Jerry is considering taking out a loan of $100,000 to make an investment. The details for the loan are as follows:

• Repayments will be made on a half-yearly basis. However, the first repayment will be made exactly THREE months from today.

• The loan will be repaid by 10 level repayments

• The interest charged on the loan is 6% p.a. compounding monthly

Using this information, answer the following questions.

a) Let r_q denote the equivalent effective quarterly (i.e. a 3-month period) rate charged on Jerry’s loan. Write down an equation that can be solved to obtain the effective quarterly rate (your equation should involve r_q). You do NOT need to solve this equation. (1 mark)

b) Explain whether or not we should use the interest rate r_q in an annuity formula when working with Jerry’s loan. (1 mark)

c) State clearly the type of effective rate that should be used in an annuity formula for Jerry’s loan as described above (e.g. “effective monthly rate”, “effective weekly rate”). Then determine the value of this effective rate. Express your answer as a percentage to 4 decimal places. (1 mark)

d) Draw a cash flow diagram, and then determine the size of the level repayment for Jerry’s loan. (2 marks)

Jerry intends to use the money from his loan (and his personal savings if necessary) to make an investment in his friend Elaine’s business. In return, Elaine has predicted the following returns on Jerry’s investment:

• Four half-yearly payments of $10,000, the first being exactly 6 months from today,

• Four half-yearly payments of $20,000, the first being exactly 6 months after the final $10,000 payment described above, and

• Two half-yearly payments of $25,000, the first being exactly 6 months after the final $20,000 payment described above

(i.e. there are 10 half-yearly payments in total, the final one occurring exactly 5 years from today). Whilst Elaine is a trusted friend and a capable businesswoman, Jerry has some doubts.

To make a more conservative assessment of the investment, Jerry will expect a higher rate of return. Jerry decides that he requires the following return on his investment:

• 12% p.a. effective for the first 2 years of the investment, and

• 15% p.a. compounding quarterly for the remaining term of the investment.

Using this information, answer the following questions.

e) For the two rates given above, determine the equivalent effective half-yearly rates. Clearly label your answers. (2 marks)

f) Using your results from part e), draw a cash flow diagram, and then determine the maximum price Jerry would be willing to pay for this investment. 

g) Jerry makes an offer to Elaine of the price calculated in part f). Elaine thinks for a moment and gives a counter-offer of $115,000. Write down an equation that can be solved to determine Jerry’s return on his investment using Elaine’s new price, expressed as an effective half-yearly rate. You do NOT need to solve this equation