(a) Using the constant annual effective rate 2.3443% , determine the accumulated amount in your account from 1st january 2011 to 31 December 2020 (ten years in total) according to the following deposit patterns.

i) Deposit $1,000 at the start of every month. ii) Deposit $1,000 at the start of the first month and keeps increasing by $100 in the subsequent months until the monthly deposit achieves $3,000, then remains at $3,000 in all the following months. iii) Deposit $3,000 at the start of the first month and keeps decreasing by $100 in the subsequent months until the monthly deposit achieves $1,000, then remains at $1,000 in all the following months. iv) Deposit $1,000 monthly (at the start of each month) in year 2011, $1100 monthly in year 2012, $1200 monthly in year 2013, , and $1,900 monthly in year 2020. v) Deposit $1,000 monthly (at the start of each month) in the first five years, and deposits $2,000 monthly in the second five years.

(b) Convert the constant annual effective rate 2.3443% to force of interest , determine the accumulated amount in your account at 31 December 2020 (ten years in total) according to the following deposit patterns. i) Deposit $5,000 per annum continuously over 10 years. ii) Deposit $5,000 continuously for the first year and keeps increasing by $1000 in the subsequent years until the yearly continuous rate of deposit achieves $9,000, then remains at $9,000 in all the following years. iii) Deposit $9,000 continuously for the first year and keeps decreasing by $500 in the subsequent years until the yearly continuous rate of deposit achieves $5,500, then remains at $5,500 in all the following years. iv) Deposit $6,000 continuously over year 2011, $6600 continuously over year 2012, $7200 continuously over year 2013, , and $11,400 continuously over year 2020. e) An insurance company is planning to set up a reserve account for the claim payments in the coming year. The proposed modelling approach is as follows: The average number of claims per year (denoted by n ) and average claim amount (denoted by Z ) in the past ten years are obtained. Using the average cash rate c in (a) as the effective rate of interest per year in the evaluation. The reserve is calculated as the total present value of n payments of Z evenly spread over the coming year. Comment on the appropriateness of the proposed modelling approach.