Suppose that you make a sequence of 31 equal monthly deposits into an account paying a nominal rate of 6.7% convertible quarterly. If the balance in the account 7 months after the final deposit is $8, 000, how large is each deposit

Each Deposit is $228.2

Explanation:

First we need to calculate the effective Interest rate

Effective annual interest rate = [ 1 + APR/n ]^n - 1

Where

APR = 6.7%

n = months ina quarter = 4 months

Placing values in the formula

Effective annual interest rate = [ 1 + 6.7%/4 ]^4 - 1

Effective annual interest rate = 6.87%

Now calculate the monthly effective rate using following formula

Effective monthly rate = [ 1 + Effective annual interest rate ]^(1/n) - 1

Where

n = Months in a year = 12 years

Placing values in the formula

Effective monthly rate = [ 1 + 6.87% ]^(1/12) - 1

Effective monthly rate = 0.55523%

Use following formula to calculate the amount of each deposit

Balance after 7 months of last deposit = Future value of annuity x ( 1 + r )^7

Balance after 7 months of last deposit = [ Monthly deposit x ( 1 + r )^n - 1 / r ] x ( 1 + r )^7

where

Balance after 7 months of last deposit = $8,000

r = 0.55523%

n = 31 months

Placing values in the formula

$8,000 = [ Monthly deposit x ( 1 + 0.55523% )^31 - 1 / 0.55523% ] x ( 1 + 0.55523% )^7

$8,000 = [ Monthly deposit x 33.726 ] x 1.03952

$8,000 / 1.03952 = Monthly deposit x 33.726

$7,695.86 = Monthly deposit x 33.726

Monthly deposit = $7,695.86 / 33.726

Monthly deposit = $228.19

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