Chapter 5, Problem 60QP

Summary Introduction

To determine: Whether the policy is worth purchasing.

Introduction:

The future value of cash flow is the accumulated value that includes an interest after a specified period. It is important to take an effective decision at present or to assess the investment potentiality. The total of the future values in each cash flow is known as the multiple cash flow.

Answer to Problem 60QP

The policy is not worth purchasing.

Explanation of Solution

Given information:

An insurance company offers a new policy to their customers. The children’s parents or grandparents will buy the policy at the time of the child’s birth. The parents can make 6 payments to the insurance company. The six payments are as follows:

• On the 1^st birthday, the payment amount is $800.
• On the 2^nd birthday, the payment amount is $800.
• On the 3^rd birthday, the payment amount is $900.
• On the 4^th birthday, the payment amount is $900.
• On the 5^th birthday, the payment amount is $1000.
• On the 6^th birthday, the payment amount is $1000.

After the 6^th birthday of the child, the payment will not be made. At the time when the child is 65 years, he or she gets $150,000. The interest rate for the first 6 years is 9% and for the rest of the years is 5.5%.

Note: From the given information, it is essential to compute the future value of the premiums for the comparisons of the promised cash payments at 65 years. Thus, it is necessary to determine the premiums’ value at 6 years first, as the rate of interest varies at that time.

Time line of the payments:

Formula to calculate the future value is as follows:

$\text{Future value}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}$

Note: PV denotes the present value, r denotes the rate of discount and t denotes the number of years.

Compute the future value for the five years is as follows:

$\begin{array}{c}{\text{Future value}}_1=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$800{\left(1+0.09\right)}^5\\ =\$700{\left(1.09\right)}^5\\ =\$1,230.90\end{array}$

Hence, the future value of 1st year is $1,230.90.

$\begin{array}{c}{\text{Future value}}_2=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$700{\left(1+0.09\right)}^4\\ =\$800{\left(1.09\right)}^4\\ =\$1,129.27\end{array}$

Hence, the future value of 2^nd year is $1,129.27.

$\begin{array}{c}{\text{Future value}}_3=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$900{\left(1+0.09\right)}^3\\ =\$900{\left(1.09\right)}^3\\ =\$1,165.53\end{array}$

Hence, the future value of 3^rd year is $1,165.53.

$\begin{array}{c}{\text{Future value}}_4=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$900{\left(1+0.09\right)}^2\\ =\$900{\left(1.09\right)}^2\\ =\$1,069.29\end{array}$

Hence, the future value of 4th year is $1,069.29.

$\begin{array}{c}{\text{Future value}}_5=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$1,000\left(1+0.09\right)\\ =\$1,000\left(1.09\right)\\ =\$1,090\end{array}$

Hence, the future value of 5^th year is $1,090.

Compute the value of 6^th year is as follows:

$\begin{array}{c}\text{Value at year 6}=\$1,230.90+\$1,129.27+\$1,165.53+\$1,069.29+\$1,090+\$1,000\\ =\$6,684.98\end{array}$

Note: The value of 6^th year is calculated by adding all the computed future values and the 6^th year’s value, that is, $1,000.

Hence, the value of 6^th year is $6,684.98.

Compute the future value of the lump sum at the 65^th birthday of the child is as follows:

$\begin{array}{c}\text{Future value}=\text{PV}{\left(1+\text{r}\right)}^{\text{t}}\\ =\$6,684.98{\left(1+0.055\right)}^{59}\\ =\$6,684.98{\left(1.055\right)}^{59}\\ =\$157,396.57\end{array}$

Note: The number of years is 59, because after the 6^th birthday of the child the payments will not be paid.

Hence, the future value of the lump sum at the 65^th birthday of the child is $157,396.57.

From the above calculation of the future value, it can be stated that the policy is not worth purchasing as the deposit’s value in the future is $157,396.57, but the contract will pay off at $150,000. The premium’s amount to $7,396.57 is more than the payoff of the policy.

Note: The present value of the two cash flows can be compared.

Formula to calculate the present value of the premiums is as follows:

$\text{Present value}=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}$

Compute the present value of the premiums as follows:

$\begin{array}{c}\text{Present value }=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\$800}{{\left(1.09\right)}^1}+\frac{\$800}{{\left(1.09\right)}^2}+\frac{\$900}{{\left(1.09\right)}^3}+\frac{\$900}{{\left(1.09\right)}^4}+\frac{\$1,000}{{\left(1.09\right)}^5}+\frac{\$1,000}{{\left(1.09\right)}^6}\\ =\$3,986.04\end{array}$

Hence, the present value of the premiums is $3,986.04.

The today’s value of the $150,000 at the age of 65 is as follows:

$\begin{array}{c}\text{Present value}=\frac{\text{Cash flow}}{{\left(1+\text{r}\right)}^{\text{t}}}\\ =\frac{\frac{\$150,000}{{\left(1+0.055\right)}^{59}}}{{\left(1.09\right)}^6}\\ =\frac{\frac{\$150,000}{23.54480611}}{1.677100111}\\ =\$3,798.72\end{array}$

The cash flow of the premiums is still higher. At the time of zero, the difference is $187.32$\left(\$3,986.04-\$3,798.72\right)$. At the time of comparing the streams of two or more cash flows, the cash flows with the highest value at one time will have the highest value at another time.