Introduction Let's continue with Leland's model for the risk-neutral valuation of a bank, this time focusing on the market risk of bank stock in an idealized setting. An idealized bank Let's model the risk-neutral value of the equity of a idealized fractional reserve bank with shareholder limited liability and deposit insurance. The bank's assets consist of cash reserves and loans; and the bank's liabilities consist of deposits. The cash reserves are a regulatory requirement, and must be at least a fraction Rmin (the reserve ratio) of the book value of deposits. Note that cash reserves contribute to book value, but not to market value: they generate no income for shareholders. Cash reserves are essentially the property of the depositors. Let the book value of the deposits be D and say the bank pays interest at a constant annual rate e. The market value of the deposits is the value of a perpetuity paying out constant annual income eD. If the long- term risk-free interest is r, the value of this perpetuity is E K = D- which should, in principle, be sufficient to defease the obligations of the owners of the bank. The proceeds of the deposits are available to make loans. These loans are risky. The bank will charge an annual interest rate 8 + y expecting that interest income will be partly offset by other-then-temporary impairments being recognized at a rate of per year, leaving a net loan revenue rate of 8 per year. Let's say that the value of the performing loans is initially So; and that St for t > 0 fluctuates like a geometric brownian motion with annual volatility rate o: If the impairments are lower than expected, St increases. If they are higher than expected, St decreases. The bank can manage this variation to some extent through diversification and careful underwriting, but this variation is still the most signifigant source of risk to the bank's shareholders. 1 The limited liability and deposit insurance is reflected by a real option, a perpetual put with strike price K on the loans. We know from the previous assignment that the definition of the bank's default stopping timer is min {r > 0: S = L} where the exercise boundary is K L= 1+ and 2r -T 1 + + 02 04 in terms of o, the volatility of the loans. Assume that So starts at So = D = K" Note that we know that market value and book value on the loans is equal at this moment, because the bank has paid and lent exactly this much. As time goes by, the market value will continue to be St but the book value might be different. The value of equity is Et = St + Pt-K Note that Et > 0 for 0 L, the risk-neutral value of equity (as a fraction of the fixed K) as a function of assets is Et 77 St + 1 K (1+7)+7 K St = Problems Define "long coverage" and "short coverage" for a stock as the ten-day (two-week) 99% value-at-risk for a long and short position in the stock as a fraction of the value of the posiiton. For these problems, let's work with a idealized bank as described above with r = 0.02/yr, e = 0.02/yr, 8 = 0.03/yr, o = 0.02/yr /2 and Rmin = 0.08. You can assume that the drift in the asset value is negligible over this period. 1. Calculate this bank's book ratio (2 points) 2. Calculate the long and short coverage for this bank's stock (3 points each). 3. Calculate the bank's implied credit spread from its interest expenses and the market value of its liabil- ities (2 points) Introduction Let's continue with Leland's model for the risk-neutral valuation of a bank, this time focusing on the market risk of bank stock in an idealized setting. An idealized bank Let's model the risk-neutral value of the equity of a idealized fractional reserve bank with shareholder limited liability and deposit insurance. The bank's assets consist of cash reserves and loans; and the bank's liabilities consist of deposits. The cash reserves are a regulatory requirement, and must be at least a fraction Rmin (the reserve ratio) of the book value of deposits. Note that cash reserves contribute to book value, but not to market value: they generate no income for shareholders. Cash reserves are essentially the property of the depositors. Let the book value of the deposits be D and say the bank pays interest at a constant annual rate e. The market value of the deposits is the value of a perpetuity paying out constant annual income eD. If the long- term risk-free interest is r, the value of this perpetuity is E K = D- which should, in principle, be sufficient to defease the obligations of the owners of the bank. The proceeds of the deposits are available to make loans. These loans are risky. The bank will charge an annual interest rate 8 + y expecting that interest income will be partly offset by other-then-temporary impairments being recognized at a rate of per year, leaving a net loan revenue rate of 8 per year. Let's say that the value of the performing loans is initially So; and that St for t > 0 fluctuates like a geometric brownian motion with annual volatility rate o: If the impairments are lower than expected, St increases. If they are higher than expected, St decreases. The bank can manage this variation to some extent through diversification and careful underwriting, but this variation is still the most signifigant source of risk to the bank's shareholders. 1 The limited liability and deposit insurance is reflected by a real option, a perpetual put with strike price K on the loans. We know from the previous assignment that the definition of the bank's default stopping timer is min {r > 0: S = L} where the exercise boundary is K L= 1+ and 2r -T 1 + + 02 04 in terms of o, the volatility of the loans. Assume that So starts at So = D = K" Note that we know that market value and book value on the loans is equal at this moment, because the bank has paid and lent exactly this much. As time goes by, the market value will continue to be St but the book value might be different. The value of equity is Et = St + Pt-K Note that Et > 0 for 0 L, the risk-neutral value of equity (as a fraction of the fixed K) as a function of assets is Et 77 St + 1 K (1+7)+7 K St = Problems Define "long coverage" and "short coverage" for a stock as the ten-day (two-week) 99% value-at-risk for a long and short position in the stock as a fraction of the value of the posiiton. For these problems, let's work with a idealized bank as described above with r = 0.02/yr, e = 0.02/yr, 8 = 0.03/yr, o = 0.02/yr /2 and Rmin = 0.08. You can assume that the drift in the asset value is negligible over this period. 1. Calculate this bank's book ratio (2 points) 2. Calculate the long and short coverage for this bank's stock (3 points each). 3. Calculate the bank's implied credit spread from its interest expenses and the market value of its liabil- ities (2 points)