Question

You are offered a special set of annuities by your insurance company, whereby you will receive $23,600 a year for the next 7 years, nothing for the next 10 years, and then $40,000 a year for the following 12 years. How much would you be willing to pay for these annuities today, if the annuities are paid at the beginning of each year? Assume the annuities have a beta of 1.47, the expected return on the market is 12.4% and the risk free rate is 2.4%.

Answer 1

Required Return = Risk-free Rate + Beta * (Return on Market - Risk-free Rate)
Required Return = 2.40% + 1.47 * (12.40% - 2.40%)
Required Return = 2.40% + 1.47 * 10.00%
Required Return = 17.10%

Annual Payment for first 7 years = $23,600
Annual Payment for next 10 years = $0
Annual Payment for next 12 years = $40,000

Payments are made at the beginning of each year

Present Value = $23,600 + $23,600/1.1710 + … + $23,600/1.1710^5 + $23,600/1.1710^6 + $40,000/1.1710^17 + $40,000/1.1710^18 + … + $40,000/1.1710^27 + $40,000/1.1710^28
Present Value = $23,600 * 1.1710 * (1 - (1/1.1710)^7) / 0.1710 + $40,000 * (1/1.1710)^17 * 1.1710 * (1 - (1/1.1710)^12) / 0.1710
Present Value = $23,600 * 4.579852 + $40,000 * 0.397462
Present Value = $123,982.99

You should pay $123,982.99 or $123,983 for these annuities.