hazard rate – online resources

is decent —

Average failure rate is the fraction of the number of units that fail during an interval, divided by the number of units alive at the beginning of the interval. In the limit of smaller time intervals, the average failure rate measures the rate of failure in the next instant for those units surviving to time t, known as instantaneous failure rate.

is more mathematical.

http://www.omdec.com/articles/reliability/TimeToFailure.html has short list of jargon

ccy swap^ IRS^ outright FX fwd

Ccy swap vs IRS

– Similar – exchange interest payments

– diff — Ccy swap requires a final exchange of principal. FX rate is set on deal date

Ccy swap vs outright fx fwd?

– diff — outright involves no interest payments

– similar — the far-date principal exchange has an FX rate. Rate is set on deal date

– diff — the rate is the spot rate on deal date for ccy swap, but in the outright deal, the fwd rate as of the deal date

Ccy swap vs FX swap? Less comparable. Quite a confusing comparison.

hazard rate – my first lesson

Imagine credit default is caused only by a natural disaster (say hurricane or tsunami). For a brief duration ΔT (measured in Years), we assume the chance of disaster hitting is λ*ΔT, with a constant [1] λ .

Pr(no hit during    A N Y   5-year period)
= Pr (surviving 5 years)
= Pr (no default for the next 5 years from now)
= Pr (T > 5) = exp(-5λ) , denoted V(5) on P522 [[Hull]]

, where T :=  # of years from now to next hit.

This is an exponential distribution. This λ is called the hazard rate, to be estimated from market data. Therefore it has a term structure, just like the term structure of vol.

More generally,  λ could be assumed a function of t, i.e. time-varying variable, but a slow-moving variable, just like the instantaneous vol. In a noisegen, λ  and vol function as configurable parameters.

In http://www.financial-risk-manager.com/risks/credit/edf.html, λ is denoted “h”, which is assumed constant over each 12-month interval.

“Hazard rate” is the standard terminology, and also known as “default intensity” or “failure rate”.

I feel hazard rate is perhaps the #1 or among top 3 applications of
conditional probability,
conditional distribution,
conditional expectation

So the big effort in studying the conditional probability is largely to help understand credit risk.

conditional independence, learning notes

Remember — Discrete is always easier to understand …

Q: does conditional independence imply unconditional independence?

Simple example – among guys, weight is unrelated to income – conditional independence, but removing the condition, among all genders weight has a bearing on income.

However, in this math problem the terminology can be confusing – Given W = w, the 2 random variables X and Y are conditionally iid exp(w) distributed. W itself follows a G(alpha, beta) distribution. Are X and Y unconditionally independent?

I feel the key is the symbol “w”, which is neither a variable nor a number, but rather a configurable parameter. In the noisegens, this w is a constant, like 39.8. However, as operator of the noisegen, we could set this parameter and potentially modify the distribution(s).

If for all w values, X and Y are independent, then I believe X and Y are unconditionally independent.

intuitive – stdev(A+B) when independent ^ 100% correlated

(see also post on linear combo of random variables…)

Develop quick intuitions — Quiz: consider A + B under independence assumption and then under 100% correlation assumption. When is variance additive, and when is stdev additive?

(First, recognize A+B is not a regular variable like “A=3, B=2, so A+B=5”. No, A and B are random variables, from 2 noisegens. A+B is a derived random variable that’s controlled from the same 2 noisegens.)

If you can’t remember which is which, remember independence means good diversification[intuitive], lower dispersion, lower spread-out around the expected return, thinner bell, lower variance and stdev.

Conversely, remember strong correlation means poor diversification [intuitive] , magnified variance/stdev.

–Case: 100% correlated, then A+B is exactly a multiple of A [intuitive], like 2*A or 2.4*A. If you think of a normal (bell) or uniform (rectangle) distribution, you realize 2.4*A is proportionally magnified horizontally by a factor of 2.4, so the width of the distribution increases by 2.4, so stdev increases by 2.4. In Conclusion, stdev is additive.

–Case: independent
“variance is additive” applicable in the multi-period iid context.

simple rule — variance of independent[1] A + B is the sum of the variances.

[1] 0 correlation is sufficient

–Case: generalized — http://www.stat.ucla.edu/~hqxu/stat105/pdf/ch01.pdf P27 Eq5-36 is a good generalized formula.

V(A+B) = V(A) + V(B) + 2 Cov(A,B)  …. easiest form

2*Cov(A,B) := 2ρ V(A)V(B)

V( 7A ) = 7*7 V(A)

notation tips for probability puzzles

* There are many alternative notations for “probability of A and B”. I prefer p(A . B) — good for hand writing and computers

* There are many alternative notations for “probability of not-A”. I prefer p(A’ ) — good for computers. How about p(!A)? Alien to many mathematicians.

* Favor shortest abbreviations for event names. For example, probability of “getting two 6’s in 2 consecutive dice tosses” should NOT be written as p(66), but as p(K) by denoting the event as K.
* Avoid numbers in event short names — things like 2 Pr(3) very ambiguous. If feasible, avoid number subscripts too.
* Favor single letters including greek letters and Chinese characters. If feasible, avoid any subscript.

Venn diagram – good at showing mutual exclusion between 2 events.
tree diagram – good at showing cond prob between 2 events
tree diagram – good at showing independence between 2 events

Note — Mutual exclusion => independence,  but not vice versa.

Independence is intuitive most of the time but can be non-intuitive when you are deep into a tough puzzle.

Independence can be counter-intuitive and not captured in tree diagrams. Let h denote “salary above 100,000”; f=female. The 2 events happen to be indie in one firm but not another. In general, we have to assume not-independent.

Tossing a coin in the morning vs afternoon. Toss Should be independent of timing, but actual observation may not prove it.