Question

In: Statistics and Probability

Find the interest rate needed for the sinking fund to reach the required amount. Assume that...

Find the interest rate needed for the sinking fund to reach the required amount. Assume that the compounding period is the same as the payment period.

​$30,000 to be accumulated in 10 ​years; annual payments of ​$2322

The interest rate needed is approximately___​%.

​(Round to two decimal places as​ needed.)

Solutions

Expert Solution

Answer

Back-up Theory

If a sum, P, is invested every year (at the start of the year) for T years at an interest rate of r% per annum, the amount at the end of T years is:

A = P{(1 + i)T + (1 + i)T-1 + (1 + i)T-2 + (1 + i)T-3 + ……. + (1 + i)1}

= P(1 + i){(1 + i)T - 1)}/{(1 + i) – 1}

= P[(1 + i){(1 + i)T - 1)}/i]

Now, to work out the answer,

Here we have: A = 30000 and P = 2322 and hence A/P = 12.92 and T = 10

So, [(1 + i){(1 + i)10 - 1)}/i] = 12.92

Or, (1 + i)11 - (1 + i) = 12.92i

Or, (1 + i)11 – 13.92i = 1.

Using Excel, i is found to be close to 0.0465 [Details of calculations are given at the end]

Thus, interest rate required is approximately, 4.65% Answer

Details of calculations

i

(1 + i)

(1 + i)^11

13.92i

d

0.1

1.1

2.85311671

1.392

1.461117

0.15

1.15

4.6523914

2.088

2.564391

0.05

1.05

1.71033936

0.696

1.014339

0.04

1.04

1.53945406

0.5568

0.9826541

0.045

1.045

1.62285305

0.6264

0.996453

0.046

1.046

1.64001768

0.64032

0.9996977

0.047

1.047

1.6573472

0.65424

1.0031072

0.0465

1.0465

1.64866174

0.64728

1.0013817

DONE


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