AFNSS model
DNS and AFNS models, and their multi-factor extensions
Diebold and Li (2006) along with Diebold et al. (2006) pioneered the dynamic Nelson-Siegel (DNS) model. By incorporating dynamics of state variables into the Nelson-Siegel (1987) model, they demonstrated its empirical success.
Subsequently, Christensen et al. (2011) strengthened the DNS model's theoretical foundation by enforcing the arbitrage-free (AF) restriction, giving rise to the well-known 3-factor Arbitrage-free Nelson-Siegel (AFNS) model.
Naturally, the 4-factor DNSS model is expected to be transformed into an affine arbitrage-free model, however, Christensen et al. (2009) show that it is not possible to obtain the AF class of DNSS model with matching the Svensson loadings exactly.
To avoid this difficulty, they add a second slope factor and derive the arbitrage-free generalized Nelson-Siegel (AFGNS) model. In fact, the AFGNS model is a 5-factor model since it is the AF class of the dynamic generalized Nelson-Siegel (DGNS) model. Therefore, a four-factor arbitrage-free Nelson-Siegel-Svensson (AFNSS) model has not yet been developed.
Introducing 4-factor AFNSS model
Since the DNSS model is popular but lacks theoretical rigor, it's crucial to develop an essential AFNSS model, even with some modifications, while preserving most characteristics of the DNSS model. To achieve this objective, I derive an AFNSS model by applying a slight modification to the mean-reversion matrix under the risk-neutral Q measure.
Despite this slight modification, the model retains a highly similar functional form, nearly identical empirical fits, and comparable estimated yield curve factors to the DNSS model. Details on this new AFNSS model can be found in the following working paper, which is currently under the review process.
| Lee, S.-H. (2024), An Arbitrage-Free Nelson-Siegel-Svensson Term Structure model, working paper. https://ssrn.com/abstract=4769455 |
Now, we can summarize a comprehensive overview of both dynamic and arbitrage-free classes of Nelson-Siegel models, alongside their multi-factor extensions, as can be seen in the following table.
| Number of factors | Dynamic model | Arbitrage-Free model |
|---|---|---|
| 3-factor | DNS | AFNS |
| 4-factor | DNSS | AFNSS |
| 5-factor | DGNS | AFGNS |
AFNSS model structure
The AFNSS model can be expressed as a state-space model as follows.
\begin{align} y_t (\boldsymbol{\tau}) &= B(\boldsymbol{\tau}) X_t - \frac{A(\boldsymbol{\tau})}{\boldsymbol{\tau}} + \epsilon_t (\boldsymbol{\tau}) \\ dX_t &= K^P (\theta^P - X_t) dt + \Sigma dW_t^P \end{align}
where other terms except the factor loading matrix and the following yield-adjustment term are the same as the AFNS model, with appropriate dimensions.
The yield-adjustment term in the AFNSS model is as follows.
R code
The R code for the independent-factor AFNSS model is implemented as follows. I use the so-called DRA data of Diebold et al. (2006), which is representative U.S. yield curve data. The DRA data can be downloaded from the following link:
"http://econweb.umd.edu/~webspace/aruoba/research/paper5/DRA%20Data.txt".
#================================================================# # R code for Arbitrage-free Nelson-Siegel-Svensson (AFNSS) model #---------------------------------------------------------------- # written by Sang-Heon Lee (shlee725@gmail.com) # # Reference: # # Lee, S.-H. (2024), An Arbitrage-Free Nelson-Siegel-Svensson # Term Structure model, https://ssrn.com/abstract=4769455 # # Comments are welcome. #================================================================# library(expm) graphics.off(); rm(list = ls()) # remove all files from your workspace setwd("your folder") # Yield-adjustment term AFNSS.YAT.analytic.func <- function(s, la, tau) { n <- la[1]; m <- la[2]; t <- tau n2 <- n^2; m2 <- m^2 vs <- rep(NA, 10) vs[1] <- sum(s[1,]^2) vs[2] <- sum(s[2,]^2) vs[3] <- sum(s[3,]^2) vs[4] <- sum(s[4,]^2) vs[5] <- sum(s[1,]*s[2,]) vs[6] <- sum(s[1,]*s[3,]) vs[7] <- sum(s[1,]*s[4,]) vs[8] <- sum(s[2,]*s[3,]) vs[9] <- sum(s[2,]*s[4,]) vs[10] <- sum(s[3,]*s[4,]) rs <- vr <- matrix(NA, length(t), 10) sS <- function(la) { return((1-exp(-la*t))/(la*t)) } sC <- function(la) { return((1-exp(-la*t))/(la*t)-exp(-la*t)) } fS <- function(la) { return(exp(-la*t))} fC <- function(la) { return(la*t*exp(-la*t)) } vr[,1] <- t^2/6 vr[,2] <- (1/n2)*(1/2-sS(n)+0.5*sS(2*n)) vr[,3] <- (1/n2)*(1/2 + fS(n) - (1/8)*fC(2*n) - (3/4)*fS(2*n) - 2*sS(n) + (5/4)*sS(2*n)) vr[,4] <- (1/(n2*m2))*(((n-m)^2)/2 + (1/t)*(n+m-n^2/m-m^2/n) + ((n-m)/t)*((n/m)*fS(m) - (m/n)*fS(n)) + n^2/2*sS(2*m) + m^2/2*sS(2*n) - n*m*sS(n+m)) vr[,5] <- (1/n)*(t/2 - (1/n)*sC(n)) vr[,6] <- (1/n)*(t/2 + t*fS(n) - (3/n)*sC(n)) vr[,7] <- (1/m-1/n)*(t/2) - (1/m2)*sC(m) + (1/n2)*sC(n) vr[,8] <- (1/n2)*(1 + fS(n) - (1/2)*fS(2*n) - 3*sS(n) + (3/2)*sS(2*n)) vr[,9] <- (1/n )*((1/n)*( 2*sS(n) - sS(2*n) - 1) + (1/m)*(1 - sS(n) - sS(m) + sS(n+m))) vr[,10] <- (1/n2)*(3*sS(n) - sS(2*n) - 1 - fS(n)) + (1/(m*(n+m)))*(sS(n+m) - fS(n+m)) + (1/(n*m))*(1 + fS(n) - sS(m) - 2*sS(n) + sS(n+m)) - (1/(2*n2))*( sS(2*n) - fS(2*n)) for(i in 1:10) rs[,i] = vs[i]*vr[,i] return(rowSums(rs)) } # factor loading matrix AFNSS.factor_loading <- function(lambda, maty){ la1 <- lambda[1]; la2 <- lambda[2] col1 <- rep.int(1,length(maty)) col2 <- (1-exp(-la1*maty))/(la1*maty) col3 <- col2-exp(-la1*maty) col4 <- (1-exp(-la2*maty))/(la2*maty)-col2 return( cbind(col1,col2,col3,col4)) } # parameter restrictions AFNSS.trans <- function(b0) { b1 <- b0 b1[ 9:12] <- b0[ 9:12]^2 # sigma b1[13:29] <- b0[13:29]^2 # H # lambda1 > lambda2 b1[30] <- b0[30]^2 + b0[31]^2 b1[31] <- b0[31]^2 return(b1) } # negative log likelihood function to be minimized AFNSS.objf<-function(para_un) { nmat <- length(maty); nobs<- nrow(yield) # parameter restrictions para_con <- AFNSS.trans(para_un) kappa <- diag(para_con[1:4]) theta <- para_con[5:8]; sigma <- diag(para_con[9:12]) H <- diag(para_con[13:29]) lambda <- para_con[30:31] Phi1 <- expm(-kappa*dt) Phi0 <- (diag(nf)-Phi1)%*%theta # Conditional and Unconditional covariance matrix : Q, Q0 m <- eigen(kappa) eval <- m$values; evec <- m$vectors; ievec<-solve(evec) Smat <- ievec%*%sigma%*%t(sigma)%*%t(ievec) Vdt <- Vinf <- matrix(0,nf,nf) for(i in 1:nf) { for(j in 1:nf) { Vdt[i,j] <-Smat[i,j]*(1-exp(-(eval[i]+eval[j])*dt))/(eval[i]+eval[j]) Vinf[i,j]<-Smat[i,j]/(eval[i]+eval[j]) }} # Analytical Covariance matrix Q <- evec%*%Vdt%*%t(evec); Q0 <- evec%*%Vinf%*%t(evec) # factor loading matrix B <- AFNSS.factor_loading(lambda, maty); tB <- t(B) # Yield adjustment term C_yat <- AFNSS.YAT.analytic.func(sigma, lambda, maty) #------------------------------------------------ # Eigenvalue check #------------------------------------------------ eval <- eigen(kappa)$values if(is.complex(eval)) { # Check if the real part of all eigenvalues is positive all_positive <- all(Re(eval) > 0) # Not all eigenvalues have positive real parts if (!all_positive) return(100000000) } else { # real # Check if all eigenvalues are positive all_positive <- all(eval > 0) # Not all eigenvalues are positive if (!all_positive) return(100000000) } #------------------------------------------------ # Penalty for two lambda for feasible values #------------------------------------------------ la1 <- lambda[1]; la2 <- lambda[2] # 1) maximum loading maturity is less than max(maturity) if(max(B[,nf]) == B[nmat,nf]) return (100000000); # tau-star f_slo <- function(la,m) { return( (1-exp(-la*m))/(la*m) ) } f_cur <- function(la,m) { return( (1-exp(-la*m))/(la*m)-exp(-la*m) ) } # intersection of C1 and C2 : maximum of AC2 f_err <- function(m, la1, la2) {return(f_cur(la2,m)-f_cur(la1,m))} f_sqr <- function(m, la1, la2) {return(f_err(m, la1, la2)^2)} tau_star <- uniroot(f_err, c(0.001, 10000), tol = 0.000001, la1 = la1, la2 = la2) # 2) tau-star is larger than (tau(la1) + 1 year) if(1.8/la1 + 1 > tau_star$root) return (100000000) #------------------------------------------------ # initialization prevX <- theta; prevV <- Q0 loglike <- 0 for (t in 1:nobs) { # prediction Xhat <- Phi0+Phi1%*%prevX; Vhat <- Phi1%*%prevV%*%t(Phi1)+Q # measurement error y_real <- yield[t,] # measurement of y y_fit <- B%*%Xhat - C_yat # prediction of y v <- y_real - y_fit # error # updating ev <- B%*%Vhat%*%tB + H ev <- (ev+t(ev))/2 # symmetric if(is.nan(log(det(ev)))) return(100000000) evinv <- solve(ev) # invpd(ev) KG <- Vhat%*%tB%*%evinv # Kalman Gain prevX <- Xhat+KG%*%v # E[X|y_t] updated mean prevV <- Vhat-KG%*%B%*%Vhat # Cov[X|y_t] updated cov # log likelihood function loglike <- loglike - 0.5*(nmat)*log(2*pi)-0.5*log(det(ev))-0.5*t(v)%*%evinv%*%v } return(-loglike) } #=============================================================================== # Main ; Independent-factor AFNSS model estimation #=============================================================================== dt <- 1/12; nf <- 4 #fn <- "http://econweb.umd.edu/~webspace/aruoba/research/paper5/DRA%20Data.txt" fn <- "DRA_Data.txt" df <- read.delim(fn, header=T) yield <- as.matrix(df[,2:18]/100) maty <- c(3,6,9,12,15,18,21,24,30,36,48,60,72,84,96,108,120)/12 nmat <- length(maty); nobs <- nrow(yield) #------------------------------------------------ # Initial guesses : A, U, Q, H, lambda1, lambda2 #------------------------------------------------ init_par <- c( # A 0.143419606590641, 0.622832889743488, 1.66358997751309, 0.785371023388317, # U 0.0847478248037573, -0.0262498570577886, 0.0118544154577742, -0.0267016207522752, # Q 0.108981665021336, 0.150855823055516, 0.189855730677439, 0.138272221344484, # H 0.00212569804819625, -0.000310448155097482, 0.000948212689065406, 0.00102597509422836, 0.000800782770285079, 0.000677617287103599, 0.00070423191117115, 0.000778897072760653, -0.000682023248952573, 0.000653880954570606, 0.000872068445231712, -0.000682421393917689, 0.000966013422859042, 0.00112871797156858, 0.000959837834967723, 0.00118698263107226, 0.00156353540645325, # lambda1, lambda2 1.05303433998221, 0.538730960886413 ) #------------------------------------------------ # estimation by numerical optimization #------------------------------------------------ m<-optim(init_par, AFNSS.objf, control=list(maxit=3000,trace=2), method=c("Nelder-Mead")) prev_likev <- m$value; i <- 1 repeat { print(paste(i,"-th iteration")) m<-optim(m$par,AFNSS.objf,control=list(maxit=3000,trace=2), method=c("Nelder-Mead")) if (abs(m$value - prev_likev) < 1e-4) break prev_likev <- m$value; i <- i + 1 } #------------------------------------------------ # estimated parameters and log-likelihood value #------------------------------------------------ AFNSS.trans(m$par) AFNSS.objf(m$par) | cs |
The log-likelihood is estimated at 30932, and parameters are estimated as appropriate values.
> > #------------------------------------------------ > # estimated parameters and log-likelihood value > #------------------------------------------------ > AFNSS.trans(m$par) [1] 1.434196e-01 6.228329e-01 1.663590e+00 7.853710e-01 [5] 8.474782e-02 -2.624986e-02 1.185442e-02 -2.670162e-02 [9] 1.187700e-02 2.275748e-02 3.604520e-02 1.911921e-02 [13] 4.518592e-06 9.637806e-08 8.991073e-07 1.052625e-06 [17] 6.412530e-07 4.591652e-07 4.959426e-07 6.066806e-07 [21] 4.651557e-07 4.275603e-07 7.605034e-07 4.656990e-07 [25] 9.331819e-07 1.274004e-06 9.212887e-07 1.408928e-06 [29] 2.444643e-06 1.399112e+00 2.902310e-01 > AFNSS.objf(m$par) [,1] [1,] -30932.58 > | cs |
Filtered estimates of yield curve factors are as follows:
As a reference, the estimated DNSS model shows the following yield curve factors with a log-likelihood value of 30896.
Reference
Christensen, J. H. E., F. X. Diebold, and G. D. Rudebusch (2009) An arbitrage-free generalized Nelson-Siegel term structure model, Econometrics Journal 12-3, 33–64.
Christensen, J. H. E., F. X. Diebold, and G. D. Rudebusch (2011) The affine arbitrage-free class of Nelson–Siegel term structure models, Journal of Econometrics 164-1, 4–20.
Diebold, F. X. and C. Li (2006) Forecasting the term structure of government bond yields, Journal of Econometrics 130-2, 337–364.
Diebold, F. X., G. D. Rudebusch, and S. B. Aruoba (2006) The macroeconomy and the yield curve: A dynamic latent factor approach, Journal of econometrics 131-1, 309–338.
Nelson, C. R. and A. F. Siegel (1987) Parsimonious modeling of yield curves, The Journal of Business 60-4, 473–489.
Svensson, L. E. O. (1995) Estimating forward interest rates with the extended Nelson–Siegel method, Sveriges Riksbank, Quarterly Review 3, 13–26.
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