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Peterson Institute for International Economics | 1750 Massachusetts Ave., NW | Washington, DC 200366/17/2016 1
Estimating Optimal Capital
Requirements for Banks
William R. Cline
Senior FellowPeterson Institute for International Economics
Modigliani-Miller Equity Cost as
Function of Debt/Equity Leverage
𝑖𝑗 = 𝜌 + 𝜌 − 𝑟𝐷𝑗
𝑆𝑗
𝑖𝑗 = unit cost of equity
𝜌 = sectoral capitalization rate
𝑟 = interest rate
𝐷𝑗 = debt
𝑆𝑗 = shareholder equity
M&M Test for 51 Large US Banks 2001-13
Earnings yield: Adj. R2 = 0.088
eyt = 6.63 + 0.0513 Rt–1 -1.89 D0810; (19.5) (1.62) (–7.2)
R: debt/equity D: dummy
Net income/ Book equity Adj. R2 =0.268
NIt/Et–1 = 7.206 + 0.636 Rt–1 -5.823 D0810; (10.0) (9.4) (–10.5)
Losses from a Banking CrisisOutput
Timet0t-1 t1 t2
Qt-1
Q*
Qt0
A
B
Deriving the benefits curve
Baseline damage: 𝐷0 = 𝑃𝑐𝑟0𝜆0
Crisis probability: 𝑃𝑐𝑟𝑘 = 𝐴𝑘𝛾, 𝛾 < 0
Benefit: 𝐵 = −(𝑃𝑐𝑟𝑘 − 𝑃𝑐𝑟0)𝜆0
= −𝐴𝜆0 𝑘𝛾 − 𝑘0𝛾
Marginal benefit: 𝑑𝐵
𝑑𝑘= −𝐴𝜆0𝛾𝑘
𝛾−1
BCBS Schedule of Crisis Probability
For Alternative Capital Ratios
Benefits of higher capital ratios
0.000
0.005
0.010
0.015
0.020
0.0
39
3
0.0
5
0.0
6
0.0
7
0.0
8
0.0
9
0.1
0.1
1
0.1
2
0.1
3
0.1
4
0.1
5
0.1
6
0.1
7
0.1
8
0.1
9
0.2
0.2
1
0.2
2
0.2
3
0.2
4
0.2
5
TCE/TA
Fraction of GDP
Impact on cost of capital
to the economy
Banks: 𝑧 = 𝑧0 + (𝑘 − 𝑘0)(𝜌𝐵 − 𝑟𝑑)(1 − 𝜇)
Non-banks: 𝑟𝑁𝐵,0 + 𝜃 × (𝑧 − 𝑧0)
Economy:
𝑤 = 𝜙𝐵 𝑧 + 𝑆𝑓 + 𝜙𝑁𝐵𝑟𝑁𝐵 + 𝜙𝑓𝜌𝑓
Proportional change 𝑣 =𝑤𝑘
𝑤0− 1
Cost of higher capital cost to
economy
𝐶 =𝑣 × 𝛼 × 𝜎
(1 − 𝛼)
= output elasticity with respect to capital
= elasticity of substitution, capital & labor
If =0.33, = 0.5, w0= 0.1, w = 0.01, and
v =0.1, then:
𝐶 = 0.025
Marginal cost to economy from
higher k is constant
𝑑𝐶
𝑑𝑘=𝑑𝐶
𝑑𝑣×𝑑𝑣
𝑑𝑤×𝑑𝑤𝑑𝑧
×𝑑𝑧𝑑𝑘
=𝛼𝜎
1−𝛼
1
𝑤0𝜙𝐵 + 𝜃𝜙𝑁𝐵 𝜌𝐵 − 𝑟𝑑 1 − 𝜇
≡ 𝜓
Benefits and costs of additional
bank capital
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.0
39
3
0.0
5
0.0
6
0.0
7
0.0
8
0.0
9
0.1
0.1
1
0.1
2
0.1
3
0.1
4
0.1
5
0.1
6
0.1
7
0.1
8
0.1
9
0.2
0.2
1
0.2
2
0.2
3
0.2
4
0.2
5
CostBenefit
ratio of capital to total assets
fraction of GDP
Simulation parameters
Parameter Concept Low OCR Base High OCR
Loss from crisis 0.3 0.64 1.0
B Equity cost to banks 0.13 0.10 0.07
M&M offset 0.35 0.45 0.60
Nonbank spillover 0.7 0.5 0.2
Capital elasticity 0.43 0.40 0.33
Substitution elast. 0.8 0.5 0.4
Frequency, optimal capital ratio
0
100
200
300
0.0
5
0.0
6
0.0
7
0.0
8
0.0
9
0.1
0.1
1
0.1
2
optimal capital/asset ratio
Optimal Capital Requirements(tce/rwa %)
Great Recession change, net income/assets,
and log asset size, 50 large US banks
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
0 1 2 3 4 5 6 7 8