debtequity

We do not consider the case R c > R E for which c i > 0 and e = 0, since banks must have positive
equity in equilibrium. If R E = R c then we assume investors only invest in equity (i.e. c i = 0) and
this is without loss of generality under assumptions that we will introduce shortly.
There is a unit mass of each of the three types of agents and a unit mass of banks, to which we
now turn our attention.
Banks
Banks collect the deposits (d), the equity (e) and a part (c b ) of the lending of cash funds
(the rest finances the government) and invest the proceeds K = d + e + c b in risky projects with
payoff ã per unit of capital at date 1. The random variable ã is as described in Section 2 with
support on the interval [a, ∞) and with continuous density f(a) extended to the interval [0, ∞). The
safe part Ka of their date 1 payoff can be interpreted as the safe component which can be pledged as
collateral for borrowing from cash funds (akin to the senior tranche of the banks securitized assets).
Cash funds will lend an amount in excess of this sure component i.e. R c c b > Ka only if they are
sure to recover their funds. In this section we consider two possibilities, with or without implicit
insurance for the cash funds. If there is implicit insurance, cash funds believe that the government
will reimburse their loans if the banks default and this belief is realized. In this case they may
accept to lend more than the value of the safe collateral Ka, i.e. they accept risky collateral. We
call the insurance "implicit" because there is no explicit contract or insurance premium attached to
it. An example of implicit insurance is the belief that the government will bailout too-big-to-fail
banks if they are in difficulty. However if the government makes it credible that it will not intervene
in case of banks’ default, then there is no implicit insurance, and the cash funds will not lend to the
banks more than the sure collateral Ka. As for the depositors, we assume that they are explicitly
insured (FDIC) 3 . In addition banks provide payment services to depositors which cost them µ per
unit of spending by a depositor at date 1. To recoup some of the cost incurred by the taxpayers to
pay the banks’ debts in case of bankruptcy, the government charges an insurance premium π per
unit of debt at date 1. 4
Introducing cash funds into the model serves to capture the change in banking from traditional
banking based on deposits to modern banking based on securitization of assets and collateralized
borrowing on the wholesale money market. Such loans are safe as long as the collateral retains
its value: when collateral is at risk of losing value (low return on bank assets) the Central Bank
3 Large uninsured deposits enter as “implicitly” insured cash funds loans since these loans are unsecured and
uninsured. The implicit insurance of such deposits was made explicit after the financial crisis when temporarily (up
to December 2012) all the non-interest-bearing accounts of banks were insured for an unlimited amount.
4 If FDIC were a standard insurance company the premium would only be charged on the value of the insured
deposits: in the model we follow the practice in the US, by which the FDIC premium is charged on all the bank’s
debt.
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