Chapter 11, Problem 1CCOMP

1.

To determine

Compute the issue (sale) price on January 1 when the annual market interest rate is 5%.

Answer to Problem 1CCOMP

Particulars	Amount in $
Present value (2)	613,910
Present value of annuity (3)	386,087
Issue price	999,997

(Table 1)

Therefore, the issue price on January 1when annual market interest rate is 5% is $999,997.

Explanation of Solution

Present value:

Present value is the current value of an amount that is to be paid or received in future. Present value is determined by using the formula:

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

Annuity:

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

Working Notes:

Compute the issue (sale) price on January 1 when the annual market interest rate is 5%:

$\begin{array}{c}\text{Interest payment =Issue price }\times \text{ interest rate}\\ \text{=\$1,000,000}\times \text{5\%}\\ \text{=\$50,000}\end{array}$ (1)

$\begin{array}{c}\text{Present Value = }\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\times \text{Amount}\\ \text{=}\frac{1}{{\left(1+0.05\right)}^{10}}\times 1,000,000\\ =0.61391\times 1,000,000\\ =\$613,910\end{array}$ (2)

$\begin{array}{c}\text{Present Value of Annuity= }\frac{\text{1}-{\left(1+\text{i}\right)}^{-\text{n}}}{\text{i}}\times \text{Amount}\\ =\frac{1-{\left(1+0.05\right)}^{-10}}{5\%}\times \$50,000(1)\\ =7.72173\times \$50,000\\ =\$386,087\end{array}$ (3)

2.

To determine

Compute the issue (sale) price on January 1 when the annual market interest rate is 4%.

Answer to Problem 1CCOMP

Particulars	Amount in $
Present value (4)	675,560
Present value of annuity (5)	405,545
Issue price	1,081,105

(Table 2)

Therefore, the issue price on January 11when annual market interest rate is 4% is $1,081,105.

Explanation of Solution

Present value:

Present value is the current value of an amount that is to be paid or received in future. Present value is determined by using the formula:

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

Annuity:

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

Working Notes:

Compute the issue (sale) price on January 1 when the annual market interest rate is 4%:

$\begin{array}{c}\text{Present Value = }\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\times \text{Amount}\\ \text{=}\frac{1}{{\left(1+0.04\right)}^{10}}\times 1,000,000\\ =0.67556\times 1,000,000\\ =\$675,560\end{array}$ (4)

$\begin{array}{c}\text{Present Value of Annuity= }\frac{\text{1}-{\left(1+\text{i}\right)}^{-\text{n}}}{\text{i}}\times \text{Amount}\\ =\frac{1-{\left(1+0.04\right)}^{-10}}{4\%}\times \$50,000(1)\\ =8.11090\times \$50,000\\ =\$405,545\end{array}$ (5)

3.

To determine

Compute the issue (sale) price on January 1 when the annual market interest rate is 6%.

Answer to Problem 1CCOMP

Particulars	Amount in $
Present value (6)	558,390
Present value of annuity (7)	368,005
Issue price	926,395

(Table 3)

Therefore, the issue price on January 11when annual market interest rate is 6% is $926,395.

Explanation of Solution

Present value:

Present value is the current value of an amount that is to be paid or received in future. Present value is determined by using the formula:

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

Annuity:

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

Working Notes:

Compute the issue (sale) price on January 1 when the annual market interest rate is 6%:

$\begin{array}{c}\text{Present Value = }\frac{1}{{\left(1+\text{i}\right)}^{\text{n}}}\times \text{Amount}\\ \text{=}\frac{1}{{\left(1+0.06\right)}^{10}}\times 1,000,000\\ =0.55839\times 1,000,000\\ =\$558,390\end{array}$ (6)

$\begin{array}{c}\text{Present Value of Annuity= }\frac{\text{1}-{\left(1+\text{i}\right)}^{-\text{n}}}{\text{i}}\times \text{Amount}\\ =\frac{1-{\left(1+0.06\right)}^{-10}}{6\%}\times \$50,000(1)\\ =7.36009\times \$50,000\\ =\$368,005\end{array}$ (7)