An insurance company is offering a new policy to its customers. Typically, the policy is bought by a parent or grandparent for a child at the child’s birth. The details of the policy are as follows: The purchaser (say, the parent) makes the following six payments to the insurance company:
| First birthday: | $ | 790 | |
| Second birthday: | $ | 790 | |
| Third birthday: | $ | 890 | |
| Fourth birthday: | $ | 850 | |
| Fifth birthday: | $ | 990 | |
| Sixth birthday: | $ | 950 | |
| After the child’s sixth birthday, no more payments are made. When the child reaches age 65, he or she receives $290,000. The relevant interest rate is 10 percent for the first six years and 7 percent for all subsequent years. |
Find the future value of the payment at the child's 65th birthday. (Do not round intermediate calculations and round your final answer to 2 decimal places. (e.g., 32.16)) |
| Future value | $ |
Explanation:
We need to find the FV of the premiums to compare with the cash payment promised at age 65. We have to find the value of the premiums at year 6 first since the interest rate changes at that time. So: |
| FV1 = $790(1.10)5 = $1,272.30 |
| FV2 = $790(1.10)4 = $1,156.64 |
| FV3 = $890(1.10)3 = $1,184.59 |
| FV4 = $850(1.10)2 = $1,028.50 |
| FV5 = $990(1.10)1 = $1,089.00 |
| Value at Year 6 = $1,272.30 + 1,156.64 + 1,184.59 + 1,028.50 + 1,089.00 + 950 |
| Value at Year 6 = $6,681.03 |
| Finding the FV of this lump sum at the child’s 65th birthday: |
| FV = $6,681.03(1.07)59 = $361,814.88 |
The policy is not worth buying; the future value of the deposits is $361,814.88, but the policy contract will pay off $290,000. The premiums are worth $71,814.88 more than the policy payoff. |
| Note, we could also compare the PV of the two cash flows. The PV of the premiums is: |
| PV = $790/1.10 + $790/1.102 + $890/1.103 + $850/1.104 + $990/1.105 + $950/1.106 |
| PV = $3,771.27 |
| And the value today of the $290,000 at age 65 is: |
| PV = $290,000/1.0759 = $5,354.95 |
| PV = $5,354.95/1.106 = $3,022.73 |
The premiums still have the higher cash flow. At time zero, the difference is $748.54. Whenever you are comparing two or more cash flow streams, the cash flow with the highest value at one time will have the highest value at any other time. |
Here is a question for you: Suppose you invest $748.54, the difference in the cash flows at time zero, for six years at a 10 percent interest rate, and then for 59 years at a 7 percent interest rate. How much will it be worth? Without doing calculations, you know it will be worth $71,814.88, the difference in the cash flows at time 65! |
| Calculator Solution: |
Note: Intermediate answers are shown below as rounded, but the full answer was used to complete the calculation. |
| Value at Year 6: |
| Enter | 5 | 10% | $790 | ||||||||||||
N | I/Y | PV | PMT | FV | |||||||||||
| Solve for | $1,272.30 | ||||||||||||||
| Enter | 4 | 10% | $790 | ||||||||||||
N | I/Y | PV | PMT | FV | |||||||||||
| Solve for | $1,156.64 | ||||||||||||||
| Enter | 3 | 10% | $890 | ||||||||||||
N | I/Y | PV | PMT | FV | |||||||||||
| Solve for | $1,184.59 | ||||||||||||||
| Enter | 2 | 10% | $850 | ||||||||||||
N | I/Y | PV | PMT | FV | |||||||||||
| Solve for | $1,028.50 | ||||||||||||||
| Enter | 1 | 10% | $990 | ||||||||||||
N | I/Y | PV | PMT | FV | |||||||||||
| Solve for | $1,089.00 | ||||||||||||||
| So, at Year 5, the value is: $1,272.30 + 1,156.64 + 1,184.59 + 1,028.50 + 1,089.00 + 950 = $6,681.03 |
| At Year 65, the value is: |
| Enter | 59 | 7% | $6,681.03 | ||||||||||||
N | I/Y | PV | PMT | FV | |||||||||||
| Solve for | $361,814.88 | ||||||||||||||
The policy is not worth buying; the future value of the deposits is $361,814.88 but the policy contract will pay off $290,000.00. |