# Risk management in banks

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1 Econ 4335 Lecture 13 University of Oslo April 26, 2010

2 Types of risk Credit risk (default) Exchange rate risk Market risk Counterparty risk Operational risk

3 Mitigating risks Risk pricing Diversification Hedging Risk measurement Good procedures

4 The expected cost of default Consider a marketable loan (e.g. a bond) Promised repayments: C 1, C 2,..., C n at t 1, t 2,...t n Risk-free interest rate: r, continuous compounding Value if all repayments on time: P 0 = n C k e rt k (1) k=1 Value of loan to risk-neutral investors when positive probability of default: n P N = E[C k ]e rt k (2) k=1 Expected cost of default: P 0 P N.

5 The yield to maturity The yield to maturity is the rate of interest that the buyer of a loan implicitly gets if there is no default. If the loan above is sold for the price P, the yield to maturity R is defined by n C k e Rt k = P (3) k=1 If the market is dominated by risk-neutral investors, P = P N and n n C k e Rt k = E[C k ]e rt k = P N (4) k=1 k=1 The credit spread is defined as s = R r. Alternative measure of credit risk Measure of expected cost of default when investors are risk neutral

6 Application to banking Marginal funding rate replaces the risk-free interest rate Risk-neutral bank in a competitive environment then sets repayments that make P N = L. The yield to maturity on the loan is then R. Example All repayments made in period n. Probability of no default in period n is p n. No recovery if default. The banks sets the C n that satisfies P N = p n C n e rn = L The return to maturity is given by C n e Rn = P N = L Solution: C n = Le rn /p n = Le Rn, R = r (1/n) ln p n

7 The spread depends positively on default probabilities depends negatively on recovery rate s = R r (5) depends in complicated way on time path of cash flow and default probabilities if the probability of default per unit of time is constant, λ (Poisson process) and the recovery rate is zero, s = λ Higher spread for borrowers with high debt-to-asset ratios Higher spreads for borrowers with more risky assets or incomes

8 The spread Banks with pricing power will add a profit margin Should risk-averse banks add a risk premium?

9 A digression on accounting Default probabilities often start low and increase as the loan matures In the beginning the spread is usually more than enough to cover actual losses Loan loss provisions are usually not made before a loss is more likely than not (either on an individual loan or a group of loans) Fast growing banks tend to show high profits to begin with Their accounts may exaggerate their solidity Reform proposal: Use spread to make provisions in advance

10 Another digression on accounting Spread on bonds sometimes exceed default probabilities by a wide margin In many bond market liquidity is low Statutory regulations severely limits demand for bonds that are rated beyond a certain levels Bond markets sometimes freeze Market prices reflect forced sales ( fire sales ) Value on the basis of hold to maturity or market price?

11 Back to banks management of credit risk Suppose large number of small borrowers default probabilities for the individual borrowers are independent the spread is set to cover expected default loss loan loss provisions properly reflect the spread and expected losses Conclusion no reason for a risk-averse bank to charge more for the loan no need to keep equity in order to offset loan losses. Or: What about model and statistical uncertainty?

12 Why equity is needed Some borrowers are large Losses are highly correlated Specialized banks Accounts do not always reflect all expected losses

13 The Basel accords Old approach: Minimum capital requirement relative to total liabilities Basel I, 1988: Capital requirement 8 per cent of risk-weighted assets. Four different risk weight depending on type of borrower, collateral and maturity Range from 0.0 for OECD-states, via 0.2 for OECD banks, 0.5 for residential mortgages (below 100 per cent), to 100 for loans to businesses Some off-balance sheet items included Basel II Banks may use their own risk models to produce weights More types of risk included (market risk)

14 The Basel accords Capital requirements can be satisfied with equity, subordinated loans or hybrid capital. Tier 1: Minimum 4 per cent. Equity and some hybrids Tier 2: Other hybrids and subordinated loans. Detailed rules differ from country to country. Capital adequacy rules are supplemented by limits on credits to single customers.

15 Academic criticism of Basel I The risk weights do not represent the actual risks well It is macro risk factors that matter, not individual risks Fixed weights can never represent more than one macro risk factor well (F&R8.1.3)

16 The demand for reserves The bank s expected profits Π(R) = r L (D R) + rr r p E[max(0, x R)] r D D (6) R = level of reserves r = interest rate on reserves r p = penalty interest rate x = net withdrawals

17 Cost of liquidity shortages D C(R) = E[max(0, x R)] = r p (x R)f (x)dx (7) D C (R) = r p f (x)dx = r p Pr[ x R] < 0 (8) R C (R) = r p f (R) 0 (9) C(R) decreasing and convex R

18 Choice of R First order condition for R Π(R) = r L (D R) + rr C(R) r D D (10) r L + r C (R) = 0 Pr[ x R] = r L r r p (11) Probability of borrowing at penalty rate equals liquidity premium r L r over penalty rate r p

19 Marginal costs and benefits Insert R = D L in definition of profits: Π = r L (L) + r(d L) C(D L) r D D (12) Marginal cost of loans / Marginal benefit from deposits r C (D L) = r + r p Pr[ x R] (13) Charge more for loans and pay more for deposits Marginal costs and gains falls with D L Can be incorporated in monopolistic bank model

20 Liquidity management in practice Wholesale deposits more volatile Reserves mainly in interbank market Premium on longer interbank loans Prudent banks who borrow a lot in interbank market must also have deposits there or unused credit lines Longer term interbank loans carry a liquidity premium

21 can in principle be avoided by two methods floating interest rates on all deposits and loans every loan with fixed interest rate is financed with fixed interest rate bonds or time deposit with the same duration Reason that banks assume interest rate risk: Depositors prefer more liquid deposits Deposit financing is cheaper than bond financing Some borrowers are willing to pay more for fixed-rate loans Reserves will be needed

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