The Scenario

You are a financial planner who is investigating the options for a 45 year old man who wishes to have enough money to retire by the time he is 65. His current financial arrangements are as follows:

• He is paying off a mortgage on the property he lives in. The payments are currently $3,500 at the end of each month, which is enough to have the property paid off exactly in 10 years at the current interest rate of 4% p.a. effective.

• His salary has just been increased to $180,000 p.a., which is taxed at a rate of 30% before being paid to him. This is currently being paid on a monthly basis at the same time as the mortgage payment is due (i.e. at the end of each month).

• Currently each month, $3,500 of the net salary is going towards the mortgage, $3,500 is being used in expenses and any additional amount is being invested in a bank account earning 3% p.a. compounded monthly. These payments are being made on the same day the net salary is received.

• The current bank account balance is $100,000.

• He expects his expenses to increase by 0.4% each month, including next month (i.e. the expense at the end of the 1st month will be 3500 × 1.004).

• He expects his salary to increase at a rate of 5% p.a. going forward. This increase is processed on a yearly basis, with the next increase to occur after 12 further salary payments at the recently increased rate.

• Interest income in the bank account is taxed at a rate of 20% immediately on the interest income earned each month.

• He intends to live in his home when he is retired and live off the proceeds of his bank account in retirement.

a) Calculate the expected amount the man will have in his bank account at age 65 under the basis above. Assume all interest rates remain constant.

The first thing you notice about this man’s financial affairs is that it would most likely be beneficial for him to pay off the mortgage as quickly as possible. This would involve closing the bank account down immediately and making a lump sum contribution to the home loan, and then continuing the $3,500 per month repayments until the loan is paid off. Again any additional amount of salary not used is credited to the bank account (which starts from a balance of zero).

b) Calculate how much more the man will have at age 65 if he takes this approach.

The client mentions to you he is extremely risk averse and does not want to invest in anything where the cash-flows aren’t known in advance. Based on this you decide to only recommend fixed interest securities, with the intention of investigating the impact of investing the current bank account balance in these securities.

Two options which you are considering are as follows:

Bond A – A zero coupon bond redeemable in 20 years with a gross yield of 7.5% p.a. effective.

Bond B – Pays half yearly coupons of 8% p.a. Redemption is at par on any coupon date between 15 and 20 years from now at the decision of the issuer. This bond has a minimum gross yield of 10% p.a. effective.

c) Assuming the man invests in either Bond A or Bond B (but not both at the same time); calculate the nominal amount of each bond which can be purchased using the current bank account balance of $100,000.

Should the client decide to invest in one of the bonds above, you decide to recommend that the mortgage continue to be paid off with the $3,500 per month for 10 years. You also discover that the government is planning to increase taxes on higher salaries. Any salary over $250,000 p.a. will be taxed at 40% (i.e. the first $250,000 p.a. of salary is taxed at 30%, the amount over $250,000 p.a. is taxed at 40% - tax is paid at a constant rate each month during each year assuming that the monthly salary is to be earned over the whole year). The client is also subject to a tax of 20% on coupons and 30% on capital gains, payable immediately it is incurred, with coupons to be invested in the bank account when they are received. The gross yields of the bonds are unchanged.

d) For both bond purchase options, calculate how much money the man will have in the bank account at age 65. Assume that Bond B is redeemed at the most advantageous date for the issuer.

e) Which of the four alternatives discussed above should the man choose such that he has the largest bank balance at the age of 65? Give a short explanation for your answer.

**Subject Business Mathematical Finance**