i) Using a 'discrete' derivative and an interest rate of 2.75%, solve for the Macaulay duration of a life annuity for a 65 year old person that cannot live beyond 95 years.

ii) Assuming a maximum possible life of 90 years, what is the approximate implied interest rate for a 'life income' of $220,000/yr. priced at $2.75 million for a person retiring at age 65?

iii) Derive the Macaulay duration formula for a par bond using continuously compounded interest rates.

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You are in the market for a house. Your effective all-in market borrowing rate for a second mortgage with a 5 year term from a bank is 2.69%. The vendor of one of the houses you are considering purchasing is willing to undertake a $600,000 second mortgage, with a 5 year term at 1.69%, and a 25 year amortization period. The asking price on the house is $900,000. What adjustment to the sales price of the house is warranted if, as part of the purchase, you take up the vendor's second mortgage offer?