Integrating by parts the expression for expected unit cost of effective investment, Holmstrom and Tirole, show that at the optimum, the threshold liquidity shock is equal to the expected unit cost of effective investment
From which it follows that
And that, the wait and see approach to liquidity shocks is sub optimal.
We relax one of the initial assumptions and allow for verifiable, exogenous and deterministic income at the intermediate stage, the new threshold is
Assuming free cash flow, then the optimal re-investment rule is
If the intermediate income is a function of the borrower’s effort i.e. endogenously determined then the continuation rule becomes,
being an increasing function of the intermediate income.
Finding: firms require insurance against liquidity shocks, as long as they cannot pledge the full value of their activity to new investors. Secondly: firms should fully hedge, if it is costless to do so, even if the idiosyncratic shocks are outside their control. Assume a unit hedging cost and hedging ratio and exogenous shock to date 1 income
Then the firm has enough liquidity to continue at date 1, If and only if
If date 1 income is, per unit cost of investment at date 0 becomes
, and investors break even constraint in expectation, implies that investors total outlay is equal to their benefit. [similar to Eqn (1)]
And just like in Eqn (3), entrepreneur’s utility is equal to the project NPV
If we define the unit cost of effective investment (cost of obtaining, in expectation, one unit of unliquidated investment) as
Then the entrepreneur objective function is to maximize net utility (social surplus of the project)
The extent of hedging , and liquidity hoarding are determined by
Hedging ratio is invariant to changes in variables that affect only date 2 total benefits , and pledgeable income. Normalizing the model to a uniform distribution
where is the variance of .
Note that , giving us
, since we are minimizing
In the uniform case, the optimal hedging ratio decreases with the unit cost of hedging. If we substitute into (8) and set the least cost, we have
The threshold liquidity shock is depends on date 1 income .Furthermore an increase in the cost of hedging, reduces the hedging ratio, decreases the hoarding of liquidity and raises the cost.
Increase in r, increases amount of short term debt per unit of investment
Hedging may or may not depend on factors that affect liquidity management like short and long run leverage.
Banking and risk management:
1996 amendment to 1986 Basle Accord imposed extra Capital Adequacy Requirements on banks trading books, by differentiating between credit and market risk; by specifying a Value At Risk of 99%, adding the CAR for the trading book to that for the banking book etc.
Basic model assumptions:
Trading book serves as hedge for the banking book
and are the income shocks at date 1 on the banking and trading book respectively.
Hypothesis 1: Both income shocks are observed by regulators.
Expropriation of bank surplus when both shocks are favorable.
Penalties assessed when realizations of both shocks are positively related
Bank is rewarded if the realization of both shocks are negatively rewarded
Punishments are hard to implement when the bank is under capitalized. Hypothesis 2: Only the sum of the shocks is observed by regulators. Since regulators cannot differentiate between shocks, and have no way of knowing if the trading book is being used to hedge risks or gamble, banks have the incentive to carry out transfers between the two books to avoid punishment or minimize charges, leading to “double moral hazard”;
They exert effort to shift the distribution of income towards higher values.
Take risks or hedge against the uncertain income, thus distorting the riskiness of the income distribution.
Hypothesis 3: Only the shock to trading book income is observed. According to Holmstrom and Tirole, this is a situation is which information about the banking and trading books accrues at different frequencies. More specifically shocks to the trading book reveal themselves faster than shocks to the banking book, by the very nature of the two portfolios.