STAT2032/STAT6046 Financial Mathematics

Question 1

[5 + 5 = 10 Marks]

a.   At the start of a given year, an investor deposited $20,000 with the bank. The accumulated amount of the investor’s account was $20,596.21 midway through the year and $21,183.70 at the end of the year. The bank credits interest at a variable force of interest which is a      linear function of time: () =  +  for 0 ≤  ≤ 2.

Find the amount in the bank at the end of 2 years.

b.   On 1 November 2016, Adam was receiving 2 separate annuity payments from an investment company:

    $200 per annum payable annually on 1 February each year until 2027.

    $400 per quarter which increases by $40 every quarter, where the first payment is

made on 1 January 2017 and the last payment is made on 1 January 2030 .

Adam wants to combine the two payments into a level monthly payment which will start on 1 November 2016 and will continue until the end of 2027. Assume the effective rate of       interest that applies over the entire period to be 6% per annum.

What is the value of this new monthly payment?

Question 2

[5 + 5 + 2 = 12 Marks]

A computer manufacturer is to develop a new chip to be produced from 1 January 2017 until 30    June 2025. The cost of development comprises $9 million to be paid on 1 January 2017 and $12    million paid continuously during 2018. From 1 January 2019 the chip will be ready for sales and it is assumed that income will be received quarterly in arrears at a rate of $8 million per annum.

a.   Evaluate the Net Present Value at an effective risk discount rate of 9% p.a.

b.   Calculate the discounted payback period at an effective rate of 9% p.a.

c.   Explain why the discounted payback period increases if we significantly increase the effective rate of interest.

Question 3

[8 Marks]

Jill entered a home loan contract of $450,000 10 years ago which is to repaid by level monthly        payments paid in arrears over a total term of 30 years. The interest rate for the first 10 years of the   loan was 9% per annum compounded monthly and then an effective rate of 6% for the remainder of the term.

Jill now decides to increase her monthly payment at a rate of 0.5% per month from the very next repayment. In how many months will the loan be repaid under the new repayment structure?

Question 4

[6 Marks]

A 1-year forward contact is issued on 1 April 2016 on a stock with a price of $15 on that date.       Equal amounts of dividends are to be paid on 1 September 2016 and 1 January 2017. The risk-free 5-month and 9-month spot rates are 5% p.a. and 6% p.a. respectively. The 3-month risk-free          forward rate from 1 January 2017 is 5.5% p.a. The price of the forward contract is $12.60.

Evaluate the value of the dividends paid (under the principles of no arbitrage).

Question 5

[4 + 3 = 7 Marks]

A savings fund earns interest at a constant effective annual rate which follows a uniform                   distribution between the values of 6% and 10% per annum (both rates inclusive) for each of the next 20 years. Jack deposits P at the start of each year for the next 20 years.

a.   If the expected value of the accumulation is $49,423 then find the value of P to the nearest dollar.

b.   Find the probability that the accumulated value of the 20 deposits will be greater than $52,000. Assume that standard deviation of the accumulation is $1662 and that the    accumulation is approximately normally distributed.

Question 6

[4 + 3 = 7 Marks]

A personal investment fund’s value and cash-flow transactions at various time over a 3-year period are given in the table below.

a.   If the time weighted rate of return is 13% per annum over the period from 1 April 2014 to 1

January 2017 then evaluate the missing cash-flow of X.

b.   Evaluate the money weighted rate of return for the calendar year 2015.

Question 7

[5 + 5 = 10 Marks]

In a certain country, zero-coupon bonds are the only fixed interest securities available. All such bonds are redeemable at par on maturity.

An investor owes $10 million in 12 years’ time. He wishes to purchase a combination of an 8 year and a 15 year zero coupon bond to pay for his liability. The current force of interest is 6% per       annum for all maturities.

a.   Find the face value of the two zero coupon bonds above that would satisfy the first two conditions of immunisation.

b.   Find the approximate surplus of assets over liabilities if the force of interest falls by 1% using both duration and convexity of assets and liabilities.

Question 8

[6 + 4 = 10 Marks]

A 20-year $1million bond can be redeemed at the option of the issuer at any of the coupon paying dates in the last 4 years of the term of the bond for the following redemption amounts:

    At 110% of the face value if redeemed in years 17 and 18.

    At 120% of the face value if redeemed in years 19 and 20.

Coupons are paid quarterly at the rate of 8% per annum. John is interested in buying this bond and   wishes to achieve a net redemption yield of 5% per annum compounded quarterly. John is subjected to a tax on income of 20% and tax on capital gains of 10%.

a.   What is the price (to the nearest dollar) that John should pay for this bond?

b.   If he sells this bond for $1.2mllion immediately after receiving his 16th coupon what is the holding period effective gross-yield achieved by John?

(Assume that John has paid the price calculated in part a. above to purchase the bond).