– Assignment 1 –
MATH3090/7039: Financial mathematics
Assignment 1
Semester I 2024
Due Tuesday March 19 1pm
MATH3090 total marks 24 marks
MATH7039 total marks 30 marks
Submission:
ˆ Submit onto Blackboard softcopy (i.e. scanned copy) of (i) your assignment solutions, as well
as (ii) Matlab/Python code for Problem 3. Hardcopies are not required.
ˆ Include all your answers, numerical outputs, figures, tables and comments as required into
one single PDF file.
ˆ You also need to upload all Matlab/Python files onto Blackboard.
General coding instructions:
ˆ You are allowed to reuse any code provided/developed in lectures and tutorials.
Notation: “Lx.y” refers to [Lecture x, Slide y]
Assignment questions - all students
1. (6 marks) a. (3 marks) Suppose a company issues a zero coupon bond with face value
$10, 000 and which matures in 20 years. Calculate the price given
(i) an 8% discrete compound annual yield, compounded annually,
(ii) an 8% continuous annual yield,
(iii) a nonconstant yield of y(t) = 0.06 + 0.2te−t2 .
b. (3 marks) A 10 year $10, 000 government bond has a coupon rate of 5% payable quarterly
and yields 7%. Calculate the price.
2. (6 marks) Consider the cash flow
C0 = −3x, C1 = 5, C2 = x
(at periods 0, 1, 2 respectively) for some x > 0.
a. (3 marks) Apply the discount process d(k) = (1 + r)−k so that the present value is
P =
2∑
k=0
d(k)Ck.
What is the range of x such that P > 0 when r = 5%?
b. (3 marks) The IRR (internal rate of return) is r such that P = 0. For what range of x,
will there be a unique, strictly positive IRR?
MATH 3090/7039 – 1 – Kazutoshi Yamazaki
– Assignment 1 –
Cashflows (Ci) Times (ti)
2.3 1.0
2.9 2.0
3.0 3.0
3.2 4.0
4.0 5.0
3.8 6.0
4.2 7.0
4.8 8.0
5.5 9.0
105 10.0
Table 1: Bond cashflows
3. (8 marks) In this question, consider a bond with the set of cashflows given in Table 1. Here,
note that the face value F is already included in the last cashflow. Let y be the yield to
maturity, ti be the time of the i
th cashflow Ci, and PV = 100 be the market price of the
bond at t = 0. Assume continuous compounding. Then, y solves
PV =
∑
i
Cie
−yti . (1)
a. (3 marks) Write out the Newton iteration to compute yn+1 from yn (see L2.49). Specif-
ically, clearly indicate the functions f(y) and f ′(y).
b. (5 marks) Implement the above Newton iteration in Matlab using the stopping criteria
|yn+1 − yn| < 10−8.
Fill in Table 2 for y0 = 0.05 (add rows as necessary).
In addition, try with larger values for y0 and observe the accuracy and convergence
speed. How does the performance change?
n yn |yn − yn−1|
0 . . . N/A
1 . . . . . .
2 . . . . . .
3 . . . . . .
...
...
...
Table 2: Output
4. (4 marks) In the Constant Growth DDM model, the present value of the share is
PV =
∞∑
t=1
Dt
(1 + k)t
, (2)
where D1, D2, . . . are (non-random) dividends and k > 0 is the required rate of return.
Suppose D0 > 0, k > 0 and g > 0.
Derive the formula for the present value (2) when
Dt = D0(1 + g)
⌈t/2⌉, t = 1, 2, . . . ,
MATH 3090/7039 – 2 – Kazutoshi Yamazaki
– Assignment 1 –
where ⌈x⌉ is the smallest integer greater than or equal to x. What is the condition of g so
that the PV is finite? To get full marks, you need to write an explicit expression (without
summation).
Assignment questions - MATH7039 students only
6. (3 marks) In Q4, derive the formula for the present value (2) if
Dt = D0(1 + g)
max(t,10), t = 1, 2, . . . .
What is the condition of g so that the PV is finite? To get full marks, you need to write an
explicit expression (without summation).
7. (3 marks) Recall that the discount rate corresponding to a simple interest rate r when maturity
is T is given by
d(T ) =
r
1 + rT
.
See L2.16.
Suppose r = 3%. Let
f(T ) = d(0) + Td′(0) +
T 2
2
d′′(0)
be the second-order (Taylor) approximation and
ε(T ) = ln
( |d(T )− f(T )|
T 3
)
be a (log) normalised error. Complete the following table:
T d(T ) f(T ) ε(T )
10 · · · · · · · · ·
5 · · · · · · · · ·
1 · · · · · · · · ·
You can use Matlab but you do not need to submit the code for this problem.
MATH 3090/7039 – 3 – Kazutoshi Yamazaki