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**Unformatted text preview: **CREDIT RISK AND OPTION PRICING (Adapted form Pennacchi, G. (1992), Lecture Notes,
University of Illinois College of Commerce and Business Administration.) Bank Loans and Option Pricing Goal: to calculate a “fair” interest rate on a bank loan to a
borrower with the following characteristics. (A1) Loan 1s a single payment (bullet) loan paying the
principal plus interest of $X 1n 1: periods ~ (A2) Right after the loan is made, at time t, the borrower -. -
has assets worth $A(t) that back the loan. Comment: $A(t) is the time t-wvalue of those assets that
the bank can legally seize (foreclose on) if the borrower- defaults on the loan at time t + 7:. For a securedloan, $A -
is the value of collateral. For an unsecured loan to an all eguity firm $A is the
entire firm s assets For a mortgage loan, EBA 1s the value of the house (A3) The variance of the rate of return on the.$A of
assets is 0'2 per unit time. (A4) The assets have a payout rate of u. Examples (1) If loan is to an all equity ﬁrm, u may be the dividend
payout rate to stockholders. (2) If assets are physical capital, it could represent the
depreciation rate. (A5) The riskwfree rate of interest = 7" per unit time. Let D(t) denote that (market) value of this loan at time t.
Then the loan’s value at maturity, D(t + r) is D(t + a = Minix, Av + at
: X m. MaX[0, X — AU + T)] Equation“) says the valueiof a risky lean at maturity equals the value of a risk'vfr‘iee loan, X, minus the value of
a put option on the value oi” the assets backing the loan,
A(t + r), and having an exerCise price of X, cf. writing
put option and writing (selling) loan insurance. Limited—liability of the borrower essentially means that
the borrower has purchased the right (option) to sell the
assets backing the loan, AU), back to the bank at the price X. Thus the value of the risky loan prior to maturity is the
value of a risk-free loan minus the put option. D0) = Xe“ - P(A(t), X, r) (2)
: Xe” - {Xe‘m N(-d2) ﬂ A61” N(*d1)}
= Xe“ mag) + Adm N(—d1) If this loan is made at its initial fair value, DU), then its
fair yield to maturity is defined as D0) = Xe'R". Hence R : iii-[L]
1:" ‘ DU) (3)77 A Numerical Example Calculate the breakweven interest rate a bank would have
to charge on the following ﬁrm’s loan. (a) The loan has a two year maturity with no interim
payments. 1' = 2 years
(b) The payment at maturity (principal + interest) is $4
million
. =$4 (c) The bank’s marginal cost of funds (cost of two year ﬁxed rate ﬁnancing) including its marginal cost of
monitoring, is 7%. I R: 0 07
(d) The value of the ﬁrm 5 assets backing the loan
currently equals $7 million. The ﬁrm IS not expected to
make any payments out of these assets over the life of - _ the loan. A: 7 .r
(e) The annualised variance of the rate of return on these assets of the ﬁrm is 20%.
02 = 0.20 SOLUTION: D(t) m {RISK FREE LOAN} — {PUT OPTION} = Xe” — {Xe‘rr N(-—d2) — Ae‘m N(-d1)} d =
I 0%.
ln[l]+ 0.'(7)7:7+—Q'2— x2
d ..... 4 2
1 00.2035
511: 0.5596+0.3§;00 21.42. . 0.6325
d2 2 d1 ~m/?-..-4=1.42 #06325 d 2 m 0.79
N(d1) = 0.9222 N909) = 0.0778 DU) 2: 4.50M2 —— {46”0'O7X2(0.2148)m 7.(1)(0.0778)} = 3.48 ~ {0.75 — 0.54} D(t) = 3.27 3.279;"?2 m 4 R 21111 mi]
2 3.27 R 210.08 BREAK EVEN INTEREST RATE
m 10.1% SAY "EECOLLATERALIZED LOAN *
, _::::> REPAY LOAN WITH CERTAINTY BUT ALSO
gRETAIN PUT TO SELL COLLATERAL TO LENDER. A AWHEN WILL YOU SELL?? .
‘ ~'7j-‘§SELL WHEN COLLATERAL AT MATURITY <
' LOAN VALUE , ...

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- Spring '17
- Interest, Mortgage loan