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