Question $3$: Paul, a retiree aged $60,$ has an investment pool of HK $$1,000,000.$ One option for his retirement income is to convert his pool into an annuity plan from HKMC Annuity. Under this plan, Paul receives a fixed income for life as follows: Monthly annuity payment: HK $$5,100$ Annual annuity equivalent: HK $$61,200$ This annuity is designed for a $60-$year$-$old retiree and continues until he reaches the age of $100.$ In comparison, if Paul were to invest his HK $$1,000,000$ in a time deposit at a $3\%$ per annum interest rate, his money would grow at that rate $($without any periodic withdrawal$).$ You are required to answer the following questions to help Paul make a wise choice: $($a$)$ Draw a cash flow diagram illustrating the annuity plan. $($b$)$ Discuss whether the annuity plan provides a reasonable income and value given his $40-$year payout period with the initial investment of HK $$1,000,000.$ It is assumed that Paul can live until the age of $100.($c$)$ Determine the time $($in years$)$ it takes for the accumulated value of the annuity payments to equal the initial investment of HK$$1,000,000.$ This is the break$-$even point where the total value received from the annuity matches the original fund when compounded at the prevailing time deposit interest rate.