If we were to explain to people with basic math background, the
arithmetic on P524-525 could be expanded into a 5-pager. It's a good
example worth study.
There are 2 parts to the math. Using bond prices, Part A computes the
"expected" (probabilistic) loss from default to be $8.75 for a
notional/face value of $100. Alternatively assuming a constant hazard
rate, Part B computes the same to be $288.48*Q. Equating the 2 parts
gives Q =3.03%.
Q3: How is the 7% market yield used? Where in which part?
Q4: why assume defaults happen right before coupon date?
%%A: borrower would not declare "in 2 days I will fail to pay that
coupon" because it may receive help in the 11th hour.
--The continuous discounting in Table 23.3 is confusing
Q: Hull explained how the 3.5Y row in Table 23.3 is computed. But Why
discount to the T=3.5Y and not discounting to T=0Y ? Here's my long
answer.
The "risk-free value" (Column 4) has a confusing meaning. Hull
mentioned earlier a "similar risk-free bond" (a TBond). Right before
the 3.5Y moment, we know this risk-free bond is scheduled to pay all
cash flows at future times T=3.5Y, 4Y, 4.5Y, 5Y. That's 4 coupons +
principal. We use risk-free rate 5% to discount all 4+1 cash flows to
T=3.5Y. We get $104.34 as the value of the TBond cash flows
"discounted to T=3.5Y"
Column 5 builds on it giving the "loss due to default@3.5Y, discounted
to T=3.5Y". Iin Column 6, This value is further discounted from 3.5Y
to T=0Y.
---
Part B computes a PV relative to the TBond's value. Actually Part A is
also relative to the TBond's value.
In the model of Part B, there are 5 coin flips occurring every
mid-year at T=0.5Y 1.5Y 2.5Y 3.5Y 4.5Y with Pr(default_0.5) =
Pr(default_1.5) = … = Pr(default_4.5) = Q. Concretely, imagine that
Pr(flip = Tail) is 25%. Now Law of total prob states
100% = Pr(d05) + Pr(d15) + Pr(d25) + Pr(d35) + Pr(d45) + Pr(no d). If
we factor in the amount of loss at each flip we get
Pr(d05) * $65.08 + Pr(d15) * $61.20 + Pr(d25) * $57.52 + Pr(d35) *
$54.01 + Pr(d45) * $50.67 + Pr(no d, no loss) + $0 == $288.48*Q
