An insurance company is offering a new policy to its customers. Typically, the policy is bought by a

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.