Finding the interest rate and the number of years
The future value and present value equations also help in finding the interest rate and the number of years that correspond to present and future value calculations.
If a security currently worth $12,800 will be worth $18,807.40 five years in the future, what is the implied interest rate the investor will earn on the security-assuming that no additional deposits or withdrawals are made?
О 8.00% О 6.40% О 1.47% О 0.29%
If an investment of $45,000 is earning an interest rate of 8.00%, compounded annually, then it will take _______ for this investment to reach a value of $56,687.04-assuming that no additional deposits or withdrawals are made during this time.
Which of the following statements is true-assuming that no additional deposits or withdrawals are made?
It takes 10.50 years for $500 to double if invested at an annual rate of 5%.
It takes 14.21 years for $500 to double if invested at an annual rate of 5%.
| Solution: | ||
| 1st | Answer is 1st option 8.00% | |
| Working Notes: | ||
| Future value = Present Value (1+r)^t | ||
| Future value =$18,807.40 | ||
| present value = $12,800 | ||
| t= period = 5 years | ||
| r= interest implied = ?? | ||
| Future value = Present Value (1+r)^t | ||
| $18,807.40 = $12,800 (1+ r)^5 | ||
| 1+ r = (18807.40/12800 )^(1/5) | ||
| 1+ r = 1.0800000007 | ||
| r= 1.08-1 | ||
| r= 0.08 | ||
| r= 8% | ||
| 2nd | ||
| Answer is 3.00 years | ||
| Working Notes: | ||
| Future value = Present Value (1+r)^t | ||
| Future value =$56,687.04 | ||
| present value = $45,000 | ||
| t= period = ??? | ||
| r= interest implied = 8% | ||
| Future value = Present Value (1+r)^t | ||
| $56,687.04 = 45,000 (1+ 0.08)^t | ||
| 56687.04/45000 = (1+ 0.08)^t | ||
| taking log on both side | ||
| (1.08)^t =1.259712 | ||
| Log(1.08)^t = Log(1.259712) | ||
| t x Log(1.08) = Log(1.259712) | ||
| t = Log(1.259712)/Log(1.08) | ||
| t= 3.00 | ||
| t= 3.00 years | ||
| 3rd. | ||
| TRUE is 2nd option "It takes 14.21 years for $500 to double if invested at an annual rate of 5%." | ||
| Working Notes: | ||
| lets check 1st equation | ||
| 10.50 years for 500 to double @ 5% | ||
| Future value = Present Value (1+r)^t | ||
| Future value = 500 x 2 = 1000 | ||
| present value = 500 | ||
| t= period = 10.50 | ||
| r= interest implied = 5% | ||
| Future value = Present Value (1+r)^t | ||
| FV = 500 x (1+.05)^10.50 | ||
| 1000= 500 x (1+.05)^10.50 | ||
| 1000 is not equal to 834.5601522 | ||
| Hence | 1st is FALSE | |
| lets check 2nd equation | ||
| 14.21 years for 500 to double @ 5% | ||
| Future value = Present Value (1+r)^t | ||
| Future value = 500 x 2 = 1000 | ||
| present value = 500 | ||
| t= period = 14.21 | ||
| r= interest implied = 5% | ||
| Future value = Present Value (1+r)^t | ||
| FV = 500 x (1+.05)^14.21 | ||
| 1000=$1,000.16 | ||
| $1,000 = $1,000 | ||
| Hence | 2nd equation TRUE | |
| Please feel free to ask if anything about above solution in comment section of the question. | ||
The future value and present value equations also help in finding the interest rate and the number of years that correspond to present and future value calculations.