Calculate the following with work shown.

a$.$ What is the amount of the annuity purchase required if you wish to receive a fixed payment of $$200,000$ for $20$ years? Assume that the annuity will earn $6$ percent per year.

b$.$Calculate the annual cash flows $($annuity payments$)$ from a fixed$-$payment annuity if the present value of the $20-$year annuity is $$1$ million and the annuity earns a guaranteed annual return of $6$ percent. The payments are to begin at the end of the current year.

c$.$You deposit $$10,000$ annually into a life insurance fund for the next $10$ years, at which time you plan to retire. Instead of a lump sum, you wish to receive annuities for the next $20$ years. What is the annual payment you expect to receive beginning in year $11$ if you assume an interest rate of $9\%$ for the whole time period?

d$.$ An insurance company$$s projected loss ratio is $78\mathrm{.}4$ percent and its expense ratio is $23\mathrm{.}9$ percent. It estimates that dividends to policyholders will add another $5\%.$ What is the minimum yield on investments required in order to maintain a positive operating ratio?