Question

1. Danny is saving towards his vacation. He has determined that the total cost including airfare an accommodation will amount to $5,000.
   1. If he sets aside $400 per month from his salary, starting from the end of this month, in an account that pays interest of 9% per annum, will he have enough to make the trip at the end of the year?    
   2. How much more will he have in the account if he made his first deposit of $400 today? What kind of annuity is this?

Answer 1

Answer :(a.) Calculation of Amount of Money he will deposit $400 per month :

Calculation of Future value of Annuity

Future Value of Annuity = Periodic Payment * {[(1+ r)^n - 1 ] / r }

where

rate of interest is 9%/12 or 0.75% or 0.0075

n is the number of payments i.e 12

Periodic payment is 400

Future Value of Annuity = 400 * {[(1+ 0.0075)^12 - 1 ] / 0.0075 }

= 400 * {0.09381 / 0.0075}

= 400 * 12.4578

= 5003.03

(b.) Calculation of Amount  if he made his first deposit of $400 today

Future Value of Annuity if the payments are made in the beginning of year it will be treated as annuity due :

Future Value = Periodic Payment * {[(1+r)^n - 1] / r } * (1 + r)

where

rate of interest is 9%/12 or 0.75% or 0.0075

n is the number of payments i.e 12

Periodic payment is 400

Future Value of Annuity = 400 * {[(1+ 0.0075)^12 - 1 ] / 0.0075 } * (1 + 0.0075)

= 400 * {0.09381 / 0.0075} * 1.0075

= 400 * 12.4578 * 1.0075

= 5040.56

He will have 40.56 (5040.56 - 5000) more as compared to trip expenses.

But as compared to Ordinary annuity 37.52 more money in the account.

Future Value of Annuity if the payments are made in the beginning of year it will be treated as Annuity due