Description
• “The correlations between financial quantities are notoriously unstable” —
Paul Wilmott, quant guru.
• “Co-association between securities is not measurable using correlation…
Anything that relies on correlation is charlatanism” — Nassim Nicholas Taleb,
hedge fund manager.
• “The most dangerous part is when people believe everything coming out of
it” — David Li, inventor of the Gaussian copula model for CDO valuation.
• “The corporate CDO world relied almost exclusively on this copula-based
correlation model” — Darrell Duffie, Stanford University finance professor.
• “…it was not obvious that a pool of mortgage-backed securities rated BBB
could be transformed into a new security that is mostly rated triple-A. But math
made it so.” — from the Final Report of the National Commission on the Causes
of the Financial and Economic Crisis in the United States. The link to the full
report is www.gpo.gov/fdsys/pkg/GPO-FCIC/pdf/GPO-FCIC.pdf.
1. Read Wired Magazine’s 2-23-09 article Recipe for Disaster: The Formula
that Killed Wall Street, by Felix Salmon. You can find it at www.wired.com/
2009/02/wp-quant/. The first four quotes above appear in this article.
• The code RiskyAnnuity.cpp values a promised stream of monthly cash flow
in the amount of $100/month for 30 years (the term of a typical mortgage). The
paying entity’s time to default is modeled as exponentially distributed with a
mean that depends on the entity’s credit (see table below). If it enters default its
promised payments cease. The code computes the expected present value at a
risk-free rate of 3% compounded continuously. Based on this computed present
value, a continuously compounding yield-to-maturity (YTM) of the promised
cash flow is computed. This is done for six credit ratings of the entity. Run the
code and verify that the resulting YTMs are:
Credit
Expected
Years to
Default YTM (%)
Spread
Over
Risk-free (bps)
Risk-free NA 3.00 0
AAA 200 3.50 50
AA 100 4.00 100
A 67 4.50 150
BBB 50 5.00 200
Speculative 33 6.00 300
• Here we assign a credit rating to the five tranches of a Collateralized Debt
Obligation (CDO) using a very simple copula-based correlation model. The
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collateral of the CDO will consist of 20 entities, referred to as names, each
promising payment into the CDO of $100 per month for 30 years. The names
each have a credit rating of BBB and their times to default Ti
, 1 ≤ i ≤ 20, are
exponential random variables with mean 50 years. If and when a name enters
default it stops making its promised payment into the CDO. Each tranche is
promised $100 × 20 ÷ 5 = $400 per month from the CDO. Tranche number 1 is
paid first, then tranche number 2 is paid with remaining funds, and so forth in
what is called a cash flow waterfall. As names default the lower tranches are the
first to lose out and may eventually receive no cash flow at all.
2. Use Monte Carlo simulation to value the steam of cash flow to each of the five
tranches. For each realization of the CDO, generate uniforms U1, U2, . . . , U20
from a Gaussian copula model as follows. Each Ui = Ψ(Ni), where Ni ∼
Normal (0, 1) and
Cov (Ni
, Nj ) =
ρ if i 6= j
1 if i = j,
where the correlation ρ is a parameter of the model. Then put Ti = −50 ln(Ui);
this is the time of default for the i
th name. Using that the i
th name pays $100
into the CDO at month m if and only if Ti > m/12.0, calculate the monthby-month stream of cash flow into the CDO. Calculate how it is then awarded
to the each of the five tranches and discount the flows to each tranche at the
risk-free rate (3% annually, compounded continuously) to estimate the expected
present value (EPV) of the flows going to each of the five tranches. Convert this
EPV into a yield-to-maturity, just as in the code RiskyAnnuity.cpp. Based on
the table above, assign an implied credit rating to each tranche. Do this exercise
for values of ρ ranging from 0 to 1 in steps of 0.1.
3. Explain your results in problem 2.
