MATH3090/7039: Financial mathematics Assignment 1 Semester I 2023

Submission:

❼ Submit onto Blackboard softcopy (i.e. scanned copy) of (i) your assignment solutions, as well

as (ii) Matlab code for Problems 3 and 7. Hardcopies are not required.

❼ Include all your answers, numerical outputs, figures, tables and comments as required into

one single PDF file.

❼ You also need to upload all Matlab files onto Blackboard.

General coding instructions:

❼ 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.  (1 mark) What is the price of a 90 day T-bill with the face value of $100, 000 and which is yielding 4% simple interest?

b.  (3 marks)  Suppose a company issues a zero coupon bond with face value $100, 000 and which matures in 10 years. Calculate the price given

(i) a 6% discrete compound annual yield, (ii) a 6% continuous annual yield,

(iii) a nonconstant yield of y(t) = 0.04 + 0.002te−t .

c.  (1 mark) A 5 year $100, 000 government bond has a coupon rate of 5% payable semian- nually and yields 6%. Calculate the price.

d.  (1 mark)  Repeat c for the case it is payable quarterly.

2.  (3 marks)  Recall that the discount rate corresponding to a simple interest rate r when maturity

is T is given by

r     

d(T) =

f(T) = d(0) + Td\(0)

be the first-order (Taylor) approximation and

ε(T) =

You can use Matlab but you do not need to submit the code for this problem.

3.  (6 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 ith  cashflow Ci, and PV = 100 be the market price of the bond at t = 0. Assume continuous compounding. Then, y solves

PV = 工 Cie−yti   .

i                                  (1)

Table 1: Bond cashflows

a.  (1 mark) Write out the Newton iteration to compute yn+1  from yn  (see L2.49). Specifi- cally, 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| < 10−6 .

Fill in Table 2 for y0 = 0 (add rows as necessary).

In addition, try with larger values for y0  and observe the accuracy and convergence speed. How does the performance change?

Table 2: Output

4.  (3 marks) In the Constant Growth DDM model in L.3.15, suppose the dividend at time t is instead

Dt = D0(1 + g)「t/2⌋,    t = 0, 1, . . . ,

where「x⌋ is the largest integer less than or equal to x.  Derive the formula for the present value. What is the condition of g so that the PV is finite?

5.  (3 marks)  Show that a geometric random variable X with success probability p > 0 such that P(X = k) = (1 − p)k−1p,    k = 1, 2, . . .                                       (2)

has mean 1/p.

6.  (3 marks) In the pricing of a coupon-paying bond in L3.36, suppose the default time is X , which is geometrically distributed according to (2).  We assume recovery is zero.  In other words, at each t = 1, 2, . . . ,T, coupon  is received if t < X and nothing is received otherwise. Face value  is received at T if T < X and nothing is received otherwise. Derive the formula of the present value of the bond.

Assignment questions - MATH7039 students only

7.  (3 marks)  Use Newton’s method to compute z ∗  such that     ( 1 + )z *  − 1 = 0.1051.

Clearly indicate the functions f(y) and f\(y) (you may need to choose f wisely). Fill in the following table with your selection of y0  (just iterate 5 times).

What is the smallest number (integer) of compounding periods per year n∗   such that the effective annual yield exceeds 10.51% if the nominal annual yield is 10%?

8.  (3 marks) A sequence (an; n = 1, 2, . . .) converges (i.e. limn→∞ an  exists) if n !→ an  increases and there exists M ∈ R such that an  < M for all n (an  is bounded from above). To confirm that the Euler’s number e exists, do the following. Suppose

en  := ( 1 + )n ,    n ≥ 1.

a.  (1 mark)  Using the binomial theorem, show

en =之(n)  ( 1 − )( 1 − ) ··· ( 1 − ) .

b.  (1 mark)  Show that en+1  > en  for all n ≥ 1.

c.  (1 mark)  Show that the right hand side of (3) is less than 3 for all n ≥ 1.