Interest Rate and Credit Risk Modelling
Yas Suttakulpiboon
June 21, 2023
Yield Curve Behavior
Non-negativity
Mean Reversion
Change in level
Change in slope
Change in curvature
Yield Curve Behavior in Data
# install.packages("YieldCurve")
library(YieldCurve)## Loading required package: xts## Warning: package 'xts' was built under R version 4.0.2## Loading required package: zoo##
## Attaching package: 'zoo'## The following objects are masked from 'package:base':
##
## as.Date, as.Date.numeric# Call Fed Yield Curve Dataset
data(FedYieldCurve)
YC <- FedYieldCurve
YC.diff <- diff(YC)
# Time series plot of all interest rates over time
plot(YC)
# YC as of 1981-12-31, 1990-03-31, 1998-07-31, 2006-11-30
par(mfrow = c(2,2))
plot(as.numeric(YC[1,]), type = "l")
plot(as.numeric(YC[100,]), type = "l")
plot(as.numeric(YC[200,]), type = "l")
plot(as.numeric(YC[300,]), type = "l")
# Time Series Plot of the short-term rate (3-Month Rate)
par(mfrow = c(3,1))
plot(YC[,1])
plot(diff(YC[,1]))
hist(diff(YC[,1]))
# Time Series Plot of the short-term rate (6-Month Rate)
par(mfrow = c(3,1))
plot(YC[,2])
plot(diff(YC[,2]))
hist(diff(YC[,2]))
# Time Series Plot of the short-term rate (1-Year Rate)
par(mfrow = c(3,1))
plot(YC[,3])
plot(diff(YC[,3]))
hist(diff(YC[,3]))
# Time Series Plot of the short-term rate (2-Year Rate)
par(mfrow = c(3,1))
plot(YC[,4])
plot(diff(YC[,4]))
hist(diff(YC[,4]))
# Time Series Plot of the short-term rate (3-Year Rate)
par(mfrow = c(3,1))
plot(YC[,5])
plot(diff(YC[,5]))
hist(diff(YC[,5]))
# Time Series Plot of the short-term rate (5-Year Rate)
par(mfrow = c(3,1))
plot(YC[,6])
plot(diff(YC[,6]))
hist(diff(YC[,6]))
# Time Series Plot of the short-term rate (7-Year Rate)
par(mfrow = c(3,1))
plot(YC[,7])
plot(diff(YC[,7]))
hist(diff(YC[,7]))
# Time Series Plot of the short-term rate (10-Year Rate)
par(mfrow = c(3,1))
plot(YC[,8])
plot(diff(YC[,8]))
hist(diff(YC[,8]))
# Correlation of all the rate change (discard row 1)
cor(YC.diff[-1,]) ## R_3M R_6M R_1Y R_2Y R_3Y R_5Y R_7Y
## R_3M 1.0000000 0.9328258 0.8329279 0.7165065 0.6486443 0.5609318 0.5062884
## R_6M 0.9328258 1.0000000 0.9581396 0.8703235 0.8089202 0.7326576 0.6825867
## R_1Y 0.8329279 0.9581396 1.0000000 0.9561221 0.9129480 0.8474693 0.8016415
## R_2Y 0.7165065 0.8703235 0.9561221 1.0000000 0.9855402 0.9439361 0.9045949
## R_3Y 0.6486443 0.8089202 0.9129480 0.9855402 1.0000000 0.9753412 0.9454168
## R_5Y 0.5609318 0.7326576 0.8474693 0.9439361 0.9753412 1.0000000 0.9875644
## R_7Y 0.5062884 0.6825867 0.8016415 0.9045949 0.9454168 0.9875644 1.0000000
## R_10Y 0.4724634 0.6479118 0.7640692 0.8660014 0.9118204 0.9674999 0.9879781
## R_10Y
## R_3M 0.4724634
## R_6M 0.6479118
## R_1Y 0.7640692
## R_2Y 0.8660014
## R_3Y 0.9118204
## R_5Y 0.9674999
## R_7Y 0.9879781
## R_10Y 1.0000000What Drives the Yield Curve Movement: The Theory
There are several theories and hypotheses that attempt to explain the shape and behavior of the yield curve:
Expectations Theory: This theory suggests that the shape of the yield curve reflects market participants’ expectations of future interest rates. It posits that long-term interest rates are an average of current and expected short-term interest rates. If investors expect rates to rise, the yield curve may be upward-sloping; if they expect rates to fall, it may be downward-sloping.
Liquidity Preference Theory: This theory asserts that investors demand a premium (higher yield) for holding longer-term bonds to compensate for the increased risk associated with tying up their funds for an extended period. This explains why longer-term yields tend to be higher than short-term yields, resulting in an upward-sloping yield curve.
Market Segmentation Theory: According to this theory, different market participants have preferences for specific maturities based on their investment needs and risk appetite. As a result, the supply and demand dynamics in each maturity segment can cause variations in yields, leading to a segmented or humped yield curve.
Preferred Habitat Theory: Building upon the market segmentation theory, this theory suggests that investors may be willing to move across different maturity segments (or “habitats”) if they are compensated adequately for the perceived risk. It explains deviations from a purely segmented yield curve and helps explain transitions between different curve shapes.
Term Premium Theory: This theory focuses on the risk premium embedded in long-term bonds due to uncertainty and potential price volatility over a more extended period. It suggests that long-term yields incorporate an additional term premium to compensate investors for holding longer maturities.
Key Factors That Move the Yield Curve
Central Bank: The actions and policies of a central bank, such as changes in the benchmark interest rate or monetary policy, can significantly influence the yield curve. If a central bank raises interest rates to combat inflation, short-term rates tend to increase more than long-term rates, resulting in a steeper yield curve. Conversely, if a central bank lowers rates to stimulate economic growth, short-term rates may decrease more than long-term rates, leading to a flatter yield curve.
Country Risk / Government Default Risk: The perceived risk of a government defaulting on its debt obligations affects the yield curve. Higher country risk leads to higher borrowing costs, causing investors to demand higher yields on government bonds. As a result, the yield curve may become steeper due to increased yields on longer-term bonds, reflecting the higher risk associated with holding those securities.
Macroeconomic Condition: The overall state of the economy, including factors like GDP growth, inflation, employment, and consumer confidence, influences the yield curve. In periods of strong economic growth and low inflation, the yield curve tends to steepen as investors anticipate higher future interest rates. Conversely, during economic downturns or periods of low inflation, the yield curve may flatten or even invert, with short-term rates exceeding long-term rates.
Interest rate and economic conditions in major countries: Yield differentials between countries can impact the yield curve. If interest rates rise in major countries, investors may shift their capital to take advantage of higher yields, leading to an increase in domestic bond yields and a steeper yield curve. Similarly, economic conditions in major economies can affect global market sentiment and capital flows, influencing the yield curve of other countries.
Investor’s sentiments: Market participants’ perceptions and expectations can impact the yield curve. If investors anticipate higher inflation or future economic instability, they may demand higher yields on long-term bonds, resulting in a steeper yield curve. Conversely, if investors are concerned about economic contraction or deflation, they may seek the safety of longer-term bonds, causing the yield curve to flatten.
Other factors: Various other factors can influence the yield curve, including supply and demand dynamics in the bond market, changes in the regulatory environment, geopolitical events, and market liquidity conditions. For example, increased government bond issuance can put upward pressure on yields and steepen the yield curve. Changes in regulations affecting the financial sector or monetary policy tools can also impact the shape of the yield curve. Additionally, unexpected events such as geopolitical tensions or financial crises can cause shifts in investor sentiment and yield curve movements.
Note: Duration can only capture a small level change of the yield curve!
Duration measures the sensitivity of bond prices to small changes in yields, but is less accurate for larger level changes.
Duration assumes a linear relationship between bond prices and yields, which may not hold true for significant yield curve movements.
Larger shifts in the yield curve, such as steepening or flattening, can lead to discrepancies in price changes estimated by duration.
To capture larger level changes, additional measures like convexity should be considered alongside duration.
Note: Other types of duration could work too: Effective Duration, Effective Convexity
Effective Duration = (-1) * [(P1 - P2) / (P * (y2 - y1))]
Where:
P1 represents the price of the bond at time 1.
P2 represents the price of the bond at time 2.
P represents the original price of the bond.
y1 represents the yield at time 1.
y2 represents the yield at time 2.
Effective Convexity = [(P1 + P2 - 2P) / (P * (y2 - y1)^2)]
Where:
P1 represents the price of the bond at time 1.
P2 represents the price of the bond at time 2.
P represents the original price of the bond.
y1 represents the yield at time 1.
y2 represents the yield at time 2.
Note: Other types of duration could work too: Key Rate Duration and Key Rate Convexity
Key Rate Duration at a specific maturity = (-1) * [(P1 - P2) / (P * (y2 - y1))]
Where:
P1 represents the price of the bond at time 1.
P2 represents the price of the bond at time 2.
P represents the original price of the bond.
y1 represents the yield at time 1.
y2 represents the yield at time 2.
∆y = y2 - y1 represents the change in the specific key rate (at the given maturity).
Key Rate Convexity at a specific maturity = [(P1 + P2 - 2P) / (P * (∆y)^2)]
Where:
P1 represents the price of the bond at time 1.
P2 represents the price of the bond at time 2.
P represents the original price of the bond.
y1 represents the yield at time 1.
y2 represents the yield at time 2.
∆y = y2 - y1 represents the change in the specific key rate (at the given maturity).
… But What About Spread?
A bond credit spread represents the yield difference between a treasury and corporate bond of the same maturity.
Treasury bonds are considered risk-free, while corporate bonds carry higher risk and offer compensation to investors in the form of credit spreads.
Credit spreads vary based on the issuer’s credit rating, with higher quality bonds offering lower spreads and vice versa.
Fluctuations in credit spreads are influenced by economic conditions, liquidity, and investor demand.
Widening credit spreads indicate investor concern and are often seen during uncertain or worsening economic conditions.
Different bond market indexes track yields and credit spreads for various types of debt, including corporate bonds, mortgage-backed securities, and government bonds.
Component of Bond Spread: The Diagram That I Love!
Farahvash, P. (2020). Asset-liability and liquidity management. John Wiley & Sons.
Types of Spread Jargons
Government Spread (G-Spread): The difference in yield between a government bond and a benchmark, typically a risk-free treasury bond, used to measure the credit risk of non-government bonds.
Zero Volatility Spread (Z-Spread): The constant spread added to the yield curve to make the present value of a bond’s cash flows equal to its market price.
Option Adjusted Spread (OAS): The spread over the risk-free rate that takes into account the embedded optionality of a bond, such as call or put options. It represents the compensation investors demand for the risk associated with the bond’s embedded options.
Country Default Spread: The additional yield offered on bonds issued by a particular country to compensate investors for the risk of default. It reflects the difference in creditworthiness between the country and a benchmark, usually a more creditworthy country or a risk-free rate.
How to Decompose/Capture the Yield Curve Behavior using 3 factor model?
The Nelson-Seigel model is a popular framework for yield curve modeling that characterizes the term structure of interest rates using a set of parameters. The model is used for forecasting interest rates, pricing fixed-income securities, and risk management. It provides a coherent framework for understanding the dynamics of the yield curve and enables analysis of the term structure of interest rates.
The Nelson-Siegel’s model to describe the yield curve is:
\(y_t(τ) = β_{0t} + β_{1t} \frac{1-\exp(-λ τ)}{λ τ} + β_{2t} (\frac{1-\exp(-λ τ)}{λ τ} - \exp(-λ τ) )\)
The estimated parameters of the Nelson-Seigel model have economic interpretations. The level parameter represents long-term interest rate expectations, the slope parameter indicates market sentiment regarding future rate changes, and the curvature parameter reflects investor preferences for specific maturities.
Original Paper: Nelson, C.R., and A.F. Siegel (1987), Parsimonious Modeling of Yield Curve, The Journal of Business, 60, 473-489. Link: https://www.jstor.org/stable/2352957?casa_token=ds4Xy2yRfDoAAAAA:Wle4GmAv_GTm_b4kRpBoJvWK7scZ_HWeht6f7EIcTznhc9nIiIVhlWYo-ecvGoYeLKHVGqVqRtyqXXIbKP4o_8r4yDx-DU6_1BtAT_KH-9_c8jMtcz7O
Nelson-Seigel Model: The Model
## Nelson-Siegel Model
## Reading: Estimating the Yield Curve Using the Nelson-Siegel Modelby Jan Annaert et al (2010)
## Link: https://www.efmaefm.org/0EFMAMEETINGS/EFMA%20ANNUAL%20MEETINGS/2010-Aarhus/papers/Smoothed_Bootstrap_-_Nelson-Siegel_Revisited_June_2010.pdf
### Yield Calculation from NS Model: Note - tau is lambda
nelson_siegel_calculate<-function(theta,tau,beta0,beta1,beta2){
beta0 + beta1*(1-exp(-theta/tau))/(theta/tau) + beta2*((1-exp(-theta/tau))/(theta/tau) - exp(-theta/tau))
}
#install.packages("dplyr")
library(dplyr)##
## Attaching package: 'dplyr'## The following objects are masked from 'package:xts':
##
## first, last## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, unionns_data <-
data.frame(maturity=1:30) %>%
mutate(ns_yield=nelson_siegel_calculate(theta=maturity,tau=3,beta0=3,beta1=-3,beta2=-3))
head(ns_data)## maturity ns_yield
## 1 1 0.04715752
## 2 2 0.16100543
## 3 3 0.31091497
## 4 4 0.47697854
## 5 5 0.64657898
## 6 6 0.81201170nelson_siegel_calculate<-function(theta,tau,beta0,beta1,beta2){
beta0 + beta1*(1-exp(-theta/tau))/(theta/tau) + beta2*((1-exp(-theta/tau))/(theta/tau) - exp(-theta/tau))
}
ns_data <-
data.frame(maturity=1:30) %>%
mutate(ns_yield=nelson_siegel_calculate(theta=maturity,tau=3.3,beta0=0.07,beta1=-0.02,beta2=0.01))
head(ns_data)## maturity ns_yield
## 1 1 0.05398726
## 2 2 0.05704572
## 3 3 0.05940289
## 4 4 0.06122926
## 5 5 0.06265277
## 6 6 0.06376956library(ggplot2)
ggplot(data=ns_data, aes(x=maturity,y=ns_yield)) + geom_point() + geom_line()
Nelson-Seigel Model: Parameter Estimation
# Parameter Estimation
# install.packages("YieldCurve")
library(YieldCurve)
?Nelson.Siegel
data(FedYieldCurve) # Call Fed Yield Curve Dataset
NS_result <- Nelson.Siegel(as.matrix(FedYieldCurve[1:10,]),maturity=c(3,6,12,24,36,60,72,120))Nelson-Seigel Model: Yield Curve Forecasting
# Yield Curve Forecasting
library(forecast)## Warning: package 'forecast' was built under R version 4.0.5## Registered S3 method overwritten by 'quantmod':
## method from
## as.zoo.data.frame zoo#b0
b0 <- NS_result[,1]
plot(b0, type = "l")
acf(b0)
pacf(b0)
b0.mod <- auto.arima(b0)
b0.for <- forecast(b0.mod, h = 12)
#b1
b1 <- NS_result[,2]
plot(b1, type = "l")
acf(b1)
pacf(b1)
b1.mod <- auto.arima(b1)
b1.for <- forecast(b1.mod, h = 12)
#b2
b2 <- NS_result[,3]
plot(b2, type = "l")
acf(b2)
pacf(b2)
b2.mod <- auto.arima(b2)
b2.for <- forecast(b2.mod, h = 12)
#lambda or tau
b3 <- NS_result[,4]
plot(b3, type = "l")
acf(b3)
pacf(b3)
b3.mod <- auto.arima(b3)
b3.for <- forecast(b3.mod, h = 12)
#Yield Curve Forecasting
h <- 12
a <- b0.for[["mean"]][h]
b <- b1.for[["mean"]][h]
c <- b2.for[["mean"]][h]
d <- b3.for[["mean"]][h]
ns_forecast <-
data.frame(maturity=1:30) %>%
mutate(ns_yield=nelson_siegel_calculate(theta=maturity,tau=d,beta0=a,beta1=b,beta2=c))
ggplot(data=ns_forecast, aes(x=maturity,y=ns_yield)) + geom_point() + geom_line()
Nelson-Seigel Model: Yield Curve Scenario Analysis
#Bond Sensitivities
nelson_siegel_sensitivities<-function(tau=3,beta0=0.08,beta1=-0.03,beta2=-0.01,nominal_rate=100,coupon_rate,maturity){
f_i <- nominal_rate*rep(coupon_rate,maturity)
f_i[length(f_i)]<-f_i[length(f_i)]+nominal_rate
theta_i <- 1:maturity
r_i <- 0
for(i in 1:maturity){
r_i[i] <- nelson_siegel_calculate(theta=i,tau=tau,beta0=beta0,beta1=beta1,beta2=beta2)
}
beta0_sens <- -sum(f_i*theta_i*exp(-theta_i*r_i))
beta1_sens <- -sum(f_i*theta_i*(1-exp(-theta_i/tau))/(theta_i/tau)*exp(-theta_i*r_i))
beta2_sens <- -sum(f_i*theta_i*((1-exp(-theta_i/tau))/(theta_i/tau) - exp(-theta_i/tau))*exp(-theta_i*r_i))
return(c(Beta0=beta0_sens,Beta1=beta1_sens,Beta2=beta2_sens))
}
nelson_siegel_sensitivities(coupon_rate=0.05,maturity=2)## Beta0 Beta1 Beta2
## -192.51332 -141.08199 -41.27936nelson_siegel_sensitivities(coupon_rate=0.05,maturity=7)## Beta0 Beta1 Beta2
## -545.4198 -224.7767 -156.7335nelson_siegel_sensitivities(coupon_rate=0.05,maturity=15)## Beta0 Beta1 Beta2
## -812.6079 -207.1989 -173.0285What about Vasicek and CIR?
Those are one factor model only concerns about the mean reversion and level shift
They CANNOT capture the slope and curvature change
However, you can use use them for long term interest projection.
We can discuss about this again once we study option pricing models