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日期:2019-06-10 09:48

Financial Markets Instruments

25877

Autumn 2019

Assignment 2

Due Date:

Friday June 14, 2019

You may use EXCEL (or MATLAB or R ) to help you complete the

assignment.

Total Marks : 70 (Questions 1 – 7, each worth 10 marks).

2

Question 1.

i. The current exchange rate is USD0.80 per AUD and the 6-month interest rates are 4% in the

US and 6% in Australia. Find the 6-month forward exchange rate for AUD into USD.

ii. The 90-day bank bill rate is 6% per annum and the 180-day rate is 6.5% per annum. The

Australian 90-day bank bill futures maturing in 90-days quoted as 93.00.

(a) Show that there is an arbitrage opportunity.

(b) By constructing the appropriate arbitrage portfolio, determine the profit per $100, 000

face value of the 90-day bank bill.

(c) Given that the bank bill rates are correct, what should the bank bill futures quote be?

Question 2.

In this question all interest rates are assumed to be continuously compounded.

(a) Show that the instantaneous forward rates

are given in terms of the T-maturity

risk-free rates, r(0, T), by

(b) Given that the yield curve is

find the instantaneous forward rates,

(c) If the yield curve is changed to

find the corresponding instantaneous forward rates,

(d) Plot and compare the yield curves represented by

(e) Plot and compare the instantaneous forward rates

(f) Based on the results of (d) and (e), what can you say about the potential effects of small

perturbations (or errors) in the yield curve on the (instantaneous) forward curve?

Hint: Use high resolution time grids when plotting the yield curves in (d) and particularly

when plotting the forward curves in (e).

Note also that in sin(t) in (c), t should be given in radians, not degrees. ]

3

Question 3.

Consider a situation where BHP needs a 3-year floating rate loan, and Rio Tinto needs a 3-year

fixed rate loan. They have been quoted the following 3-year semi-annually compounded rates:

FIXED FLOATING

BHP 6.3% BBSW + 0.3%

Rio Tinto 7.6% BBSW + 0.6%

Assume BBSW is 6% p.a.

(a) Design a swap that will net a bank, acting as an intermediary, 30 basis points and provide

the remaining benefit in the ratio 2 : 1 between BHP and Rio Tinto. The bank assumes all

default risk.

(b) If the principle of the loan is AUD$40 million, then what is the value of the swap (with

the bank) to BHP?

Question 4.

Consider at date T0, a 6-month LIBOR standard swap contract with maturity 6 years. Suppose

that the swap nominal amount is $1 million and the rate of the fixed leg is 6%. At date T0, zero

coupon-rates are as given in the table below:

Maturity (yrs) ZC Rates (%)

(a) What is the price formula for this plain vanilla swap?

(b) Compute the discount factors corresponding to the zero-coupon rates in the table above.

(c) Give the price of this swap.

(d) What is the swap rate such that the price of this swap is zero?

(e) An investor has bought 100,000 5-year bonds with a 7.2% annual coupon rate and a face

value of $1,000. The investor fears an increase in interest rates. Carefully explain how

interest rate swaps can be used to protect the bond portfolio and determine how many

swaps are needed.

(f) Assume there is a parallel shift upwards in the zero-coupon yield curve of 0.3%. What is

the investor’s new position with and without the hedge?

4

Question 5.

The following bonds are available

Maturity

year

Face value Coupon

rate (pa)

Yield to

maturity

2.0 $ 150,000 4.50% 3.60%

2.0 $ 180,000 4.00% 3.56%

1.5 $ 200,000 3.50% 3.41%

1.0 $ 200,000 3.00% 3.34%

All the bonds pay semi-annual coupons.

(a) Find the prices of each of the four bonds.

(b) Write down the values of the four bonds in terms of the prices of four zero coupon bonds,

each with face value $1 and with maturities of 0.5, 1.0, 1.5 and 2.0 years, respectively.

(c) Use the answer to Parts (a) and (b) to determine a 4 × 4 system of linear equations for the

prices of these zero coupon bonds.

(d) Solve these linear equations to find the prices of the zero coupon bonds. [Hint: it’s

probably easier to do this using Excel, Matlab, Maple, Mathematica or another suitable

piece of software rather than to do it by hand.]

(e) From Part (d), determine the semi-annual interest rates for 6 months, 1 year, 18 months

and 2 years.

(f) How sensitive are these interest rates to small changes (or errors) in the quoted bond

yields? [Hint: see what happens if you add or subtract 10 basis points, for example, from

some or all of the bond yields].

5

Question 6.

Given below are the short term deposit rates, futures prices and swap rates for Friday May 17,

2019.

Deposit Rates 90-day Bank Bill Futures Quarterly Swaps

Maturity Rate

(% p.a.) Settle Price Maturity

(yrs) Swap Rate (% p.a.)

1d 1.65 Jun-19 98.51 1.00 1.2278

1w 1.605 Sep-19 98.70 2.00 1.2292

1m 1.67 Dec-19 98.75 3.00 1.3022

3m 1.605 Mar-20 98.79 4.00 1.4645

6m 1.775 Jun-20 98.80 5.00 1.5345

Use the above data and Excel to answer the questions below, and attach the printouts of the

spreadsheet containing the forward rates and their maturities.

(a) Construct the 3-month forward rate curve using the naive bootstrap method. You may use

linear interpolation to compute the discount factors and the swap rates that may be

required.

(b) Apply the Frishling and Yamamura smoothing algorithm to construct the smoothed 3-

month forward rate curve. You will need to use Excel’s Solver for this part.

(c) Determine the price of a 4.5-year semiannual 8% coupon bond with face value

$10, 000, 000 according to the curves constructed in parts (a)–(b).

[You may use the Spreadsheets provided on UTSOnline to help you solve this problem].

Question 7.

Three European call options on a stock all expire in exactly 9 months’ time, and have strike prices

of $90, $100 and $110, respectively. Their market prices are $21.43, $16.20 and $12.05,

respectively, and the current market price of the stock is $100.

(a) Using only these options, create a portfolio to take advantage of a view that the stock

price will move substantially from its current value before in the next year. [This can be,

but need not be, a standard portfolio.]

(b) Construct a table show the profit from this portfolio as a function of the stock price in a

year’s time.

(c) Determine the range of stock prices at expiry for which your portfolio would result in a

profit [and remember that the cost of setting up the portfolio occurs a year before any

payoff].


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