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svetoff [14.1K]
3 years ago
8

A woman worked for 30 years before retiring. At the end of the first year of employment she deposited 5000 into an account for h

er retirement. At the end of each subsequent year of employment, she deposited 3% more than the prior year. The woman made a total of 30 deposits. She will withdraw 50,000 at the beginning of the first year of retirement and will make annual withdrawals at the beginning of each subsequent year for a total of 30 withdrawals. Each of these subsequent withdrawals will be 3% more than the prior year. The final withdrawal depletes the account. The account earns a constant annual effective interest rate. Calculate the account balance after the final deposit and before the first withdrawal.
Business
1 answer:
Mrrafil [7]3 years ago
8 0

Answer:

$797,837

Explanation:

the first withdrawal is $50,000

the second is $51,500

and so on...

the formula that used to solve the interest rate earned by the annuity is:

$50,000 x {[(1 + i)³⁰ - (1 + 3%)³⁰] / [(1 + i)³⁰ x (i - 3%)]} x (1 + i) = $5,000 x {[(1 + i)³⁰ - (1 + 3%)³⁰] / (i - 3%)}

we start to simplify the equation by cancelling  {[(1 + i)³⁰ - (1 + 3%)³⁰] / (i - 3%)}

[$50,000 x (1 + i)] / (1 + i)³⁰ = $5,000

now we cancel $5,000 on each side:

[10 x (1 + i)] / (1 + i)³⁰ = 1

now lets take away (1 + i):

10 / (1 + i)²⁹ = 1

things get a little bit more simple now:

10 = (1 + i)²⁹

²⁹√10 = ²⁹√(1 + i)²⁹

1.082636734 = 1 + i

i = 1.082636734 - 1 = 0.082636734 = 8.2636734%

now we replace i in any equation:

= $50,000 x {[(1 + 0.082636734)³⁰ - 1.03³⁰] / [(1 + 0.082636734)³⁰ x (0.082636734 - 0.03)]} x (1 + 0.082636734)

= $50,000 x  {[10.82636738 - 2.427262471] / [10.82636738 x 0.052636734]} x (1 + 0.082636734)

= $50,000 x  {8.399104909 / 0.56986462} x (1.082636734)

= $50,000 x 14.73877236 x 1.082636734

= $797,837

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Answer:

<u><em>Part a </em></u>

<u>Belmain Co.</u>

<u>Estimated Income statement for the year ended 2017.</u>

Sales ($240 x 12,000)                                                               $2,880,000

<u>Less Variable Costs :</u>

Direct Materials ($50.00 x 12,000)                                           ($600,000)

Direct Labor ($30.00 x 12,000)                                                 ($360,000)

Factory Overheads ($6.00 x 12,000)                                          ($72,000)

Sales Salaries and Commissions ( $4.00 x 12,000)                  ($48,000)

Miscellaneous selling expenses ( $1.00 x 12,000)                     ($12,000)

Supplies ($4.00 x 12,000)                                                           ($48,000)

Miscellaneous administrative expenses ($1.00 x 12,000)         ($12,000)

Contribution                                                                               $1,728,000

<u>Less Fixed Expenses :</u>

Factory overhead                                                                     ($350,000)

Sales salaries and commissions                                             ($340,000)

Advertising                                                                                 ($116,000)

Travel                                                                                            ($4,000)

Miscellaneous selling expense                                                   ($2,300)

Office and officers’ salaries                                                    ($325,000)

Supplies                                                                                        ($6,000)

Miscellaneous administrative expense                                      ($8,700)

Net Income ( Loss)                                                                     $576,000

<u><em>Part b</em></u>

0.6 or 60 %

<u><em>Part c</em></u>

Break-even sales (units) = 8,000

Break-even sales (dollars) = $1,920,000

<u><em>Part d</em></u>

<em>See attachment </em>

<u><em>Part e</em></u>

Margin of safety in dollars  =    $960,000

Margin of safety in percentage  =  33.3 %

<em><u>Part f</u></em>

Operating Leverage = 3.00

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<u>Income Statement :</u>

<em>Sales - Expenses = Income</em>

Note : I have separated Variable and Fixed Expenses

<u>Contribution Margin ratio :</u>

<em>Contribution Margin ratio = Contribution ÷ Sales</em>

                                          =  $1,728,000  ÷  $2,880,000

                                          = 0.6 or 60 %

<u>Break-even sales ( units and dollars) :</u>

<em>Break-even sales (units) = Fixed Costs ÷ Contribution per unit</em>

                                        = $1,152,000 ÷ $144.00

                                        = 8,000

<em>Break-even sales (dollars) = Fixed Costs ÷ Contribution margin ratio</em>

                                            = $1,152,000 ÷ 0.60

                                            = $1,920,000

<u>Margin of safety in dollars and as a percentage of sales :</u>

<u />

<em>Margin of safety in dollars  = Expected Sales (dollars) - Break-even sales (dollars)</em>

                                             =  $2,880,000 - $1,920,000

                                             =   $960,000

<em>Margin of safety in %       = (Expected Sales  - Break-even sales ) ÷ Expected Sales</em>

                                             = $960,000 ÷ $2,880,000

                                             = 33.3 %

<u>Operating leverage</u>

<em>Operating Leverage = Contribution ÷ Earnings Before Interest and Tax</em>

                                  =  $1,728,000 ÷ $576,000

                                  = 3.00

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