Econometrics论文模板 – Hedging Methods

The minimum variance hedge ratio was determined for a long position of Canadian Dollars which was to be hedged with a short position in US dollars in order to reduce the risk exposure of foreign currency in the corporation. Two volatility models were used to calculate the minimum variance hedge ratio namely; the Rolling Window model and the Exponentially Weighted Moving Average model. This minimum variance hedge ratio was also used to determine the optimal parameter estimates of the volatility models. These optimal values increase the accuracy of the estimates of variance of the returns using the specific models.

The first model to be applied to determine the minimum variance hedge ratio was the rolling window model. This model entails performing regression with different window lengths in order to determine the volatility of the given data. Large data increases the accuracy of the resultant variance of returns. A shorter window length increases the sensitivity of the model to the observed data. In this case, two window lengths were used. They were 50 and 254 was the optimum window length for the mode. To begin with, the returns of the GBP/USD and those of GBP/CAD were calculated by taking today’s exchange rate less yesterday’s exchange rate and then divide the answer obtained by yesterday’s exchange rate. This was the formula used to calculate the returns of each currency;   (Staff, n.d.). The window length used in the rolling window model was 50. Then the variance of the US dollars were calculated using the variance of a sample function in excel. The variance of the Canadian Dollars was also calculated using the variance of a sample function. The variance could also have been calculated using the formula for finding variance in rolling window model which is;                                   (Perera, n.d.). The covariance of the US Dollars and the Canadian Dollars was then calculated. This was done through the use of covariance function.
After obtaining the values of covariance of the US Dollars and the Canadian Dollars, the time-varying minimum variance hedge ratio was obtained. This was done by dividing the covariance of US dollars and the Canadian dollars by the variance of US dollars. This was the formula used;       (Financial Mathematics, n.d.). The minimum variance hedge ratio was used to calculate the returns of the hedged portfolio. This was obtained by subtracting the product of the hedge ratio and the returns of US Dollar from the return of the Canadian Dollar.   (SteinerConsulting, 2011).  

The minimum variance hedge ratio was also calculated when the window length was optimal, with a value of 254. The variances of the Canadian Dollars and the US Dollars were calculated using the variance of sample function in excel. The covariance of the Canadian Dollars and the US Dollars was then calculated using the covariance of sample function. This covariance was then used to calculate the minimum variance hedge ratio applying the formula; . The returns of the hedged portfolio when M=254 were also calculated applying the formula; . The variances of the hedged portfolio and the unhedged portfolio when M=50 and when M=254 were then compared. The reduction percentage of the variances of the hedged and the unhedged portfolio when M=50 was compared to the reduction percentage of the hedged and unhedged portfolio when the window length was optimum M=254. This comparison was to help determine the best strategy to minimize risks in the corporation’s holdings.

This rolling window method is a good way to forecast the volatility of the returns since it takes into account the most recent data that had been projected when estimating the variance, thus the estimate is relevant. However, it is difficult to choose a window length that is optimum as the window length greatly influences the results. If the data falls within the window length, the model gives it full weightings and if the data does not fall within the window length, it is given zero weightings. The rolling window method can therefore produce inaccurate results of the variance of returns and a better method of forecasting or a combination of the rolling window model and another model can be used in order to enhance accuracy of the results (McCracken, 2009).

The other model used to determine the minimum variance hedge ratio was the exponentially weighted moving average model. This model gives the most recent prices more weight than the past prices as the weighting decreases exponentially with each previous price. First, the return of the Canadian Dollars and the US Dollars were calculated. This was done by taking today’s exchange rate less yesterday’s exchange rate and then divide the answer obtained by yesterday’s exchange rate. The formula used to calculate the returns of each currency was;  . Then the variances of both the US Dollars and the Canadian Dollars were calculated. This was done by using the formula for calculating the variance of an exponentially weighted moving average model which is;   (Milton, 2018). The covariance of the Canadian Dollars and the US Dollars was then calculated. This was obtain by using the formula; .

The minimum variance hedge ratio was calculated by dividing the covariance of US dollars and the Canadian dollars by the variance of US dollars. The formula used was; . Using the minimum variance hedge ratio obtained, the returns of the hedged portfolio were calculated by subtracting the product of the hedge ratio and the returns of US Dollar from the return of the Canadian Dollar.   . The minimum variance hedge ratio was also calculated using the exponentially weighted moving average model when the decay factor was 0.94. 0.974684974 was the optimal value of decay factor for this data. The variances of the hedged and unhedged portfolios when the decay factor was 0.94 and when the decay factor was 0.974684974 were compared. The reduction percentage of the variances of the hedged and unhedged Canadian Dollar when the decay factor was 0.94 was compared to the reduction percentage of hedged and unhedged portfolio when the decay factor was optimum in order to determine the best strategy to reduce some risks in the corporation’s holdings.

The exponentially weighted model is a good forecasting model as it gives different weightings to the data, with the most recent data having more weight than the past data. This enhances accuracy as all the prices recorded in the data are taken into consideration and they are used in the calculation of the variance of the returns. Although the variance does not converge to a common variance over time, it is easy to use the model.

References

Financial Mathematics, p. e. p. e. 1., n.d. finance train. [Online]
Available at: https://financetrain.com/minimum-variance-hedge-ratio/
[Accessed 7 November 2018].

McCracken, T. E. C. a. M. W., 2009. Improving Forecast Accuracy by Combining Recursive and Rolling Forecasts. International Economic Review, 50(2).

Milton, A., 2018. the balance. [Online]
Available at: https://www.thebalance.com/simple-exponential-and-weighted-moving-averages-1031196
[Accessed 7 November 2018].

Perera, S., n.d. medium. [Online]
Available at: https://medium.com/making-sense-of-data/time-series-next-value-prediction-using-regression-over-a-rolling-window-228f0acae363
[Accessed 7 November 2018].

Staff, M. F., n.d. The Motley Fool. [Online]
Available at: https://www.google.com/amp/s/www.fool.com/amp/knowledge-center/how-to-calcuate-foreign-exchange-gains-or-losses.aspx

SteinerConsulting, R. N. A., 2011. Slideshare. [Online]
Available at: https://www.slideshare.net/mobile/andsteinconsult/currency-hedged-return-calculations
[Accessed 7 November 2018

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