Minimum connectedness portfolios and the market for green bonds: Advocating socially responsible investment (SRI) activity

data("bcg2022")

NAMES = colnames(bcg2022)
k = ncol(bcg2022)
Y = bcg2022[-1,]
for (i in 1:k) {
  Y[,i] = 100*diff(log(as.numeric(bcg2022[,i])))
}

# SUMMARY STATISTICS
kable(SummaryStatistics(Y, correlation="pearson"))

	US.Green.Bond	US.Black.Bond	EU.Green.Bond	EU.Black.Bond	CN.Green.Bond	CN.Black.Bond
Mean	0.017*	0.013**	0.011*	0.008	0.015***	0.015**
	(0.053)	(0.015)	(0.099)	(0.158)	(0.000)	(0.040)
Variance	0.086***	0.036***	0.053***	0.036***	0.006***	0.061***
Skewness	-0.895***	-0.808***	-1.092***	-0.824***	-1.271***	0.168**
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.019)
Ex.Kurtosis	10.586***	10.003***	8.911***	6.645***	29.503***	3.442***
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
JB	5628.833***	5013.559***	4110.425***	2288.532***	42822.697***	584.175***
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
ERS	-10.344***	-13.930***	-8.071***	-13.414***	-5.806***	-12.005***
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
Q(20)	70.917***	33.112***	27.241***	25.708***	105.537***	23.452***
	(0.000)	(0.000)	(0.001)	(0.001)	(0.000)	(0.004)
Q2(20)	469.423***	1253.746***	747.621***	728.895***	47.643***	32.941***
	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)	(0.000)
pearson	US.Green.Bond	US.Black.Bond	EU.Green.Bond	EU.Black.Bond	CN.Green.Bond	CN.Black.Bond
US.Green.Bond	1.000***	0.541***	0.480***	0.475***	0.010	0.313***
US.Black.Bond	0.541***	1.000***	0.566***	0.517***	0.027	0.025
EU.Green.Bond	0.480***	0.566***	1.000***	0.941***	0.013	0.049
EU.Black.Bond	0.475***	0.517***	0.941***	1.000***	0.003	0.055
CN.Green.Bond	0.010	0.027	0.013	0.003	1.000***	0.082***
CN.Black.Bond	0.313***	0.025	0.049	0.055	0.082***	1.000***

nlag = 1
nfore = 20
prior = BayesPrior(Y, nlag=nlag)
fit = TVPVAR(Y, configuration=list(l=c(0.99,0.99), nlag=nlag, prior=prior))
dca = TimeConnectedness(Phi=fit$B_t, Sigma=fit$Q_t, nfore=nfore, corrected=TRUE)
kable(dca$TABLE)

	US.Green.Bond	US.Black.Bond	EU.Green.Bond	EU.Black.Bond	CN.Green.Bond	CN.Black.Bond	FROM
US.Green.Bond	57.96	16.28	9.33	9.55	0.32	6.55	42.04
US.Black.Bond	14.91	52.62	17.21	13.88	0.45	0.92	47.38
EU.Green.Bond	7.43	14.10	41.86	35.83	0.25	0.53	58.14
EU.Black.Bond	7.59	11.67	36.71	43.28	0.28	0.48	56.72
CN.Green.Bond	0.95	1.16	1.09	1.12	92.17	3.51	7.83
CN.Black.Bond	13.02	4.28	1.52	1.44	1.65	78.09	21.91
TO	43.90	47.50	65.86	61.82	2.94	12.00	234.01
Inc.Own	101.86	100.12	107.72	105.10	95.12	90.09	cTCI/TCI
NET	1.86	0.12	7.72	5.10	-4.88	-9.91	46.80/39.00
NPT	2.00	3.00	5.00	4.00	0.00	1.00

Figure 3: Dynamic Total Connectedness

PlotTCI(dca, ylim=c(20,60))

Figure 4: Net Total Directional Connectedness

PlotNET(dca, ylim=c(-20,20))

Figure 5: Net Pairwise Directional Connectedness

PlotNPDC(dca, ylim=c(-10,15))

Multivariate Portfolios

k = 4
y = Y[,1:k]
date = index(y)
Q_t = fit$Q_t[1:k,1:k,]
R_t = ConditionalCorrelation(Q_t)
C_t = dca$PCI[1:k,1:k,]

# Minimum Variance Portfolio
mvp = MinimumConnectednessPortfolio(y, Q_t)
kable(mvp$TABLE)

	Mean	Std.Dev.	5%	95%	HE	p-value
US.Green.Bond	0.04	0.05	0.00	0.14	0.69	0.00
US.Black.Bond	0.35	0.10	0.23	0.57	0.25	0.00
EU.Green.Bond	0.02	0.08	0.00	0.27	0.49	0.00
EU.Black.Bond	0.59	0.14	0.16	0.69	0.25	0.00

# Minimum Correlation Portfolio
mcp = MinimumConnectednessPortfolio(y, R_t)
kable(mcp$TABLE)

	Mean	Std.Dev.	5%	95%	HE	p-value
US.Green.Bond	0.35	0.06	0.21	0.40	0.63	0.00
US.Black.Bond	0.26	0.06	0.17	0.38	0.10	0.09
EU.Green.Bond	0.11	0.08	0.00	0.22	0.39	0.00
EU.Black.Bond	0.28	0.07	0.18	0.42	0.10	0.07

# Minimum Connectedness Portfolio
mpc = MinimumConnectednessPortfolio(y, C_t)
kable(mpc$TABLE)

	Mean	Std.Dev.	5%	95%	HE	p-value
US.Green.Bond	0.34	0.07	0.18	0.40	0.63	0.00
US.Black.Bond	0.26	0.04	0.20	0.34	0.11	0.04
EU.Green.Bond	0.09	0.08	0.00	0.23	0.40	0.00
EU.Black.Bond	0.31	0.10	0.15	0.49	0.12	0.03

Weights

par(mfcol=c(4,3), oma=c(.5,.75,0,0)+0.1, mar=c(.5,.5,.5,.5)+1, mgp=c(.5, .5, 0))
for (i in 1:k) {
  plot(date, mvp$portfolio_weights[,i], type="l", las=1, ylim=c(0,0.8), xaxs="i", xlab="", ylab="", col="steelblue4", main=paste("MVP", NAMES[i])); grid(NA, NULL)
  plot(date, mcp$portfolio_weights[,i], type="l", las=1, ylim=c(0,0.8), xaxs="i", xlab="", ylab="", col="steelblue4", main=paste("MCP", NAMES[i])); grid(NA, NULL)
  plot(date, mpc$portfolio_weights[,i], type="l", las=1, ylim=c(0,0.8), xaxs="i", xlab="", ylab="", col="steelblue4", main=paste("MCoP",NAMES[i])); grid(NA, NULL)
}

Cumulative Returns

par(mfrow=c(1,1))
plot(date, mcp$cumulative_portfolio_return, type="l", las=1, xaxs="i", xlab="", ylab="", col="red")
grid(NA, NULL); abline(h=0, lty=3)
lines(date, mpc$cumulative_portfolio_return, col="steelblue1")
lines(date, mvp$cumulative_portfolio_return, col="steelblue4")
legend("topleft", c("MVP","MCP","MCoP"), fill=c("steelblue4", "red", "steelblue1"), bty="n")

Bivariate Portfolios

bpw = BivariatePortfolio(y, Q_t)
kable(bpw$TABLE)

	Mean	Std.Dev.	5%	95%	HE	p-value
US.Green.Bond/US.Black.Bond	0.14	0.12	0.00	0.38	0.61	0.25
US.Green.Bond/EU.Green.Bond	0.32	0.15	0.12	0.66	0.49	0.00
US.Green.Bond/EU.Black.Bond	0.19	0.12	0.04	0.46	0.62	0.10
US.Black.Bond/US.Green.Bond	0.86	0.12	0.62	1.00	0.06	0.00
US.Black.Bond/EU.Green.Bond	0.71	0.10	0.56	0.87	0.11	0.00
US.Black.Bond/EU.Black.Bond	0.51	0.08	0.37	0.62	0.26	0.00
EU.Green.Bond/US.Green.Bond	0.68	0.15	0.34	0.88	0.16	0.00
EU.Green.Bond/US.Black.Bond	0.29	0.10	0.13	0.44	0.40	0.04
EU.Green.Bond/EU.Black.Bond	0.06	0.20	0.00	0.79	0.32	0.91
EU.Black.Bond/US.Green.Bond	0.81	0.12	0.54	0.96	0.09	0.00
EU.Black.Bond/US.Black.Bond	0.49	0.08	0.38	0.63	0.27	0.00
EU.Black.Bond/EU.Green.Bond	0.94	0.20	0.21	1.00	0.01	0.00

Weights

par(mfcol=c(3,2), oma=c(.5,.75,0,0)+0.1, mar=c(.5,.5,.5,.5)+1, mgp=c(.5, .5, 0))
for (i in 1:k) {
  for (j in 1:k) {
    if (i>j) {
      plot(date, bpw$portfolio_weights[i,j,], type="l", las=1, ylim=c(0,1), xaxs="i", xlab="", ylab="", col="steelblue4", main=paste0(NAMES[i], '-', NAMES[j]))
      grid(NA, NULL); box(); abline(h=0, lty=3)
    }
  }
}

Cumulative Returns

par(mfcol=c(3,2), oma=c(.5,.75,0,0)+0.1, mar=c(.5,.5,.5,.5)+1, mgp=c(.5, .5, 0))
for (i in 1:k) {
  for (j in 1:k) {
    if (i>j) {
      plot(date, bpw$cumulative_portfolio_return[i,j,], type="l", las=1, xaxs="i", xlab="", ylab="", col="steelblue4", main=paste0(NAMES[i], '-', NAMES[j]), ylim=c(min(bpw$cumulative_portfolio_return), max(bpw$cumulative_portfolio_return)))
      grid(NA, NULL); box(); abline(h=0, lty=3)
    }
  }
}

Hedge Ratios

hr = HedgeRatio(y, Q_t)
kable(hr$TABLE)

	Mean	Std.Dev.	5%	95%	HE	p-value
US.Green.Bond/US.Black.Bond	0.80	0.20	0.50	1.11	0.32	0.00
US.Green.Bond/EU.Green.Bond	0.50	0.21	0.15	0.79	0.28	0.00
US.Green.Bond/EU.Black.Bond	0.60	0.20	0.27	0.94	0.26	0.00
US.Black.Bond/US.Green.Bond	0.35	0.10	0.20	0.57	0.33	0.00
US.Black.Bond/EU.Green.Bond	0.47	0.05	0.40	0.56	0.35	0.00
US.Black.Bond/EU.Black.Bond	0.50	0.08	0.36	0.63	0.30	0.00
EU.Green.Bond/US.Green.Bond	0.32	0.16	0.08	0.61	0.28	0.00
EU.Green.Bond/US.Black.Bond	0.69	0.10	0.53	0.88	0.34	0.00
EU.Green.Bond/EU.Black.Bond	1.11	0.09	0.91	1.21	0.89	0.00
EU.Black.Bond/US.Green.Bond	0.28	0.12	0.10	0.49	0.26	0.00
EU.Black.Bond/US.Black.Bond	0.51	0.11	0.33	0.70	0.29	0.00
EU.Black.Bond/EU.Green.Bond	0.78	0.09	0.62	0.97	0.89	0.00

par(mfcol=c(k,k-1), oma=c(.5,.75,0,0)+0.1, mar=c(.5,.5,.5,.5)+1, mgp=c(.5, .5, 0))
for (i in 1:k) {
  for (j in 1:k) {
    if (i!=j) {
      plot(date, hr$hedge_ratio[i,j,], type="l", las=1, xaxs="i", xlab="", ylab="", col="steelblue4", main=paste0(NAMES[i], '-', NAMES[j]), ylim=c(min(hr$hedge_ratio),max(hr$hedge_ratio)))
      grid(NA, NULL); box(); abline(h=0, lty=3)
    }
  }
}

Cumulative Returns

par(mfcol=c(k,k-1), oma=c(.5,.75,0,0)+0.1, mar=c(.5,.5,.5,.5)+1, mgp=c(.5, .5, 0))
for (i in 1:k) {
  for (j in 1:k) {
    if (i!=j) {
      plot(date, hr$cumulative_portfolio_return[i,j,], type="l", las=1, xaxs="i", xlab="", ylab="", col="steelblue4", main=paste0(NAMES[i], '-', NAMES[j]), ylim=c(min(hr$cumulative_portfolio_return), max(hr$cumulative_portfolio_return)))
      grid(NA, NULL); box(); abline(h=0, lty=3)
    }
  }
}