Valuation of CDO with equal amount

January 17, 2011

(This article was first published on My Life as a Mock Quant in English, and kindly contributed to R-bloggers)

Bank of Japan(BOJ) publish research paper regularly.
And, they issued very interesting paper about valuation of CDO recently.

(The paper is 金融危機時における資産価格変動の相互依存関係:コピュラに基づく評価 新谷 幸平、山田 哲也、吉羽 要直(sorry,japanese only!))

They introduced copula for pricing of CDO,and discussed how different CDO spreads were with using different copula for pricing.
I would like to reproduce their result (especially,P23-Table7)

The condition of calculation is following that

  • number of debt(NUM.REFDEBT):=100
  • maturity(MATURITY):=5 year
  • recovery rate(RECOVERY.RATE):=40%(constant value)
  • probability of default (DEFAULT.PROBABILITY):=5%(in 5 years)
  • parameter of nomal copula ρ:=0.15
  • parameter of clayton copula α:=0.21
They apporoximated their valuation formula for easy calculation(equation (27))
(They assumed that CDO spread were paid as discounted bond at the begging.)
I simulated valuation of CDO with their method.
The result is following that
copula/tranche Equity mezzanine senior super senior
normal 1,145.42 62.49 0.52 0.000
t(20) 1,055.28 86.07 2.18 0.004
t(6) 896.74 126.44 8.56 0.044
t(3) 733.31 165.90 23.56 0.191
clayton 857.64 135.73 12.83 0.084
This table reproduce their result(P23-Table7).
And, In senior or super senior,you can understand that the CDO spread which is evaluated by normal copula is lower than the others. It means that normal copula is inadequate in financial crisis.
I show you my programming code(by R language).
If you copy and run my source code, you can duplicate my result easily.
Before you run, please install “copula”package.

#function for CDO spread calculation
SpreadOfCDO <- function(copula, default.probability, maturity,
recovery.rate, attachment, detachment, num.path, num.refdebt)
random.copula <- rcopula(copula,num.path)
num.default <- rowSums(random.copula < default.probability)
loss.refdebt <- (1-recovery.rate)/num.refdebt*num.default
loss.tranche <- (pmax(loss.refdebt - attachment,0)-pmax(loss.refdebt - detachment,0))/(detachment-attachment)
expectation.loss.tranche <- sum(loss.tranche)/num.path
spread <- -1/maturity*log(1-expectation.loss.tranche)
################ main ##################
NUM.PATH <- 10^3
#copulas which I want to compare.
COPULA <- list(normalCopula(0.15, dim = NUM.REFDEBT),
tCopula(0.15, dim = NUM.REFDEBT, df = 20),
tCopula(0.15, dim = NUM.REFDEBT, df = 6),
tCopula(0.15, dim = NUM.REFDEBT, df = 3),
claytonCopula(0.21, dim = NUM.REFDEBT)
#define for easy programming
SpreadOfCDOWithFixedParameter <- function(copula,attachment, detachment){
RECOVERY.RATE, attachment, detachment, NUM.PATH, NUM.REFDEBT)
result <- list()
result[[1]] <- sapply(COPULA,SpreadOfCDOWithFixedParameter,0.0,0.06)
result[[2]] <- sapply(COPULA,SpreadOfCDOWithFixedParameter,0.06,0.18)
result[[3]] <- sapply(COPULA,SpreadOfCDOWithFixedParameter,0.18,0.36)
#tranche:super senior
result[[4]] <- sapply(COPULA,SpreadOfCDOWithFixedParameter,0.36,1)
#convert to matrix, and chenge unit to "bp"
result <- 10^4*"cbind", result)
colnames(result) <- c("equity","mezzanine","senior","super senior")
rownames(result) <- c("normal","t(20)","t(6)","t(3)","clayton")

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