Chapter E, Problem 2SSQ

To determine

Calculate the present value for the given items.

Explanation of Solution

Present Value:

Present value refers to the current value of future sum of money in lump sum or in instalments with a stated rate of interest.

$\text{Present Value = }\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\times \text{Amount}$

Annuity:

An annuity is referred as a sequence of payment of fixed amount of cash flows that occurs over the equal intervals of time.

a. 1

Calculate the present value of $90,000 deposited in the savings account for 10 years, annual interest rate of 8% compounded annually.

$\begin{array}{c}\text{Present value of the amount deposited = }\left(\begin{array}{l}\text{Amount deposited }\times \\ \text{Present value factor for 10 years at 8 \% interest}\end{array}\right)\\ =\$90,000\times 0.463\\ =\$41,670\end{array}$

Therefore, the present value of the amount deposited is $41,670.

a. 2

Calculate the present value of $90,000 deposited in the savings account for 10 years, annual interest rate of 8 % compounded semiannually.

$\begin{array}{c}\text{Present value of the amount deposited = }\left(\begin{array}{l}\text{Amount deposited }\times \\ \text{Present value factor for 10 years at 12 \% interest}\end{array}\right)\\ =\$90,000\times 0.456\\ =\$41,040\end{array}$

Therefore, the present value of the amount deposited is $41,040.

Note:

When present value is compounded semiannually, the number of years will be doubled and the rate of interest will decrease by half of the given interest rate.

a. 3

Calculate the present value of $90,000 deposited in the savings account for 10 years, annual interest rate of 8 % compounded quarterly.

$\begin{array}{c}\text{Present value of the amount deposited = }\left(\begin{array}{l}\text{Amount deposited }\times \\ \text{Present value factor for 10 years at 8 \% interest}\end{array}\right)\\ =\$90,000\times 0.453\\ =\$40,770\end{array}$

Therefore, the present value of the amount deposited is $40,770.

b.

Calculate the present value of $1,000 amount received at the end of each year for 8 years, if the money earns an interest at the rate of 10% compounded annually.

$\begin{array}{c}\begin{array}{l}\text{Present value of the}\\ \text{ amount deposited}\end{array}\}\text{ = }\left(\begin{array}{l}\text{Amount deposited }\times \\ \text{Present value of annuity factor for 8 years at 10 \% interest}\end{array}\right)\\ =\$1,000\times 5.335\\ =\$5,335\end{array}$

Therefore, the present value of the amount deposited is $5,335.

c.

Calculate the present value of $600 amount received semiannually for the next fifteen years, if the money earns an interest at the rate of 8% compounded semiannually.

 $\begin{array}{c}\begin{array}{l}\text{Present value of the}\\ \text{ amount deposited}\end{array}\}\text{ = }\left(\begin{array}{l}\text{Amount deposited }\times \\ \text{Present value of annuity factor for 30 years at 4 \% interest}\end{array}\right)\\ =\$600\times 17.292\\ =\$10,375.20\end{array}$

Therefore, the present value of the amount deposited is $10,375.20.

Note:

When present value is compounded semiannually, the number of years will be doubled and the rate of interest will decrease by half of the given interest rate.

d.

Calculate the present value of $500,000 for the next 10 year, if the annual interest at the rate of 10% compounded annually.

$\begin{array}{c}\begin{array}{l}\text{Present value of the}\\ \text{ amount deposited}\end{array}\}\text{ = }\left(\begin{array}{l}\text{Amount deposited }\times \\ \text{Present value factor for 10 years at 10 \% interest}\end{array}\right)\\ =\$500,000\times 0.386\\ =\$193,000\end{array}$

Therefore, the present value of the amount deposited is $193,000.

e.

Calculate the present value of $2,500 received each half year for the next 10 years, with the annual interest at the rate of 12% compounded semiannually.

$\begin{array}{c}\begin{array}{l}\text{Present value of the}\\ \text{ amount deposited}\end{array}\}\text{ = }\left(\begin{array}{l}\text{Amount deposited }\times \\ \text{Present value of annuity factor for 20 years at 6\% interest}\end{array}\right)\\ =\$2,500\times 11.470\\ =\$28,675\end{array}$

Calculate the present value of $85,000 deposited at the end of 10 years, with the annual interest at the rate of 12% compounded semiannually.

$\begin{array}{c}\begin{array}{l}\text{Present value of the}\\ \text{ amount deposited}\end{array}\}\text{ = }\left(\begin{array}{l}\text{Amount deposited }\times \\ \text{Present value factor for 20 years at 6 \% interest}\end{array}\right)\\ =\$85,000\times 0.312\\ =\$26,520\end{array}$

Therefore, the present value of the amount deposited is $\left[\$28,675+\$26,520\right]$$55,195.

Note:

When present value is compounded semiannually, the number of years will be doubled and the rate of interest will decrease by half of the given interest rate.