# Applying Dual Time Dynamics to Credit Risk Models

Dual time Dynamics attempts to model the loss behaviour of a portfolio of retail loans as a function of calendar time and portfolio maturation time, as well as portfolio risk profile and the impact of the macro-economy as an exogenous factor.

DTD decomposes vintage level data to months-on-books (maturation), calendar date (exogenous) and vintage (quality). It follows the same concept in Age-Cohort-Period model which is commonly used in sociology, epidemiology and other population and demography studies.
In credit risk stress testing models, we would like to model delinquency or default rates to forecast them in different macroeconomic scenarios. In this context, estimating time effect is our main interest but it is contaminated by vintage quality and loan ages. More precisely, since observation date (t) minus date of loan issue (v) is equal to loan age (a) ( a = t — v), it is not possible to estimate these factors based on a unique solution.

I worked on adapting the framework for application to stress testing models across a range of retail products. Following is the R code

```##############loading libraries#########################

library(data.table)
library(ggplot2)
library(gam)
library(plotly)
library(plot3D)

#######Number of Obligations and Time Points#######

Nobl <- 1000
T_final <- 200 #####time upto which we are generating asset price series######

#######Generating Unemployment Data (Macro economic variable) for all 1000 customers#######################

unemployment_series <- c(0.08)
while(length(unemployment_series)<T_final){
curr_unemp <- (unemployment_series[length(unemployment_series)] + rnorm(1, 0, 0.01))
if(curr_unemp > 0.04 & curr_unemp < 0.12){
unemployment_series <- c(unemployment_series, curr_unemp)
}
}
ggplot(data.table(arp = unemployment_series,
pt = 1:T_final)) + geom_line(aes(pt, arp))

#############Generating Obligations Data##########

data_customer <- data.table(oblig_id = rep(paste(“ID”, seq(1,Nobl,1), sep=’_’), T_final),
vintage = rep(1:T_final, each=Nobl),
unemployment = rep(unemployment_series, each=Nobl),
age = -1)

#############Generating Age###########

for (i in 1:20){
curr_cust_list <- paste(“ID”, seq((50*(i-1)+1),(50*(i))), sep=”_”)
data_customer[oblig_id %in% curr_cust_list, age := vintage-1–9*(i-1)]
}
data_customer[age < 0, age:=NA]
#######20 clusters, 1st 50 customers with all 200 quaters,2nd 50 customers originated at 9th quater, etc.
###########Generating Default For DtD Modeling (randomly aloting defaults)########

data_customer[, is_default := -1]
data_customer[is.na(age), is_default := NA]
for (age_curr in unique(data_customer\$age)){
if(is.na(age_curr)){
data_customer[age==age_curr, is_default := NA]
}else if(age_curr <= 25){
data_customer[age==age_curr, is_default := rbinom(nrow(data_customer[age==age_curr]),
1, (0.01 + 9*age_curr*0.0004))]
}else{
data_customer[age==age_curr, is_default := rbinom(nrow(data_customer[age==age_curr]),
1, 0.1)]
}
}

View(data_customer[, mean(is_default),age])
ggplot(data_customer[, list(prob_def=mean(is_default)),age]) + geom_line(aes(age, prob_def))

#######GAM model for DtD########

gam_model <- gam(is_default ~ s(age, df = 3) + s(unemployment, df=3) + s(vintage, df=3),
family = binomial, data = data_customer, na.action=na.omit)

#########Generating Asset Prices##############

sigma_i <-runif(Nobl, 0.03, 0.05)
A_i <- sigma_i *100000 #########asset_price####
B_i <- A_i — 1000 ##########default_threshold_price#####

############generating mu_i for each customer for each quater########

GetReturnRate <- function(dt, group){
dt\$prob_def <- 1/(1+exp(-predict(gam_model, newdata=dt)))
mu_i <- (log(B_i[group]/A_i[group]) + dt[age>0]\$age*(sigma_i[group]²)/2 -
sigma_i[group]*sqrt(dt[age>0]\$age)*qnorm(dt[age>0]\$prob_def))/dt[age>0]\$age
dt\$mu_i <- c(rep(NA, nrow(dt[is.na(age) | age==0])), mu_i)
return(dt)
}
data_customer <- data_customer[, GetReturnRate(.SD, .GRP), oblig_id]

########generating asset price using merton’s formula#####

GetAssetPrice <- function(dt, group){
dt\$asset_price <- exp(log(A_i[group]) + dt\$mu_i*dt\$age — dt\$age*(sigma_i[group]²)/2 +
sigma_i[group]*sqrt(dt\$age)*rnorm(nrow(dt), 0, 1))
dt[age==0]\$asset_price <- A_i[group]
dt\$default_threshold <- B_i[group]
return(dt)

}

data_customer <- data_customer[, GetAssetPrice(.SD, .GRP), oblig_id]
data_customer <- data_customer[, default_or_not := as.numeric(asset_price < default_threshold)]
CreateRowData <- function(series){
lst <- as.list(series)
names(lst) <- paste0(‘T’, (0:(length(series)-1)))
return(lst)
}

data_cust_asset<- data_customer[,CreateRowData(asset_price), oblig_id]
data_cust_def_or_not<- data_customer[,CreateRowData(default_or_not), oblig_id]

########Plotting#######

ViewPricePlot <- function(Nobl_no){
curr_obgl <- paste(‘ID’, Nobl_no, sep=’_’)
curr_dt <- data_customer[oblig_id == curr_obgl][!is.na(age)]
p <- ggplot(curr_dt) + geom_point(aes(age, asset_price), size=0.2) +
geom_line(aes(age, asset_price)) +
geom_hline(aes(yintercept=unique(curr_dt\$default_threshold), color=’red’))+
scale_color_identity(guide=’legend’, labels=c(‘default threshold’))+ ggtitle(paste(“Asset Price for”, Nobl_no, “obligation”))
return(p)
}

ViewPricePlot(600) #######view price plot of 600th customer#####

ggplot(data_customer[!is.na(age), list(prob_def=mean(asset_price<default_threshold)),age]) +
geom_line(aes(age, prob_def))
def_rate <- data_customer[!is.na(age), list(prob_def=mean(asset_price<default_threshold)),.(vintage, age)]
Sys.setenv(“plotly_username”=”aditi_tiwari”)
Sys.setenv(“plotly_api_key”=”Ow0zdx8YcWbyQsdeyJQs”)
p <- plot_ly(def_rate, x = ~vintage, y = ~age, z = ~prob_def,
type = ‘scatter3d’, mode = ‘surface’,
opacity = 1, line = list(width = 6,
color = ~vintage,
reverscale = T))
p
View(data_customer[oblig_id==’ID_100'])library(data.table)
```