1. Currency Exposure. VaR for currency positions. Hedged and unhedged positions
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1 RISK MANAGEMENT [635-0]. Currency Exposure. ar for currency positions. Hedged and unhedged positions Currency Exposure Currency exposure represents the relationship between stated financial goals and exchange rate movements, instruments and currency selection and portfolio structure. Currency Risk Measures Statistical Measures A number of different statistics can be applied to measure currency risk. These include variances, standard deviations, percentiles, coefficients of variation. Currency log return distribution is very close to symmetric normal distribution. Thus variance or standard deviation are good currency risk measures. ariance for the currency portfolio is computed using the correlation between returns on exchange rate (appreciation and depreciation rates and returns on foreign assets. The variance of currency returns is also used to compute risk diversification effects. ar, EaR, CFaR variance and covariance method In this approach log currency returns (continuously compounded returns are used in temporal and cross-section aggregation. It is assumed that log returns are normally distributed. In a portfolio all returns are multivariate normally distributed. The sum of normal random variables is itself normally distributed. The entire distribution of returns can be characterized by two parameters: mean and variance. These parameters are usually forecasted using historical data. Such assumptions are not suitable for portfolios that contain options. Financial return distributions often have fat tails. Extreme returns movements occur more frequently than implied by normal distribution and the peak of the return distribution is higher and narrower (leptokurtotic distribution.
2 RISK MANAGEMENT [635-0] f(r ( r < -,65σ + µ 5% P = 0 0 r = -,65σ + µ % 5% Return Figure. Return Distribution We want to find the 5 th percentile of return. There is a 5 percent probability that an observed return at time t is less than.65 times its standard deviation plus its mean. Assuming that mean=0 (one day horizon we have ( { r,65σ } 5% P t = t t- where -,65 5 th percentile of the standard normal distribution σ t t- standard deviation of continuously compounded returns for time t prepared at time t-. f( t = e,65σ t t t 95% 5% t t- alue ar Figure. alue at Risk ar of a single asset can be calculated as follows ar,65σ t t ( = [ ] t where e Euler s constant (e=,783, t- portfolio value at time t-. e or, using the common approximation (3 ar,65σ t t t
3 RISK MANAGEMENT [635-0] Problem. ar one asset Suppose you hold a $5 million position and current spot exchange rate is 4 PLN/USD. Expected return over a -day horizon period is equal to 0. Expected volatility is equal to %. (a What is the current market value of this position. What is the forecasted value? (b Make a forecast of the -day return on the portfolio, such that there is 5% chance that the realized return will be less than forecasted return. (c Calulate the portfolio's "worst case" value based on the forecasted rate (b. (d Calculate AR. (e Calculate ar using simple approximation. Solution (a The current mark-to-market value is 00 million zloties. r The forecasted future value is = (b We assume that the ralized return will be less than the forecasted return with a 5% probability. P r < r = 5% ( ( < -,65σ + µ 5% P r We assume that the expected return over a day horizon is equal to 0. µ = 0 0 = 0 0 The forecasted return is r = -,65σ = -,645% 0 (c The portfolio "worst case" value is r = = 98,369 million zloties 0e (d The alue at Risk is ar = - = r ( e 0e =,63 million zloties (e ar is approximately equal to α/ t t σ 0 ar = t σ 0 5,00%,645 0, ,645 million zloties 3
4 RISK MANAGEMENT [635-0] Problem. ar two assets Suppose you hold a long currency position $ 5 million in the 0-year US bond and the current spot exchange rate is 4,00 PLN/USD. Currency return is a random variable which follows conditional normal distribution with the following parameters: mean = 0; standard deviation,0%. (a What is the current value of this position in zloties? (b What is your ar over a -day horizon period, given there is a 5%,,5%, or 0,5% chance of understating the realized loss. (c Calculate AR, when you estimate the expected volatility of a bond equal to be 0,5%, and the correlation between the return on the PLN/USD exchange rate and the 0-year US bond to be -0,5. Solution (a The currency exposure is 00 million PLN. (b ar is appriximately equal to t α σ tt α/ t- ar 5,00%,645,0% 00,645 million PLN,50%,960,0% 00,960 million PLN 0,50%,576,0% 00,576 million PLN (c We have the following information. standard deviation of the currency return =,0%. 0-year US bond standa deviation = 0,5% 3. correlation coefficient = -0,5 The variance of returns is equal to = 0,008% The standard deviation is = 0,866% We compute ar using formula ar t ασr H t t α ar t σ σ = σ + σ + ρ α/ σ rh t- ar 5,00%,645 0,866% 00,44 million PLN,50%,960 0,866% 00,697 million PLN 0,50%,576 0,866% 00,3 million PLN α t t rh t rz rd rzrd σ rz σ σ rd σ rz ρ rz r d rd σ rh 4
5 RISK MANAGEMENT [635-0] Currency Gap Statistical measures including ar are not sufficient currency risk measures. It is also very important to calculate currency gap. All assets and liabilities in a given currency and also all positions in derivatives (currency forward, futures, options and swaps must be included in calculating currency gaps. Long and short position An investor has a long position in a currency when he holds more assets than liabilities in this currency. A short position exists when foreign liabilities exceed foreign assets. (a Long Position (b Short Position Foreign assets Foreign Liabilities Foreign Assets Foreign Liabilities USD 9 mln USD 60 USD USD 6 mln USD 9 mln Figure 3. Long and Short Currency Position In Figure 3 a, net long position is equal to $3 million. An investor faces the risk that the foreign currency will fall in value against zloty (dollar depreciates, zloty appreciates. The value of such position decreases in zloties. When dollar appreciates (zloty depreciates net long position in a currency is worth more in zloties. In Figure 3 b, net short position is equal to -$30 million. An investor faces the risk that the foreign currency will rise in value against zloty (dollar appreciates, zloty depreciates. The value of such position decreases in zloties. When dollar depreciates (zloty appreciates net short position in a currency is worth more in zloties. Table. Profits and Losses on Foreign Currency Positions Foreign currency Long Position Short Position Appreciation Profit Loss Depreciation Loss Profit 5
6 RISK MANAGEMENT [635-0] Currency gap in a given currency Currency gap in a given currency j measured in domestic currency (zloties is L = AN + ZN (4 j [ j j] t-, j AN j net assets (assets - liabilities in currency j, ZN j net FX bought (FX bought FX sold, S t-,j spot exchange rate. Total currency gap S Total currency gap is calculated as follows: n (5 Lw = [ AN j + ZN j] j= S t,j Gains or losses on foreign currency positions depend on currency gaps ad foreign exchange rate movements (currency returns. Change in value of position in currency j is (6 W L r j = j dj where: r dj currency return (rate of appreciation, rate of depreciation, Total change in value for the overall currency portfolio depends on all currency gaps and all currency returns. n (7 W = Wj = L jr j= n j= dj Problem 3. Currency Gap The foreign assets are 0 million USD, foreign liabilites million USD. Net FX bought 3 million USD. The foreign exchange rate is 4,00 PLN/USD. (a Calculate the net currency position. (b What is the FX loss/gain as a result of the exchange rate shocks +-0%. Solution (a Net position (b USD PLN FX Return W W 3,6-0% ,0 0% ,4 0%
7 RISK MANAGEMENT [635-0] Problem 4. Spot Position. Forward position Today's spot exchange rate is 4,00 zł/usd. The treasurer of a company expects to receive a payment $00000 from a foreign customer in the next 60 days. Two-month forward contracts are quoted at a rate 4,05 PLN/USD. (a Describe the current currency position. (b Indicate whether a company should use a long or short forward contract to hedge against currency risk. (c Discuss the unhedged position, forward position and hedged position at expiration (i if the spot rate is 4,30 PLN/USD. (ii if the spot rate is 3,90 PLN/USD. Solution (a A company takes a long currency position. (b The company has to go short a forward contract, or sell $00000 at a price of 4,05 PLN/USD. A two-month hedge would be constructed by selling dollars with two months to expiration. Selling currency forward eliminates the currency risk. (c Expiration date (I The spot exchange rate is 4,30 PLN/USD Unhedged position The company receives the payment from a foreign customer x 4,30 = PLN Positive exchange rate adjustments are equal to PLN (extra profits before tax. Forward contract The company is obliged to sell $00000 to the dealer and is paid 4,05 PLN per USD for a total payment x 4,05 = PLN If non-delivery forward contract was agreed, the dealer receives the difference between the spot and the forward rate exchange rate. The transaction is settled in cash x 0,5 = 5000 PLN The net effect for the company from the forward transaction (loss with a minus sign is: K = (forward rate - spot rate x amount As the forward rate is lower than the spot rate, the company suffers a loss K = (4,05 PLN/USD - 4,30 PLN/USD x USD = PLN. Hedged position A typical hedging situation is when long (or short curence position is supplemented with a short (or long forward position. The company locks in the exchange rate at which it will convert the dollars. Regardless of the exchange rate at expiration the company has a gain equal to 5000 PLN. 7
8 RISK MANAGEMENT [635-0] (ii The spot exchange rate is 3,90 PLN/USD Unhedged position The company receives the payment from a foreign customer x 3,90 = zł Negative exchange rate adjustments are equal to (losses before tax. Forward contract The company is obliged to sell $00000 to the dealer and is paid 4,05 PLN per USD for a total payment x 4,05 = zł If non-delivery forward contract was agreed, the dealer pays the difference between the spot and the forward rate exchange rate. The transaction is settled in cash x 0,5 = 5000 zł The net effect for the company from the forward transaction (loss with a minus sign is: K = (forward rate - spot rate x amount As the forward rate is larger than the spot rate, the company has a gain K = (4,05 PLN/USD - 3,90 PLN/USD x USD = 5000 PLN. Hedged position The company locks in the exchange rate at which it will convert the dollars. Regardless of the exchange rate at expiration the company has a gain equal to 5000 PLN. 8
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