Question

2. A couple would like to take a cruise to Alaska, 15 months from now. They have $1500 in a savings account. The savings account pays a nominal rate of 1%, compounded monthly. a. If they start investing $110 today for 15 months how much will they have 15 months from today? b. If they start investing $110 a month from now for 15 months how much will they have 15 months from today? c. If they need $3900 for the trip, how much would they need to invest each month, if they make 15 monthly equal investments starting a month from now?

Answer 1

Answer:

a.- $ 3,529.82  

b.-  $ 3,512.11

c.-  $   132,77

Explanation:

In each case, we must calculate the value of their current savings and the additional investment.

The saving are the same for each scenario so let's calculate that first:

$Principal \: (1+ r)^{time} = Amount$

Principal 1,500.00

time 15 years

rate 0.01000

$1500 \: (1+ 0.01)^{15} = Amount$

Amount 1,741.45

Then we add the funds generated from the investment:

a.- 110 annuity due for 15 month:

$C \times \frac{(1+r)^{-time} -1}{rate}(1+r) = FV$

C  $ 110

time 15 months

rate 0.01

$110 \times \frac{(1+0.01)^{15} -1 }{0.01} = FV$

FV $1,788.3651

We add the savings and get a total of:  $ 3,529.82  

b.- 110 ordinary annuity

$C \times \frac{(1+r)^{time} -1}{rate} = FV$

C  $ 110

time 15 months

rate 0.01

$110 \times \frac{(1+0.01)^{15} -1}{0.01} = FV$

FV $1,770.6585

Plus, original savings of 1,741.45 =  3,512.11  

c.-

If they need 3,900 then the fund must cover the difference between these and the savings future value:

3,900 - 1,741.45 = 2,158.55

Now we calculate the PMT, considering the payment are at the beginning:

$FV \div \frac{(1+r)^{time} -1 }{rate}(1+r) = C$

FV  $ 2,158.55

time  15

rate           0.01

$2158.55 \div \frac{(1+0.01)^{15} -1}{0.01} (1+0.01) = C$

C  $ 132.770

Answer 2

The future value of annuity is calculated when we want to know the amount that we will be receiving on the investment made today at a future date. The values calculated for (a) $1788.3651 (b)$1771 and (c)$132.770

What is Future Value of Annuity?

Future annuity value is the group of repeated payments for a specific future date, deducted a certain refund rate, or a discount rate. The higher the discount rate, the greater the annuity amount.

As per the given information,

Let's calculate the amount of savings  that will be invested each year:

$\rm\,Amount = P (1 + r)^{n}$

Principal is equal to $1,500

Time/ Number of periods is equal to 15 months

Rate is 1% is 0.01

$\rm\,Amount = 1,500(1 + 0.01)^{15}\\\rm\,Amount = 1,741.45$

a) Calculation of Future Value if they invest $110 today for 15 months:

$FV(\rm\,due) = A[\dfrac{(1+r)^{n} -1} {r}](1 + r)\\\\A= \ 110\\\\\rm\,R = 0.01\\\\N = 15\\\\FV(due) = 110[\dfrac{(1+0.01)^{15} -1} {0.01}](1 + 0.01)\\\\FV(\rm\,due) = \ 1,788.3651$

Now, we add the original savings and get the total of $3529.82

b) Calculation of Future Value of ordinary annuity if they start investing $110 a month from now for 15 months from today:

$\rm\,FV = A[\dfrac{(1+r)^{n} -1} {r}]\\\\A = \ 110\\\\\rm\,R = 0.01\\\\n = 15\\\\\FV = 110[\dfrac{(1+0.01)^{15} -1} {0.01}]\\\\\rm\,FV = \ 1,771$

Adding the amount of original savings, total savings will be $3512.45

c)If they need $3,900 then the fund must cover the difference between these and the savings future value:

$\ 3,900 - \ 1741.45 = \ 2,158.55$

Now we need to calculate the periodic payment(PMT) by applying the formula of Future Value of annuity due.

$FV(\rm\,due) = PMT [\dfrac{(1+r)^{n} -1} {r}](1 + r)\\\\FV(Due) = \ 2158.55\\\\PMT = ?\\\\\rm\,R = 0.01\\\\N = 15\\\\\ 2158.55 = PMT[\dfrac{(1+0.01)^{15} -1} {0.01}](1 + 0.01)\\\\PMT = \dfrac{\ 2158.55 }{[\dfrac{(1+0.01)^{15} -1} {0.01}](1 + 0.01)}\\\\PMT = \ 132.770$

Hence, they have to invest 132.770 dollars if they $3,900 for the trip.

To learn more about Future Value of Annuity, refer to the link:

https://brainly.com/question/26662056