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MATH3090/7039: Financial mathematics
创建日期：2024-05-06 16:28:06
所属分类：数学mathematics
标签： MATH3090
Financial mathematics

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MATH3090/7039: Financial mathematics

Assignment 2
Semester I 2022
Due 5pm May 5 Weight 10 %
Total marks 30 marks
Submission: Submit a copy of your assignment and your code on Blackboard.
Notation: “Lx.y” refers to [Lecture x, Slide y]
Assignment questions - all students
1. (8 marks) (Investment)
You are considering a 5-year investment project which is expected to cost $1, 000, 000. In
each year, you have decided that there are 3 possible states of the economy: good, average,
and poor. In each individual year there is a 35% chance of the economy being good and a
15% chance of it being poor. You forecast the following net cashflows for the project:
Economy Year 1 Year 2 Year 3 Year 4 Year 5
Good 300,000 350,000 400,000 350,000 250,000
Average 250,000 275,000 325,000 275,000 175,000
Poor 200,000 225,000 250,000 225,000 150,000
You have arranged the following sources of funding:
(i) $200,000 from a 5-year fixed interest loan whose annual loan payments are $48,126.91.
(ii) $250,000 from a 5-year zero-coupon bond with a face value of $350,000.
(iii) $300,000 from an ordinary share issue where a dividend of $18,000 will be paid in one
year and it is expected to grow at 3% per annum.
(iv) $250,000 from a 5-year coupon-paying bond issue whose coupon rate is 7% and face
value is $250,000.
Should you invest in the project? (Use discrete compounding.)
Note: For the part involving Newton method, choose y0 = 0.0655 and apply only one iteration.
2. (10 marks) (Yield curve modelling)
Consider a binomial model of the yield curve over 3 years where y0,1 = 5%. The probability
of an up movement in 1-period forward rates for year t = 2, 3 is pt = 0.4 + 0.1t, and 1-period
forward rates can go up by a factor of u = 1.6 or down by a factor of d = 0.9. Calculate the
zero-coupon bond yield curve and the implied 1-period forward rates embedded in this yield
curve.
3. (12 marks) (Yield curve and swap pricing)
Assume that you observe the following yield curve for for government’s coupon paying bonds.
ˆ There are a total of 30 bonds.
MATH 3090/7039 – 1 – Roxane Foulser-Piggott
– Assignment 2 –
Table 1: Bond prices
k prices k prices k prices
1 98,828.9817 11 89,083.9301 21 81,633.1317
2 97,812.0511 12 87,944.5962 22 81,287.3700
3 96,937.8969 13 86,976.3584 23 81,087.7608
4 96,159.9962 14 85,928.4188 24 80,919.5136
5 95,269.2339 15 84,982.5065 25 80,780.7515
6 94,353.5669 16 84,248.2589 26 80,669.7282
7 93,276.0334 17 83,540.8304 27 80,584.8196
8 92,237.8837 18 82,911.5228 28 80,524.5143
9 91,261.0455 19 82,417.0923 29 80,487.4060
10 90,214.0597 20 82,009.0742 30 80,472.1856
ˆ For the k-th bond, k = 1, . . . , 30, the maturity is k years.
ˆ The face value is $100,000 and the coupon rate for the k-th bond, k = 1, . . . , 30, is 4%.
ˆ The prices of the bonds are given in the following table
Assume that all the coupon payments are made annually. Use continuous compounding.
a. (4 marks) Modify the Newton iteration program that you developed for Assignment 1 to
compute the yield to maturity (YTM) for each bond. Submit a table similar to Table 1
with the “price” column being replaced by the “YTM” column filled with the computed
YTMs.
Plot YTMs vs maturities and comment. You may find the Matlab function plot useful.
For the initial guess, choose 10%. Use the stopping criteria
|yn+1 − yn| < 10−6.
b. (4 marks) Taking as input the computed YTMs in part (a), implement in a Matlab pro-
gram to compute spot zero-coupon bond yield curve embedded in the observed coupon-
paying bond yield curve, and the implied one-year forward rates.
Submit Table 2 filled with computed values.
Table 2: Table for Question 2 (b)
period spot forward
1 . . . . . .
2 . . . . . .
...
...
...
30 . . . . . .
c. (4 marks) Suppose you enter into a 30-year vanilla fixed-for-floating swap on a notional
principal of $1,000,000 where you pay the fixed rate of 6.15% and the counter-party pays
the yield curve plus 1%.
Code in Matlab a program to compute the swap value. Submit a table of results, similar
to the table on L5.12.
MATH 3090/7039 – 2 – Roxane Foulser-Piggott
– Assignment 2 –
No more questions will be added.
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