Chapter13 Raroc Calculation 6w4e5v

Chapter 13 Risk-Adjusted Return on Capital Models

Definition of RAROC • • • •

AdjustedIncome CapitalatRisk

RAROC = If RAROC > Hurdle rate then value adding. ROA = RORAC = AdjustedIncome AssetsLent

AdjustedIncome Risk  basedCapital Re quirement

• EVA = economic value added = Adjusted income – ROE x K. Invest if  0. Saunders & Allen Chapter 13

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The Numerator: Adjusted Income • • • • •

= Spread (direct income on loan) + + Fees (directly attributable to loan) – - Expected Loss (EDF x LGD) – - Operating Costs (allocated to loan) Then multiply the entire amount by 1 – the marginal tax rate. Saunders & Allen Chapter 13

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The Denominator: Capital at Risk • Market-based approach (BT model) – Measure the maximum adverse change in the market value of the loan resulting from an increase in the credit spread – Use duration model to measure price effects.

• Experientially-based approach (BA model) – Calculate UL using a multiple x LGD x exposure x standard deviation of default rates. Saunders & Allen Chapter 13

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The Market-based Approach to Measuring Capital at Risk L

=

-DL

(Dollar capital risk the exposure or loss the amount)

x

L

(Duration of the loan

x

R/(1+RL)

(13.9)

(Risk amount or

(Expected discounted change in

loan exposure)

credit premuim or risk factor on loan)

• If DL=2.7, L=$1m, R=1.1%, R=10%, then: L = -$ 27,000

Saunders & Allen Chapter 13

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Figure 13.1 Estimating the change in the risk .

Frequency

1% of All AAA Bonds

 Risk (R )

1%

0

1.1%

Saunders & Allen Chapter 13

3.5%

 Risk (R )

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The Experientially-based Approach to Capital at Risk Measurement • If 99.97% VAR (AA rating) and normal distribution, then the multiplier is 3.4. • But, most banks use a large multiplier because loan distributions are not normal. • BA uses multiplier = 6. • If LGD=.5, Exposure=$1m, Loan =.00225, then UL=6 x .00225 x .5 x $1m = $27,000 (same as market-based approach) Saunders & Allen Chapter 13

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Calculating the RAROC of the Loan Example • • • • • • •

Spread = .2% x $1m = $2,000 Fees = .15% x $1m = $1,500 EL = .1% x $.5m = ($500) Tax rate = 0% Adjusted Income = $3,000 RAROC = $3,000/$27,000 = 11.1% If cost of capital < 11.1% then make loan. Saunders & Allen Chapter 13

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The RAROC Denominator and Correlations Ri – RF = i (Rm - RF) (13.15) where Ri = the return on a risky asset, RF = the risk-free rate, Rm = the return on the market portfolio, i = the risk of the risky asset, and (13.16) i = im/2m = imim/2m = imi/m where im = covariance between the returns on risky asset i and the market portfolio m, m = standard deviation of the return on the market portfolio, im = correlation between the returns on the risky asset i and the market portfolio Ri - RF imi RAROC

=

Rm - RF m = Hurdle Rate

(13.18)

Saunders & Allen Chapter 13

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Incorporating Unsystematic Risk • •

• •

Equation (13.18) is the traditional Sharpe ratio for a loan. But, if all idiosyncratic risk is diversified away, then no need for RAROC. RAROC deals with untraded and unhedgeable assets (loans). Banks specialize in info-intensive relationship lending that cannot be hedged in capital markets. Risk of loan should be divided into: (1) liquid, hedgeable market risk component and (2) illiquid, unhedgeable component. The correlation of the unhedgeable component with the bank’s portfolio will determine the loan’s price. So different banks (with different portfolio correlations) will have different pricing (credit risk). Froot & Stein (1998): Loan’s hurdle rate =market price of the loan’s traded risk + bank shareholders’ cost of capital to cover nontradeable risk. The second term is idiosyncratic. Saunders & Allen Chapter 13

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