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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Answer to a math 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!
Nash
4.987 Answers
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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