Solutions to Practice Problems - Springer

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Solutions to Practice Problems

CHAPTER 1

1.1Original 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

1.2Given

US$1

US$1.4560

A$=US$0.5420

¥.

£12325

1 (a) Calculate the cross rate for pounds in yen terms.

¥? £

£ ¥.

£. .¥

1 1 12325

1 14560 12325 179US$1.4560

US$1 .45 (b) Calculate the cross rate for Australian dollars in yen terms. ¥? ¥. ..¥   A$

A$ US$0.5420

US$1 A$1 1 12325

1 05420 12325 6680.

(c) Calculate the cross rate for pounds in Australian dollar terms.

A$ £

£ US$1.4560

US$0.5420 A$1

A$? ./. £1 1

1 14560 05420 26863.330

1.3(a) Calculatetherealizedprofitorlossasanamountindollarswhen

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 P17,283,945purchasedatarateofRial1=P0.5080andthatcould now be sold at a rate of R1 = P0.5072.

Unrealised profit Proceeds of potential sale

Cost  of purchase of pesos 

17 283 945

0507217 283 945

0,, .,, . 5080

34 077 17863 34 023 51378

,,. ,,.

R53,664.85

= 53,664.85 0.5072 =P27,218.81

1.4Calculate 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 14510936,, . ,, ., 0 000 14620 14510

9 360 000 00110,(. .)

,, .  US$102,960

1.5Calculate 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

50600020 000 000

4,, .,, 72000

395 25692 423 72881.

,. ,. US$28471.90

SOLUTIONS331

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

 2 000 675
10092
365
3403,
. $. (b) Calculate the future value of the investment in 2.1(a).

FV P I

 2 000 3403

2 03403,.

$, .

Alternatively,

FV P rt

    () , . ,.1

2 000 1

675
10092
365

2 000 1017014

2 03403$, .

2.2Calculate the future value of $1,000 compounded semi-annually at

10% p.a. for 100 years.

FV P i

P i n FV n     () , ./ . ,1 1 000

010 2 005

100 2 200

1 000( . )

,(. ) $, , .1005

100 000 106125

17 292 58082

200
4  

2.3An interest rate is quoted as 4.80% p.a. compounding semi-annually.

Calculate the equivalent interest rate compounding monthly.

11210048

21048576

1 12 1048

122
    r r. . / . 576 1003961

00475 475

112/
. ..% rp.a.

332SOLUTIONS

2.4Calculate 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. FV FV 6 9

1 0045 180 360 1025

1 00425 270 360    (. /).(. /)

10525
360

90103188

1022501

00367 367

69
. . . ..% , r  p.a.

2.5Calculate 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.

PV 10 000 000

1 00525 48 551 52587

12 ,, (./),,.

2.6Calculate 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 is5.75% 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

2.7Calculate the forward interest rate for a period from 4 years fromnow 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-annuallycompounding. Express the forward rate in continuouslycompounding terms.

10055

210056

2

1282148

8059
05     .. . . . e e r r

12423811032009

2 1032009 0063014 630..

ln( . ) . . %   rp.a.

SOLUTIONS333

CHAPTER 3

3.1Show the cash flows when $2,000,000 is borrowed from one month

till six months at a forward interest rater 1,6 of 5% p.a.

3.2Show the cash flows when2,000,000 are purchased three months

forward against US dollars at a forward rate of1 = US$0.8560.

3.3Prepare a net exchange position sheet for a dealer whose localcurrency is the US dollar who does the following five transactions.Assumingheorsheissquarebeforethefirsttransaction,thedealer:

1. Borrows7,000,000 for four months at 4.00% p.a.

2. Sells7,000,000 spot at1 = 0.8500

3. Buys ¥500,000,000 spot at US$1 = ¥123.00

4. Sells ¥200,000,000 spot against euro at1 = ¥104.50

5. Buys4,000,000 one month forward at1 = US$0.8470.

¥

NEP NEP

Transaction 1 -

93,333.3393,333.33

Transaction 2 -

7,000,000.007,093,333.33 Transaction 37,093,333.33 500,000,000 500,000,000 Transaction 41,913,875.605,179,457.74 -200,000,000 300,000,000 Transaction 54,000,000.001,179,457.74300,000,000 The dealer"s net exchange position is long ¥ 300,000,000 and short 1,179,457.74.

334SOLUTIONS

US$ Spot US$

1 month

2,000,000.00

US$

6 months

-2,041,666.67

2,000,000(1 + 0.05 - 5/12

)

¬SpotUS$

¬

3 monthsUS$

2,000,000.00 0.8560 -1,712,000.00



3.4Show the cash flows when US$1,000,000 is invested from three

months for six months at a forward rater 3,9 of 3.5% p.a.

3.5Show the cash flows when ¥4,000,000,000 is sold against euros forvalue 3 November at an outright rate of1 = ¥103.60.

CHAPTER 4

4.1The 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%

FV 3 = (1 + 0.055 × 3/12) = 1.013750 FV 1 = (1 + 0.050 × 1/12) = 1.004167 (b) Assumethisgapwasopenedonaprincipalamountof$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%

SOLUTIONS335

US$ Spot US$

3 months-1,000,000.00

US$

9 months

1,017,500.00

1,000,000(1 + 0.035 × 6/12

)

Spot¥

3 Nov ¥

38,610,038.61 103.60 -4,000,000,000

  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?

Profit = 1,014,626.74 - 1,013,750.00 = $876.74

4.2The 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

months

Current interest

rates% p.a.Expected interestrates% 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. (a) Calculate the break-even interest rate at which it will need to be able to borrow $10,000,000 for 2 months in one month"s time. (. /)( /)(. /) (/).10041121 212 1003312

1 6 1007    

b b5 1003333 1004153

0004153 6 0024917

249/. .

.. .%  b p.a. (b) Assumingtheyieldcurvemovesaccordingtoexpectation,calcu- late the profit or loss which will be realized on closing the gap.

336SOLUTIONS

$

Today1,004,166.67-1,004,166.67

1,004,166.67 1,004,166.67

$

2 months

1,014,626.74-1,013,750.001,004,166.67(1 + 0.0625 × 2/12

)

876.74Profit

1,014,626.74 1,014,626.74

Profit = 10,083,500 - 10,075,000 = -$8,500

Note : The gap would result in a loss because the 2 month rate at which the corporation expects to borrow 3.00% p.a. is greater than the break-even rate 2.49% p.a.

4.3The crown yield curve is currently normal and expectations are thatit will become steeper with the pivotal point at 6 months as reflectedbelow:

Months Current

ratesExpected rates 3months from now

3 5.0% 4.5%

6 5.5% 5.5%

9 6.0% 6.5%

12 6.5% 7.5%

Two gapping strategies are contemplated:

(a) Borrowing C1,000,000 for 3 months and lending C1,000,000 for 6 months. Strategy (a) would result in a profit of C3,609.38.

SOLUTIONS337

$Today

10,000,000.00 -10,000,000.00

$

1 month

-10,033,333.33

1,000,000(1 + 0.04 × 1/12)

$

3 months

10,075,000.00

1,000,000(1 + 0.03 × 3/12)

Closing negative gap

$Today

10,033,333.33-10,033,333.33

10,033,333.33 10,033,333.33

$

2 months

10,075,000.00-10,083,500.0010,033,333.33(1 + 0.03 × 2/12

) -8,500.00Profit

10,075,000.00 10,075,000.00

(b) BorrowingC1,000,000for3monthsandlendingC1,000,000for12 months. Strategy (b) would result in a profit of C3,140.63.

338SOLUTIONS

Opening a negative gap

CToday

1,000,000.00 -1,000,000.00

C

3 months

-1,012,500.00

1,000,000(1 + 0.05 × 3/12)

C

12 months

1,065,000.00

1,000,000(1 + 0.065 × 12/12

)

Closing negative gap

CToday

1,012,500.00-1,012,500.00

1,012,500.00 1,012,500.00

C

9 months

1,065,000.00

-1,061,859.381,012,500(1 + 0.065 × 9/12)

3,140.63

Profit

1,065,000.00 1,065,000.00

Opening a negative gap

CToday

1,000,000.00 -1,000,000.00

C

3 months

-1,012,500.00

1,000,000(1 + 0.05 × 3/12)

C

6 months

1,027,500.00

1,000,000(1 + 0.055 × 6/12

)

Closing negative gap

CToday

1,012,500.00-1,012,500.00

1,012,500.00 1,012,500.00

C

3 months

1,027,500.00

-1,023,890.631,012,500(1 + 0.045 × 3/12 )

3,609.38Profit

1,027,500.00 1,027,500.00

Assuming that interest rates move according to expectations and that the gap is closed after 3 months, which strategy will prove more profitable? Strategy (a) would be more profitable. It would result in a larger profit at an earlier date. Profit under Strategy (a) = C3,609.38 received after 6 months Profit under Strategy (b) = C3,140.63 received after 12 months To draw an exact comparison calculate the present value in each case. Strategy (a) PV = 3,609.38 /(1 + 0.055 × 6/12) = C3,512.78 Strategy (b) PV = 3,140.63 /(1 + 0.065 × 12/12) = C2,948.95

4.4On 1 July a company borrows $10,000,000 at a three month floatingrateof3.75%p.a.(360daysperyearbasis).Thisdebtwillberolledon1 October (92 days). The company also placed $10,000,000 on depositmaturing on 3 January (186 days) also at a rate of 3.75% p.a.

(a) Is the gap which the company has opened positive or negative? The company has opened a negative gap by borrowing for a shorter period than it has lent. (b) Would the company like the 3 month rate on 1 October to be higher or lower than at present? The company needs to borrow at the 3 month rate on 1 October so it would like the rate to be lower. The break-even rate would be: r 1

1 00375 186 360

1 00375 92 3601

Oct, 3 Jan

  (. /) (. /)   360

186 92371.%p.a.

(c) Calculate the profit or loss if the company rolls the floating rate borrowing for 94 days from 1 October at exactly 3.75% p.a.

SOLUTIONS339

$1 October10,095,833.33-10,095,833.33

10,095,833.33 10,095,833.33

$

3 January

10,193,750.00-10,194,688.3710,095,833.33(1 + 0.0375 × 94/360

) -938.37Profit

10,193,750.00 10,193,750.00

The company would lose $938.37 because it had to borrow $10,095,833.33 at 3.75% p.a. which is higher than the break-even rate of 3.71% p.a.

4.5The dollar yield curve is currently:

1 month 5.00%

2 months 5.25%

3 months 5.50%

Interest rates are expected to fall.

(a) Which two money market transactions should be performed to open a gap 3 months against 1 month?

Borrow dollars for 1 month at 5. 00%, and

Lend dollars for 3 months at 5.50%

(b) Assuming the gap was opened on a principal amount of $1,000,000 and after 1 month rates have fallen such that the yield curve is then:

1 month 4.75%

2 months 5.00%

3 months 5.25%

What money market transaction should be performed to close the gap?

Borrow dollars for 2 months at 5.00%.

(c) How much profit or loss would have been made from opening and closing the gap?

CHAPTER 5

5.1A bank quotes £1 = US$1.4020/1.4025.

(a) Thebankwillbuydollarswhereitsellspounds;thatis,at1.4025.

340SOLUTIONS

$1 month -1,004,166.67

FV = 1,000,000 (1 + 0.05 × 1/12)

+1,004,166.675.00% $

3 months

+1,013,750.00

FV = 1,000,000 (1 + 0.055 x 3/12)

-1,012,534.73

FV = 1,004,166.67 (1 + 0.05 x 2/12

)

1,215.27Profit

1,013,750.00 1,013,750.00

(b) A customer could sell pounds at the bank"s bid rate; that is, at

1.4020.

(c) At customer could sell dollars where it buys pounds; that is; at

1.4025.

5.2BankAcallsandasksBankBforapricefordollar/yen.BankBquotes

US$1 = ¥125.40/125.50. At what rate can Bank A sell yen? Bank A can sell yen where it buys dollars. That is at Bank Bs offer rate, 125.50.

5.3A customer in Crownland asks a bank for a crown/dollar quote. Thebank quotes C1 = $1.4935/1.4945.

(a) 1,000,000 × 1.4945 = $1,494,500 (b) 1,000,000 × 1.4935 = $1,493,500 (c) 1,494,500 - 1,493,500 = $1,000 (d) 1,000,000/1.4935 = C669,568.13 (e) 1,000,000/1.4945 = C669,120.11 (f) C669,568.13 - 669,120.11 = C448.02

5.4A bank quotes overnight dollars at 4.25/4.50% p.a.

(a) A customer could borrow dollars at 4.50% p.a. (b) A customer could invest dollars at 4.25% p.a.

5.5A bank quotes 7 day francs at 4.50/4.75% p.a. There are 365 days peryear.(a) Interest = 1,000,000 × 0.0475 × 7/365 = F910.96(b) Interest = 1,000,000 × 0.0450 × 7/365 = F863.01(c) 910.96 - 863.01 = F47.95

5.6A broker has dollar/yen prices from three banks:

Bank A US$1 = ¥125.60125.65

Bank B US$1 = ¥125.62 125.67

Bank C US$1 =¥125.63125.68

The broker price is:125.63 125.65.

5.7A bank quotes F1 = $1.2130/1.2140. A customer calls and sells thebank F10,000,000 at its bid rate 1.2130. The bank would like to squareitsposition(ifpossibleataprofit).Ifanotherbankcallsaminutelaterasking for a price, which of the following rates should the first bankquote?

Rate A F1 = $1.2125 1.2135

Rate B F1 = $1.2130 1.2140

Rate C F1 = $1.2135 1.2145

SOLUTIONS341

5.8Bank A quotes NZ$1 = US$0.42200.4225

Bank B quotes NZ$1 =US$0.42260.4231

What arbitrage opportunity exists? How much profit could be made by performing this arbitrage on a principal amount of

NZ$10,000,000?

Buy NZ$10,000,000 from Bank A at 0.4225 and sell NZ$10,000,000 to

Bank B at 0.4226.

Profit = US$ 4,226,000 - 4,225,000 = US$1,000

5.9US$1 = S$ 1.7050 1.7060

1 = US$0.8490 0.8500 A Singaporean exporter wants to sell euro and buy Singapore dollars. What is the break-even rate for euros in Singapore dollar terms? S US US S S$? $. $$. $ ..      1

1 08490

17050
1

1 08490 17050

11

1 14475

S$ .

5.10US$1 = M$ 3.8010 3.8030

£1 = US$1.4470 l.4480

What bid and offer rates should a bank quote for pounds against ringittinMalaysiantermstomakeatenpointspreadoneithersideof the break-even rates?

342SOLUTIONS

Market

US$

¬0.8490

¬ Bank

Customer

S$ US$

S$1.7050

Market

  BID M US US M

M$? £

£$.

$$. £$ ..    1

1 14470

1 38010

1

14470 38010

1    1

1 55000

00010

1 55000£$.

.

£$.M

Less spread

M OFFER M US US M

M$? £

£$.

$$. £$ ..    1

1 14480

1 38010

1

14480 38010

1    1

1 55067

00010

1 55077£$.

.

£$.M

Less spread

M

5.11A bank calls four other banks for dollar/Swiss franc rates.

Bank A $ 1 = SF 1.2430 1.2433

Bank B $ 1 = SF 1.2430 1.2432

Bank C $ 1 = SF 1.2431 1.2433

Bank D $ 1 = SF 1.2430 1.2433

The bank wishes to sell Swiss francs. With which bank and at what rate should it deal?

SOLUTIONS343

Market

US$

¬1.4470

¬ Bank

Customer

M$ US$

M$3.8010

Market

£ £ The bank should buy dollars at the lowest offer rate which is 1.2432 from Bank B.

5.12US$1 = ¥104.50 104.601 = US$0.8550 0.8555

A Japanese importer wants to buy euros and sell yen. What is the break-even rate for euros in yen terms? ¥? $. $¥. ¥ ..       1

1 08555

1 10460

1

1 08555 10460

1US US 1

18949¥.

5.13AcustomercallsandwantstobuyHongKongdollarsagainstAustra-

lian dollars. What rate should a bank quote for Hong Kong dollars in terms of Australian dollars to ensure a one point profit?

US$1 = HK$ 7.7360 7.7370

A$1 = US$0.5240 0.5245

AHK HK US US A HK A$? $ $. $ $. $ $ $ .  1

77360 1

05420 1

1 111
77

360 05420

1 02467

02465

. $$. $.HK A

Less spread A

CHAPTER 6

6.1Spot rate £1 = US$1.5000

3 month US$ interest rate 2.50% p.a. (91/360)

3 month £ interest rate 3.00% p.a. (91/365)

(a) 3 month forward rate f  150001 0025 91 360

1 003 91 365.(. /)

(. /) (b) 3 month forward margin f-s= 1.4983 - 1.5000 = -0.0017

344SOLUTIONS

6.2Spot rate1 = ¥107.00

7 month euro3.50% p.a. (212/360)

7 month yen0.35% p.a. (212/360)

(a) 7 month forward rate f  107001 00035 212 3601 00350 212 360105.(. /)(. /).06 (b) 3 month forward margin f-s= 105.06 - 107.00 = -1.94

6.3Spot rate1 = US$0.8490 0.8500

5 month3.00 3.10% p.a. (152/360)

5 month US$ 1.90 1.95% p.a. (152/360)

A customer wishes to buy dollars five months forward. What rate should a bank quote to make 2 points profit?

Customer wants to buy dollars and sell euros.

Quoting bank is buying euros forward.

Quoting bank sells euros spot at 0.8490.

Quoting bank has to borrow euros at 3.10% p.a. and lend dollars at

1.90% p.a.

f  084901 0019 152 360

1 0031 152 36008448.(. /)

(. /). To make 2 points profit the bank lowers its bid rate by 2 points

Quoted rate = 0.8448 - 0.0002 = 0.8446

6.4Spot rate1 = US$0.8490 0.8500

5 month3.00 3.10% p.a. (152/360)

5 month US$ 1.90 1.95% p.a. (152/360)

A customer wishes to sell dollars five months forward. What rate should a bank quote to make 2 points profit?

Customer wants to sell dollars and buy euros.

Quoting bank is selling euros forward.

f  085001 00195 152 3601 00300 152 360084.(. /)(. /).63 To make 2 points profit the bank increases its offer rate by 2 points

Quoted rate = 0.8463 + 0.0002 = 0.8465

SOLUTIONS345

6.5Spot rate A$1 = US$0.5100 0.5105

2 year A$ interest rate 5.00% 5.20% p.a. (semi-annually)

2 year US$ interest rate 4.50% 4.70% p.a. (semi-annually)

The break-even 2 year forward bid and offer rates: Bid f f(./) .(./) .1 0052 2 05100 1 0045 2 05031
2222
   Offer f f(./) .(./) .1 005 2 05105 1 0047 2 05075
2222
  

2 year forward rates: A$/US$ 0.5031/0.5075

6.6Spot rate1 = US$0.8780 0.8785

Overnight US$ interest rate 2.25% 2.375% p.a. (3/360)

Overnightinterest rate 3.25% 3.375% p.a. (3/360)

Calculate the break-even bid and offer rates to 5 decimal places for outright value tomorrow. Bid tom tom(. /).(. /) .1 002375 3 360 08780 1 00325 3 360

08  

7806 Offer tom tom(. /).(. /) .1 00225 3 360 08785 1 003375 3 360

08  

7858

Outright value tomorrow1 = US$0.87806/0.87858

6.7A trader has done the following 3 transactions:

US$ amount ¥ amount Rate Maturity

+10,000,000 -1,075,000,000 107.50 Spot -2,000,000 +210,610,000 105.30 6 months -5,000,000 +512,000,000 102.40 1 year Calculate the traders yen Net Exchange Position in NPV terms and marked-to-market profit or loss given the current rates:

Spot US$/¥ 110.30

6 month dollar interest rate 4.20% p.a.

6 month yen interest rate 0.30% p.a.

346SOLUTIONS

1 year dollar interest rate 4.10% p.a.

1 year yen interest rate 0.45% p.a.

¥ Amount PV

 

1 075 000 000

210 610 000

512 000 0001 075 000 0

,,, ,, ,,,,, 00

11 075 000 000

210 610 000

1 0003 2210 284 57

 ,,, ,, ./,, 3

512 000 000

1 00045509 706 322,,

.,, 

Net exchange position = -355,009,105

Close out value = 355,009,105 /110.30 = $3,218,577.56

US$ Amount PV

 10 000 000 10 000 000 10 000 00000

2 000 0002,, ,, ,,.

,,,

000 000

1 0042 21 958 86386

5 000 000

5 000 000,

./,,. ,, ,,   

1 00414 803 07397 .,,.

Counter value = $3,238,062.17

MTM profit Counter value Close out value

3 238 062,,. ,,. $, .17 3 218 57756

19 48461

US

6.8Calculate the 1 year, 2 year and 3 year zero coupon discount factors

given the following par curve:

1 year 2.50% p.a.

2 years2.40% p.a.

3 years2.60% p.a.

df df 1 2 100

10250975610

1 0024 0975610

10240

 . .. .. ..

953697

1 0026 0975610 0953697

1026092576

3 df .(. . ) ..8

6.9SpotNZ$ 1 = US$ 0.3940/0.3950

Overnight NZ$4.00%/4.15% (1/365)

Overnight US$2.00%/2.15% (1/360)

Quote your bid and offer rates outright value tomorrow.

SOLUTIONS347

Bid t t(. /).(. /) .1 0040 1 365 03940 1 0020 1 360

039402   

 Offer t t(. /).(. /) .1 00415 1 365 03950 1 00215 1 360

039502  



Outright value tomorrow NZ$1 = US$0.39402/0.39502

6.10Spot US$1 = Yen 107.00

2 year dollars 6.00%/6.25%

2 year yen 1.75%/2.00%

Interest paid semi-annually in arrears.

Calculate the break-even bid and offer rates for the 2 year forward margins . Bid f f(./) .(./) .1 00625 2 10700 1 00175 2 9796
2222
  348SOLUTIONS US$

Spot¥

+

107.00

-- 6.25%

1.75%+

US$

2 years¥

- ? ++ -

NZ$TomUS$

+

0.39402-

-

4.000%

2.150%

+ NZ$

SpotUS$

-

0.3940+

+

4.000%

2.150%

-

Forward margin bid rate = 107.00 - 97.96 = -9.04

Offer f f(./) .(./) .1 006 2 10700 1 002 2 9893
2222
   Forward margin offer rate = 107.00 - 98.93 = -8.07

2 year forward margin: Yen 9.04/8.07

CHAPTER 7

7.1An Australian importer has an obligation to pay ¥1,000,000,000 in 3

months" time. Calculate the cost in Australian dollars if the expected spot rate at maturity is A$1 = ¥ 65.20/65.30.

AcostA$,,,

.$, , .1 000 000 000

652015 337 42331

7.2A New Zealand exporter is due to receive US$4,560,000 in 2 months.The exporter considers the alternatives of remaining unhedged andselling the US dollars spot upon receiving them, or hedging byforward selling the US dollar receipts.

Spot rate NZ$1 = US$0.4200 0.4205

2 month NZ$3.75 3.85% p.a. (62/365)

2 month US$2.65 2.75% p.a. (62/360)

(a) Calculate the forward rate at which the exporter could hedge. The exporter needs to buy NZ$ at the bank"s forward offer rate.

Forward offer rate

s 0.4205 Bank buys NZ$ spot to cover its forward sale to the importer r C 3.75% Bank lends NZ$ at the market bid rate r T 2.75% Bank borrows US$ at the market offer rate t 62/365 and 62/360 f  042051 00275 62 3601 00375 62 36504198.(. /)(. /). (b) If the expectation is that in 2 months" time the spot rate will be NZ$1 = US$0.41/4550, should the exporter hedge or remain unhedged?

SOLUTIONS349

The exporter would buy NZ$ at 0.4155 if unhedged. This would prove cheaper than buying them forward at 0.4198. Accord- ingly, the exporter should remain unhedged. (c) Calculate the break-even rate between being hedged and unhedged? The break-even rate will be the forward rate, 0.4198. Conse- quently, the exporter should buy the NZ$ forward at 0.4198 if, but only if, the expected spot offer rate is 0.4198 or higher.

7.3An Indonesian exporter expects to receive US$4,000,000 in 5 monthstime.

Spot USD/IDR 10,200 10,400

5 month dollars 2.50% 2.60% p.a. (150/360)

5 month rupiah 25.00% 26.00% p.a. (150/360)

(a) At what rate could the exporter hedge its dollar receivables? Exporter would sell dollars at the forward bid rate f   10 200

1 025 150 360

1 026 150 360

1114180,

(. /) (. /) ,. (b) How many rupiah would the exporter receive from the proceeds if it hedged?

Hedged rupiah proceeds

4 000 000 111418044 567,, ,. , ,200 000, (c) If the exporter elected not to hedge and at the end of the 5 months the spot rate turned out to be 10,600/10,700, how many rupiah would the exporter receive?

Unedged rupiah proceeds

4 000 000 10 60042 400 00,, , , , 0 000,

7.4An Australian exporter will be receiving US$5,000,000 in one years

time.

Spot A$1 = US$0.5720/25

1 year forward margin 50/45

(a) What will the A$ proceeds be if it is hedged? Exporter sells US$ /buys A$ at the outright offer rate:

350SOLUTIONS

0.5725



0.0045

0.5680

A proceedsA$,,

.$, , .5 000 000

056808 802 81690

(b) If at the end of the year the spot rate is A$1 = US$0.5625/30, what would the A$ proceeds be if unhedged?

A$ proceeds if unhedged

5 000 000

056308 880 99467,,

.$, , .A (c) Would the exporter be better off hedged or unhedged? The A$ proceeds would turn out to be greater if the exporter remained unhedged in this case.

7.5A company requires US$8,000,000 for 9 months. Two alternatives areconsidered:

1. Borrowing dollars domestically at an interest rate of 3.50% p.a.

(272/360)

2. Borrowing euros at an interest cost of 4.00% p.a. (272/360)

(a) Calculate the effective borrowing cost if the spot rate at draw down is1 = US$0.8650, and at repayment of principal and interest is1 = US$0.8540.

08540 086501 272 360

1 004 272 360

227..
(/) (. /) .%   r rp.a. (b) Which of the alternatives involves the lower cost? It would have turned out cheaper to borrow euro unhedged at

2.27% p.a. than to borrow dollars at 3.50% p.a.

7.6A Thai borrower has to choose between borrowing baht orborrowing dollars.

Spot US$1 THB 35.7020 35.7030

3 month dollars 3.10% 3.20% p.a. (90/360)

3 month baht 15.50% 15.75% p.a. (90/360)

Calculate the break-even exchange rate between borrowing baht directly and borrowing US dollars on an unhedged basis. The borrower could borrow baht at 15.75% p.a. or borrow US dollars at 3.20% p.a. and sell the dollars spot for bath at 35.7020.

SOLUTIONS351

Break-even rate3570201 01575 90 360

1 00320 90 360368133.(. /)

(. /).   The borrower will be better off borrowing US dollars provided the spotrateremainsbelow36.8133butworseoffifthespotrateatmatu- rity is above 36.8133.

7.7Unhedged foreign currency investmentsA funds manager has US dollars to invest for six months.

Spot rates

US$1 = ¥120.00

£1 = US$1.5000

The funds manager considers three alternatives:

1. Investing the dollars directly at 2.50% p.a.

2. Selling the dollars to buy yen to invest unhedged at 0.50% p.a.

3. Sellingthedollarstobuypoundstoinvestunhedgedat3.20%p.a.

(a) Calculatetheeffectiveyieldontheunhedgedyenandunhedged pound investments if the spot rates at maturity turn out to be

US$1 = ¥120.00 and £1 = US$1.4850.

1. Invest in dollarsy

1 = 2.50%

2. Sell dollars (buy yen) at 120.00

Invest in yen at 0.50%

6 months later buy dollars at 120.00

1201 0005 6 12

1 100 6 12120

050
2 (. /) (/ /) .%   y yp.a.

3. Buy pounds (sell dollars) at 1.5000

Invest pounds at 3.20%

6 months later sell pounds at 1.4850

150001 100 6 12

1 0032 6 1214850

117
3 .(/ /) (. /)..%  y yp.a. (b) Which of the three alternatives would have yielded the highest return on the investment? Investing in dollars yielding 2.50% p.a. would have produced the highest return.

352SOLUTIONS

7.8Break-even rate on unhedged investment

Spot rate US$1 = ¥116.50 116.60

6 month dollars 2.00% 2.25% p.a. (181/360)

6 month yen 0.10% 0.20% p.a. (181/360)

A funds manager has US dollars to invest for six months. (a) If the funds manager elects to use the dollars to buy yen for an offshore investment, what is the break-even future spot rate?

Sell USD spot for yen at 116.50

Invest yen for 6 months at 0.10%

Alternative yield on USD 2.00%

Break-even rate 

116501 0001 181 360

1 002 181.(. /)

(. /) .360

11539

(b) If at maturity of the yen investment, the spot rate turns out to be US$1 = ¥113.30/113.40, calculate the effective yield. At maturity the investor would need to buy dollars/sell yen at

113.40. Ify= effective yield

116501 0001 181 360

1 181 36011340

554.
(. /) (/). .%   y yp.a.

7.9A money market manager considers investing in Malaysian ringgit

asawaytoearnahigheryield.ThespotrateiscurrentlyfixedatUS$/ M$ 3.8000. If the money manager can access a 3 month ringgit fixed depositrateof8.50%p.a.,whatwouldbetheeffectiveyieldindollars if on maturity of the deposit the pegged exchange rate had been broken and the spot rate was then 4.0000/4.0100?

380001 0085 3 12

131240100

1289.
(. /) (/). .%   r rp.a. The fall in the value of the ringgit against the US dollar has much more wiped out the interest rate benefit from investing in ringgit rather than dollars.

7.10AnAustralianexporterwithreceiptsofUS$5,000,000eachquarterfor3 years could hedge its foreign exchange risk by doing 12 separateforward deals in which it would sell US$5,000,000 against dollars atthe different forward rates for each of the 12 maturities.

SOLUTIONS353

Based on a spot rate A$1 = US$0.5205 and the relevant interest rates the following forward rates and zero coupon discount factors apply:

Years Forward US$ cash flow A$ cash flow zcdf

0.25 0.5177 5,000,000.00 9,658,103.15 0.9895

0.50 0.5151 5,000,000.00 9,706,853.04 0.9792

0.75 0.5128 5,000,000.00 9,750,390.02 0.9688

1.00 0.5108 5,000,000.00 9,788,566.95 0.9586

1.25 0.5099 5,000,000.00 9,806,805.92 0.9476

1.50 0.5089 5,000,000.00 9,825,112.99 0.9370

1.75 0.5080 5,000,000.00 9,843,488.53 0.9266

2.00 0.5070 5,000,000.00 9,861,932.94 0.9163

2.25 0.5057 5,000,000.00 9,886,796.18 0.9051

2.50 0.5045 5,000,000.00 9,911,785.11 0.8918

2.75 0.5032 5,000,000.00 9,936,900.68 0.8806

3.00 0.5019 5,000,000.00 9,962,143.85 0.8673

The par forward rate is that rate for which the net present value of the Australian dollar cash flows is the same as the net present value for the 12 separate forward deals. If the first estimate of the par forward rate is 0.5088 being the average of the forward rates:

Years US$ A$ at forwards PV(forward) A$ at par

forwardPV (par forward)

0.25 5,000,000.00 9,658,103.15 9,556,693.07 9,827,044.03 9,723,860.06

0.50 5,000,000.00 9,706,853.04 9,504,950.50 9,827,044.03 9,622,641.51

0.75 5,000,000.00 9,750,390.02 9,446,177.85 9,827,044.03 9,520,440.25

1.00 5,000,000.00 9,788,566.95 9,383,320.28 9,827,044.03 9,420,204.40

1.25 5,000,000.00 9,806,805.92 9,292,929.29 9,827,044.03 9,312,106.92

1.50 5,000,000.00 9,825,112.99 9,206,130.87 9,827,044.03 9,207,940.25

1.75 5,000,000.00 9,843,488.53 9,120,976.47 9,827,044.03 9,105,738.99

2.00 5,000,000.00 9,861,932.94 9,036,489.15 9,827,044.03 9,004,520.44

2.25 5,000,000.00 9,886,796.18 8,948,539.23 9,827,044.03 8,894,457.55

2.50 5,000,000.00 9,911,785.11 8,839,329.96 9,827,044.03 8,763,757.86

2.75 5,000,000.00 9,936,900.68 8,750,434.74 9,827,044.03 8,653,694.97

3.00 5,000,000.00 9,962,143.85 8,640,167.36 9,827,044.03 8,522,995.28

Total109,726,138.77109,752,358.49

If the par forward rate was 0.5088, the net present value of the par forward would be greater than the net present value of the 12 sepa- rateforwardsimplyingthatthebreak-evenparforwardrateisworse (that is, higher) than 0.5088.

354SOLUTIONS

Break-even par forward rate05088109 752 35849

109 726 1387705089.,,.

,,..

CHAPTER 8

8.1Spot rates: US$1 = ¥121.30 121.35

1 year swap 5.17 5.01

(a) Atwhatratecanacustomerbuyyenoutrightoneyearforward?

Customer can sell dollars at the bid rate

Outright bid rate = 121.30 - 5.17 = 116.13

(b) What is the benefit or cost to a customer of buying dollars 1 year forward and selling dollars spot in a pure swap?

Customer will sell dollars spot at 121.32

Customer will buy dollars 1 year at 116.31

Benefit of the swap to customer

= Cost of swap to the bank = 5.01 (c) At what rates would a customer deal if it bought dollars 1 year forward and sold dollars spot in an engineered swap?

Customer would sell dollars spot at 121.30

Customer would buy dollars 1 year at 116.34

Benefit of the swap to the customer

= Cost of the swap to the bank = 4.96

8.2Spot rates US$1= SF1.2735 1.27401 month swap rates 0.0030 0.0025

(a) What is the 1 month outright bid rate?

Outright bid rate = 1.2735 - 0.0030 = 1.2705

(b) What is the 1 month outright offer rate?

Outright offer rate = 1.2740 - 0.0025 = 1.2715

A customer wants to buy dollars spot and sell dollars 1 month forward (c) What is the benefit or cost of an engineered swap to the customer? The customer would buy dollars spot at 1.2740 and sell dollars forward at 1.2705.

SOLUTIONS355

The cost of the engineered swap to the customer = 1.2740 -

1.2705 = 0.0035.

(d) Whatisthebenefitorcostofapureswapifbasedonaspotrateof

1.2740?

The cost of a pure swap to the customer = 1.2740 - 1.2710 =

0.0030

8.3A company needs to borrow Singapore dollars for one year.

Spot rate US$1 = S$ 1.7500

1 year forward US$1 = S$ 1.7320

1 year interest rate US$1 3.25% p.a.

Calculate the effective cost of generating Singapore dollars for one year through a swap.

175001

1 0032517320

219.
() (.). .%  r rp.a.

8.4An American company wants to borrow Canadian dollars for 6

months.

Spot US$1 = C$1.3540 1.3550

6 month US$ 5.50% 5.75%

6 month C$ 8.00% 8.50%

6 month swap rate 148 168

Is it cheaper to borrow the Canadian dollars directly or to borrow US dollars and swap them into Canadian dollars?

Cost to borrow C$ directly 8.50% p.a.

Borrow US$ 5.75% p.a.

Swap US$ into C$ by:

Selling US$ spot at 1.3545

Buying US$ 6 months at 1.3545+0.0168 = 1.3713

Let c = effective cost:

135451612

1 00575 6 1213713

830.
(/) (. /). .   c cp.a. It would be cheaper to raise the Canadian dollars through a swap.

8.5A fund manager has euros to invest for three months and considers

two alternatives:

356SOLUTIONS

1. Investing euros directly at 3.5% p.a.

2. Swapping euros into US dollars and investing the dollars.

Which alternative provides the higher effective yield given the prevailing market rates.

Spot1 = US$0.8860

3 month US$ 3.00 3.25% p.a. (90/360)

3 month swap 11 10

10% withholding tax applies to interest earned from a direct invest-

ment in euro.

After WHT yield on direct euro investment

= 3.50 × (1 - 0.1) = 3.15% p.a. Alternatively, swap the euro into US dollars (sell euro spot at 0.8860 andbuyeuroforwardat0.8850)andlendUSdollarsat3.00%p.a.Let y= effective yield with swap:

08850 088601 003 90 360

1 90 360

346..
(. /) (/) .%   y yp.a. Investing through the swap earns a higher yield because it avoids withholding tax.

8.6Market rates are

5 month US$ interest rates 3.25% p.a. 3.35% p.a. (153/360)

5 month ¥ interest rates 0.20% p.a. 0.30% p.a. (153/360)

Spot rateUS$1 =¥ 123.40 123.50

5 month swap rates- 1.63

- 1.53

5 month outright forward US$1 =¥121.77121.97

rates A customer called a bank late in the afternoon and asked for a rate at which to sell US dollars 5 months forward. Hoping to make two points profit, the bank quoted a forward bid rate US$1 = ¥121.75. The customer agreed to deal and sold the bank US$10,000,000. The bank was then long US$10,000,000/short ¥1,217,500,000 and had mismatched cash flows on the 5 months date. Using T-accounts, show how the bank could hedge its position withaspotdealandaswap.Howmuchprofitwouldthebankmake?

SOLUTIONS357

The 2 points profit equals ¥200,000 due in 5 months time.

8.7Three months ago a Japanese importer purchased US$10,000,000

threemonthsforwardatanoutrightrateof130.00tohedgeexpected US dollar payments. The original forward contract is maturing in two days time, that is, today"s spot value date. The ship has been delayed and the importer will not be required to make the US dollar paymentforafurthermonth.Thecurrentinter-bankratescenariois:

Spot US$1 = ¥125.00 125.05

1 month dollars 3.15% 3.25% (30/360)

1 month yen 0.20% 0.25% (30/360)

1 month swap rate 29 31

Calculate the break-even forward rate for an historic rate rollover. TheimporterneedstosellUS$10,000,000spotandbuyUS$10,000,000 one month forward. If this was done at market rates the forward leg would be done at 125.00 - 0.31 = 124.69. It would be necessary to borrow ¥50,000,000 for 1 month at 0.25% p.a. to cover the cash short- fall on the spot date. The HRR forward rate would be:

1 296 910 417

10 000 00012969,,,

,,. as shown in the cash flow diagram opposite.

358SOLUTIONS

US$Spot¥

-10,000,000.00 123.40 1,234,000,000

10,000,000.00 123.40 -1,234,000,000

10,000,000.00 -10,000,000.00 1,234,000,000 1,234,000,000

US$

5 months¥

10,000,000.00121.75-1,217,500,000

-10,000,000.00 121.77 1,217,700,000

Profit200,000

10,000,000.00 10,000,000.001,217,700,000 1,217,700,000

8.8Spot US$1 = ¥123.56/123.61

Today is Friday 24 May. Spot value is Tuesday 28 May.

Swap rates:

O/N 2.0/1.9

T/N 0.4/0.3

S/W 7.0/6.0

24252627282930311234

Tod Tom Spot 1 week

(a) At what rate can a customer buy US$ outright value today (24 May)? Outright value today offer rate = 123.61 + 0.02 + 0.004 = 123.634 (b) AtwhatswapratecouldacustomerbuyUS$valuetodayandsell

US$ value 4 June in a pure swap?

1 week over today swap bid rate = 2.0 + 0.4 + 7.0

= 9.4 points Forexample,thecustomercouldbuyUS$spotat123.60(say)and sell US$ value 4 June at 123.60 - 0.094 = 123.506.

SOLUTIONS359

Bank"s cash flows with market

US$

Spot¥

10,000,000130.00-1,300,000,000

-10,000,000 125.00 1,250,000,000

0.25% 50,000,000

10,000,000 10,000,0001,300,000,000 1,300,000,000

US$

1 month¥

10,000,000124.69-1,246,900,000

P + I -50,010,417

1,296,910,417

Bank"s cash flows with importer

US$

Spot¥

10,000,000130.00-1,300,000,000

-10,000,000 130.00 1,300,000,000

10,000,000 10,000,0001,300,000,000 1,300,000,000

US$

1 month¥

10,000,000129.69-1,296,900,000

CHAPTER 9

9.1US dollar interest rates are higher than yen rates, so the swaps curve

isnegative.Overthenextmonth,dollarinterestratesareexpectedto rise relative to yen rates and the dollar is expected to appreciate against the yen.

Current rates Expected rates (1 month from now)

Tenor in

months Swap rates Exchange rates Swap rates Exchange rates

Spot 123.00 125.00

1 -0.20122.80-0.25 124.75

2 -0.40 122.60 -0.50124.50

3 -0.60122.40-0.75 124.25

(a) Whatgap(threemonthsagainstonemonth)shouldbeopenedto take advantage of the expected movement in rates?

Buy dollars 1 month at 122.80 20 points benefit

Sell dollars 3 months at 122.40

60points cost

Cost of opening gap 0.40

40points net cost

(b) How much profit would be generated on a principal amount of US$1,000,000 if rates move as expected? Assume that when the gap is closed, the 2 month yen interest rate is 0.30% p.a.

One month later...

Profit = ¥101,100 = US$812.05 (at 124.50)

The profit can be thought of as:

360SOLUTIONS

$Spot¥

1,000,000 122.80

125.00122,800,000

-1,000,000 +125,000,000

0.30%-2,200,000

1,000,0001,000,000125,000,000125,000,000

$2 months¥

1,000,000 122.40

+1,000,000 124.50 -124,500,000 +2,201,100

Profit101,100

1,000,0001,000,000125,400,000124,500,000

122,400,000

Benefit of closing gap

 cost of opening gap¥500,000400,000 

¥100,000

plus interest from lending

¥2,200,000 for two months

¥ 1,100 ¥101,100

CHAPTER 10

10.1Abankwritesaeuroput/USdollarcallfor10,000,000facevalue.The

strike price is1 = US$0.9000; time to expiry 4 months and the premium 2.00%. (a) Calculate the premium in US dollars if the current spot rate is1 = US$0.9100 Premium = 10,000,000 × 0.02 × 0.9100 = US$182,000 (b) Calculate the pay-out if the spot rate at expiry turns out to be1 = US$0.8950. Pay-out = 10,000,000 (0.9100 - 0.8950) = US$150,000 (c) What would the spot rate at expiry need to be for the pay-out to break-evenwiththefuturevalueofthepremiumgiventhatthe4 month dollar interest rate is 3.00% p.a. (120/360)? FV(Premium) = 182,000 (1 + 0.03 × 120/360) = US$183,820

Ifb= break-even rate,

10 000 000 09100 183 820

08916,,(. ) ,

. b b

10.2Use a 3-step binomial model to calculate the premium of a 3 monthUS$ call/S$ put given:

Spot rates= 1.7000

Forward ratef= 1.6940

Strike pricek= 1.7100

Face value US$1,000,000

3 month US$ interest rate 3.0% p.a. (90/360)

3 month S$ interest rate 1.6% p.a. (90/360)

up-down movement S$0.0200 per month +/- drift

Drift = (1.6940 - 1.7000)/3 = -0.0020

SOLUTIONS361

Today 1 month 2 months 3 months Pay-Off p E(PO)

1.7540 0.04401/80.0055

1.7360

1.7180 1.7140 0.00403/80.0015

1.7000 1.6960

1.6780 1.6740 0 3/80.0000

1.6560

1.6340 0 1/80.0000

0.0070

PremiumS per US 

00070 1 0016 90 360 0006972./(. /)$. $

SperUS$, $, ,6 972 1 000 000

10.3Identify the arbitrage opportunity available given the following

prices. Articulate the actions that need to be taken to profit through the above arbitrage. Calculate the profit that could be made on a face value of £10,000,000.

Spot rate£1 = US$1.7000

1 year forward rate £1 = US$1.6950

1 year £ call (k = 1.7200) premium US$0.0230

1 year £ put (k = 1.7200) premium US$0.0480

1 year US$ interest rate4.0% p.a. (360/360)

PV F K

cp()(. .)/(.) $. .   16950 17200 1 004 00240 00US

230 00480 00250.$.US

To make a profit: pay 240 points and receive 250 points. Sell 1.72 put and buy 1.72 call =buy £ forward at 1.7200 sell £ forward at 1.6950 loss0.0250

PV loss = 0.0240

Net premium 0.0250

Profit0.0010per £

Profit on £10,000,000 = 10,000,000 × 0.0010 = US$10,000

10.4(a) Use the modified Black-Scholes model to calculate the premium

of a European US$ call with strike price of ¥105.00 given:

Spot US$/¥ 110.00

Expected volatility 15% p.a.

Time to expiry 3 months (90/360)

US$ interest rate 6.50% p.a. (90/360)

362SOLUTIONS

¥ interest rate1.00% p.a. (90/360)

Implied forward rate108.55

cS Nd K Nd d

SK r y t

t d ytrt      ee() ( ) ln( / ) () 12 1 122
2   ln( / ) ()SK r y t tdt  122
1  

Use theztables provided in the Appendix:

t r y    015 0075

1 001 1 000995

1 0065

14 .. ln( . ) . ln( . ) . . . ln( / ) ln(     

1 0062975

0997516

0984380

1e e rt yt

SK10 105 0046520

1 2 000995 0062975 05

2 /). (/)(. . .    ry t(. ) ) . . (. . )/.015 025 0010444

0046520 0010444 007

2 1  d5 04810

0481017 0075 04060

06844 01

2 1    . ... () . .(d

Nd06879 06844 068475

06554 06 06591 06

2 ..). (). .(. .  Nd554 065762

110 0984380 068475 105 0997516 0).

.. ..    c65762

7415 6888

527..

. (b)

Use Black"s model:

cFNdKNd d FK t t dd t rt     e[() ()] ln( / ) 12 1 122
21
   to calculate the premium of the same option as in (a):

SOLUTIONS363

t r rt     

015 0075

1 001 1 000995

099751

14 .. ln( . ) . .e6

10855 105 0033251

12 05015

22
ln( / ) ln( . / ) . /.(.)FK t025 0002813

0033251 0002813 0075 04810

1 2 .. (. . )/. .  d d   0481017 0075 0406006844 01 06879 06 1 ... () . .(. .Nd844 068475

06554 06 06591 06554 0657

2 ). (). .(. . ).   Nd62

0997156 10855 068475 105 065762

527c.(.. .).asin(a)

(c) Use put-call parity to calculate the premium of the 105.00 put with the same data as in (a). pcFK rt        () .(. .). .e

527 10855 10500 0997516 173

CHAPTER 11

11.1An exporter with the identical exposure as in Example 11.2 enters

into a participating collar to hedge euro receivables. The exporter buys a euro put/dollar call with the strike of 0.8762 for1,000,000 at a premium of 1.0% and writes a euro call/dollar put with the strike of

0.9000 for600,000 at a premium of 1.84%.

(a) Calculatethefuturevalueofthenetpremiumpayableindollars. Net premium payable = 1,000,000 × 0.01 - 600,000 × 0.0184 =1.040 Note : premium received > premium paid

Net premium receivable =1,040 = US$ 936

FV(Net premium receivable) = 936 × (1+ 0.03 × 90/360) = US$943.02

364SOLUTIONS

(b) Calculate the proceeds from selling1,000,000 if the spot rate at maturity is: (i) 0.8662 Proceeds = 1,000,000 × 0.8762 + 943.62 = US$877,143.62 (ii) 0.8862 Proceeds = 1,000,000 × 0.8862 + 943.62 = US$887,143.62 (iii) 0.9062 Proceeds = 600,000 × 0.9000 + 400,000 × 0.9062 + 943.62 = US$903,423.62

11.2AforeigncurrencyborrowerwiththesameexposureasinExample11.3

constructs a participating option to hedge Swiss franc liabilities. The borrower buys a US dollar put/Swiss franc call for SF 25,395,300 with a strike of 1.2300 at a premium of 3.0% and writes a US dollar call/Swiss franc put for SF 12,697,650 with a strike 1.2300 at a premium of 2.4%. (a) Calculatethefuturevalueofthenetpremiumpayableindollars.

Put premium

25 395 300

12500003,,

..US$609,487.20

Call premium

12 697 650

125000024,,

..US$243,794.88

Net premium payableUS$365,692.32

FV (Net premium)365,692.32(1+0.05/2)US$374,834.63

SOLUTIONS365

850,000875,000900,000925,000950,000

0.8524 0.8613 0.8703 0.8792 0.8881 0.8970 0.9059

Spot rate at maturity 1 = US$x

Dollar proceeds from 10,000,000

(b) Calculate the dollar cost of repaying the Swiss franc loan prin- cipal plus interest if the spot rate at maturity is: (i) 1.2000

Put is exercised and call lapses

CostUS 25 395 300

12300374 83463 21 021 42000,,

.,. $,,. (ii) 1.2400

Put lapses and call is exercised

Cost 12 697 650

1230012 697 650

12400374 83463,,

.,, .,. US$,,.20 938 16763 (iii) 1.3000

Put lapses and call is exercised

Cost 12 697 650

1230012 697 650

13000374 83463,,

.,, .,. US$,,.20 465 55039 (c) Calculate the effective borrowing cost in percent per annum of the Swiss franc loan if the spot rate at maturity is:

Effective borrowing cost

US cost$,, ,,20 000 000

20 000 000200

(i) 1.2000: 21 021 420 20 000 000

20 000 000200 1021,, ,,

,,.%p.a. (ii) 1.2400: 20 938 16763 20 000 000

20 000 000200 938,,. ,,

,,.%p.a. (iii) 1.300: 20 465 55039 20 000 000

20 000 000200 466,,. ,,

,,.%p.a.

11.3A funds manager with the same exposure as in Example 11.5 buys a

collarbybuyingadollarcallat110.00for¥1,111,152,778atapremium of 3.25% and writing a dollar put at 109.00 for ¥777,806,945 at a premium of 2.00%.

Net premium 

777 806 945 002 1111152 778 00325,, . ,,, .

¥, ,

$,.20 556 326

186 87569US

FV(Net premium)US 

 186 87569 1 005365

360, . .$196 34926,.

(a) Calculate the effective yield if the spot rate at maturity is:

366SOLUTIONS

Effective yieldUS proceeds$,,

,,10 000 000

10 000 0003

65
360
(i) 100.00; call lapses and put is exercised

US proceeds$,,

.,,, 777 806 945

10900333 345 833

100196 34926

10 272 95260

277.
$, , . .% US

Effective yield p.a.

(ii) 110.00; call and put both lapse

US proceedsUS$,,,

.,. $,1111152 778

11000196 34926 9 905 03963

095,.
.%Effective yield p.a. (iii) 120.00; exercise call, put lapses

US proceedsUS$,,,

.,. $,1111152 778

11000196 34926 9 905 03963

095,.
.%Effective yield p.a. (b) If the spot rate at maturity is 114.00, calculate the effective yield percent per annum versus being:

Effective yield = -0.95% p.a. (again)

(i) unhedged

US proceedsUS

Eff$ ,,, .$, , .1111152 778

114009 746 95419

ective yield p.a.253.% (ii) invested in dollars

Effective yield = 0.05 × 365/360 = 5.07% p.a.

(iii) hedged with a bought dollar call (strike 110.00)

US proceedsUS$,,,

.,. $,1111152 778

11000344 93788 9 756 45101

240,.
.%Effective yield p.a. If the spot rate at maturity turned out to be 114.00, the best outcome would have occurred if the investor was invested in US dollars.

11.4A 2 for 1 strategy refers to the practice of buying the option requiredto hedge an underlying exposure and selling twice the face value of

SOLUTIONS367

the opposite type of option (call or put) usually to earn enough premium to make the net premium zero. One month ago, a foreign exchange trader bought £10,000,000 against US dollars at an outright 4 month forward rate of 1.4800. The spot rate has since risen to 1.5150 and the 3 month forward rate is now 1.5100. The 3 month (90/360) dollar interest rate is 3.00% p.a. The trader considers buys a sterling put (strike 1.5100) premium

2.0% for face value £10,000,000 and sells a sterling call (strike 1.5200)

premium 1.0% for twice the face value (£20,000,000). (a) Calculate the future value of the net premium in dollars. Net premium = 10,000,000 × 0.02 - 20,000,000 × 0.01 = 0 (b) Calculate the profit if the spot rate at expiry is: US$ cost of buying £10,000,000 at 1.4800 =US$14,800,000 FV(US$14,800,000) = 14,800,000 × (1 + 0.03 × 3/12) = US$14,911,000 This assumes that short-term pound interest rates are around

3.00% p.a.

Profit = Proceeds of sale of £10,000,000 under 2 for 1:

14,911,000

(i) 1.4500: put exercised, calls lapse

Proceeds

10,000,0001.5100US$15,100,000

Profit

15,100,00014,911,000US$189,000 (ii) 1.5000: put exercised, calls lapse

Proceeds

10,000,0001.5100US$15,100,000

Profit

15,100,00014,911,000US$189,000 (iii) 1.5500: put lapses, calls are exercised

Trader sells £20,000,000 at 1.5200:

US$ proceeds

US$30,400,000

Trader needs to buy £10,000,000 at 1.5500:

US$ cost

US$15,500,000

Profit

30,400,00015,500,00014,911,000US$11,000 (c) Draw the profit profile showing profit against various possible exchange rates at expiry.

368SOLUTIONS

CHAPTER 12

12.1Calculate the premium of an option that will pay US$1,000,000 if the

A$/US$spotrateisbelow0.5300in90daystimegiventhefollowing:

Current spot rate A$/US$ 0.5540

3 month LIBOR 3.25% p.a. (90/360)

Expected probability of spot being below 0.5300 24%

Digital put premium

ANd rt(())1 1 2 Here: A Nd r t   US$ , , (). . /1 000 000 1024
00325

90 360

2

PremiumUS

1 000 000 024

1 00325 90 360236 1623,, .

./$,.6

12.2Power option

Calculate the premium of a call with a pay-out equal to (X- 105.00) 3 assuming the binomial tree as shown in Exhibit 10.3. The 6 month yen interest rate is 0.50% p.a. and the current spot rate is US$1 =

¥100.00.

SOLUTIONS369

-300,000-200,000-100,0000100,000200,000300,000400,000

1.4000 1.4300 1.4600 1.4900 1.5200 1.5500

Spot rate at expiry

Profit in US$

Outcome Pay-outProbability Expected pay-out

118 13

3 = 1,197 1/64 ¥34.33 112 7
3 = 3436/64 ¥32.16 106 1
3 = 115/64 ¥ 0.23

100 020/64 0

94 015/64 0

88 06/64 0

82 01/64 0

Expected pay-out¥66.72

Premium 6672

1 0005 6 126655.

./¥. If the face value of the power option is US$1,000,000:

Premium

US ¥, , $,66 720 000

667 200

12.3Improving forward

A Japanese importer needs to buy US dollars at a future date. The spot rate is currently US$1 = ¥122.00 and the market forward rate is

120.30. A bank offers the importer a deal in which the rate at which

the importer will buy US dollars on the forward date will be either

121.00 if the spot rate remains above 115.00 or 118.00 if the spot rate

falls below 115.00 prior to the maturity date.

How does the bank engineer the improving forward?

Method 1

Buy a 121.00 call that knocks-out at 115.00 and sell a 121.00 put that knocks-out at 115 Buy a 118.00 call that knocks-in at 115.00 and sell a 118.00 Put that knocks-in at 115.00 If the spot never reaches 115.00, the importer has a bought 121 call and a sold 121 put = 121 forward If the spot reaches 115.00. the importer has a bought 118 call and a sold 118 put = 118 forward and the 121 forward knocks out.

Method 2

BuyUSdollarsforwardat120.30andbuyadigitalputwithapay-out of ¥3.00 if the spot rate falls below 115.00. The premium of the digital put must be equal to the present value of ¥0.70. If the spot rate never reaches 115.00, the importer has effectively bought dollars at 120.30 + 0.70 = 121.00. If the spot reaches 115.00,

370SOLUTIONS

theimportercollectsthe¥3.00pay-outfromthedigitalputtoachieve an effective rate = 120.30 + 0.70 - 3.00 = 118.00. Notice it is possible to construct the same pay-off using a forward and a digital as with four barrier options.

12.4Currency linked noteAninvestorplacesUS$1,000,000ondepositatafixedrateof3.5%p.a.for 6 months (180/360) and purchases a one-touch either side digitaloption with a pay-out of US$10,000 if the US$/¥ spot rate remainswithin a range of 120.00 to 130.00 for the entire 6 months. Thepremium of the option is US$2,948.40.

Calculate the effective yield if:

Interest on deposit US1 000 000 0035 180 360 17, , . / $ ,500

Digital pay out USp$, .%,

,.%10 000 3510 00017 500200 .a.

FV(Premium)US 

2 94840 1 0035 180 360 3 0003,. ( . /) $, .

503 000

17 500060%,

,.%p.a. (a) The spot rate remains within the range

Digital is exercised

Effective yield = 3.50% + 2.00% - 0.60% = 4.90% p.a. (b) The spot rate does not remain within the range

Digital is not exercised

Effective yield = 3.50% - 0.60% = 2.90% p.a.

CHAPTER 14

14.1Market scenario:

Spot1 = US$0.9250

6 month euro 3.50% p.a. (180/360)

6 month dollars 2.75% p.a. (180/360)

f 092501 00275 2

1 0035 209216.(./)

(./). A dealer purchased10,000,000 at a 6 month outright forward rate of

0.9216 and has not covere7d the position.

SOLUTIONS371

(a) Calculate the 2 standard deviations stressed rate if spot rate changes are assumed to be normally distributed and volatility is expected to be 9.2% p.a.

Stressed rate = 0.9250e

-2(0.092)×90/360 = 0.8834 (b) Calculate the value at risk

VaR = 10,000,000(0.9250 - 0.8834) = US$416,000

14.2Delta hedgingOn a day when the spot rate was US$1 = ¥123.50 a bank sold a US$call/¥ put with face value US$10,000,000 and strike price 122.50. Apricing model displayed the following premiums for the sold call:

Spot rate Premium

122.50 ¥2.08

123.00 ¥2.31

123.50 ¥2.57

124.00 ¥2.84

(a) Calculate the average delta between 123.00 and 124.00. What transaction should the bank do to delta hedge?

Average delta

284 231

124 123053...

Thebanklosesmoneyonthesoldcallasthespotraterises,soto delta hedge the bank needs to buy US$5,300,000 against yen. One week later the spot rate has fallen to 123.00 and the pricing model displays the following premiums:

Spot rate Premium

122.50 ¥2.08

123.00 ¥2.31

123.50 ¥2.57

(b) Calculate the revised average delta. What transaction should the bank do to adjust its delta hedge?

Average delta

257 208

1235 1225049..

... To be delta neutral the bank needs to hold US$4,900,000. There - fore, to adjust the delta hedge the bank would need to sell

US$200,000.

Note : The bank would realize a loss as a result of adjusting the delta hedge. It purchased US$200,000 at 123.50 and sold them at 123.00 for arealizedlossof¥100,000=US$813.Thisoffsetsomeofthepremium

372SOLUTIONS

received by the bank when it sold the option which was 10,000,000 ×

2.57 = ¥25,700,000 = US$208,097.

14.3Credit riskTwo months ago a bank purchased A$10,000,000 from XYZ Limitedat the then 5 month forward rate of A$1 = US$0.5230. The prevailingrates today are:

Spot A$/US$ 0.5620

3 month swap rate -0.0010

3 month US$ rate 3.20% p.a. (90/360)

3 month credit risk factor 5.0%

Calculate in US$ NPV terms:

(a) The bank"s marked-to-market exposure on XYZ Limited

Close out rate

FV(MTM)

05610

10 000 000 05610 05.

, , ( . . 230 380 000

380 000 1 0032 90 360 376)$,

,/( . /) $  US

MTM US ,984

(b) The bank"s estimated potential exposure on XYZ Limited Estimated potential exposure = 5,230,000 × 0.05 = US$261,500.00 (c) The bank"s pre-settlement risk on XYZ Limited

PSR MTM PE

US  376 984 261 500

638 484,,

$, XYZ Limited wishes to sell more US dollars to the bank for the same value date. (d) How large a deal can be done if the bank"s PSR limit on XYZ is

US$2,000,000?

Available credit line

2,000,000638,484US$1,361,516 CRF 5.0%

Maximum deal size

1,361,516100/5US$27,230,320

SOLUTIONS373

APPENDIX

Cumulative Standard Normal

Distribution (

01,)

N(z)=p(Z

See table overleaf.

375

376APPENDIX

z 0.000.010.020.030.040.050.060.070.080.09

0.00.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.10.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

0.20.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.30.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.40.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.50.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.60.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.70.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.80.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

0.90.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.00.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.10.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.20.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.30.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177

1.40.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.50.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.60.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545

1.70.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633

1.80.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706

1.90.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

2.00.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.10.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

2.20.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890

2.30.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916

2.40.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.50.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.60.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964

APPENDIX377

z 0.000.010.020.030.040.050.060.070.080.09

2.70.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974

2.80.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981

2.90.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.00.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

3.10.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993

3.20.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995

3.30.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997

3.40.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

3.50.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998

3.60.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.70.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.80.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.90.99995 0.99995 0.99996 0.99996 0.99996 0.99996 0.99996 0.99996 0.99997 0.99997

4.00.99997 0.99997 0.99997 0.99997 0.99997 0.99997 0.99998 0.99998 0.99998 0.99998

5.00.9999997 0.9999997 0.9999997 0.9999998 0.9999998 0.9999998 0.9999998 0.9999998 0.9999998 0.9999998

6.00.9999999990 0.9999999991 0.9999999991 0.9999999992 0.9999

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