Chapter M, Problem 15P

Cash Flow Amounts On January 1, 2019, Philip Holding invests $40,000 in an annuity to provide 8 equal semi-annual payments. Interest is 10%, compounded semiannually.

Required:

Compute the equal semiannual amounts that Philip will receive assuming that the first withdrawal is to be received on:

July 1, 2019

January 1, 2019

July 1, 2022

January 1, 2024

To determine

Determine the equal semiannual amount that would be received by Person P, if the first withdrawal is to be received on July 1, 2019.

Explanation of Solution

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

Cash flow occurs during the first day of each time period is known as an annuity due, whereas cash flow occurs during the last day of each time period is known as an ordinary annuity.

Determine the equal semiannual amount that would be received by Person P, if the first withdrawal is to be received on July 1, 2019.

Present value (PV_O) – $40,000 investment

Interest rate (i) – 5% per half year $\left(\frac{\text{10\%per}\text{year}}{\text{12}}\text{×6months in a semiannual}\right)$

Number of period (n) – 8 semi-annual withdrawals (principal and interest) beginning at the end of 1^st semi-annum. Here, the first withdrawal (first cash flow) occurs during the last day of semi year. Hence, it is an ordinary annuity.

$\begin{array}{c}{\text{PV}}_{\text{O}}=\text{Cash flow}\times \left({\text{p}}_{\text{On,i}}\right)\\ \$40,000=\text{Cash flow}\times \left({\text{p}}_{\text{On=8,i=5\%}}\right)\\ \$40,000=\text{Cash flow}\times \text{6}\text{.463213}\\ \text{Cash flow=}\frac{\$40,000}{\text{6}\text{.463213}}\end{array}$

$\text{Cash flow=}\$6,188.87$

Hence, the equal semiannual amount that would be received by Person P is $6,188.87, if the first withdrawal is to be received on July 1, 2019.

Note:

• Present value of ordinary annuity of $1: n = 8, i =5% is taken from the table value (Table 4 at the end of the time value money module).

To determine

Determine the equal semiannual amount that would be received by Person P, if the first withdrawal is to be received on January 1, 2019.

Explanation of Solution

Determine the equal semiannual amount that would be received by Person P, if the first withdrawal is to be received on July 1, 2019.

Present value of annuity due (PV_D) – $40,000 investment

Interest rate (i) – 5% per half year $\left(\frac{\text{10\%per}\text{year}}{\text{12}}\text{×6months in a semiannual}\right)$

Number of period (n) – 8 semi-annual withdrawals (principal and interest) beginning at the first day of the semi-annum. Here, the first withdrawal (first cash flow) occurs during the first day of semi year. Hence, it is an annuity due.

$\begin{array}{c}{\text{PV}}_{\text{D}}=\text{Cash flow}\times \left({\text{p}}_{\text{Dn,i}}\right)\\ \$40,000=\text{Cash flow}\times \left({\text{p}}_{\text{Dn=8,i=5\%}}\right)\\ \$40,000=\text{Cash flow}\times \text{6}\text{.786373}\\ \text{Cash flow=}\frac{\$40,000}{\text{6}\text{.786373}}\end{array}$

$\text{Cash flow=}\$5,894.16$

Hence, the equal semiannual amount that would be received by Person P is $5,894.16, if the first withdrawal is to be received on January 1, 2019.

Note:

• Present value of annuity due of $1: n = 8, i =5% is taken from the table value (Table 5 at the end of the time value money module).

To determine

Determine the equal semiannual amount that would be received by Person P, if the first withdrawal is to be received on July 1, 2022.

Explanation of Solution

This is a deferred annuity, since an investment of $40,000 accrues interest for 7 semi annum before the withdrawals begin. Present value factor for a single sum of $40,000 for 6semi annum of deferral should be taken into consideration.

Determine the equal semiannual amount that would be received by Person P, if the first withdrawal is to be received on July 1, 2022.

$\begin{array}{l}{\text{PV}}_{\text{Deferred}}=\text{Cash flow}\times \left({\text{p}}_{\text{On=8,i=5\%}}\times {\text{p}}_{\text{k=6,i=5\%}}\right)\\ \$40,000=\text{Cash flow}\times \left(\text{6}\text{.463213}\times 0.746215\right)\\ \$40,000=\text{Cash flow}\times 4.8229465\\ \text{Cash flow}=\$8,293.69\end{array}$

Hence, the equal semiannual amount that would be received by Person P is $8,293.69, if the first withdrawal is to be received on July 1, 2022.

Note:

• i stands for interest rate for each of the stated time periods
• n stands for number of time periods
• Present Value _Deferred stands for present value of deferred annuity
• Present value of ordinary annuity of $1: n = 8, i =5% is taken from the table value (Table 4 at the end of the time value money module).
• Present value of single sum of (P_k) $1 for 6 period of deferral: k = 6, i = 5% is taken from the table value (Table 3 at the end of the time value money module).

To determine

Determine the equal semiannual amount that would be received by Person P, if the first withdrawal is to be received on January 1, 2024.

Explanation of Solution

This is a deferred annuity, since an investment of $40,000 accrues interest for 10 semi annum before the withdrawals begin. Present value factor for a single sum of $40,000 for 9semi annum of deferral should be taken into consideration.

Determine the equal semiannual amount that would be received by Person P, if the first withdrawal is to be received on January 1, 2024.

$\begin{array}{l}{\text{PV}}_{\text{Deferred}}=\text{Cash flow}\times \left({\text{p}}_{\text{On=8,i=5\%}}\times {\text{p}}_{\text{k=9,i=5\%}}\right)\\ \$40,000=\text{Cash flow}\times \left(\text{6}\text{.463213}\times 0.644609\right)\\ \$40,000=\text{Cash flow}\times 4.1662453\\ \text{Cash flow}=\$9,600.97\end{array}$

Hence, the equal semiannual amount that would be received by Person P is $9,600.97, if the first withdrawal is to be received on January 1, 2024.

Note:

• i stands for interest rate for each of the stated time periods
• n stands for number of time periods
• Present Value _Deferred stands for present value of deferred annuity
• Present value of ordinary annuity of $1: n = 8, i =5% is taken from the table value (Table 4 at the end of the time value money module).
• Present value of single sum of (P_k) $1 for 9 period of deferral: k = 9, i = 5% is taken from the table value (Table 3 at the end of the time value money module).