Question

You have done a great job, and your boss offers you a few different possible gifts. Using a discount rate of 10%, which present would you choose? A) A single payment of $75,000 received in 5 years B) $8,000 per year for 10 years, starting two year from now C) One payment of $47,500 received today D) A growing perpetuity that pays $2000 each year starting one year from now. The perpetuity grows at a rate of 1.5% per year. E) All the presents stated are equally good!

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To determine which present is the most valuable, we need to calculate the present value of each option using the given discount rate of 10%.

A) Present value of the single payment of $75,000 received in 5 years:

Using the formula for present value of a single payment:

PV = \frac{FV}{(1 + r)^n}

where PV is the present value, FV is the future value, r is the discount rate, and n is the number of periods.

PV = \frac{\$75,000}{(1 + 0.1)^5} = \$54,432.69

B) Present value of $8,000 per year for 10 years, starting two years from now:

Using the formula for present value of an ordinary annuity:

PV = P \times \left(\frac{1 - (1 + r)^{-n}} {r}\right)

where PV is the present value, P is the annuity payment, r is the discount rate, and n is the number of periods.

PV = \$8,000 \times \left(\frac{1 - (1 + 0.1)^{-10}} {0.1}\right) = \$56,877.62

C) Present value of the one payment of $47,500 received today:

The present value is the same as the future value since we are receiving it today.

PV = \$47,500

D) Present value of the growing perpetuity that pays $2000 each year starting one year from now, with a growth rate of 1.5% per year:

Using the formula for present value of a growing perpetuity:

PV = \frac{P}{r - g}

where PV is the present value, P is the payment, r is the discount rate, and g is the growth rate.

PV = \frac{\$2,000}{0.1 - 0.015} = \$23,809.52

Since the present values are different for each option, we can conclude that all the presents stated are not equally good.

Answer: Option B) $8,000 per year for 10 years, starting two years from now, has the highest present value of $56,877.62.

A) Present value of the single payment of $75,000 received in 5 years:

Using the formula for present value of a single payment:

where PV is the present value, FV is the future value, r is the discount rate, and n is the number of periods.

B) Present value of $8,000 per year for 10 years, starting two years from now:

Using the formula for present value of an ordinary annuity:

where PV is the present value, P is the annuity payment, r is the discount rate, and n is the number of periods.

C) Present value of the one payment of $47,500 received today:

The present value is the same as the future value since we are receiving it today.

D) Present value of the growing perpetuity that pays $2000 each year starting one year from now, with a growth rate of 1.5% per year:

Using the formula for present value of a growing perpetuity:

where PV is the present value, P is the payment, r is the discount rate, and g is the growth rate.

Since the present values are different for each option, we can conclude that all the presents stated are not equally good.

Answer: Option B) $8,000 per year for 10 years, starting two years from now, has the highest present value of $56,877.62.

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