4a. Derive the formula for the future value of an ordinary annuity with $1 payme

Question

4a. Derive the formula for the future value of an ordinary annuity with $1 payments, 12 payment periods, and 4.6% interest per payment period. [The first line below is partially completed for you. You need to complete that line, and then perform the subtraction of equations, showing the cancellations that occur. Then solve for FV.]

   = (1.046)FV

–

($1)(1.046)11 + ($1)(1.046)10 + …+ ($1)(1.046)3 + ($1)(1.046)2 + ($1)(1.046) + $1   = FV __________________________________________________________________________________

4b. Your answer to 4a should be a formula for FV. Use the formula that you derived to calculate the future value of the annuity described in 4a.

Explanation / Answer

4(a) N= 12 , RATE OF INTEREST 4.6% i.e. i = 4.6 /100 = 0.046 ,INT FOR 11 (12-1) PERIOD, 10 PERIOD,9 PERIOD.................... AND SO ON UP TO 1 PERIOD + LAST PAYMENT WITH NO INTEREST

NOW, FUTURE VALUE = ($1)(1.046)(11) + ($1)(1.046) (10) +($1)(1.046)(9)+ ($1)(1.046)(8)+($1)(1.046)(7)+ ($1)(1.046)(6) +($1)(1.046) (5) + ($1) (1.046)(4) + ($1) (1.046)(3)+($1)(1.046)(2)+($1)(1.046)(1) + ($1)

= ($1)(1.046)[11+10+9+8+7+6+5+4+3+2+1] +($1)

= ($1)(1.046)[(11(11+1) /2] +($1)

= ($1)(1.046) [ 11X6] + ($1)

= ($1) ( 69.036 ) + ($1) = $ 70.036

4(b) FORMULA FOR CALCULATION OF F.V. = (a) (1 + r/100) [ (n-1) + (n-2)+ (n-3)+..................+1] + (a)

= (a) (1+ r/100)[n(n-1)/2] + (a)

HERE (a) is installment, n=no. of payment ,r = rate of interest

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