Debt Beta, Correlation, and Bond Volatility
Summary
The document clarifies the relationship between a bond's beta, its volatility, and its correlation with the market portfolio. The question derives debt beta from expected debt and market returns, then uses the regression-beta identity to infer debt volatility. It raises a sign concern based on an assumed negative correlation between debt and the market.
The answer explains that beta equals correlation multiplied by the ratio of the asset's volatility to the market's volatility. Since volatility is nonnegative, beta and correlation must have the same sign. Thus, a positive debt beta cannot coexist with a negative market-debt correlation under this definition; the issue is the assumed sign, not a negative volatility estimate. The exchange does not estimate a bond's correlation or volatility and gives no empirical guidance on debt risk, so the identity resolves the algebraic confusion without supplying market inputs.
Key ideas
- Regression beta is covariance with the market divided by market variance.
- Beta equals correlation multiplied by the ratio of the asset's volatility to market volatility.
- Because volatilities are nonnegative, beta and correlation have the same sign.
- A positive debt beta is incompatible with a negative market-debt correlation under this identity.
- The formula alone does not provide the correlation or volatility inputs.
Tags
Full text
# Risk of bond calculation
# Risk of bond calculation
I am studying a course and I am a bit confused on how to find the a bonds $\sigma$. My course mentions the following: Once calculated the expected returns on the bond $\mathrm{E}(r_d)$, we can calculate the debt beta:
$\mathrm{E}(r_d) = r_f + \beta_d(\mathrm{E}(r_M) − r_f))$
$\beta_d = \mathrm{E}(r_d − r_f)/\mathrm{E}(r_M − r_f)$
where $r_f$ and $r_M$ are the usual risk-free rate and market portfolio and where:
- $\mathrm{E}(r_d)$: expected return on debt
- $r_f$: return on riskless debt
- $(\mathrm{E}(r_M) − r_f))$: return on equity market portfolio If we know the Beta of debt, we can estimate the volatility of debt:
$\beta_d = \rho(r_M, r_d) \times (\sigma_d/\sigma_M)$
$\sigma_d = (\beta_d \times \sigma_M)/\rho(r_M, r_d)$
There is still an assumption that needs to be made between the correlation between the market portfolio and debt. But I believe this is negative. If I use this formula and the $\beta_d$ is positive the $\sigma_d$ will be negative and this doesn't make sense to me. The only way that $\sigma_d$ is positive is if the $\beta_d$ is negative. Am I wrong? Can someone please help me understand? Thanks
## Answer by mbison (score 2)
https://quant.stackexchange.com/a/75265
the answer to your question really just boils down to the definition of correlation and regression Beta. You can not mathematically have a positive Beta and a negative correlation. This is mathematically impossible.
$\beta = \frac{cov(y,x)}{var(x)} = \frac{\rho \sigma_y \sigma_x}{\sigma_x^2}$.
i.e. $\beta = \rho \frac{\sigma_y}{\sigma_x} $.
Since volatilities are always positive the $\beta$ has the same sign as $\rho$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.