Suppose that in Problem 23.12 the correlation between the...

Suppose that in Problem 23.12 the correlation between the S\&P 500 Index (measured in dollars) and the FTSE 100 Index (measured in sterling) is 0.7 , the correlation between the S\&P 500 Index (measured in dollars) and the dollar/sterling exchange rate is 0.3 , and the daily volatility of the S\&P 500 index is $1.6 \%$. What is the correlation between the S\&P 500 index (measured in dollars) and the FTSE 100 index when it is translated to dollars? (Hint: For three variables $X, Y$, and $Z$, the covariance between $X+Y$ and $Z$ equals the covariance between $X$ and $Z$ plus the covariance between $Y$ and $Z$.)

Key Concepts

Linear Transformation of Random Variables

A linear transformation, such as converting an asset's return from one currency to another, involves a combination of the asset return and the exchange rate change. The covariance of a sum of random variables equals the sum of their covariances with a third variable. This property is used to understand how the correlation structure changes when an index is re-expressed in another currency.

Asset Return Volatility

Volatility, often measured as the standard deviation of asset returns, represents the degree of variation or dispersion of returns around the mean. In financial markets, understanding volatility is crucial as it quantifies the risk associated with an asset’s return, and is a key component in asset pricing, portfolio construction, and risk management.

Covariance and Correlation

Covariance and correlation are statistical measures that describe the degree to which two variables move together. Covariance indicates the direction of the linear relationship between variables, while correlation standardizes this measure, providing a dimensionless value between -1 and 1 that indicates both strength and direction of the relationship. In asset pricing and finance, these measures are essential for understanding the co-movement of asset returns.

Exchange Rate Impact on Asset Returns

When dealing with international assets, exchange rates introduce additional risk that affects the asset's return when translated into another currency. This conversion embeds both the asset’s local return and the fluctuation in the exchange rate, altering the asset’s overall risk profile and correlation with other assets denominated in a different currency.

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