Empirical Evaluation of Hybrid Defaultable Bond Pricing ... - risklab

with
B(t, T ) = 1 ³
−âr(T
1 − e
−t)´
,
â r
C d (t, T ) = 1 ³
−âs(T
1 − e
−t)´
,
â s
D d (t, T ) = b µ
su 1 − e
−â u (T −t)
+ e−â u(T −t) − e −â
s(T −t)
,
â s â u
â u − â s
E d (t, T ) = b µ
r 1 − e
−â w (T −t)
+ e−â w(T −t) − e −â
r(T −t)
â r â w − â r
A d (t, T ) =
Proof.
See appendix.
â w
− b µ
sw 1 − e
−â w (T −t)
â s
Z T
t
â w
+ e−â w(T −t) − e −â
s(T −t)
,
â w − â s
1 ¡ σ
2
2 s C d (l, T) 2 + σ 2 uD d (l, T) 2 + σ 2 wE d (l, T) 2 + σ 2 rB(l, T) 2¢
−θ r (l)B(l, T) − θ s C d (l, T) − θ w E d (l, T) − θ u D d (l, T)dl.
Based on the formulas for defaultable and non-defaultable zero-coupon bonds
we can easily calculate the yields to maturity R (t, T ) and R d (t, T ) of defaultable
and non-defaultable bonds as well as the yield spread S (t, T ) of defaultable
bonds to non-defaultable bonds:
and
R (t, T ) = − A(t, T )
T − t + B(t, T )
T − t r + E(t, T )
T − t w,
R d (t, T ) = − Ad (t, T )
T − t
+ Ed (t, T )
T − t
+ B(t, T )
T − t r
w + Cd (t, T )
T − t
s + Dd (t, T )
u,
T − t
S (t, T ) = A(t, T )
T − t − Ad (t, T )
T − t
µ E d
(t, T ) − E(t, T )
+
w
T − t
+ Cd (t, T )
T − t
s + Dd (t, T )
u.
T − t
R (t, T ) depends on the GDP growth rate w with a positive weight E(t, T )/(T −
t). The model considers the empirically observable relationship between interest
rates and the general condition of the economy in a reasonable way.
S (t, T ) depends on the GDP growth rate w with a negative weight (E d (t, T ) −
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