Collateral affects return risk: evidence from the euro bond market

Abstract

Covered bonds and senior bonds are prominent securities in the euro bond market. Senior bonds are unsecured, while covered bonds are secured—backed by collateral. Our results show that the presence of collateral reduces the total risk in individual bonds by more than 70%. Compared to diversified portfolios of senior bonds, diversified portfolios of covered bonds have a significantly lower level of systematic risk. However, the fraction of systematic risk to total risk is higher for covered bonds. By decomposing the variance of bond returns, we find that around 33% of the risk in senior bonds is systematic, versus 53% in covered bonds. Both types of bonds contain instrument-specific risk.

Appendices

Appendix

Decomposition of risk

Let $r_t=R_t-R_{\mathrm{f}}$ be the period t return in excess of the risk-free rate. By regressing the bond excess return on the portfolio excess return, we get the following relationship:

$$\begin{aligned} r_{it}=\beta _{ip}r_{{\mathrm{p}}t}+{\tilde{\epsilon }}_{it}, \end{aligned}$$

(6)

where ${\tilde{\epsilon }}_{it}$ is the error term.

Campbell et al. (2001) present a simplified model that permits a variance decomposition on an appropriate aggregate level, without having to keep track of covariances and without having to estimate betas. The model, generally called a “market-adjusted model,” drops the beta coefficient $\beta _{ip}$ from Eq. (6)

$$\begin{aligned} r_{it}=r_{{\mathrm{p}}t}+\epsilon _{it}. \end{aligned}$$

(7)

From Eq. (7), we have that $\epsilon _{it}$ is the difference between the bond return $r_{it}$ and the portfolio return $r_{{\mathrm{p}}t}$. Comparing Eqs. (6) and (7), we have

$$\begin{aligned} \epsilon _{it}={\tilde{\epsilon }}_{it}+(\beta _{ip}-1)r_{{\mathrm{p}}t}. \end{aligned}$$

The residual $\epsilon _{it}$ equals the residual in Eq. (6) only if the bond beta $\beta _{ip}=1$ or the portfolio return $r_{{\mathrm{p}}t}=0$. Calculating the variance of the bond return yields

$$\begin{aligned} \begin{aligned} \text {Var}(r_{it})&=\text {Var}(r_{{\mathrm{p}}t})+\text {Var}(\epsilon _{it})+2\text {Cov}(r_{{\mathrm{p}}t},\epsilon _{it})\\&=\text {Var}(r_{{\mathrm{p}}t})+\text {Var}(\epsilon _{it})+2(\beta _{ip}-1)\text {Var}(r_{{\mathrm{p}}t}), \end{aligned} \end{aligned}$$

where taking account of the covariance term once again introduces the bond beta into the variance decomposition. Although the variance of an individual bond return contains covariance terms, the weighted average of variances across bonds in the portfolio is free of the individual covariances. This result follows from the fact that the weighted sums of the betas equal unity (note that we outline the methodology using general weights)

$$\begin{aligned} \sum \limits _iw_{it}\beta _{ip}=1. \end{aligned}$$

Consequently, we have

$$\begin{aligned} \sum \limits _iw_{it}\text {Var}(r_{it})=\text {Var}(r_{{\mathrm{p}}t})+\sum \limits _iw_{it}\text {Var}(\epsilon _{it})=\underbrace{\sigma _{{\mathrm{p}}t}^{2}}_\text {portfolio}+\underbrace{\sigma _{\epsilon t}^{2}}_\text {idiosyncratic}, \end{aligned}$$

(8)

where $\sigma _{{\mathrm{p}}t}^2\equiv \text {Var}(r_{{\mathrm{p}}t})$ and $\sigma _{\epsilon t}^2\equiv \sum _iw_{it}\text {Var}(\epsilon _{it})$. The weighting and summing have removed the covariance terms. We use the residual $\epsilon _{it}$ in Eq. (7) to construct a measure of average bond-level risk that does not require any estimation of betas. We interpret the weighted average individual bond variance, the left-hand side of Eq. (8), as the expected volatility of a randomly drawn bond (with the probability of drawing bond i equal to its weight $w_{it}$). This measure of expected volatility reflects two components, a portfolio factor and an idiosyncratic factor.

The variances of excess returns on the two portfolios (covered bond portfolio and senior bond portfolio) include the impact of a market-wide factor and an instrument-specific factor. To identify the market-wide component, we cannot eliminate the betas as in the decomposition described above because covered bonds and senior bonds do not form the total market, that is, their betas do not necessarily add up to 1. We adapt Houston and Stiroh (2006) and decompose portfolio volatility ex post using the one-factor market model for each of the two bond portfolios

$$\begin{aligned} r_{{\mathrm{p}}t}=\beta _{pm}r_{mt}+\eta _{{\mathrm{p}}t}, \end{aligned}$$

where $r_{mt}$ is the daily excess return for the overall market. By construction, we have

$$\begin{aligned} \text {Var}(r_{{\mathrm{p}}t})=\beta _{pm}^{2}\text {Var}(r_{mt})+\text {Var}(\eta _{{\mathrm{p}}t}). \end{aligned}$$

(9)

We substitute Eq. (9) into Eq. (8) and define $\sigma _{\eta t}^2\equiv \text {Var}(\eta _{{\mathrm{p}}t})$. The final decomposition of bond return variance is now

$$\begin{aligned} \begin{aligned} \sum \limits _iw_{it}\text {Var}(r_{it})&=\text {Var}(r_{{\mathrm{p}}t})+\sum \limits _iw_{it}\text {Var}(\epsilon _{it})\\&=\beta _{pm}^{2}\text {Var}(r_{mt})+\text {Var}(\eta _{{\mathrm{p}}t})+\sum \limits _iw_{it}\text {Var}(\epsilon _{it})\\&=\underbrace{\beta _{pm}^{2}\text {Var}(r_{mt})}_\text {market-wide}+\underbrace{\sigma _{\eta t}^{2}}_\text {instrument}+\underbrace{\sigma _{\epsilon t}^{2}}_\text {idiosyncratic}. \end{aligned} \end{aligned}$$

(10)

The left-hand side of Eq. (10) shows total risk (average individual bond risk). The right-hand side shows the three components of risk; (market-wide) systematic risk, instrument-specific risk, and idiosyncratic risk.

Overview of banks in the sample

Table 8 Overview of issuers in the sample