To determine the total amount payable to Bob and Lisa, we need to compute the annuities payable by each. Lisa paid her contributions at the beginning of the year implying that it is an annuity due. We assume that Bob paid the contributions at the end of the year. The total amount at the age of sixty-five will be;
FV = C X [ ]
For Bob, = 2,000 X [ ]
= 2000 x 118.93343
= $237,866.85
For Lisa = C X [ ] x (1+ i )
= 2,000 X [ ]x 1.07
= 2000 x 201.41 x 1.07
= $43100.98
At 32 years, Lisa total investment will be $43100.98 however the amount will continue to earn compound interest at the rate of 7% for the next 33 year until Lisa is 65 years old.
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Amount = P (1 + r)^n
= 43,100.98 (1.07)^33
= 43,100.98 x 9.32534
= $401931.28
Total deposits by the two for the period
Bob =2000 x 33 = $66,000
Lisa = 2000 x 13 = 26,000
| Bob | Lisa | |
| Age at start of investment | 32 | 20 |
| Initial starting balance | 0 | 2000 |
| Annual contribution | 2000 | 2000 |
| Age at retirement | 65 | 65 |
| Interest rate % | 7 | 7 |
| Years to maturity | 33 | 45 |
| Total contribution | 66000 | 26000 |
| Future amount at retirement | 237,866.85 | 43100.98 |
From the above analysis, it is evident that Lisa will receive a larger future value compared to Bob. Even though she invested for a shorter duration and less money compared to Bob, the initial investment has continued to earn compound with the total amount at the end of the twelve years being the principal.
Although the two make an equal annual contribution, the additional twelve years that Lisa invested the earned her more interest and increased her future value. From this example, we can deduce that annuities with similar contributions and interest rates result in different future amounts depending on the duration of the investment. Even though Lisa discontinued her annual contribution, the invested amount continued to earn interest irrespective of whether additional contributions are made.
