a. 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 10%. Th

a. To calculate the annual cash flows from a fixed-payment annuity with a present value of $1 million and a guaranteed annual return of 10%, we can use the formula for the present value of an annuity: PV = C * (1 - (1 + r)^-n ) / r where PV is the present value, C is the annual payment or cash flow, r is the discount rate or guaranteed annual return, and n is the number of years in the annuity. Rearranging the formula to solve for C, we get: C = PV * r / (1 - (1 + r)^-n ) Substituting the given values into the formula, we get: C = $1 million * 10% / (1 - (1 + 10%)^-20 ) = $132,149.90 Therefore, the annual cash flows or annuity payments for a 20-year fixed-payment annuity that has a present value of $1 million and earns a guaranteed annual return of 10% are $132,149.90. b. To calculate the annual cash flows from a fixed-payment annuity with a present value of $1 million and a guaranteed annual return of 10% that begins at the end of five years, we can use the same formula as in part (a), but we need to discount the present value to the end of the fifth year first. Using the formula for the future value of a single sum, we can find the value of $1 million in five years at a discount rate of 10%: FV = PV * (1 + r)^n where FV is the future value, PV is the present value, r is the discount rate, and n is the number of years. Substituting the given values into the formula, we get: FV = $1 million * (1 + 10%)^5 = $1,610,510.51 This means that $1 million today is worth $1,610,510.51 in five years at a discount rate of 10%. Now we can use the same formula as in part (a), but with the present value of $1,610,510.51 and the number of years reduced by five: C = $1,610,510.51 * 10% / (1 - (1 + 10%)^-15 ) = $207,377.48 Therefore, the annual cash flows or annuity payments for a 20-year fixed-payment annuity that has a present value of $1 million and earns a guaranteed annual return of 10% and begins at the end of five years are $207,377.48. c. To calculate the amount of the annuity purchase required to receive a fixed payment of $200,000 for 20 years at a guaranteed annual return of 10%, we can use the formula for the present value of an annuity again: PV = C * (1 - (1 + r)^-n ) / r but this time we need to solve for PV. Substituting the given values into the formula, we get: $200,000 = PV * 10% / (1 - (1 + 10%)^-20 ) Solving for PV, we get: PV = $1,730,731.54 Therefore, the amount of the annuity purchase required to receive a fixed payment of $200,000 for 20 years at a guaranteed annual return of 10% is $1,730,731.54.