Collateralized Borrowing and Default Risk

Abstract

We study how margin requirements in the collateralized borrowing affect banks’ risk exposure. In a model where a firm’s asset value and margin requirement follow correlated geometric Brownian motions, we derive analytic expressions for firm’s default probability and debt value. Our results show that variations in margin requirements, reflecting funding liquidity shocks in the short-term collateralized lending market, can lead to a significant increase in firms’ default risks, in particular for those firms heavily relying on short-term collateralized borrowing. Moreover, our results imply that reducing margin in liquidity crises can be very effective to restore market lending confidence.

Appendix

Proof of Proposition 3

We first calculate the value of short-term debt $D_S$,

$$\begin{aligned} \begin{array}{lll} &{} D_S(V,B)\\ =&{} \mathbb {E}\left[ \int \limits _0^{T\wedge \tau } C_S e^{-r_fs}ds\right] + \mathbb {E}\left[ e^{-r_fT}\chi _{\{T<\tau \}}S\right] + \frac{S}{S+L}\mathbb {E}\left[ e^{-r_f\tau }\bar{V}_{\tau }\chi _{\{\tau \le T\}}\right] \\ =&{} T_1 + T_2 + T_3. \end{array} \end{aligned}$$

(A.1)

The first term is

$$\begin{aligned} \begin{array}{lll} T_1 &{}=&{} \mathbb {E}\left[ \int \limits _0^{T\wedge \tau } C_k e^{-r_f s}ds \right] \\ &{}=&{} \frac{C_S}{r_f}\left( 1 - e^{-r_f (T\wedge \tau )} \right) \\ &{}=&{} \frac{C_S}{r_f} -\frac{C_k}{r_f}\mathbb {E}\left[ e^{-r_f T}\chi _{\{\tau> T\}}\right] - \frac{C_S}{r_f}\mathbb {E}\left[ e^{-r_f \tau }\chi _{\{\tau \le T\}}\right] \\ &{}=&{} \frac{C_S}{r_f} \left( 1- e^{-r_fT}P(\tau >T) \right) - \frac{C_S}{r_f} \int \limits _0^T e^{-r_fs} P(\tau \in ds), \end{array} \end{aligned}$$

where the survival probability $P(\tau >T)$ and the default density function are given in (4) and (5). The second term can be computed as

$$\begin{aligned} T_2 = \mathbb {E}\left[ e^{-r_fT}\chi _{\{T<\tau \}}S\right] = S e^{-r_fT} P(\tau >T). \end{aligned}$$

The last term is

$$\begin{aligned} \begin{array}{lll} T_3 &{}=&{} \frac{RS}{S+L}\mathbb {E}\left[ e^{-r_f\tau }\bar{V}_{\tau }\chi _{\{\tau \le T\}}\right] \\ &{}=&{} \frac{RS}{S+L} \mathbb {E}\left[ e^{-r_f\tau _m}\bar{V}_{\tau _m} \chi _{\{\tau _m<\tau _i<T\}} + e^{-r_f\tau _i}\bar{V}_{\tau _i} \chi _{\{\tau _i<\tau _m<T\}} \right. \\ &{} &{} \left. + e^{-r_f\tau _m}\bar{V}_{\tau _m} \chi _{\{\tau _m<T<\tau _i\}} + e^{-r_f\tau _i}\bar{V}_{\tau _i} \chi _{\{\tau _i<T<\tau _m\}}\right] \\ &{}=&{} \frac{RS}{S+L} \left( \int \limits _0^T \int \limits _0^u e^{-r_fv}\bar{V}_v P_{v<u}(\tau _m\in dv,\tau _i\in du) + \int \limits _0^T \int \limits _u^T e^{-r_fu}\bar{V}_u P_{u<v}(\tau _m\in dv,\tau _i\in du) \right. \\ &{}&{} \qquad \qquad \left. + \int \limits _T ^\infty \int \limits _0^T e^{-r_fv}\bar{V}_v P_{v<u}(\tau _m\in dv,\tau _i\in du) + \int \limits _0^T \int \limits _T^\infty e^{-r_fu}\bar{V}_u P_{u<v}(\tau _m\in dv, \tau _i\in du) \right) , \end{array} \end{aligned}$$

(A.2)

where the joint default density is given in (6) and (7). For the default time $v<u$, we know

$$\bar{V}_v = e^{\lambda _m v}S,$$

and for $u<v$

$$\bar{V}_u = e^{\lambda _i u}B.$$

The long-term debt value will be the same by replacing principal and coupon. The total firm value is the unlevered firm value plus tax shields minus bankruptcy costs

$$\begin{aligned} \begin{array}{ll} &{} v(V,B) \\ = &{} V_0 + \mathbb {E}\left[ \int \limits _0^{T\wedge \tau }\iota (C_S+C_L)e^{-r_fs}ds\right] -(1-R)\mathbb {E}\left[ e^{-r_f\tau }\bar{V}_\tau \chi _{\{\tau \le T\}}\right] \\ = &{} V_0 \frac{\iota (C_S+C_L)}{r_f} \left( 1- e^{-r_fT}P(\tau >T) \right) - \frac{\iota (C_S+C_L)}{r_f} \int \limits _0^T e^{-r_fs} P(\tau \in ds) \\ &{} -\,(1-R) \left( \int \limits _0^T \int \limits _0^u e^{-r_fv}\bar{V}_v P_{v<u}(\tau _m\in dv,\tau _i\in du) \right. \\ &{} \qquad \qquad + \int \limits _0^T \int \limits _u^T e^{-r_fu}\bar{V}_u P_{u<v}(\tau _m\in dv,\tau _i\in du) \\ &{} \qquad \qquad + \int \limits _T ^\infty \int \limits _0^T e^{-r_fv}\bar{V}_v P_{v<u}(\tau _m\in dv,\tau _i\in du) \\ &{} \qquad \qquad \left. + \int \limits _0^T \int \limits _T^\infty e^{-r_fu}\bar{V}_u P_{u<v}(\tau _m\in dv, \tau _i\in du) \right) . \end{array} \end{aligned}$$

(A.3)

Finally, equity value is calculated as total firm value net the debt value

$$\begin{aligned} \begin{array}{lll} E(V,B)= & {} v(V,B) - D(V,B). \end{array} \end{aligned}$$

(A.4)

$\square$