Economi Capital. Tiziano Bellini. Università di Bologna. November 29, 2013

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1 Economi Capital Tiziano Bellini Università di Bologna November 29, 2013 Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

2 Outline Framework Economic Capital Structural approach to default. The asymptotic risk factor (ASRF) approach. Concluding remarks. Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

3 Structural approach to default Banking Assets and Liabilities Stylized Balance Sheet Figure: Stylized Commercial bank balance sheet. Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

4 Structural approach to default Firm Valuation and Default Definition Introduction to Merton Model 1 Figure: Default probability in Merton model. 1 Merton, R.C. (1974). On the pricing of corporate debt. The risk structure of interest rates, Journal of Finance, 29, Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

5 Structural approach to default Asset Valuation Asset and Liability We have not used any option pricing formulae. In fact, there is no theoretical reason why we need them to determine default probabilities, but there is instead a practical one: for a typical firm, we cannot observe the market value of assets. Option pricing theory can help as it implies a relationship between the unobservable (A t, σ) and observable variables. For publicly traded firms, we observe the market value of equity, which is given by the share price multiplied with the number of outstanding shares. At maturity T, we can establish the following relationship between equity value and asset value E t = max(0, A T L). (1) Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

6 Structural approach to default Asset Valuation Equity Holders Call Option Figure: Pay-off to equity and bondholders at maturity T of equation (1). Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

7 Structural approach to default Asset Valuation Option Price Estimation If the firm pays no dividends, the equity value can be determined, considering that r is the logaritmic risk free rate of return, through the standard Black Scholes call option formula where and E t = A t Φ(d 1 ) Le r(t t) Φ(d 2 ), (2) d 1 = ln(a t/l) + (r + σ 2 /2)(T t) σ, (3) T t d 2 = d 1 σ T t. (4) In order to solve the equation we need to estimate σ and A t using historical information. A practical solution can be found exploiting an iterative approach where the initial values are set in a way that A t k corresponds to the market value of equity, while L t k is equal to the book value of liabilities. Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

8 The Asymptotic Risk Factor (ASRF) Approach One Factor Model Asset Returns The asymptotic risk factor (ASRF) approach assumes that the bank credit portfolio consists of a large number of relatively small exposures. It is derived from an adaptation of the single asset model of Merton (1974). We distinguish between idiosyncratic risk associated with individual exposures and systematic risks. The default distribution of a firm is a Bernoulli distribution, derived from the distribution of the value of the firm asset returns. We start by assuming that the normalised asset return R i of firm i in the credit portfolio is driven by a single common factor Y and an idiosyncratic noise component ɛ i R i = ρ i Y + 1 ρ i ɛ i, (5) where Y and ɛ i are i.i.d. N(0, 1), meaning that R i is considered to have a standardised Gaussian distribution. Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

9 The Asymptotic Risk Factor (ASRF) Approach One Factor Model Default vs. Non Default We define a binary random variable Z i for each firm, which takes on value 1 (meaning that the ith obligor defaults) with probability p i and value 0 with probability 1-p i. According to the theory of Merton (1974), we have Z i = 1 R i Φ 1 (p i ) Z i = 0 R i > Φ 1 (p i ), (6) where Φ 1 is the cumulative distribution function of the standard Gaussian distribution. The parameter p i is the unconditional default probability of obligor i. Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

10 The Asymptotic Risk Factor (ASRF) Approach One Factor Model Conditional Probability If the outcome of the systematic risk factor was known, we could calculate the conditional probability of default by P(Z i = 1 Y = y) = P(R i Φ 1 (p i ) Y = y) = P( ρ i Y + 1 ρ i ɛ i Φ 1 (p i ) Y = y) = P(ɛ i < Φ 1 (p i ) ρ i Y 1 ρi Y = y) = Φ ( Φ 1 (p i ) ) ρ i y. (7) 1 ρi Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

11 The Asymptotic Risk Factor (ASRF) Approach Credit Portfolio Portfolio Assumptions Vasicek (1977) 2 showed that under certain conditions, Merton single asset model can be extended to a model for the whole portfolio. The portfolio model used in the advanced IRB approach (Gordy, 2003) 3 is very similar to Vasicek model. Inputs supplied by the bank include the exposure at default (EAD), the probability of default (PD), the loss given default (LGD) and the effective remaining maturity (M). 2 Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, Gordy, M. (2003). A risk-factor foundation for risk-based capital rules. Journal of Financial Intermediation, 12, Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

12 The Asymptotic Risk Factor (ASRF) Approach Credit Portfolio Portfolio Losses Assume that we have a portfolio with N clients with different exposures E i, asset correlations ρ i, probabilities of default p i and loss given defaults LGD i. Define the exposure weight w i = E i / N i=1 E i of client i, and let the portfolio loss per euro of exposure be given by L = N w i LGD i Z i. (8) i=1 It can be shown that as the number of clients in the portfolio N, the α-percentile of the resulting portfolio loss distribution approaches the asymptotic value q α (L) = ( N Φ 1 (p i ) + ) ρ i Φ 1 (α) w i LGD i Φ. (9) 1 ρi i=1 Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

13 The Asymptotic Risk Factor (ASRF) Approach Credit Portfolio Unexpected Losses Losses above the expected levels are usually referred to as unexpected losses (UL). Institutions know that these losses will occur now and then, but they cannot know in advance the time of their arrival and their amount. Banks are in general expected to cover their EL on an ongoing basis, e.g. by provisions and write-offs, because it represents just another cost component of the lending business. According to this concept, capital is only needed for covering unexpected losses. Hence, in Basel II, the banks are only required to hold capital against UL. Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

14 The Asymptotic Risk Factor (ASRF) Approach Credit Portfolio IRB Economic Capital Banks needed capital per euro of exposure is given by C α = q α (L) E(L) = ( N Φ w i LGD i [Φ 1 (p i ) + )] ρ i Φ 1 (α) p i, 1 ρi i=1 (10) since the expected loss per euro of exposure for the whole credit portfolio is given by E(L) = N w i LGD i p i. (11) i=1 In Basel II, the confidence level α is set to 99.9%, i.e. an institution is expected to suffer losses that exceed its economic capital once in a thousand years on average. Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

15 Concluding Remarks Summary Conclusions We introduced the structural approach to default. We summarized the distance to default idea. We investigated Basel II IRB economic capital equation. Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16

16 Concluding Remarks References References Aas, K. (2005). The Basel II IRB approach for credit portfolios: A survey, SAMBA, 33 Gordy, M. (2003). A risk-factor foundation for risk-based capital rules. Journal of Financial Intermediation, 12, Merton, R.C. (1974). On the pricing of corporate debt. The risk structure of interest rates, Journal of Finance, 29, Vasicek, O. (1977). An equilibrium characterization of the term structure. Journal of Financial Economics, 5, Tiziano Bellini (Università di Bologna) Economi Capital November 29, / 16