Solutions to Practice Problems CHAPTER 1 1.1 Original exchange rate Reciprocal rate Answer (a) €1 = US$0.8420 US$1 = €? 1.1876 (b) £1 = US$1.4565 US$1 = £? 0.6866 (c) NZ$1 = US$0.4250 US$1 = NZ$? 2.3529 1.2 Given US$1 = ¥.123 25 £1 = US$1.4560 A$ = US$0.5420 (a) Calculate the cross rate for pounds in yen terms. ¥?= £1 £1 = US$1.4560 US$1 = ¥.123 25 £.1=´ 14560 123 . 25 = ¥ 179.45 (b) Calculate the cross rate for Australian dollars in yen terms. ¥?= A$1 A$1 = US$0.5420 US$1 = ¥.123 25 A$1=´= 0..¥ 5420 123 25 66. 80 (c) Calculate the cross rate for pounds in Australian dollar terms. A$? = £1 £1 = US$1.4560 US$0.5420= A$1 A$1== 14560./.£ 0 5420 2. 6863 330 SOLUTIONS 331 1.3 (a) Calculate the realized profit or loss as an amount in dollars when C8,540,000 are purchased at a rate of C1 = $1.4870 and sold at a rate of C1 = $1.4675. Realised profit= Proceeds of sale of Crowns -Cost of purchase of Crowns =´-´8, 540 , 000 14675 . 8 , 540 , 000 14 . 870 = $,166 530 (b) Calculate the unrealized profit or loss as an amount in pesos on P17,283,945 purchased at a rate of Rial 1 = P0.5080 and that could now be sold at a rate of R1 = P0.5072. Unrealised profit= Proceeds of potential sale -Cost of purchase of pesos =-17,, 283 945 17,, 283 945 0. 5072 0.5080 =-34,,. 077 178 63 34 ,,. 023 513 78 = R53,664.85 = 53,664.85´ 0.5072 = P27,218.81 1.4 Calculate the profit or loss when C$9,360,000 are purchased at a rate of C$1 = US$1.4510 and sold at a rate of C$1 = US$1.4620. Realised profit=- Proceeds of sale of C$ Cost of purchase of C$ =´-´9,, 360 000 14620 . 9 ,, 360 000 14510 . = 936, 0,(..) 000´- 14620 14510 =´9,, 360 000 0 . 0110 = US$102,960 1.5 Calculate the unrealized profit or loss on Philippine pesos 20,000,000 which were purchased at a rate of US$1 = PHP47.2000 and could now be sold at a rate of US$1 = PHP50.6000. Unrealised profit= Proceeds of potential sale -Cost of purchase of pesos =-20,, 000 000 20,, 000 000 50. 6000 47. 2000 =-395,. 256 92 423 ,. 728 81 =-US$28471.90 332 SOLUTIONS CHAPTER 2 2.1 (a) Calculate the interest earned on an investment of A$2,000 for a period of three months (92/365 days) at a simple interest rate of 6.75% p.a. IPrt=´´ 675. 92 =´´2, 000 100 365 = $.34 03 (b) Calculate the future value of the investment in 2.1(a). FV=+ P I =+2,. 000 34 03 = $,2 034 . 03 Alternatively, FV=+ P()1 rt æ 675. 92 ö =+´2, 000ç 1 ÷ è 100 365 ø =´2,. 000 1017014 = $,2 034 . 03 2.2 Calculate the future value of $1,000 compounded semi-annually at 10% p.a. for 100 years. FV=+ P()1 i n P = 1, 000 i ==010./ 2 005 . n =´=100 2 200 \=FV 1, 000(1005+ . )200 = 100,(.) 000 106125 4 = $,17 292 , 580 . 82 2.3 An interest rate is quoted as 4.80% p.a. compounding semi-annually. Calculate the equivalent interest rate compounding monthly. æ r ö12 æ 0. 048 ö2 ç1+ ÷ =+ç1 ÷ =1048576. è 12 ø è 2 ø 1+=r/ 12 1048 . 576112/ = 1003961. r ==0..% 0475 4 75 p. a. SOLUTIONS 333 2.4 Calculate the forward interest for the period from six months (180/ 360) from now to nine months (270/360) from now if the six month rate is 4.50% p.a. and the nine month rate is 4.25% p.a. =+ ´ = FV6 (.1 0 045 180 /). 360 1025 =+ ´ = FV9 (.1 0 0425 270 /) 360 10525. 360 103188. r =´ -1 69, 90 102250. ==0..% 0367 3 67 p. a. 2.5 Calculate the present value of a cash flow of $10,000,000 due in three years time assuming a quarterly compounding interest rate of 5.25% p.a. 10,, 000 000 PV = = 8,,. 551 525 87 (.1+ 0 0525 /) 4 12 2.6 Calculate the price per $100 of face value of a bond that pays semi- annual coupons of 5.50% p.a. for 5 years if the yield to maturity is 5.75% p.a. Coupon t 5.50% cf YTM df 5.75% PV 0.5 2.75 0.970874 2.67 1.0 2.75 0.942596 2.59 1.5 2.75 0.915142 2.52 2.0 2.75 0.888487 2.44 2.5 2.75 0.862609 2.37 3.0 2.75 0.837484 2.30 3.5 2.75 0.813092 2.24 4.0 2.75 0.789409 2.17 4.5 2.75 0.766417 2.11 5.0 102.75 0.744094 76.46 97.87 2.7 Calculate the forward interest rate for a period from 4 years from now till 4 years and 6 months from now if the 4 year rate is 5.50% p.a. and the 4 and a half year rate is 5.60% p.a. both semi-annually compounding. Express the forward rate in continuously compounding terms. æ 0.. 055 ö8 æ 0 056 ö9 ç1+ ÷ e 05. r =+ç1 ÷ è 2 ø è 2 ø 1282148. e 05. r = =1032009. 1242381. r =´2ln( 1032009 . ) = 0 . 063014 = 6 . 30 % p. a. 334 SOLUTIONS CHAPTER 3 3.1 Show the cash flows when $2,000,000 is borrowed from one month till six months at a forward interest rate r1,6 of 5% p.a. US$ Spot US$ 1 month 2,000,000.00 US$ 6 months –2,041,666.67 2,000,000(1 + 0.05 – 5/12) 3.2 Show the cash flows when €2,000,000 are purchased three months forward against US dollars at a forward rate of €1 = US$0.8560. € ¬ Spot US$ € ¬ 3 months US$ 2,000,000.00 0.8560 –1,712,000.00 3.3 Prepare a net exchange position sheet for a dealer whose local currency is the US dollar who does the following five transactions. Assuming he or she is square before the first transaction, the dealer: 1. Borrows €7,000,000 for four months at 4.00% p.a. 2. Sells €7,000,000 spot at €1 = 0.8500 3. Buys ¥500,000,000 spot at US$1 = ¥123.00 4. Sells ¥200,000,000 spot against euro at €1 = ¥104.50 5. Buys €4,000,000 one month forward at €1 = US$0.8470. ¥ NEP NEP Transaction 1 –€93,333.33 €93,333.33 Transaction 2 –€7,000,000.00 €7,093,333.33 Transaction 3 €7,093,333.33 500,000,000 500,000,000 Transaction 4 €1,913,875.60 €5,179,457.74 –200,000,000 300,000,000 Transaction 5 €4,000,000.00 €1,179,457.74 300,000,000 The dealer’s net exchange position is long ¥ 300,000,000 and short €1,179,457.74. SOLUTIONS 335 3.4 Show the cash flows when US$1,000,000 is invested from three months for six months at a forward rate r3,9 of 3.5% p.a. US$ Spot US$ 3 months –1,000,000.00 US$ 9 months 1,017,500.00 1,000,000(1 + 0.035 × 6/12) 3.5 Show the cash flows when ¥4,000,000,000 is sold against euros for value 3 November at an outright rate of €1 = ¥103.60. € Spot ¥ € 3 Nov ¥ 38,610,038.61 103.60 –4,000,000,000 CHAPTER 4 4.1 The dollar yield curve is currently: 1 month 5.00% 2 months 5.25% 3 months 5.50% Interest rates are expected to rise. (a) What two money market transactions should be performed to open a positive gap 3 months against 1 month? Borrow dollars for 3 months at 5.50%, and Lend dollars for 1 month at 5.00% FV3 = (1 + 0.055 × 3/12) = 1.013750 FV1 = (1 + 0.050 × 1/12) = 1.004167 (b) Assume this gap was opened on a principal amount of $1,000,000 and after 1 month rates have risen such that the yield curve is then: 1 month 6.00% 2 months 6.25% 3 months 6.50% 336 SOLUTIONS What money market transaction should be performed to close the gap? Lend dollars for 2 months at 6.25%. (c) How much profit or loss would have been made from opening and closing the gap? $ Today 1,004,166.67 –1,004,166.67 1,004,166.67 1,004,166.67 $ 2 months 1,014,626.74 –1,013,750.00 1,004,166.67(1 + 0.0625 × 2/12) 876.74 Profit 1,014,626.74 1,014,626.74 Profit = 1,014,626.74 – 1,013,750.00 = $876.74 4.2 The dollar yield curve is currently inverse and expectations are that one month from now the yield curve will be 50 basis points below current levels, as reflected in the following table. Tenor in Current interest Expected interest months rates% p.a. rates% p.a. 1 4.0 3.5 2 3.5 3.0 3 3.0 2.5 A corporation borrows $10,000,000 for one month and lends $10,000,000 for three months to open a negative gap position.