Some time ago a company entered into a forward contract to...

Some time ago a company entered into a forward contract to buy $£ 1$ million for $$\$ 1.5$$ million. The contract now has 6 months to maturity. The daily volatility of a 6-month zero-coupon sterling bond (when its price is translated to dollars) is $0.06 \%$ and the daily volatility of a 6-month zero-coupon dollar bond is $0.05 \%$. The correlation between returns from the two bonds is 0.8 . The current exchange rate is 1.53. Calculate the standard deviation of the change in the dollar value of the forward contract in 1 day. What is the 10-day $99 \%$ VaR? Assume that the 6-month interest rate in both sterling and dollars is $5 \%$ per annum with continuous compounding.

Key Concepts

Volatility Scaling

Volatility scaling refers to the method of converting risk measures from one time horizon to another, typically using the square-root-of-time rule. This rule is based on the assumption that price changes are independent and identically distributed over time, allowing practitioners to scale daily standard deviations to multi-day horizons. Proper scaling is important to ensure that risk metrics such as VaR are accurately represented over the target time frame.

Value at Risk (VaR)

Value at Risk (VaR) is a statistical measure used to estimate the potential loss in value of a portfolio over a defined period for a given confidence interval. VaR aggregates the effects of various risk components, including asset volatility and correlations among assets, to provide a single number that represents the maximum expected loss. The technique often uses the square-root-of-time rule to scale daily volatility to longer time periods, such as 10 days.

Correlation of Returns

The correlation of returns quantifies how returns on different assets move in relation to each other. In the context of assessing risk on contracts that derive value from multiple underlying assets, such as bonds in different currencies, incorporating correlation helps in accurately measuring the combined volatility. A high positive correlation will tend to amplify the overall risk, whereas a low or negative correlation could reduce it.

Bond Price Volatility

Bond price volatility measures the extent to which the price of a bond fluctuates due to changes in market interest rates and other risk factors. In risk analysis, particularly with zero-coupon bonds, this volatility is critical as it affects the valuation of bonds and any derivative instruments based on them. Volatility is generally expressed as an annualized percentage change over a specified period (e.g., daily or 6-month periods).

Forward Contracts

A forward contract is a binding agreement to buy or sell an asset at a predetermined price on a future date. In financial risk management, forward contracts are used both for hedging against future price movements and for speculative purposes. Understanding how these contracts are priced, which involves current spot prices, interest rates, and time to maturity, is key to evaluating their risk exposure.

Currency Exchange Risk

Currency exchange risk refers to the potential for loss due to fluctuations in the exchange rate between two currencies. When a forward contract involves transactions in different currencies, movements in the exchange rate can significantly impact the contract’s value when converting between currencies. Understanding this risk is important when assessing both hedging strategies and the overall risk profile of cross-currency financial instruments.

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