Inferring S&P 500 Volatility from a Minimum-Variance Portfolio

Summary

The document asks whether the annual standard deviation of the S&P 500 can be recovered from the standard deviation of the global minimum-variance portfolio and their correlation. One response correctly notes that correlation alone normally requires covariance information, while another supplies an additional portfolio-theory property: the global minimum-variance portfolio has equal covariance with every efficient portfolio.

Assuming the S&P 500 is efficient, its covariance with the minimum-variance portfolio is taken to equal the latter portfolio’s variance. Substituting that covariance and the stated correlation into the correlation formula yields an implied S&P 500 annual volatility of 22%. This result depends on the efficiency assumption and the stated property applying to the portfolios in question. Without that assumption or some equivalent covariance information, the given volatility and correlation would not determine the S&P 500 volatility. The exchange is a compact worked derivation, not an empirical estimate or a general method for arbitrary indexes.

Key ideas

• Correlation and one portfolio's volatility are generally insufficient to infer another portfolio's volatility.
• The global minimum-variance portfolio has equal covariance with efficient portfolios under the stated result.
• Assuming the S&P 500 is efficient lets the covariance be set equal to the minimum-variance portfolio's variance.
• Under that assumption, the example derives an annual S&P 500 volatility of 22%.

Tags

# Correlation between two indexes


# Correlation between two indexes












The Global Minimum Variance has an annual return standard deviation of 9.9%. Its correlation with the Standard & Poor's 500 Index is 0.45. What is the annual return standard deviation of the S&P 500?

I know that correlation is given by

Corr(A,B) = Cov(A, B)/ [StDev(A)*StDev(B)]

But in this problem, the covariance isn't given. Does that mean there's not enough information to answer the question?

## Answer by AK88 (score 1)

https://quant.stackexchange.com/a/30774

Not enough information. Where did you get this problem from?

## Answer by Alex C (score 0)

https://quant.stackexchange.com/a/30779

Here is a clue: The GMVP has a fundamental property: it has the same covariance with every other efficient portfolio. You can assume that the S&P500 is an efficient portfolio. For details see here Covariance of a GMV portfolio with any asset This common covariance is usually denoted $\frac{1}{C}$ where $C = 1^T {\Sigma}^{-1} 1 $

We proceeds as follows:

What is the covariance of the GMVP with itself: That's just the standard deviation squared, i.e. ${0.099}^2$

What is the covariance of the GMVP with the S&P500: By the above theorem it is the same number i.e. ${0.099}^2$

Now we apply the definition of correlation which you gave and we have $0.45 = \frac{{0.099}^2}{0.099 \sigma_{SP}}$. From this equation $\sigma_{SP}$ is found to be 0.22 or 22%

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.