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  1. 013A foreign equity index has 16% volatility in its local currency, and that currency has 8% volatility against the rupee. What is the volatility of an unhedged position if the two correlate at plus 0.3, and if they correlate at minus 0.3?Currency and global returnsHardGlobal investingMulti-asset

    Try it first

    With a correlation of minus 0.3, is the unhedged position riskier than the hedged one?

    Show the worked solution

    About 19.9% at plus 0.3 and 15.6% at minus 0.3. The unhedged return is roughly the local return plus the currency return, so the variances add with a correlation term: 16 squared plus 8 squared, plus or minus 2 x 0.3 x 16 x 8. That is 396.8 or 243.2, with square roots of 19.9% and 15.6%. With negative correlation the unhedged position is less volatile than the hedged one at 16%.

    How can adding a second risk reduce the total?

    Think of a shop that sells umbrellas and sunglasses. Each product's sales swing a lot with the weather, but in opposite directions, so the till is steadier than either product alone. When two sources of return tend to move against each other, combining them lowers risk even though each one is volatile on its own. For an Indian investor holding foreign shares, the currency is the second source. If the foreign currency tends to strengthen against the rupee when that market falls, it cushions the loss, and the unhedged position is steadier: 15.6% instead of 16%.

    Currency adds to or offsets local risk, depending on the sign of the correlationCorrelation +0.3local market 16currency 8unhedged: 19.916 x 16 + 8 x 8 + 2 x 0.3 x 16 x 8 = 396.8square root: 19.9% against 16.0% hedgedCorrelation -0.3local market 16currency 8unhedged: 15.616 x 16 + 8 x 8 - 2 x 0.3 x 16 x 8 = 243.2square root: 15.6% against 16.0% hedged
    Drawn as vectors, local risk of 16 and currency risk of 8 combine to 19.9 when they correlate at plus 0.3, but to only 15.6 at minus 0.3, which is below the 16 of a fully hedged position.
    The relationship
    σunhedged=σL2+σX2+2ρ σLσX=256+64±76.8\sigma_{unhedged}=\sqrt{\sigma_L^2+\sigma_X^2+2\rho\,\sigma_L\sigma_X}=\sqrt{256+64\pm 76.8}
    \sigma_Lthe index's volatility in local currency, 16%
    \sigma_Xthe currency's volatility against the rupee, 8%
    \rhothe correlation between the two, plus or minus 0.3
    What it says in wordsThe unhedged variance is the two variances plus twice the covariance, and the covariance changes sign with the correlation.

    So should a global portfolio hedge its currency?

    The puzzle gives the risk side of the answer, not the whole decision. A hedge removes the currency's volatility, but it also removes the currency's correlation with the market, and when that correlation is negative the hedge adds risk rather than cutting it. The rest of the decision is cost, which depends on the interest rate gap between the two currencies, and the investor's own liabilities in rupees. Say the limitation: correlations are measured on history and tend to shift in a crisis, so the minus 0.3 that makes the unhedged position look safer is the number least likely to hold when it matters. The formula also ignores the small cross term from multiplying the two returns.

    Where candidates lose it

    Most candidates say hedging always lowers risk, because the hedge removes a volatile exposure. That is true only when the currency correlates positively with the local market. With a negative correlation the currency is itself a hedge.

    The arithmetic trap is adding 16 and 8 to get 24, which assumes perfect correlation. Add variances, include the covariance term with its sign, then take the root.

    What the interviewer asks next

    • At what correlation is the unhedged volatility exactly 16%?
    • What does it cost to hedge, and what drives that cost?
    • Why do some investors hedge their foreign bonds but not their foreign equities?