Question

1) A sinking fund is established to discharge a debt of $20,000 in 10 years. If deposits are made at the end of each 6-month period and interest is paid at the rate of 4%, compounded semiannually, what is the amount of each deposit? (Round your answer to the nearest cent.)

2) If $1500 is deposited at the end of each quarter in an account that earns 4% compounded quarterly, after how many quarters will the account contain $70,000? (Round your answer UP to the nearest quarter.)

Answer 1

1) A sinking fund is established to discharge a debt of $20,000 in 10 years. If deposits are made at the end of each 6-month period and interest is paid at the rate of 4%, compounded semiannually, what is the amount of each deposit? (Round your answer to the nearest cent.)

The semiannual deposits in sinking fund can be calculated with the help of following formula

FV = PMT *{(1+i) ^n−1} / i

Where,

Future value of deposits FV = $20,000

PMT = semiannual deposits =?

n = N = number of payments = 10 years *2 = 20 semiannual payments

i = I/Y = interest rate per year = 4% or 4%/2 =2% semiannual

Therefore,

$20,000 = Semiannual deposits *{(1+2%) ^20−1} / 2%

Semiannual deposits = $823.13

2) If $1500 is deposited at the end of each quarter in an account that earns 4% compounded quarterly, after how many quarters will the account contain $70,000? (Round your answer UP to the nearest quarter.)

Time period of quarterly deposits can be calculated with the help of following formula

FV = PMT *{(1+i) ^n−1} / i

Where,

Future value of deposits FV = $70,000

PMT = quarterly deposits =$1500

n = N = number of payments or number of months for payments =?

i = I/Y = interest rate per year = 4% or 4%/4 = 1% per quarter

Therefore,

$70,000 = $1500 *{(1+1%) ^n−1} / 1%

Or $70,000/ $1500 = {(1+1%) ^n−1} / 1%

Or ($70,000/ $1500) * 1% = {(1+1%) ^n−1}

Or 0.467 +1 = = (1+1%) ^n

Or 1.467 = (1+1%) ^n

Taking natural log from both sides

n= ln (1.467) / ln (1+0.01)            [use the natural log function either in excel or in calculator]

= 0.389/ 0.00995

= 38.49 or 38 payments

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