Answer:
The best alternative will be of 180,000 today.
Explanation:
We calculate the present value of the second and third alternatives and compare with the cash received today:
.2. A 20-year annuity of $16,000 beginning immediately
[tex]C \times \frac{1-(1+r)^{-time} }{rate} = PV\\[/tex]
C 16,000
time 20
rate 0.07
[tex]16000 \times \frac{1-(1+0.07)^{-20} }{0.07} = PV\\[/tex]
PV $169,504.2279
3.- A 10-year annuity of $50,000 beginning at age 65.
[tex]C \times \frac{1-(1+r)^{-time} }{rate} = PV\\[/tex]
C $ 50,000
time 10 years
rate 0.07
[tex]50000 \times \frac{1-(1+0.07)^{-10} }{0.07} = PV\\[/tex]
PV $351,179.0770
This start at age 65 currently he's 55 so we bring it to present:
[tex]\frac{Maturity}{(1 + rate)^{time} } = PV[/tex]
Maturity $ 351,179.08
time 10 years
rate 0.07
[tex]\frac{351179.07704663}{(1 + 0.07)^{10} } = PV[/tex]
PV 178,521.64
As non of the alternatives is better than 180,000 today we pick this alternative.
Answer: John Roberts should take the lump sum of $180 000
Explanation:
option 1
lump sum = $180 000
option 2
Present Value = P(1-(1+r)^-n)/r
Present Value = 16000(1-(1+0.07)^-20)/0.07
Present Value = 169504.22789 = 169504.23
The present value of option 2 = $169504
option 3
Present Value = P(1-(1+r)^-n)/r
Present Value = 50000(1-(1+0.07)^-10)/0.07
present value (at age 65) = 351179.07643
the annuity of option 3 will start in future when john is 65 years old. the present value of $ 35179.07643 is the present of the annuity at the age of 65. john is 55 years old,We need to discount the present value of $ 35179.07643 to determine how much is option 3 future annuity worth today.
Present value= 351179.07643
N = 65 - 55 =10
present value (at age 55) = (present value at 65)/(1+r)^n
present value (at age 55) = 351179.07643/(1+0.07)^10
present value (at age 55) = 178521.63491 = 178521.64
