Chapter B, Problem 18E

To determine

Time Value of Money:

The value of the money changes with the change in time. If an individual deposits his savings in the bank then, the amount will increase at the specified interest rate. But if he invests that same amount in different avenues then he may get loss or more profit.

Future Value:

The future value is that value of an investment which will be realizable in future. When amount is invested today at a specific rate, its future value will be more than the present value of money invested.

Present Value:

Present value is that value of money which measures the worth of a future amount in today’s value adjusted for interest and inflation. It is used in finance for the valuation of future value, stock and bond pricing.

Annuity:

An annuity refers to a series of fixed cash flows, received or paid by a person at a specific frequency. These cash flows occur for a specified period of time.

a.

To identify: The present value of an investment.

Explanation of Solution

The items required for the calculation of present value of an investment are future value of investment and discount factor.

The future value is $60,000. (Given)

The interest rate is 9%. (Given)

The time period is 4 years. (Given)

The discount factor at 9% in 4 years is 0.7084. (From Table B.1)

Formula to calculate the present value of an investment,

  $\text{Present}\text{value}=\text{Future}\text{value}\text{of}\text{the}\text{investment}\times \text{Discount}\text{factor}$

Substitute $60,000 for future value of the investment and 0.7084 for discount factor.

  $\begin{array}{c}\text{Present}\text{value}=\$60,000\times 0.7084\\ =\$42,504\end{array}$

The present value is $42,504.

Hence, the present value of the investment is $42,504 .

b.

To determine

To identify: The invested amount today to have a certain amount in future.

Explanation of Solution

To compute the amount needs to deposit today, the present value of investment needs to calculate.

The items required for the calculation of present value of an investment are future value of investment and discount factor.

The future value is $15,000. (Given)

The interest rate is 8%. (Given)

The time period is 2 years. (Given)

The discount factor at 8% in 2 years is 0.8573. (From Table B.1)

Formula to calculate the present value of an investment,

  $\text{Present}\text{value}=\text{Future}\text{value}\text{of}\text{the}\text{investment}\times \text{Discount}\text{factor}$

Substitute $15,000 for future value of the investment and 0.8573 for discount factor.

  $\begin{array}{c}\text{Present}\text{value}=\$15,000\times 0.8573\\ =\$12,859.5\end{array}$

The present value is $12,859.50.

Hence, the amount needs to deposit today is $12,859.50 .

c.

To determine

To analyze: The better option between the given situations.

Explanation of Solution

In order to analyze the better option between to have $463 today or $1,000 after 10 years, the present value of $1,000 needs to calculate.

Compute the present value of investment $1,000

The future value is $1,000. (Given)

The interest rate is 9%. (Given)

The time period is 10 years. (Given)

The present value factor is 0.4224. (Table B.1)

Formula to calculate the present value,

  $\text{Present}\text{value}=\text{Future}\text{value}\text{of}\text{the}\text{investment}\times \text{Discount}\text{factor}$

Substitute $1,000 for future value and 0.4224 for discount factor.

  $\begin{array}{c}\text{Present}\text{value}=\$1,000\times 0.4224\\ =\$422.40\end{array}$

The present value of investment is $422.40.

The present value of $1,000 is less so it is better to select the option to take $463 now and then invests to have more than $1,000 in future.

Hence, it is better to have $463 today.

d.

To determine

To identify: The value of cost of sticker in future.

Explanation of Solution

The items required for the calculation of future value are initial investment and future value factor.

The present cost or initial investment is $90. (Given)

The interest rate is 5%. (Given)

The time period is 8 years. (Given)

The future value factor is 1.4775. (Table B.2)

Formula to calculate the future value of an investment,

  $\text{Future}\text{value}=\text{Present}\text{value}\text{of}\text{the}\text{investment}\times \text{Discount}\text{factor}$

Substitute $90 for present value of investment and 1.4775 for discount factor.

  $\begin{array}{c}\text{Future}\text{value}=\$90\times 1.4775\\ =\$132.98\end{array}$

The future value is $132.98.

Hence, the value of striker in 8 years would be $132.98 .

e.

To determine

To identify: The cost of house in 8 years.

Explanation of Solution

To calculate the price of house after 8 years it is required to calculate its future value.

The items required for the calculation of future value are current price and future value factor.

The present price of house is $158,500. (Given)

The interest rate is 10%. (Given)

The time period is 8 years. (Given)

The future value factor is 2.1436. (Table B.2)

Formula to calculate the future value of an investment,

  $\text{Future}\text{value}=\text{Current}\text{price}\times \text{Discount}\text{factor}$

Substitute $158,500 for current price and 2.1436 for discount factor.

  $\begin{array}{c}\text{Future}\text{value}=\$158,500\times 2.1436\\ =\$339,760.60\end{array}$

The future value is $339,760.60.

Hence, the price of house in 8 years would be $339,760.60 .

f.

To determine

To identify: The present value of investments.

Explanation of Solution

To compute the amount that should be invest today, the present value of an investment and present value of an annuity needs to calculate.

Compute the present value of an investment

The future value is $10,000. (Given)

The interest rate is 6%. (Given)

The time period is 10 years. (Given)

The present value factor is 0.3855. (Table B.1)

Formula to calculate the present value,

  $\text{Present}\text{value}=\text{Future}\text{value}\text{of}\text{the}\text{investment}\times \text{Discount}\text{factor}$

Substitute $10,000 for future value and 0.3855 for discount factor.

  $\begin{array}{c}\text{Present}\text{value}=\$10,000\times 0.3855\\ =\$3,855\end{array}$

The present value of investment is $3,855.

Compute the present value of an annuity

The annuity amount is $400. (Given)

The present value annuity factor at 6% interest rate for 10 years is 7.3601. (From Table B.3)

Formula to calculate the present value of an annuity,

  $\text{Present}\text{value}=\text{Annuity}\text{value}\times \text{Present}\text{annuity}\text{value}\text{factor}$

Substitute $400 for annuity value and 7.3601 for present annuity value factor.

  $\begin{array}{c}\text{Present}\text{value}=\$400\times 7.3601\\ =\$2,944\end{array}$

The present value of annuity is $2,944.

Compute the total amount that should be invested today

Calculated,

The present value of annuity is $2,944.

The present value of investment is $3,855

Formula to calculate the total invested amount,

  $\text{Invested}\text{amount}=\text{Present}\text{value}\text{of}\text{annuity}+\text{Present}\text{value}\text{of}\text{investment}$

Substitute $2,944 for present value of an annuity and $3,855 for present value of investment.

  $\begin{array}{c}\text{Invested}\text{amount}=\$2,944+\$3,855\\ =\$6,799\end{array}$

Hence, the invested amount is $6,799.

g.

To determine

To identify: The present value of lotteries amount received as an annuity.

Explanation of Solution

Compute the present value of an annuity

The annuity amount is $500,000. (Given)

The present value annuity factor at 6% interest rate for 20 years is 11.4699. (From Table B.3)

Formula to calculate the present value of an annuity,

  $\text{Present}\text{value}=\text{Annuity}\text{value}\times \text{Present}\text{annuity}\text{value}\text{factor}$

Substitute $500,000 for annuity value and 11.4699 for present annuity value factor.

  $\begin{array}{c}\text{Present}\text{value}=\$500,000\times 11.4699\\ =\$5,734,950\end{array}$

The present value of annuity is $5,734,950.

Hence, the present value of annuity amount received is $5,734,950.