such circumstances compared to the use of historical cost accounting. The notion that market prices cannot be trusted to value assets in times of crisis has a long history. In his influential book, Lombard Street, on how central banks should respond to crises, Bagehot (1873) argued that collateral should be valued weighting panic and pre-panic prices. To better understand the role of different accounting methods during crises, we present a model with a banking sector and an insurance sector based on Allen and Gale (2005a) and Allen and Carletti (2006). Initially the two sectors operate in autarky. There is no way to diversify across the two sectors because of a lack of instruments to do so. Banks invest in loans, long and short assets. They use the returns of the short asset to satisfy the claims of depositors withdrawing early, and are always solvent despite the risk of their loans. The insurance companies insure firms against the possibility of their assets being damaged before maturity. The optimal arrangement is such that the insurance companies offer partial insurance to the firms and invest the premiums they collect initially in the short asset. There is aggregate risk in the insurance sector. The insurance companies do not have enough funds to repair all damaged assets when an extreme bad shock is realized and so they go bankrupt. The failure of the insurance sector does not involve deadweight costs though and it does not spill over to the banking sector because the two sectors have only the short asset in common. The equilibrium changes when credit risk transfer is introduced. The effect of this financial innovation crucially depends on the existence of markets for liquidating the long asset and the accounting method in use. When there is no liquidation market, introducing credit risk transfer is beneficial. It improves risk sharing across the two sectors without creating contagion because insurance companies still hold only the short term asset. It is not worth it for them to use the long asset because it pays off nothing in the case of bankruptcy. In contrast, when a market for liquidating the long asset is introduced, the insurance companies find it optimal to finance the credit risk transfer by investing in the long asset because they can now liquidate it for a positive amount when needed. This implies that the creation of markets helps avoid waste. However, markets are incomplete and this can lead to asset-price volatility. In order to induce some market participants to hold liquidity to purchase assets, there must be states in which asset prices are low so the participants can make a profit and cover the opportunity cost of holding cash in the other states. The low prices are determined by the endogenous amount of cash in the market 3

4 rather than the future earning power of the asset. If assets are valued at their historical cost, this volatility does not lead to contagion. Banks assets are not affected by the low prices. They remain solvent and can continue operating until assets mature. Then, introducing markets and valuing assets atthehistoricalcostimproveswelfare. Itallowstheinsurancecompanies toholdthemoreprofitable long asset and does not create unnecessary and costly contagion when they go bankrupt. In contrast, when assets are priced according to market values, then these low prices can cause a problem of contagion from the insurance sector to the banking sector. Even if banks would be solvent if they were allowed continue, the current values of their assets is lower than the value of their liabilities. Banks are then declared insolvent and forced to sell the long term assets. This worsens the illiquidity problem in the market and reduces prices even further. The overall effect of this contagion is to significantly lower welfare compared to what would happen with historic cost accounting. This result has important implications for the debate on the optimal accounting system. In particular, it stresses the potential problems arising from the use of mark-to-market for securities traded in incomplete markets with scarce liquidity. In this sense, the accounting-induced contagion that we describe could emerge in the context of many financial institutions and markets and our results have to be interpreted just as a specific exampleof it. Our paper is related to a number of others. Planta, Sapra, and Shin (2004) show that, while an historical cost regime can lead to some inefficiencies, mark-to-market pricing can lead to increased price volatility and suboptimal real decisions. Their analysis suggests the problems with mark-to-market accounting are particularly severe when claims are (i) long-lived, (ii) illiquid, and (iii) senior. The assets of banks and insurance companies are particularly characterized by these traits. This provides an explanation of why banks and insurance companies have been so vocal against the move to mark-tomarket accounting. In the current paper an additional reason for banks and insurance companies to be disturbed by mark-to-market accounting is provided. Mark-to-market accounting can induce contagion where historic cost accounting would not. Other papers analyze the implications of mark-to-market accounting from a variety of perspectives. O Hara (1993) focuses on the effects of market value accounting on loan maturity and finds that this accounting system increases the interest rates for long-maturity loans, thus inducing a shift to shorter- 4

5 term loans. In turn this shortens the liquidity creation function of banks and exposes borrowers to excessive liquidation. In a similar vein, Burkhardt and Strausz (2006) suggest that market value accounting reduces asymmetric information, thus increasing liquidity and intensifying risk-shifting problems. Finally, Freixas and Tsomocos (2004) show that market value accounting worsens the role of banks as institutions smoothing intertemporal shocks. Differently, our paper focuses on market incompleteness to show how market value accounting can lead to contagion and so may not be desirable. Allen and Carletti (2006) also analyze how financial innovation can create contagion across sectors and lower welfare relative to the autarky solution. However, while Allen and Carletti (2006) focus on the structure of liquidity shocks hitting the banking sector as the main mechanism generating contagion,wefocushereontheimpactofdifferent accounting methods and show that mark-to-market accounting can lead to contagion in situations where historical cost accounting does not. The rest of the paper proceeds as follows. Section 2 develops a model with a banking and an insurance sector. Section 3 considers the equilibrium when there are no liquidation markets, focussing first on the autarky equilibrium and then on the potential beneficial effectofcreditrisktransfer. Section 4 shows how the implications of introducing a market for liquidating the long asset depend on the accounting system in use. In particular, it shows that a liquidation market and mark-to-market accounting can lead to aparetoreductioninwelfarecomparedtohavingthemarketandhistoric cost accounting. Finally, Section 5 contains concluding remarks. 2 The model The model is based on the analyses of crises and systemic risk in Allen and Gale (1998, 2000, 2004a-b) and Gale (2003, 2004), and particularly in Allen and Gale (2005a) and Allen and Carletti (2006). A standard model of intermediation is extended by adding an insurance sector. The two sectors face risks which are not perfectly correlated so that there is scope for diversification. Credit risk transfer allows this scope to be exploited. There are three dates t =0, 1, 2 and a single, all-purpose good that can be used for consumption or investment at each date. The insurance and the banking sectors consist of a large number of competitive institutions and their lines of business do not overlap. This is a necessary assumption, 5

6 since the combination of insurance and intermediation activities in a single financial institution would eliminate the need for markets and the feasibility of mark-to-market accounting. There are two securities, one short and one long. The short security is represented by a storage technology: one unit at date t produces one unit at date t +1. The long security is a simple constant-returns-to-scale investment technologythattakestwoperiodstomature: oneunitinvestedinthelong security at date 0 produces only R>1 units of the good at date 2. We can think of these securities as being bonds or any other investment that is common to both banks and insurance companies. In addition to these securities, banks and insurance companies have distinct direct-investment opportunities and different liabilities. Banks can make loans to firms. Each firm borrows one unit at date 0 and invests in a risky venture that produces B units of the good at date 2 with probability β and 0 with probability 1 β. There is assumed to be a limited number of such firms, so that they take all the surplus and give banks a repayment b ( B), as we describe more fully below. Banksraisefundsfromdepositors,whohaveanendowmentofoneunit of the good at date 0 and none at dates 1 and 2. Depositors are uncertain about their preferences: with probability λ they are early consumers, whoonlyvaluethegoodatdate1, and with probability 1 λ they are late consumers, who only value the good at date 2. Uncertainty about time preferences generates a preference for liquidity and a role for the intermediary as a provider of liquidity insurance. The utility of consumption is represented by a utility function U(c) with the usual properties. We normalize the number of depositors to 1. Since banks compete to raise deposits, they choose the contracts they offer to maximize depositors expected utility. If they failed to do so, another bank could step in and offer a better contract to attract away all their customers. Insurance companies sell insurance to a large number of firms, whose measure is also normalized to one. Each firm has an endowment of one unit at date 0 and owns an asset that produces A units of the good at date 2. With probability α the asset suffers some damage at date 1. Unless this damage is repaired, at a cost of φ, the asset becomes worthless and will produce nothing at date 2. Firms can then decide whether to buy insurance against the probability of incurring the damage at date 1 in exchange for a premium at date 0. The insurance companies collect the premiums and invest them at date 0 in the securities in order to pay the firms damages at date 1. 6

7 The owners of the firms consume at date 2 and have a utility function V (C) with the usual properties. Similarly to the banks, the insurance companies operate in competitive markets and thus maximize the expected utility of the owners of the firms. If they did not do this, another firm would enter and attract away all their customers. Finally, we introduce a class of risk neutral investors who potentially provide capital to the banking and insurance sectors. Investors have a large (unbounded)amountofthegoodw 0 asendowmentatdate0 and nothing at dates 1 and 2. They provide capital to the intermediary through the contract e =(e 0,e 1,e 2 ),wheree 0 0 denotes an investor s supply of capital at date t =0, and e t 0 denotes consumption at dates t =1, 2. Although investors areriskneutral,weassumethattheirconsumptionmustbenon-negative at each date. Otherwise, they could absorb all risk and provide unlimited liquidity. The investors utility function is then defined as u(e 0,e 1,e 2 )=ρw 0 ρe 0 + e 1 + e 2, where the constant ρ is the investors opportunity cost of funds. This can represent their time preference or their alternative investment opportunities that are not available to the other agents in the model. We assume ρ> R so that it is not worthwhile for investors to just invest in securities at date 0. This has two important implications. First, since investors have a large endowment at date 0 and the capital market is competitive, there will be excess supply of capital and they will just earn their opportunity cost. Secondly, the fact that investors have no endowment (and non-negative consumption) at dates 1 and 2 implies that their capital must be converted into assets in order to provide risk sharing at dates 1 and 2. For the purposes of illustrating the scope for diversification and the role of the accounting rules, the structure of uncertainty is one that allows for some diversification and some aggregate risk. In particular, β and α take two values each, β H and β L and α H and α L. Note also that β and α are independent so that we have four possible states, (β H,α H ), (β H,α L ), (β L,α H ), (β L,α L ). We denote them simply as HH, HL, LH, and LL, respectively. This simple structure is enough to illustrate our main points. All uncertainty is resolved at the beginning of date 1. Banks discover whether loans will pay off or notatdate2. Depositors learn whether they are early or late consumers. Insurance companies learn which firms have damaged assets. 7

8 3 Equilibrium without liquidation markets The purpose of this section is to illustrate how the sectors work in isolation. We use this as a benchmark for considering the interaction between financial innovation and accounting methods. The most transparent way to analyze this complex interaction and illustrate what is possible is using numerical examples. The first case considered is when the banking sector and the insurance sector are autarkic and operate separately. It is initially assumed that there are no markets so that the long asset and the loans cannot be liquidated for a positive amount at date 1. Hence if a bank or insurance company goes bankrupt at date 1, the proceeds from the long asset and the loans are The banking sector in autarky We assume the following values: The return on the long asset is R =1.1. Depositors have utility function U(c) =Ln(c) and become early consumers with probability λ =0.5. Loans yield B =3when successful. The probability of success depends on whether state H or L is realized. We assume that each state is equally likely so that each of them occurs with probability 0.5. In state H the probability of the high payoff B on firms loans is β H =1;in state L it is β L =0.4. Overall, the probability that the banks loans pay off B =3 is =0.7 and the probability they pay off 0 is =0.3. Investors have an opportunity cost equal to ρ = Since they are indifferent between consumption at date 1 and date 2, itisoptimaltoset e 1 =0, invest any capital e 0 that is contributed in the long asset or loans, which have higher returns than the short asset, and make a payout e 2 to investors when loans are successful. Thus, at date 0 banks choose how much capital to raise and how much to compensate investors for it, and how to split the 1+e 0 units of funds they receive from depositors and investors to maximize their expected utility. We denote as x and y banks investment in the short and long assets, respectively, and as z their investment in loans. Since all banks are ex ante identical and compete for attracting deposits, they maximize the expected utility of depositors and solve the following problem: Max EU =0.5U(c 1 )+0.5[0.7U(c 2H )+0.3U(c 2L )] (1) 8

9 subject to x + y + z =1+e 0, (2) c 1 = x 0.5, (3) c 2H = yr + zb e 2, 0.5 (4) c 2L = yr 0.5, (5) c 1 c 2L, (6) e 0 ρ =0.7e 2. (7) The banks maximization problem can be explained as follows. Each bank has 1 unit of depositors and 0.5 of them become early consumers and 0.5 late consumers. The first term in the objective function represents the utility U (c 1 ) of the 0.5 early consumers. The bank uses the entire proceeds of the short term asset to provide them with a level of consumption c 1 = x/0.5. The second term represents the 0.5 depositors who become late consumers. With probability 0.7 loans pay off B, banks receive the repayment b and have to pay e 2 to investors so that the late consumers receive consumption c 2H = (yr+zb e 2 )/0.5. With probability 0.3 the loans pay off 0. The bank has only the return from the long asset and each late consumer gets c 2L = yr/0.5. The inequality (6) ensures that there are no runs when the bank s loans do not pay off. Preventing runs is optimal since there are no markets for liquidating the long asset or loans. The constraint (2) is the budget constraint at date 0, while the last constraint (7) is investors participation constraint. Investors must receive an expected payoff which makes them break even. As already mentioned, it is optimal to give them a repayment only when loans pay B and banks obtain b (which occurs with probability 0.7) sothatdepositors have their lowest marginal utility of consumption. Substituting the constraints (2)-(7) into the objective function (1), and noting that y =1+e 0 x z from (2), we can reduce the number of decision variables to x, z and e 0. We ignore the constraint (6) for the moment. Furthermore, given that there is a limited number of firms who want loans relative to banks, the firms obtain the surplus. This means that loans are priced so that banks are indifferent between providing loans and not providing them. Since bank capital is the marginal source of finance the amount paid on loans is b = ρ <Bin equilibrium since this just covers the marginal cost 0.7 9

10 of equity capital. In numerical calculations we assume the limited number of loans is z =0.3. Hence the banks problem becomes to choose x and e 0 to solve the following problem Max EU = 0.5U( x 0.5 )+0.35U((1 x z)r + zb e 0(b R) ) U( (1 + e 0 x z)r ), 0.5 where b = ρ/0.7. The relevant first order conditions are EU x = 0.5 x 0.35R (1 x z)r + zb e 0 (b R) 0.15 (1 + e 0 x z) =0 EU 0.35(b R) = e 0 (1 x z)r + zb e 0 (b R) (1 + e 0 x z) =0. Solving these conditions using the values of the example, we get the following solution: e 0 =0.25; e 1 =0;e 2 =0.42; x =0.5; y =0.45; z =0.3; c 1 =1.00; c 2H =1.15; c 2L =1.00; EU = We need to check that the constraint (6) is satisfied. Comparing c 1 and c 2L it can be seen that it is. Thus this is the autarkic solution for the bank. Its depositors get expected utility of One important issue concerns the role that capital is playing in the banking sector. Since the suppliers of capital are risk neutral they provide risk smoothing to the depositors in the bank. The assets their capital provides pay off when the loans do not and they only receive a payment when the loans pay off. The reason that the providers of capital do not bear all the risk is that capital is costly. In other words their opportunity cost of capital is higher than the return on the long asset. If it was the same, there would be full risk sharing and depositors would consume the same in every state. 10

11 3.2 The insurance sector in autarky We consider the insurance sector in isolation next. As already explained, insurance companies offer insurance to firms against the possibility that their assets A are damaged at date 1 and need to be repaired for cost φ. Forthe purpose of our example, we assume A =1.15 and φ =1. The probability of the assets being damaged is α H =0.5 in state H and α L =1in state L. State H occurs with probability 0.9 while state L is much less likely having probability 0.1. Finally, the utility function of the owners of the firm is V (c) =Ln(c) and recall that the endowment of each firm is 1. Similarly to the banking industry, the insurance sector is competitive. Companies maximize the expected utility of the owners of the firms they insure and do not earn any profits. The insurance contract can consist of partial or full insurance. In the case of partial insurance, companies charge apremium0.5 at date 0 and invest it in the short asset to have liquidity to repair the damaged assets at date 1. Firms still have 0.5 of their initial endowment and, given their owners consume only at date 2, theyfind it optimal to invest it in the long asset and obtain the return R. In state H insurance companies have claims α H φ =0.5 to repair the damaged assets and can use the return 0.5 from the short asset to satisfy them. All assets are repaired and the owners of the firms can then consume C 2H = A+0.5R = 1.7. In contrast in state L the insurance companies receive claims α L φ =1 but have only 0.5 in funds. Thus they cannot satisfy their claims and go bankrupt. Their assets are distributed equally among the claimants so that each firm receives 0.5 from the insurance companies liquidation of its short term assets. Firms can t repair their assets so these produce nothing. In state L the consumption of the owners of the firm is therefore C 2L = R = Their expected utility with partial insurance is EV partial = 0.9V (A +0.5R)+0.1V ( R) = 0.9V (1.7) + 0.1V (1.05) = With partial insurance there is systemic risk in the insurance industry. One possible way to avoid this is for the insurance companies to offer full insurance. In this case they charge a premium equal to 1 at date 0 and invest it in the short asset. In state H the insurance companies use 0.5 to meet their claims α H φ =0.5 and, since they operate in a competitive industry, 11

12 distribute the remaining 0.5 to firms. In state L they receive claims α L φ =1 and use all their funds to satisfy them. Thus, with full insurance, assets are repaired in both states and the owners of the firms have utility equal to EV full = 0.9V (A +0.5) + 0.1V (A) = 0.9V (1.65) + 0.1V (1.15) = Despite avoiding bankruptcy, full insurance is not optimal. The reason is that the low state is not very likely so it is better for firmstopayalower premium initially and invest the remaining endowment in the more profitable long asset rather than having the assets always repaired but not have funds to invest in the long asset. It can be shown that partial insurance dominates other strategies such as the case where firms pay partial insurance and invest the remaining amount of endowment in the short asset themselves. To sum up, the optimal scheme is for the insurance industry to partially insure firms, charge a premium equal to 0.5 at date 0 and leave firms to invest the remaining part of their endowment in the long asset. This gives the owners of the firms an expected consumption equal to EC 2 =0.9C 2H + 0.1C 2L = The assumption that the long asset cannot be liquidated at date 1 for a positive amount plays an important role here. It ensures that the firms invest their surplus in the long asset without paying it into the insurance companies. Also it implies that insurance companies do not invest in the long asset as they need funds to satisfy their claims at date 1. All this guarantees that when the insurance companies go bankrupt in state H there is not inefficient liquidation of the long asset. Note that there is no role for capital in the insurance sector so that E 0 =0. The reason is that capital providers charge a premium to cover their opportunity cost ρ. Insurance companies should invest the capital provided by investors in the short asset since it is not optimal to hold any long assets. There are already potentially enough funds from customers to hold more of the short asset but it is not worth it. If there is a premium to be paid for the capital it is even less worth it. Capital will not be used in the insurance industry unless companies are regulated to do so. 3.3 The functioning of credit risk transfer In the previous sections the banking and insurance sectors have been considered in isolation. We now consider them together and analyze the impact of 12

13 financial innovation in the form of credit risk transfer. For the moment we maintain the assumption that neither loans nor long assets can be liquidated for a positive amount at date 1. We will relax this in the next section. Since the shocks affecting the two sectors are independent, we have four states of the world describing the possible realizations of the probabilities β and α. As already mentioned, we can express these states as HH, HL, LH, and LL. Thenthefourpossiblestatesandthe(per-capita)payoffs ineach of them are as follows. Table 1 State Probability Bank Insurance Late Firms loans claims depositors owners HH =0.63 B =3 α H φ = HL =0.07 B =3 α L φ = LH = α H φ = LL = α L φ = Table 1 shows depositors in the banks who are late consumers have different payoffs in states HH and HL compared to states LH and LL. The same holds for the owners of the firms being insured. They also have different payoffs in states HH and LH as compared to HL and LL. These different payoffs imply that there is potential for risk sharing between banks and insurance companies. Credit risk transfer between the two sectors is one way of providing this risk sharing. We consider a particularly simple form of risk transfer: the banks make a payment Z HL to the insurance companies in state HL,whiletheinsurancecompaniesmakeapaymentZ LH to the banks in state LH. For simplicity we also assume that the banks depositors obtain the surplus from the credit risk transfer. The insurance companies will compete to provide the credit risk transfer that maximizes the utility of the banks depositors at the lowest cost to themselves. In equilibrium they will obtain their reservation utility, which is what they would receive in autarky. This credit risk transfer improves diversification, but notice that markets are still not complete. The question is how such transfers can be implemented and what are their effects on welfare. In state HL bank loans are successful. Banks have excess funds and use them to transfer Z HL to the insurance companies. Thus, the only difference relative to the autarky situation is that at date 2 in states 13

14 HL and LH depositors now consume c 2HL = yr + zb e 2 Z HL, (8) 0.5 c 2LH = yr + Z LH. (9) 0.5 The problem is more complicated for the insurance companies. In state LH theownersofthefirms that insure their assets with the insurance companies have plenty of funds (equal to A+0.5R), but the insurance companies themselves do not have any. They receive α H φ =0.5 claims and use all the returns of the short asset to repair the damaged assets of the firms they insure. Inorderforthemtobeabletomakethepaymentatdate2 to the banks for the credit risk transfer they must hold extra assets. They must charge a higher premium to the firms initially and reduce the part of the endowment firms hold in long assets. The insurance companies must then decide in which security, short or long, to invest this extra amount to be able to make the payment Z LH. If they invest in the short asset, they need to make an initial investment s LH = Z LH to be able to make the transfer to the banks. The insurance companies can then offer to the owners of the firms an expected utility equal to EV short = 0.63V (A + s LH +(0.5 s LH )R) (10) +0.07V (0.5+s LH +(0.5 s LH )R + Z HL ) +0.27V (A +(0.5 s LH )R) +0.03V (0.5+s LH +(0.5 s LH )R). The term in this expression can be understood as follows. The insurance companies receive an initial premium 0.5 +s LH from the firms they insure and invest it in the short asset. At date 2 they receive the transfer Z HL from the banks in state HL and pay the amount s LH = Z LH to the banks in state LH. In the other states, the insurance companies do not make any transfer and, since they operate in a competitive environment, distribute the amount s LH to the firms. The owners of the firms are left at date 1 with (0.5 s LH ), which they invest in the long asset for a return (0.5 s LH )R in each state. Then, the first term in (10) represents the date 2 utility of the owners of the firms in state HH, which occurs with probability In this case, the firms enjoy the return A of their assets which are repaired at date 14

15 1, the return from the long term asset (0.5 s LH )R, and the amount s LH the insurance companies distribute to them. The second term is the utility of the owners of the firms in state LH, which occurs with probability Now the insurance companies go bankrupt and distribute to the firms all that they have, 0.5+s LH, plus the transfer Z HL they receive from the banks. In state LH, which occurs with probability 0.27, the insurance companies use the extra amount s LH invested in the short asset to pay the transfer Z LH to the banks, and the owners of the firms are left with the returns A and (0.5 s LH )R from the assets and the long term asset, respectively. Finally, in state LL, which has probability 0.03 of occurring, the firm assets cannot be repaid and there are no transfers. The owners of the firms can enjoy utility only from the (0.5+s LH ) funds they receive from the insurance companies and the returns (0.5 s LH )R from their investment in the long asset. As we have already mentioned, we solve the problem under the assumption that banks retain the surplus from the credit risk transfer and the owners of the firms will enjoy the same level of expected as in autarky. If we maximize the expected utility of the banks while keeping the expected utility of the owners of the firms constant at we obtain optimal transfers equal to EV short =0.482, Z HL = in state HL and Z LH = in state LH. Note that in doing this optimization, we keep the portfolios of the banks thesameasbeforehereandbelow,foreaseofexposition. Strictlyspeaking with the transfers Z HL and Z LH the banks will reoptimize and have slightly different portfolios. Taking account of this change does not alter the results below. Suppose next that the insurance companies invest the extra premium they charge to the firm in the long asset. In particular, they invest an amount LH such that LH R = Z LH. This amount LH is smaller than s LH because R>1. Also the insurance companies would not suffer the opportunity cost s LH (R 1) they suffer in each state when they invest s LH in the short asset. However, given the long asset cannot be liquidated for a positive amount, the insurance companies incur a loss in states HL and LL when they do not have enough funds to repair the assets and go bankrupt. Depending on which 15

16 of these effects dominate, the insurance companies decide optimally how to invest the extra premium they charge to the firms to finance the transfer to the banks in state LH. The expected utility of the owners of the firms when the insurance companies invest the extra premium in the long asset becomes EV long = 0.63V (A + LH R +(0.5 LH )R) +0.07V (0.5+(0.5 LH )R + Z HL ) +0.27V (A + LH R +(0.5 LH )R Z LH )) +0.03V (0.5+(0.5 LH )R). The terms have a similar interpretation to before. The first term represents the utility of the owners of the firms in state HH, when the firm asset produces A and 0.5 is invested in the long asset returning a total of 0.5R at date 2. Notice that the initial investment of 0.5 in the long asset is composed now of the extra premium LH that the insurance companies invest and the amount (0.5 LH ) that the owners of the firms invest. The second term represents the utility in state HL, when the insurance companies go bankrupt, receive the transfer Z HL from the banks but incur a loss on the LH units that they have initially invested in the long asset and that they must now liquidate for 0. The third term gives the utility in state LH, when the insurance companies pay Z LH to the bank; and finallythelasttermis the utility in state LL when the insurance companies go bankrupt, cannot repair the assets A and lose the LH units that they have initially invested in the long asset and that they must now liquidate for 0. They receive the 0.5 the insurance companies distribute and (0.5 LH )R from the amount they themselves invest in the long asset. If we solve for the optimal credit risk transfer when the insurance companies invest in the long asset it is not possible to give the same utility to the owners of the firmsasinautarkywithoutmakingthebanksworseoff than their level in autarky. Thus, it is optimal for the insurance companies to finance the transfer to the banks in state LH by investing s LH = Z LH in the short asset. In this case, banks expected utility is increased relative to autarky by this credit risk transfer with Z HL =0.059 in state HL and Z LH =0.017 in state LH. It can be shown the banks obtain EU = This compares with EU = in autarky. The overall result of credit risk transfer is therefore beneficial. Despite being limited to states HL and LH 16

17 and thus not offering full diversification, credit risk transfer allows a Pareto improvement relative to autarky. If credit risk transfer was allowed in more states a further improvement would be possible. The crucial feature of this equilibrium is that, even though there is a connection between the two sectors, banks and insurance companies still invest in different securities. Banks invest in both short and long assets, but the insurance companies only invest in the short asset. It is not worth them investing in the long asset because the insurance sector is subject to systemic risk and the long asset cannot be liquidated for a positive amount. An important consequence of this is that there is no contagion between the two sectors. That is, the systemic risk of the insurance sector does not spread to the banking industry. Banks are always solvent when the insurance companies fail. 4 Equilibrium with liquidation markets One of the apparent inefficiencies in the equilibrium considered in the previous section is that the insurance companies find it optimal to invest in the short asset rather than the long asset to fund their credit risk transfer because there is no market for liquidating the long asset. We next consider the effect of creating a market for liquidating the long asset. This also introduces the possibility of differentiating between historical cost accounting and mark-to-market accounting since there will now exist a market price for the long asset. To see that there is scope for a market for the long asset suppose initially that instead of being able to liquidate the long asset at date 1 for a return of only 0, the insurance companies receive the full value LH R at date 2 in each state. In this case the expected utility of the insurance companies customers when the long asset is used to fund credit risk transfer is given by: EV long = 0.63V (A + LH R +(0.5 LH )R) +0.07V (0.5+ LH R +(0.5 LH )R + Z HL ) +0.27V (A + LH R +(0.5 LH )R Z LH )) +0.03V (0.5+ LH R +(0.5 LH )R) = The only difference from the case where there is no market is that the long 17

18 asset always returns LH R, even in states HL and LL when the insurance companies go bankrupt. This leads to an improvement with respect to the case where the insurance companies invest in the short asset and the firms owners obtain EV short =0.482 as in autarky. More importantly, this suggests that introducing financial innovation and creating a market for the long asset where the insurance companies could sell their holdings when they go bankrupt might improve welfare since it would eliminate waste. However, as we will show below, this depends crucially on which accounting method is used because markets are incomplete and therefore prices may not reflect future earning power. 4.1 Historic cost accounting We start with the simpler case where asset values are recorded at cost even if there is a market and asset prices exist. This illustrates the functioning and the potential benefit of markets in our model. The way prices are formed in our model depends on the relationship between the incompleteness of markets and asset-price volatility. As argued in Allen and Gale (2004b, 2005b), liquidity plays a crucial role in a world in which markets and contracts are incomplete. When firms and financial institutions are forced to sell assets to obtain the liquidity needed to try to satisfy their claims, the suppliers of liquidity demand a premium in the form of low asset prices in states where the demand for liquidity (or, equivalently, the supply of assets) is high. This in turn implies that the supply of liquidity is endogenously low and asset-price volatility cannot be eliminated whenever there are incomplete contracts and incomplete markets. The creation of a market for the long asset immediately raises the issue that somebody must supply liquidity to this market. In other words somebody must hold the short asset in order to have the funds to purchase the long asset supplied to the market in states HL and LL. If nobody held liquidity, then there would be nobody on the other side of the market and the price of the long asset would fall to zero at date 1. This can t be an equilibrium though because by holding a very small amount of cash somebody would be able to enter and make a large profit. In the framework considered in this paper, the group that will supply the liquidity is the investors who provide capital to the banks. In order to be willing to hold this liquidity they must be able to recoup their opportunity cost. Since in states HH and LH when there is no liquidation of assets, they end up holding the low-return short 18

19 asset throughout, they must make a significant profit in at least one of the states HL and LL when there is a positive supply of the long term asset on the market. In other words, the price of the long asset must be low in at least one of these states. To see more precisely how prices are formed, we first need to see how many units of the long asset are offered in the market. To see this, recall that the banks portfolio is such that e 0 =0.25; e 1 =0;e 2 =0.42; x =0.5; y =0.45; z =0.3. If the banks assets are evaluated at their historical cost, they are worth x + y + z =1.2. This is significantly above the total liabilities at date 1 of c 1 =1.00 so that the banks remain solvent and continue operating until date 2. They do not liquidate any assets at date 1. Given the banks remain solvent with historic cost accounting, the price inthemarketforthelongassetisdeterminedbythesalesoftheinsurance companies. In states HH and LH the insurance companies do not sell their long assets and the investors will not use their liquidity to buy any assets. The equilibrium price must then be P HH = P LH = R =1.05. The reason for this is straightforward. Given that both the banks and the insurance companies hold the long asset between dates 1 and 2 in these two states the equilibrium price must be R. This ensures that both cash and the long asset will be held and markets will clear. If P<R,theinvestorswouldwantto buy the long asset since it would provide a higher return than cash between dates 1 and 2. In contrast, if P > R, the banks and insurance companies would sell the long asset and then hold cash until date 2. The only price at which both cash and the long asset will be held between dates 1 and 2, which is necessary for the equilibrium, is R. In contrast, in states HL and LL the insurance companies go bankrupt and will liquidate their holdings of the long asset LH at a price P HL = P LL = P L. In order for the suppliers of liquidity to the date 1 market for the long asset to be willing to participate, the price P L must be low enough to allow them to cover their opportunity cost of ρ. In equilibrium it must be the case that ρ = R. (11) P L The term on the left hand side is the investors opportunity cost of capital. The firsttermontherighthandsideistheexpectedpayoff to holding the 19

20 short asset in states HH and LH, which occur with probability 0.9. The second term is the expected payoff from holding cash in states HL and LL, which occur with probability 0.1, and using it to buy 1/P L units of the long asset at date 1. Each unit of the long asset pays off R at date 2. Solving (11) gives P L =0.44. Given this value for P L, it can be shown the optimal transfers that keep the insurance companies at their reservation level of utility and maximize the banks depositors welfare is Z HL = in state HL and Z LH = in state LH. The insurance companies find it optimal to fund their transfer with the long asset. They choose LH =0.016 to provide the necessary funds. The level of utility of the banks depositors with historic cost accounting is EU HC =0.0496, which is higher than when there is no market for liquidating the long asset. Finally, consider the amount of liquidity the investors will supply to the market for the long asset. In order for the market to clear at P L =0.44 in states HL and LL the amount the suppliers of liquidity will hold, denoted as γ, is so that P L =0.44 = γ LH = γ = γ 0.016, To sum up, introducing a market for the long asset improves welfare when the banks assets are evaluated at historical cost. The price in states HL and LL does not reflect the future value of the long asset but rather it is low enough to induce the investors to participate and offer liquidity to the market. The level of liquidity is thus endogenously determined and some asset-price volatility remains. There is cash-in-the-market pricing. Despite this, introducing the market helps the economy because the insurance companies can now finance the credit risk transfer with the long asset and liquidate it for a positive amount when they go bankrupt. 20

21 The crucial feature of this equilibrium is that the accounting value of the banks assets is insensitive to the bankruptcy of the insurance companies and market prices. When evaluated at the historical cost, the value of the banks assets is enough to cover their liabilities. The banks do not have to sell the long asset and can continue until date 2. At that point their assets are enough to cover their liabilities. In contrast, if mark-to-market accounting was used, in state LL when their loans fail the banks assets would be worth only x + y P L = = This is less than their liabilities of c 1 =1.00. The banks would then be declared insolvent and their long asset would be liquidated. This would depress the prices even further and possibly lead to contagion from the insurance sector to the banking sector. We analyze this case next. 4.2 Mark-to-market and contagion We now turn to the situation where mark-to-market accounting is used and show how introducing of a market for the long asset can lead to contagion and a reduction of welfare. The difference with the case with historical cost accounting is that in state LL the banks are declared insolvent and have to sell their long term asset. The supply of the long asset on the market in state LL is now larger, as both the banks and the insurance companies need liquidity to satisfy their claims at date 2. To see how this affects the market and the pricing of the long asset, consider first the other three states HH, HL and LH. As before, in states HH and LH neither the banks nor the insurance companies sell the long asset. In state HL the insurance companies sell the long asset while the banks do not as their loans are successful and they are not forced to sell assets. Differently from before, however, the equilibrium is such that there is now excess liquidity in state HL as well. Investors keep a greater level of liquidity than with historical cost accounting to satisfy the higher demand for liquidity in state LL. This surplus of cash means that P HL = R by the same argument as for P HH and P LH. Thus, the price of the long asset at date 1 in these three states will be P HH = P HL = P LH = R. Given this, the price P LL in state LL must such that the investors supplying 21

22 liquidity to the market break even, must satisfy ρ = R P LL. (12) The terms in (12) have a similar interpretation to those in (11). The term on the left hand side is the investors opportunity cost of capital. The first term on the right hand side is the investors expected payoff to holding the short asset in states HH, HL and LH (which have a total probability of occurring equal to 0.97). The second term is their expected payoff from using the cash in state LL (which occurs with probability 0.03) tobuy1/p LL units of the longassetatdate1 for a per-unit return of R at date 2. The only difference relative to (11) is that now investors hold liquidity in all states except state LL. This means that they have to make higher profits in this state to induce them to keep cash in all the others. Solving (12), we obtain P LL = This low price provides the incentive that is needed for the investors to provide the liquidity for the market. However, it also means that the banks are forced to liquidate at date 1 in state LL. The reason is that the value of their assets is x + y P L = = 0.582, and this is less than their liabilities of c 1 =1.00. They therefore go bankrupt and their assets are liquidated in the market. Given P LL =0.183, it can be shown the optimal transfers that keep the insurance companies at their reservation level of utility and maximize the banks depositors welfare is Z HL = in state HL and Z LH = in state LH. The insurance companies find it optimal to fund their transfer with the long asset. They choose LH =0.018 to provide the necessary funds. The level of utility of the banks depositors with mark-to-market accounting is EU MTM =

23 This is clearly significantly less than the depositors expected utility with historic cost accounting EU HC = In equilibrium the total supply of long asset to the market in state LL is LH +y = = In order for the market to clear at P LL =0.183 the investors have to hold an amount γ in the short asset between dates 0 and 1 to clear the market at date 1 such that P LL = γ LH + y = γ =0.183, Solving this gives γ = It is now possible to check that there is excess liquidity in state HL as assumed above. In this state the insurance companies will go bankrupt and will liquidate their of the long asset. This will be the only supply of the asset on the market and since < 0.086, the investors will have excess liquidity P HL = R as explained earlier. Taking prices as given the actions chosen by the insurance companies and banks are optimal. If an insurance company were not to engage in credit risk transfer it would receive the same as in autarky, which is no better. If it was to use the short asset to fund its credit risk transfer it would be strictly worse off. If a bank was to choose not to do credit risk transfer, it would still be liquidated in state LL and it would now not have the benefit ofthe credit risk transfer. The expected utility of its depositors would fall. To sum up, as with historical cost accounting, introducing a market for the long asset gives the possibility to the insurance companies to invest in the long asset and liquidate it for a positive amount at date 1. This helps the insurance companies to finance the credit risk transfer. Differently from before however, this now leads to a reduction in welfare because it causes contagion from the insurance sector to the banking sector. This contagion and inefficiency arise from the mark-to-market accounting and lead to a Pareto reduction in welfare. The investors and the insurance companies have the same levels of utility as in autarky. The banks however are worse off since their assets are liquidated at a low level in state LL and they go bankrupt. The reason for the poor performance of mark-to-market accounting is that when prices are determined by liquidity rather than future payoffs theyareno longer appropriate for valuing financial institutions assets. The equilibrium prices are low to provide incentives for liquidity provision. They are not low because fundamentals are bad. This point has important implications 23

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