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