label: credit

The arithmetic on P524-525 could be expanded into a 5-pager if we were to explain to people with high-school math background…

There are 2 parts to the math. Part A computes the “expected” (probabilistic) loss from default to be $8.75 for a notional/face value of $100. Part B computes the same (via another route) to be $288.48Q. Equating the 2 parts gives Q =3.03%.

Q3: How is the 7% 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 the coupon” because it may receive help in the 11^{th} hour.

–The continuous discounting in Table 23.3 is confusing

Q: Hull explained how the 3.5Y row in Table 23.3 is computed. Why discount to the T=3.5Y and not discounting to T=0Y ?

The “risk-free value” (Column 4) has a confusing meaning. Hull mentioned earlier a “similar risk-free bond” (a TBond). At 3.5Y mark, we know this risk-free bond is scheduled to pay all cash flows at future times T=3.5Y, 4Y, 4.5Y, 5Y. We use risk-free rate 5% to discount all 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 a 3.5Y default, but discounted to T=3.5Y”. This value is further discounted from 3.5Y to T=0Y – Column 6.

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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 at T=0.5Y 1.5 2.5 3.5 4.5 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(05) + Pr(15) + Pr(25) + Pr(35) + Pr(45) + Pr(no default). If we factor in the amount of loss at each flip we get

Pr(05) * $65.08 + Pr(15) * $61.20 + Pr(25) * $57.52 + Pr(35) * $54.01 + Pr(45) * $50.67 + Pr(no default, no loss) + $0 == $288.48Q