Assignment: Financial Mathematics and Security Valuation

Outline

You have just started working for Polycorp Pyramid Financial Services, and your supervisor gives you some problems to answer/solve for clients. You should solve the following ten questions showing your full workings and explanations. Each question is worth one mark. You must have the correct answer and show your working and correct explanation to get one mark (very brief or one to two lines of answers are not acceptable). As well as solving the problems manually using the appropriate formulas, you must also solve the problems using Excel functions/calculations. An incorrect answer or an answer without the working receives zero. Normally no part marks are awarded. For questions with two or more parts, you must get all parts correct to guarantee the one mark. Include cash flow maps or tables wherever possible. Avoid rounding error.

Question 1

Now that they have accumulated a deposit of $100,000, Ed and Susie wish to use the deposit and take out a housing loan to purchase a home. The house costs $800,000. The loan is to be repaid in equal monthly instalments over a term of 30 years. Ed recalls that the interest rate quoted by the bank is an annual nominal rate of 6.0%pa. Ed has misplaced the paperwork showing the annual effective rate, so you may need to work this out. Interest is added monthly. They would like to know:

i. How much is the monthly repayment?
ii. How much interest is in the 180th repayment?
iii. How much would Ed and Susie owe the bank immediately before making the 240th repayment?
iv. Provide Ed and Susie with a repayment schedule using excel.

Question 2

Karine and Arlo are trying to establish a University Fund for their daughter Amelia, who turns 2 today. They plan for Amelia to withdraw $15,000 on her 19th birthday and $17,000, $19,000 and $21,000 on her subsequent birthdays (20th, 21st and 22nd). They wish to deposit money into an account so that, with interest it will build up enough to make the planned withdrawals. They wish to make ten equal annual deposits, and they intend to make their first deposit two year from today, and expect to earn an average return of 4.8%pa on the account.
i. How much will Karine and Arlo have to deposit each year to the account to achieve their goal?

ii. Create a schedule showing the cash inflows (including interest) and outflows of this fund. How much will be in the fund on Amelia's 17th birthday?

Question 3

Stanley has just been advised of a bequest of a lump sum of 180,000 from his Aunt's will, but it is not due to be available for him for ten years (at t = 10 he will receive $180,000). Stanley wants to receive some cash earlier than this. He is investigating using the bequest to purchase an annuity, in exchange, with the first annual cash flow of the annuity to be received at the end of year 2 (nine cash flows).
Assume that the annuity and the lump sum are of equivalent risk and 6.00% pa is the appropriate interest rate (opportunity cost of funds for Stanley). How much is the annual cash flow associated with the annuity?

Question 4

Yianni has funds in his superannuation account. He is considering using these to purchase a pension. In exchange for a lump sum payment now, Polysuper offers an annual pension over thirty years beginning with a payment of $50,000 at the end of the first year. There are thirty payments in total and the payments will increase at an annual rate of 2.00%pa. The appropriate opportunity cost of funds is j2 = 8.00%pa what is the amount of the lump sum needed today to purchase this pension?

Question 5

Your supervisor has asked you to do the following calculations:

a) A bank bill with 90 days to maturity has a price of $99,203. What is the effective annual yield implied by this price and maturity? What is the annual nominal yield? Face value is $100,000. Label your answers clearly. Provide a brief explanation of why these rates differ.

b) The All Ordinaries price index opened the year at 6,210 and closed at 6,888 by the end of the year. The equivalent accumulation index went from 56,128 to 65,425. What is the annual rate of return on each of these indices? Explain the difference.

c) Using the approach covered in your textbook calculate the geometric average annual rate of return over four years given the following annual rates, year 1 = 5.19%, year 2 = 5.87%, year 3 = 6.33%, year 4 = 6.66%. What is the arithmetic average? Explain the difference.

d) Polycorp dividends per share have increased from $6.20 to $12.80 over the last six years. Calculate the annual compounded growth rate in dividends over that period.

Question 6

Polycorp Treasury a company in the land of Zanadu is holding a parcel of Zanadu Government Bonds with a face value of $2,000,000. The bonds were issued six years and three months ago, with an initial maturity of 10 years. They pay a coupon rate of interest of 6.00% pa, with interest paid semi-annually. Currently the market yield quoted for Zanadu bonds is 4.86% pa. The convention in Zanadu financial markets is that the market yield and coupon rate are quoted as annual nominal rates. What is the current market value of the bonds?

Question 7

Polycorp has a dividend of $6.00 due in a year's time. It is expected to pay a dividend $6.50 at the end of the second year. Its dividends will grow at 5.00%pa for the following five years. Dividends are then expected to grow at 4.25%pa for another two years, after which they are expected to grow at 3.00%pa forever. Shareholders required return on equity is 10.00% pa. What is the current price (cum-dividend and ex-dividend) of Polycorp shares? D0 is $5. Explain the difference between the ex price and cum price.

Question 8

The required rate of return on the shares in the companies identified below is 11.60% pa. Calculate the current share price (ex-dividend) in each case.

(a) The current earnings per share of Alpha Ltd are $4.80. The company does not reinvest any of its earnings. Earnings are expected to remain constant.

(b) Beta Ltd's current dividend is $2.80 (D0) and dividends are expected to grow at 5.5% pa indefinitely.

(c) Gamma Ltd is not expecting to pay dividends for three years, at the end of year four a dividend of $4.80 is planned and dividends are expected to grow at 3.25% pa forever after that.

(d) Delta limited plans to pay dividends of 2.55, 2.95, and 3.55 at the end of years 4, 5, 6 respectively followed by a dividend of 4.20 pa in perpetuity after that.

Question 9

Your client wishes to insure their Ferrari. Mooncorp Insurance has quoted an annual premium to insure the car of $15,915. Mooncorp offers a 10% discount if you pay the lump sum immediately. They also offer an alternative payment method. The account can be paid in full by making 11 equal end-of-the month payments of $1,098, rather than the lump sum, with no payment in the first month (i.e. the first payment is at the end of the second month followed by ten further monthly payments). What is the effective annual opportunity cost of paying monthly? In other words, what interest rate is being charged if your client decides to use the repayment plan as opposed to the lump sum?

You must provide one complete manual trial calculation of the IRR to demonstrate that you understand the process. Failure to follow this instruction will attract a mark of zero.

Question 10

(a) What is the present value of a series of payments of $2500 every six years in perpetuity with the first payment made immediately, if the annual rate is 8.5% per annum?

(b) Polycorp debentures are selling for $110 (FV = 100) and mature in 8 years. The coupon rate is 6%pa. What is the effective annual yield on the debentures?

(c) Polycorp debentures are selling for $96 (FV = 100) and mature in ten years. The coupon rate is 5%pa, with coupons paid quarterly. What is the effective annual yield on the debentures? Your supervisor would like you to confirm your answer using the excel Rate Function (Formula).

(d) Polycorp shares are currently selling for $20 each. A month ago, they announced a Bonus share issue of one free share for every share owned (one for one Bonus Issue). The shares go ex-bonus tomorrow. "All else equal", what should happen to the share price of Polycorp when it goes ex?