[[Hull]]estimat`default probability from bond prices#learning notes

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

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