Charlie wants to retire in 15 years, and he wants to have an annuity of $50,000 a year for 20...

The correct option is (a) $167,130

Charlie wants an annuity of $50,000 for 20 years every year, and he wants to receive the first annuity payment the day he retires, i.e., the first day of the period. That means he wants to have an annuity due.

The amount that Charlie must invest today would be equal to the present value of cash flows of the annuity.

For annuity due,

{eq}\text{Cumulative Present Value Factor} = \dfrac{1 - (1 + i)^{-n}}{\dfrac{i}{1+i}} {/eq}

where,

i = interest rate

n = number of periods.

Computation of present value of annuity on the day Charlie retires

i = 8%

n = 20 (as the annuity is for 20 years)

{eq}\text{Cumulative Present Value Factor (8%, 20 years} = \dfrac{1 - (1 + 0.08)^{-20}}{\dfrac{0.08}{1+0.08}} {/eq}

{eq}\text{Cumulative Present Value Factor (8%, 20 years} = \dfrac{1 - (1.08)^{-20}}{\dfrac{0.08}{1.08}} {/eq}

{eq}\text{Cumulative Present Value Factor (8%, 20 years} = 10.6036 {/eq}

{eq}\text{Present value of annuity at t=15} = \text{Annuity Amount} * 10.6036 {/eq}

{eq}\text{Present value of annuity at t=15} = \$50,000 * 10.6036 {/eq}

{eq}\text{Present value of annuity at t=15} = \$530,180 {/eq}

To arrive at the required investment amount today, we need to discount it to t=0 further.

This time n would be 15 years as Charlie would retire in 15 years.

{eq}\text{Required Investment} = \dfrac{\$530,180}{1.08^{15}} {/eq}

{eq}\text{Required Investment} = \$167,134.85 {/eq}

Required investment (rounded to nearest $10) = $167,130