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Inferring a Risk-Free Return from Perfectly Negatively Correlated Assets

Article Quant Q&A · Author: Lorenzo De Bernardi
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Summary

The document poses a portfolio theory question: given two stocks’ expected returns and standard deviations, and a correlation of negative one, how can the risk-free rate be inferred? The proposed route is to recognize that perfect negative correlation allows a combination of the two assets whose return variability cancels. The expected return of that zero-variance portfolio then supplies the implied risk-free return under the stated assumptions.

The response offers this as a hint rather than working through the portfolio weights or calculating the resulting return. It therefore illustrates the relationship between covariance, portfolio construction, and risk-free returns, but leaves the numerical solution to the reader. The inference depends on the perfect negative correlation being exact and on the provided expected returns and volatilities applying over the same horizon; it does not address borrowing constraints, transaction costs, or whether such a portfolio is achievable in practice.

Key ideas

  • Perfect negative correlation can allow a portfolio with zero return variance.
  • The portfolio weights must balance the two assets’ standard deviations to cancel risk.
  • The expected return of the resulting risk-free portfolio implies the risk-free rate under the assumptions.
  • The response gives a hint and does not calculate the weights or the implied rate.

Tags

Portfolio constructionRisk managementStatistics
Full text
# Calculating the Risk Free Rate


# Calculating the Risk Free Rate












I have an assignment and I have to calculate the risk-free rate with the following data:

Stock A: E(R) = 10% ; Standard Deviation = 5%.

Stock B: E(R) = 20% ; Standard Deviation = 10%.

I also know that the correlation coefficient of the two assets is -1.

I have tried to use the Sharpe Ratio or the CAPM formula for risk-free rate, but without any success.

Thanks for the help!

## Answer by AlRacoon (score 4)

https://quant.stackexchange.com/a/37909

Hint: If these 2 stocks have perfect negative correlation (correlation: -1), then you can construct a risk free portfolio. What would the return on that risk free portfolio be?

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.