Question

10. The city council of the Village of Sunville has decided to replace all of its street lights in 4 years at a cost

of $412,000. Calculate how much the village needs to deposit into a sinking fund account each month, if the
account pays 18%, compounded monthly.

$5,920.44
$34, 333.33
$8,583.33
$14, 370.00

Answer 1

$5,920.44

Explanation:

The most general form to compute the amount accrued when interest is compounded with periodic contributions is given by the formula

`A = P (\left(1 + (r)/(n)\right)^(nt)-1)/((r)/(n))`

where
A = Accrued amount (principal + interest)

P = Periodic contribution to the sinking fund,

r = Annual nominal interest rate as a decimal

R = Annual nominal interest rate as a percent

r = R/100

n = number of compounding periods per unit of time

We are given A as 412,000 (amount at the end of 4 years) and asked to compute P(monthly contribution)

We have R = 18%, so r = 18/100 = 0.18

t = 4 years

n = 12 because we are compounding monthly so in 1 year we compound 12 times

Plugging these values into the equation we get

`412000 = P (\left(1 + (0.18)/(12)\right)^(12 \cdot 4)-1)/((0.18)/(12))\\\\`

We have

r/n = 0.18/12 = 0.015

1 + r/n = 1.015

nt = 12 x 4 = 48

`412000 = P\frac{ (1.015)^(48) -1 } {0.015}\\\\`

`412000 = P (1.043478)/(0.015)\\\\412000 = P \cdot 69.5652\\\\\P = (412000)/(69.5652)\\\\`

`P = 5,922.4998`

There may be differences in the given answer choices because of round off errors. The amount computed comes closest to the first answer choice

$5,920.44

Answer 2

(a) $5920.44

Explanation:

You want the monthly payment required to a sinking fund that is expected to have a value of $412,000 in 4 years if the account pays 18% interest.

Payment multiplier

A table of sinking fund payment values will tell you that the monthly payment required at an 18% interest rate for 4 years is $14.37 per thousand of account value.

Required payment

We want the account value to be 412 thousand, so the monthly payment will need to be ...

412 × $14.37 = $5,920.44

__

Additional comment

The actual payment required is $5922.50. Using a multiplier rounded to cents understates the payment because of rounding error.

If the more precise multiplier $14.375 per thousand is used, then the payment value would be correctly computed.

If you simply divide the desired $412000 into 48 equal payments, each would be $8,583.33. Since interest is earned, you know the payment is less than this amount. $5,920.44 is the only reasonable answer choice.