When buying the defaultable bonds through borrowing, one can not borrow at the rate ofrt. The total cost isrt+ht, wherehtreflects the borrowing cost in excess of
rt. The existence of ht represents a source of market friction. The model assumes
thathtis an increasing linear function of the default intensity risk λt:
hit(τ) =αi(τ)λt+δi(τ) i=c, d 0< αi < L (1.39)
This is a reasonable assumption with motivation illustrated in the following way:
Consider a simplified version of repo transaction, assume there’s a continuum of competitive cash lenders who lend out yt amount of cash if the borrowers offer 1-
dollar worth of defautable bond as collateral.19 The borrower may default on the obligation to pay back, denote ¯Nt as counting process for the borrower’s default,
assume the intensity of ¯Nt is f(λt), which is increasing in λt. This is a reasonable
assumption in the real world, the borrowers are usually buyers of the defaultable bond who can only afford to buy through borrowing, and rely on the selling of the bond to pay back their borrowing. If the bond defaults, they are unlikely to pay back and would default. Denote the wealth of the lender by wt, then the lender’s
optimization problem is:
max yt [Et(dwt)− ¯ γ 2V art(dwt)] (1.40) dwt = (wt−yt)rtdt+yt(rt+s)dt+ [(1−L)−yt]dN¯t (1.41)
wherert+s,s >0, is the exogenous rate asked by the lender for lending against the
defaultable bond as collateral.20 Thewt−ytpart of her wealth increases at the rate
ofrt, while theytpart earns interest ratert+s. If the borrower defaults, the lender
retains the bond, which is now worth 1−L but loses the yt lent to the borrower.
The optimal lending amountyt∗ is solved as:
yt∗ = s ¯ γf(λt) − 1 ¯ γ + (1−L) (1.42)
19The lender effectively buys the bond fory
t, while the borrower promises to buy back the bond
at a price higher thanyt at the end of the repo contract that reflects an interest rate higher than
the short ratert.
20
rt can be understood as the rate asked by the lender for lending against default-free bond as
Therefore, yt∗ is decreasing in f(λt), thus decreasing in λt. In other words, the
amount a borrower can borrow using defaultable bond as collateral is decreasing in the riskiness of the bond. 1−yt∗ is the so-called ‘hair-cut’asked by the lender. The hair-cut is increasing in the riskiness of the bond. The borrower borrowsyt∗ fraction of her bond purchase at the rt+srate charged by the lender, and the rest 1−yt∗
fraction at an exogenous un-collateralized rate rt+u from somewhere else. Here,
u > s >0 since un-collateralized borrowing is riskier than collateralized borrowing. So the borrower’s total borrowing cost in excess ofrtis:
ht=yt∗(rt+s) + (1−y∗t)(rt+u)−rt=u−(u−s)yt∗ (1.43)
which is a decreasing function of y∗t. Since y∗t is decreasing in λt, the borrowing
costhtis increasing in λt. With careful choice of the exogenous function f(λt), the
borrow cost has the linear functional form ofαλt+δ used in the model.
1.A2. Short-selling Costs
When short-selling the defaultable bond, one needs to borrow the bond from a bond lender. The bond lender asks for cash-collateral of a certain amount. Effectively, the short-seller buys the bond for that amount while the bond lender agrees to buy back the bond at the end of the reverse-repo contract for an amount higher than the initial cash-collateral. However, the end-of-the-day amount usually reflects an interest rate paid on the cash-collateral that is lower than the short-rate. The short- seller hence incur short-selling costs.
Assume there’s a continuum of competitive bond lenders that require Yt amount
of cash-collateral for 1-dollar worth of bond lent. The short-seller, i.e. the bond borrower, may default on the obligation to return the bond, denoteNet as counting process for the short-seller’s default, assume the intensity of Net is g(λt) which is decreasing inλt. This is a reasonable assumption in the real world, whenλt is low,
the bond price is high, the short-seller is more likely to suffer loss and hence default on the obligation to return the bond. Denote the wealth of the lender bywt, then
the lender’s optimization problem is:
max Yt [Et(dwt)− e γ 2V art(dwt)] (1.44) dwt = (wt+Yt)rtdt−Yt(rt−S)dt+ (Yt−1)dNet (1.45)
wherert−S,S >0, is the exogenous special repo rate offered by the bond lender
on the cash collateral. The wt+Yt part of her wealth increases at the rate of rt,
while the Yt part pays interest rate rt−S. If the short-seller defaults, the lender
retains the cashYt, but loses the 1-dollar worth of bond lent to the short-seller.
Yt∗= S e γg(λt) + 1 e γ + 1 (1.46)
ThereforeYt∗ is decreasing in g(λt), thus increasing in λt. The short-seller puts Yt∗
fraction of her bond sales as cash-collateral which earns interest at the rate ofrt−S
and invests the rest 1−y∗t fraction at the short ratert. So her short selling cost is:
ht=rt−[Yt∗(rt−S) + (1−Yt∗)rt] =SYt∗ (1.47)
which is an increasing function of Yt∗. Since Yt∗ is also increasing in λt, the short-
selling cost ht is increasing in λt. With careful choice of the exogenous function
g(λt), the short-selling cost has the linear functional form of αλt+δ used in the
1.9 Appendix 1.B: Proofs of Lemmas and Propositions