1) Bank Runs
There are two types of consumers E and L. Each consumer owns an initial amount of 1$ at time T = 0. The two types differ in their need to consume. Type E consumers must consume at T = 1 (early), whereas type L consumers may consume at T = 1, but can also afford to wait until T = 2 (late) instead. The utility function of type E is given by UE = √c1, the utility function of type L is given by UL = 0.5 · √c1 + c2, where c1 and c2 represent the amount of consumption in periods 1 and 2 respectively.
The initial amount of each individual can be invested in homogenous projects which run for two periods (up to T = 2) and require a 1$ investment each. In order to consume early the project has to be liquidated at T = 1 yielding X1 = 1$ (i.e. the initial investment). If the project runs until T = 2 the return is X2 = 3$.
Assume there are 100 consumers in an economy who try to maximize their own utility. The types are determined randomly and consumers learn about their own type at T = 1. The fraction of type E consumers is g = 0.25, which is known to all individuals.
a) Each consumer invests in a project at T = 0. What is the best consumption strategy of a type E and type L consumer? What is the expected utility of a consumer at T = 0?
b) Now assume the 100 individuals can form a mutually owned bank. Each one deposits 1$ and the bank uses the deposits to invest in 100 projects. In order to consume in a period, depositors have the withdraw the amount from the bank. The bank promises an amount of r1 = 1.2$ to customers who withdraw their deposit at T = 1 and an amount of r2 = 2.8$ to customers withdrawing at T = 2. The bank will liquidate the necessary number of projects to fulfill its debt servicing. Withdrawers are served sequentially in random order until the bank runs out of assets. What is the expected utility of a consumer at T = 0 if type E consumers withdraw at T = 1 and type L consumers withdraw at T = 2?
c) Verify that the strategy of type L in part b) is a Nash Equilibrium.
d) Is there a Nash Equilibrium where all type E and L consumers withdraw at T = 1? Briefly discuss the implications of this result.
e) Assume the bank can temporarily suspend servicing at T = 1 when the number of withdrawers exceeds 60% of all customers. The first 60% receive r1, while the remaining type L consumers have to come back at T = 2 to withdraw. Test whether withdrawing at T = 1 can be an optimal strategy for type L in this scenario. Briefly discuss the implications of this result.
f) Instead of suspension of service, the government now provides a deposit insurance promising that any individual who withdraws at T = 1 will receive r1 and any individual withdrawing at T = 2 will receive r2. This insurance is financed by collecting an equal tax from all individuals once the government has to step in to make payments. What is the maximum amount of taxes that will be charged? Test whether withdrawing at T = 1 can be an optimal strategy for type L in this scenario. Briefly discuss the implications of this result.
2) The Merton Model and Banks
The file bank_data.xls contains a time series of the daily market value of assets of a large German bank and the book value of debt outstanding (in bn. €). Assume on 15 July 2010 the debt has a remaining maturity of one year, i.e. it matures on 15 July 2011. The risk free rate is constant at 3% per annum. It also represents the annual growth rate μ of the assets of the firm.
The assumptions of the Merton Model are valid throughout this problem.
a) Calculate the market value of equity and the market value of debt on 15 July 2010. Use the most recent 250 asset value observations to estimate the asset volatility. The daily volatility can be estimated as the standard deviation of daily asset returns rt = ln At/At-1 scale the daily volatility by a factor of √250. To annualize the volatility,
b) Calculate the Distance-to-Default (DD) and the Probability-of-Default (PD) on that date.
c) Given the information on the assets up to 15 July 2010, simulate 1000 daily asset paths one year into the future (i.e. up to 15 July 2011). For simplicity assume there are 250 business days in a year. For each path, determine whether the firm would have defaulted according to the default mechanism of the Merton Model. Compare the observed default frequency with your estimate in b).
The bank now has the option to increase its business risk by investing in complex structured products, resulting in an asset volatility of σ = 0.4. The terms on the debt outstanding and other parameters remain unchanged.
d) Create a graph that shows the market value of equity and the market value of debt (y-axis) on 15 July 2010 as a function of asset volatility (x-axis). This can be done by evaluating the formulas for different values of volatility. Volatility should range from 0.01 to 0.6 with a step size of 0.01. In the same manner create a graph of the PD as a function of asset volatility. Briefly discuss the implications of your results. Do you see potential conflicts of interest between the stakeholders of the firm (owners, creditors, economy-wide implications)?
e) Run a simulation of asset paths as in part c), now using the high volatility of σ = 0.4. For each path store the market value of equity and the market value of debt on 15 July 2011,
i) assuming all debt matures on that date.
ii) assuming all debt is rolled over, meaning the maturity date is shifted to one year from 15 July 2011.
What is the expected market value of equity and debt in these two cases. Are they conform with your results in d)?
f) Discuss whether the default mechanism of the Merton model in part e) describes empirical evidence of bank failures and the implications of bankruptcies adequately.
Attachment:- bank_data.xls