PROBLEMS 1. Using the American term quotes from Exhibit 5.4, calculate a cross-rate matrix for the euro, Swiss franc, Japanese yen, and the British pound so that the resulting triangular matrix is similar to the portion above the diagonal in Exhibit 5.6. Solution: The cross-rate formula we want to use is: S(j/k) = S($/k)/S($/j). The triangular matrix will contain 4 x (4 + 1)/2 = 10 elements. Euro Japan (100) Switzerland U.K 2. Using the American term quotes from Exhibit 5.4, calculate the one-, three-, and six-month forward cross-exchange rates between the Canadian dollar and the Swiss franc. State the forward cross-rates in “Canadian” terms. Solution: The formulas we want to use are: FN(CD/SF) = FN($/SF)/FN($/CD) or FN(CD/SF) = FN(CD/$)/FN(SF/$). We will use the top formula that uses American term forward exchange rates. F1(CD/SF) = .8671/.9628 = .9006 F3(CD/SF) = .8686/.9624 = .9025 F6(CD/SF) = .8715/.9614 = .9065 3. A foreign exchange trader with a U.S. bank took a short position of £5,000,000 when the $/£ exchange rate was 1.55. Subsequently, the exchange rate has changed to 1.61. Is this movement in the exchange rate good from the point of view of the position taken by the trader? By how much has the bank’s liability changed because of the change in the exchange rate? ¥ 112.75 SF 1.4128 1.2531 £ .8351 .7406 .5911 $ 1.2238 1.0854 .8662 1.4655 CFA Guideline Answer: The increase in the $/£ exchange rate implies that the pound has appreciated with respect to the dollar. This is unfavorable to the trader since the trader has a short position in pounds. 4. Restate the following one-, three-, and six-month outright forward European term bid-ask quotes in forward points. Spot One-Month Six-Month Solution: One-Month Six-Month 5. Using the spot and outright forward quotes in problem 4, determine the corresponding bid-ask spreads in points. Solution: Spot One-Month Six-Month 6. Using Exhibit 5.4, calculate the one-, three-, and six-month forward premium or discount for the Canadian dollar versus the U.S. dollar using American term quotations. For simplicity, assume each month has 30 days. What is the interpretation of your results? Solution: The formula we want to use is: fN,CD = [(FN($/CD) - S($/CD/$)/S($/CD)] x 360/N 5 10 15 20 01-06 17-27 57-72 Three-Month 1.3431-1.3436 1.3432-1.3442 1.3448-1.3463 1.3488-1.3508 Bank’s liability in dollars initially was 5,000,000 x 1.55 = $7,750,000 Bank’s liability in dollars now is 5,000,000 x 1.61 = $8,050,000 Three-Month Three-Month f1,CD = [(.9628 - .9629)/.9629] x 360/30 = -.0012 f3,CD = [(.9624 - .9629)/.9629] x 360/90 = -.0021 f6,CD = [(.9614 - .9629)/.9629] x 360/180 = -.0031 The pattern of forward premiums indicates that the Canadian dollar is trading at a discount versus the U.S. dollar. That is, it becomes less expensive to buy a Canadian dollar forward for U.S. dollars (in absolute and percentage terms) the further into the future one contracts. 7. Using Exhibit 5.4, calculate the one-, three-, and six-month forward premium or discount for the U.S. dollar versus the British pound using European term quotations. For simplicity, assume each month has 30 days. What is the interpretation of your results? Solution: The formula we want to use is: fN,$ = [(FN (£/$) - S(£/$))/S(£/$)] x 360/N f1,$ = [(.6824 - .6824)/.6824] x 360/30 = .0000 f3,$ = [(.6822 - .6824)/.6824] x 360/90 = -.0012 f6,$ = [(.6821 - .6824)/.6824] x 360/180 = -.0009 The pattern of forward premiums indicates that the dollar is trading at a discount versus the British pound for maturities longer than one month. The three-month discount is larger than the six-month discount in percentage but not absolute terms. 8. A bank is quoting the following exchange rates against the dollar for the Swiss franc and the Australian dollar: An Australian firm asks the bank for an A$/SFr quote. What cross-rate would the bank quote? CFA Guideline Answer: The SFr/A$ quotation is obtained as follows. In obtaining this quotation, we keep in mind that SFr/A$ = SFr/$/A$/$, and that the price (bid or ask) for each transaction is the one that is more SFr/$ = 1.5960--70 A$/$ = 1.7225--35 advantageous to the bank. The SFr/A$ bid price is the number of SFr the bank is willing to pay to buy one A$. This transaction (buy A$—sell SFr) is equivalent to selling SFr to buy dollars (at the bid rate of 1.5960 and the selling those dollars to buy A$ (at an ask rate of 1.7235). Mathematically, the transaction is as follows: The SFr/A$ ask price is the number of SFr the bank is asking for one A$. This transaction (sell A$—buy SFr) is equivalent to buying SFr with dollars (at the ask rate of 1.5970 and then simultaneously purchasing these dollars against A$ (at a bid rate of 1.7225). This may be expressed as follows: 9. Given the following information, what are the NZD/SGD currency against currency bid-ask quotations? American Terms Bid Ask .7265 .7272 .6135 .6140 European Terms Bid Ask 1.3751 1.3765 1.6287 1.6300 Bank Quotations New Zealand dollar Singapore dollar Solution: Equation 5.12 from the text implies Sb(NZD/SGD) = Sb($/SGD) x Sb(NZD/$) = .6135 x 1.3751 = .8436. The reciprocal, 1/Sb(NZD/SGD) = Sa(SGD/NZD) = 1.1854. Analogously, it is implied that Sa(NZD/SGD) = Sa($/SGD) x Sa(NZD/$) = .6140 x 1.3765 = .8452. The reciprocal, 1/Sa(NZD/SGD) = Sb(SGD/NZD) = 1.1832. Thus, the NZD/SGD bid-ask spread is NZD0.8436-NZD0.8452 and the SGD/NZD spread is SGD1.1832-SGD1.1854. 10. Doug Bernard specializes in cross-rate arbitrage. He notices the following quotes: The resulting quotation by the bank is SFr/A$ = 0.9260—0.9271 ask SFr/A$ = (ask SFr/$)/(bid A$/$) = 1.5970/1.7225 = 0.9271 bid SFr/A$ = (bid SFr/$)/(ask A$/$) = 1.5960/1.7235 = 0.9260 Swiss franc/dollar = SFr1.5971?$ Australian dollar/U.S. dollar = A$1.8215/$ Australian dollar/Swiss franc = A$1.1440/SFr Ignoring transaction costs, does Doug Bernard have an arbitrage opportunity based on these quotes? If there is an arbitrage opportunity, what steps would he take to make an arbitrage profit, and how would he profit if he has $1,000,000 available for this purpose. CFA Guideline Answer: A. The implicit cross-rate between Australian dollars and Swiss franc is A$/SFr = A$/$ x $/SFr = (A$/$)/(SFr/$) = 1.8215/1.5971 = 1.1405. However, the quoted cross-rate is higher at A$1.1.1440/SFr. So, triangular arbitrage is possible. B. In the quoted cross-rate of A$1.1440/SFr, one Swiss franc is worth A$1.1440, whereas the cross-rate based on the direct rates implies that one Swiss franc is worth A$1.1405. Thus, the Swiss franc is overvalued relative to the A$ in the quoted cross-rate, and Doug Bernard’s strategy for triangular arbitrage should be based on selling Swiss francs to buy A$ as per the quoted cross-rate. Accordingly, the steps Doug Bernard would take for an arbitrage profit is as follows: i. ii. iii. 11. Assume you are a trader with Deutsche Bank. From the quote screen on your computer terminal, you notice that Dresdner Bank is quoting €0.7627/$1.00 and Credit Suisse is offering SF1.1806/$1.00. You learn that UBS is making a direct market between the Swiss franc and the euro, with a current €/SF quote of .6395. Show how you can make a triangular arbitrage profit by trading at these prices. (Ignore bid-ask spreads for this problem.) Assume you have $5,000,000 with which to conduct the arbitrage. What happens if you initially sell dollars for Swiss francs? What €/SF price will eliminate triangular arbitrage? Sell dollars to get Swiss francs: Sell $1,000,000 to get $1,000,000 x SFr1.5971/$ = SFr1,597,100. Sell Swiss francs to buy Australian dollars: Sell SFr1,597,100 to buy SFr1,597,100 x A$1.1440/SFr = A$1,827,082.40. Sell Australian dollars for dollars: Sell A$1,827,082.40 for A$1,827,082.40/A$1.8215/$ = $1,003,064.73. Thus, your arbitrage profit is $1,003,064.73 - $1,000,000 = $3,064.73. Solution: To make a triangular arbitrage profit the Deutsche Bank trader would sell $5,000,000 to Dresdner Bank at €0.7627/$1.00. This trade would yield €3,813,500= $5,000,000 x .7627. The Deutsche Bank trader would then sell the euros for Swiss francs to Union Bank of Switzerland at a price of €0.6395/SF1.00, yielding SF5,963,253 = €3,813,500/.6395. The Deutsche Bank trader will resell the Swiss francs to Credit Suisse for $5,051,036 = SF5,963,253/1.1806, yielding a triangular arbitrage profit of $51,036. If the Deutsche Bank trader initially sold $5,000,000 for Swiss francs, instead of euros, the trade would yield SF5,903,000 = $5,000,000 x 1.1806. The Swiss francs would in turn be traded for euros to UBS for €3,774,969= SF5,903,000 x .6395. The euros would be resold to Dresdner Bank for $4,949,481 = €3,774,969/.7627, or a loss of $50,519. Thus, it is necessary to conduct the triangular arbitrage in the correct order. The S(€/SF) cross exchange rate should be .7627/1.1806 = .6460. This is an equilibrium rate at which a triangular arbitrage profit will not exist. (The student can determine this for himself.) A profit results from the triangular arbitrage when dollars are first sold for euros because Swiss francs are purchased for euros at too low a rate in comparison to the equilibrium cross-rate, i.e., Swiss francs are purchased for only €0.6395/SF1.00 instead of the no-arbitrage rate of €0.6460/SF1.00. Similarly, when dollars are first sold for Swiss francs, an arbitrage loss results because Swiss francs are sold for euros at too low a rate, resulting in too few euros. That is, each Swiss franc is sold for €0.6395/SF1.00 instead of the higher no-arbitrage rate of €0.6460/SF1.00. 12. The current spot exchange rate is $1.95/£ and the three-month forward rate is $1.90/£. Based on your analysis of the exchange rate, you are pretty confident that the spot exchange rate will be $1.92/£ in three months. Assume that you would like to buy or sell £1,000,000. a. What actions do you need to take to speculate in the forward market? What is the expected dollar profit from speculation? b. What would be your speculative profit in dollar terms if the spot exchange rate actually turns out to be $1.86/£. Solution: a. If you believe the spot exchange rate will be $1.92/£ in three months, you should buy £1,000,000 forward for $1.90/£. Your expected profit will be: $20,000 = £1,000,000 x ($1.92 -$1.90). b. If the spot exchange rate actually turns out to be $1.86/£ in three months, your loss from the long position will be: -$40,000 = £1,000,000 x ($1.86 -$1.90). 13. Omni Advisors, an international pension fund manager, plans to sell equities denominated in Swiss Francs (CHF) and purchase an equivalent amount of equities denominated in South African rands (ZAR). Omni will realize net proceeds of 3 million CHF at the end of 30 days and wants to eliminate the risk that the ZAR will appreciate relative to the CHF during this 30-day period. The following exhibit shows current exchange rates between the ZAR, CHF, and the U.S. dollar (USD). Currency Exchange Rates Maturity Spot 30-day 90-day a. b. Calculate the following: • The CHF/ZAR cross-currency rate Omni would use in valuing the Swiss equity portfolio. • The current value of Omni’s Swiss equity portfolio in ZAR. • The annualized forward premium or discount at which the ZAR is trading versus the CHF. Describe the currency transaction that Omni should undertake to eliminate currency risk over the 30-day period. 6.2681 6.2538 6.2104 6.2789 6.2641 6.2200 1.5282 1.5226 1.5058 1.5343 1.5285 1.5115 ZAR/USD Bid ZAR/USD Ask CHF/USD Bid CHF/USD Ask CFA Guideline Answer: a. To eliminate the currency risk arising from the possibility that ZAR will appreciate against the CHF over the next 30-day period, Omni should sell 30-day forward CHF against 30-day forward ZAR delivery (sell 30-day forward CHF against USD and buy 30-day forward ZAR against USD). b. The calculations are as follows: • Using the currency cross rates of two forward foreign currencies and three currencies (CHF, ZAR, USD), the exchange would be as follows: MINI CASE: SHREWSBURY HERBAL PRODUCTS, LTD. Spot rate = 1.5343 CHF/6.2681 ZAR = 0.244779120 30 day forward ask rate 1.5285 CHF/6.2538 ZAR = 0.244411398 The premium/discount formula is: [(forward rate – spot rate) / spot rate] x (360 / # day contract) = [(0.244411398 – 0.24477912) / 0.24477912] x (360 / 30) = -1.8027126 % = -1.80% discount ZAR to CHF --30 day forward CHF are sold for USD. Dollars are bought at the forward selling price of CHF1.5285 = $1 (done at ask side because going from currency into --30 day forward ZAR are purchased for USD. Dollars are simultaneously sold to purchase ZAR at the rate of 6.2538 = $1 (done at the bid side because going from dollars into currency) --For every 1.5285 CHF held, 6.2538 ZAR are received; thus the cross currency rate is 1.5285 CHF/6.2538 ZAR = 0.244411398. dollars) • At the time of execution of the forward contracts, the value of the 3 million CHF equity portfolio would be 3,000,000 CHF/0.244411398 = 12,274,386.65 ZAR. • To calculate the annualized premium or discount of the ZAR against the CHF requires comparison of the spot selling exchange rate to the forward selling price of CHF for ZAR. Shrewsbury Herbal Products, located in central England close to the Welsh border, is an old-line producer of herbal teas, seasonings, and medicines. Its products are marketed all over the United Kingdom and in many parts of continental Europe as well. Shrewsbury Herbal generally invoices in British pound sterling when it sells to foreign customers in order to guard against adverse exchange rate changes. Nevertheless, it has just received an order from a large wholesaler in central France for £320,000 of its products, conditional upon delivery being made in three months’ time and the order invoiced in euros. Shrewsbury’s controller, Elton Peters, is concerned with whether the pound will appreciate versus the euro over the next three months, thus eliminating all or most of the profit when the euro receivable is paid. He thinks this is an unlikely possibility, but he decides to contact the firm’s banker for suggestions about hedging the exchange rate exposure. Mr. Peters learns from the banker that the current spot exchange rate is €/£ is €1.4537, thus the invoice amount should be €465,184. Mr. Peters also learns that the three-month forward rates for the pound and the euro versus the U.S. dollar are $1.8990/£1.00 and $1.3154/€1.00, respectively. The banker offers to set up a forward hedge for selling the euro receivable for pound sterling based on the €/£ forward cross-exchange rate implicit in the forward rates against the dollar. What would you do if you were Mr. Peters? Suggested Solution to Shrewsbury Herbal Products, Ltd. (Note to Instructor: This elementary case provides an intuitive look at hedging exchange rate exposure. Students should not have difficulty with it even though hedging will not be formally discussed until Chapter 8. The case is consistent with the discussion that accompanies Exhibit 5.9 of the text. Professor of Finance, Banikanta Mishra, of Xavier Institute of Management – Bhubaneswar, India contributed to this solution.) Suppose Shrewsbury sells at a twenty percent markup. Thus the cost to the firm of the £320,000 order is £256,000. Thus, the pound could appreciate to €465,184/£256,000 = €1.8171/1.00 before all profit was eliminated. This seems rather unlikely. Nevertheless, a ten percent appreciation of the pound (€1.4537 x 1.10) to €1.5991/£1.00 would only yield a profit of £34,904 (= €465,184/1.5991 - £256,000). Shrewsbury can hedge the exposure by selling the euros forward for British pounds at F3(€/£) = F3($/£) ÷ F3($/€) = 1.8990 ÷ 1.3154 = 1.4437. At this forward exchange rate, Shrewsbury can “lock-in” a price of £322,217 (= €465,184/1.4437) for the sale. The forward exchange rate indicates that the euro is trading at a premium to the British pound in the forward market. Thus, the forward hedge allows Shrewsbury to lock-in a greater amount (£2,217) than if the euro receivable was converted into pounds at the current spot If the euro was trading at a forward discount, Shrewsbury would end up locking-in an amount less than £320,000. Whether that would lead to a loss for the company would depend upon the extent of the discount and the amount of profit built into the price of £320,000. Only if the forward exchange rate is even with the spot rate will Shrewsbury receive exactly £320,000. Obviously, Shrewsbury could ensure that it receives exactly £320,000 at the end of three-month accounts receivable period if it could invoice in £. That, however, is not acceptable to the French wholesaler. When invoicing in euros, Shrewsbury could establish the euro invoice amount by use of the forward exchange rate instead of the current spot rate. The invoice amount in that case would be €461,984 = £320,000 x 1.4437. Shrewsbury can now lock-in a receipt of £320,000 if it simultaneously hedges its euro exposure by selling €461,984 at the forward rate of 1.4437. That is, £320,000 = €461,984/1.4437.