In: Statistics and Probability
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.)
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