STAT2032/6046 Financial Mathematics FINAL EXAMINATION Semester 1, 2021

(a)     Consider the following expression: k| am) .

Describe in precise terms, exactly what is meant by this expression. (3 marks)

(b)    Consider the following five expressions:

2| a , 1| a) ,  2| a , 2| a0) , and 2| a) .

Assuming that i  0,arrange the 5 expressions in order of value, from largest value to lowest value. (3 marks)

(c)     Suppose that for a particular annual effective interest rate i  0, the following is true:

1    1  2.09.

Calculate the accumulated value at 1 January 2023, of:

   $1,850 invested on 1 January 2020, plus

   $1,850 invested on 1 January 2021, plus

   $1,850 invested on 1 January 2022.

Give your answer to the nearest dollar and show all working. (4 marks)

(d)     Suppose that the force of interest is given by the following: 

Calculate to the nearest dollar, the accumulated value at t = 14, of $6,000 invested at t = 8. Show all working. (4 marks)

(e)     Consider an annuity paying $55 at the end of each year for the next 55 years. Calculate the

accumulated value at the end of 55 years, at an interest rate of:

(i)     8.5% per annum, convertible monthly. Show all working and give your answer to the nearest dollar.                                                                                          (2 marks)

.

(ii)     8.5% per annum, as simple interest. Show all working and give your answer to the

nearest dollar.                                                                                                  (3 marks)

Hint: the sum of a series 1   2    3    ..   n  

QUESTION 2

Suppose you are paying off a loan, but all that you know about the loan is the following:

   The loan was taken out exactly 2 years and 4 months ago

   The balance is now $246,615.39

   The regular monthly repayments are made in arrears (i.e. at the end of each month) and are

equal to $2,272.76, which will exactly amortise (pay off) the loan at some point in the future

   The next payment is due in exactly one month’s time

   The loan balance will be $214,852.96 in exactly three year’s time from now

(a)     Calculate the applicable interest rate on this loan (express it as an annual effective rate). Show

all working, and give your answer to 3 significant figures. (4 marks)

(b)     Calculate the initial size of the loan when the loan is first taken out. Show all working, and

give your answer to the nearest dollar. (4 marks)

(c)     Calculate the original term of the loan (that is, the length of time that the loan was planned to be paid off, when the loan was first issued). Show all working. (3 marks)

QUESTION 3

Suppose that a project has the following costs:

The project will generate revenue of $40,000 at time t =3, and then $Y at time 5, and then $Z at either time t = 7 or time t = 20.

The project can be planned to receive the revenue of $Z at either t = 7 or t = 20 (as well as receiving the revenue of $40,000 at time t = 3, and $Y at time 5).

Suppose you are a consultant who is paid to give financial advice to the project manager of the above project. The relevant interest rate for this project is 4.5% per annum (effective).

(a)     What is more likely to provide immunisation within this project – having $Z paid at t = 7, or t

= 20? Give reasons for your answer. (2 marks)

(b)     Determine appropriate values for Y and Z, that enables immunisation to occur on this project.

Show all working that proves whether immunisation occurs or not (according to the three   conditions required for immunisation). State clearly whether Z is received at t = 7 or t = 20. (10 marks)

(c)     Suppose your answer to (b) above was that the project can be immunised. The project           manager thanks you for your advice, and says “great, now that the project has its cashflows  immunised I can tell all the investors in the project that their money is safe and that the NPV will be maximised” . What are the most relevant and important things you should say in        response to this? (6 marks)

QUESTION 4

Consider the following 2 separate projects.

For both projects, income ceases after 20 years (i.e. the final income is at time t = 20).

(a)     For each project, determine the IRR and NPV, assuming that a risk discount rate of 6%

effective per year applies to NPV calculations. Give the IRR to the nearest 1%, and give the NPV to the nearest dollar. Show all working. (7 marks)

Suppose now that the initial cost for each project is funded through a loan, which has annual interest given by i  6% per year (effective).

All income from a project is paid against the loan, and then when a position of surplus is achieved,  all income and surplus in each project is invested to earn at an interest rate of i   6% effective per

year.

(b)     For each project, calculate the discounted payback period (DPP), and also calculate the

accumulated value at t = 20 (to the nearest dollar). Show all working. (7 marks)

(c)     Suppose you can only invest in one project, A or B. Based on your answers to (a) and (b), which project would you invest in and why? (2 marks)

(d)     Suppose now that you have been told that the further that income payments occur in the          future, the more uncertain they become - that is, the probability increases that they may not be paid. Does this change your answer to (c) above? If it does not, give reasons for your answer. If it does, also give reasons for your answer. (3 marks)

QUESTION 5

Suppose that the price of a $1000 nominal bond paying annual coupons of 6% per annum, is equal to $988.31. The bond has a term of 6 years and is redeemed at par.

Suppose also that the spot rates for the yield to redemption are given by

sn        4% , for  1  n  5 , with n in years.

(a)     Calculate f2,5 , and express your answer as an annual effective rate. (1 mark)

(b)    Calculate f3,6  to 3 significant figures, and express your answer as an annual effective rate.

Show all working. (5 marks)

Suppose now that an investor is wanting to purchase the above bond, but has just realised that they have to pay income tax equal to tI . The current price of $988.31 does not yet factor in the impact of income tax.

(c)     Taking into account that they now have to pay income tax, determine an expression that depends on tI , for the running yield on the bond. (4 marks)

QUESTION 6

Suppose that you invested $5,300 into an investment fund on 1-January-2021.

At 1-October-2021 it was worth $5,750. Because you saw that your investment had increased, you decided to immediately deposit another $8,500.

The investment balance then grew to $21,555 by 1-August-2022. You then immediately withdrew an amount of $X.

At 1-December-2023, the amount in the investment fund was $15,357.

(a)     Assuming that the money weighted rate of return was equal to +6.5% per annum (effective)

over the period of 1-January-2021 to 1-December-2023, what was the value of X? Give your

answer to the nearest dollar. Show all working.            (3 marks)

(b)     Assuming instead that it was the time weighted rate of return that was equal to +6.5% per

annum (effective) over the period of 1-January-2021 to 1-December-2023, what was the value

of X? Give your answer to the nearest dollar. Show all working.        (3 marks)

(c)     This question is unrelated to parts (a) and (b). Suppose that you have an amount of money to  invest, and you are considering two investment options which both have the same cost to you.

Option 1

You receive 30 annual payments, at the end of each year, with the first payment of $6,000 due in exactly one year’s time. Every annual payment after the first one increases by r % per year.

You reinvest all payments at 7% per annum (effective).

Option 2

You receive payments from a 30 year bond of face value $100,000 which is redeemable at par, with coupons of 8% per year payable every six months. The first coupon is due in six months. You          reinvest all payments at 5% per annum (effective).

In order for both options above to give the same accumulated value after 30 years, what inflation     rate r % applies to Option 1 above? Show all working and give your answer to 2 significant figures. (5 marks)

QUESTION 7

To finance a new project, you have taken out a loan of size $L.

On this loan, you will only pay interest on the loan for 10 years, then at the end of 10 years, you    will repay the entire loan with one single payment (equal to $L). The interest rate on the loan is i% per annum (effective).

The interest must be paid every 2 months, with the interest payment equal to the interest earned on the loan over the previous 2 month period.

The project earns $Y at the end of each month, for 10 years. The reinvestment rate on earnings from the project =j% per annum (effective).

(a)     Derive an expression (in terms of Y,j, L and i), that gives the accumulated value ofyour

financial position after 10 years, immediately after the loan of $L has been repaid. (8 marks)

(b)     Suppose L    95,000, i   7%, and j    2.3%. What does Y have to be in order for

the project to be profitable over 10 years? Show all working and give your answer to the nearest dollar. (4 marks)

Question 8

(Note: This is for STAT6046 students only)

The sum of (DA)   (IA)  is equal to one of the following 6 expressions. State which of the

following expressions is correct, and provide full workings / justification ofyour choice.

(n 1)an  1 , nan  , an/2 , (n 1)an  , an  , (n 1)an  .