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