Chapter 8.2, Problem 40PPS

(a)

To determine

To find: the function of retirement plan for both options.

Answer to Problem 40PPS

Thefunctions are $y\left(t\right)=5000{\left(1+\frac{0.065}{4}\right)}^{4t}$ and $y\left(t\right)=5000{\left(1+\frac{0.042}{12}\right)}^{12t}+5000{\left(1+\frac{0.023}{52}\right)}^{52t}$.

Explanation of Solution

Given information:

The retirement plan of Mrs Jackson is shown as,

  

The model formula for the ending value at time $n$ from the initial value at time $0$ is $y\left(t\right)=P{\left(1+\frac{i_a}{p}\right)}^{np}$.

For option A,

Substitute $5000$ for $P$ , $4$ for $p$ (for quarterly), $t$ for $n$ , and $0.7\%$ for $i_a$ in the formula $y\left(t\right)=P{\left(1+\frac{i_a}{p}\right)}^{np}$ to find the value of $a$.

  $y\left(t\right)=5000{\left(1+\frac{0.065}{4}\right)}^{4t}$

For option B,

Substitute $5000$ for $P$ , $12$ for $p$ (for Monthly), $t$ for $n$ , and $4.2\%$ for $i_a$ in the formula $y\left(t\right)=P{\left(1+\frac{i_a}{p}\right)}^{np}$ to find the value of $a$.

  $y\left(t\right)=5000{\left(1+\frac{0.042}{12}\right)}^{12t}\mathrm{...}\left(1\right)$

Substitute $5000$ for $P$ , $52$ for $p$ (for weekly), $t$ for $n$ , and $2.3\%$ for $i_a$ in the formula $y\left(t\right)=P{\left(1+\frac{i_a}{p}\right)}^{np}$ to find the value of $a$.

  $y\left(t\right)=5000{\left(1+\frac{0.023}{52}\right)}^{52t}\mathrm{...}\left(2\right)$

Add the equation (1) and the equation (2).

  $y\left(t\right)=5000{\left(1+\frac{0.042}{12}\right)}^{12t}+5000{\left(1+\frac{0.023}{52}\right)}^{52t}$

Therefore, the functions are $y\left(t\right)=5000{\left(1+\frac{0.065}{4}\right)}^{4t}$ and $y\left(t\right)=5000{\left(1+\frac{0.042}{12}\right)}^{12t}+5000{\left(1+\frac{0.023}{52}\right)}^{52t}$.

(b)

To determine

To find: the graph of both the functions.

Answer to Problem 40PPS

The required graphs are shown in figure (1) and (2).

Explanation of Solution

Given information:

The retirement plan of Mrs Jackson is shown as,

  

Consider the functions.

  $y\left(t\right)=5000{\left(1+\frac{0.042}{12}\right)}^{12t}+5000{\left(1+\frac{0.023}{52}\right)}^{52t}$

  $y\left(t\right)=5000{\left(1+\frac{0.065}{4}\right)}^{4t}$

Plot the graph of the function $y\left(t\right)=5000{\left(1+\frac{0.065}{4}\right)}^{4t}$ of Option A.

  

Figure(1)

Plot the graph of the function $y\left(t\right)=5000{\left(1+\frac{0.042}{12}\right)}^{12t}+5000{\left(1+\frac{0.023}{52}\right)}^{52t}$ of Option A.

  

Figure(2)

Therefore, the required graphs are shown in figure (1) and (2).

(c)

To determine

To find: the best option of retirement plan for Mrs Jackson.

Answer to Problem 40PPS

The option A is better than the option B.

Explanation of Solution

Given information:

The retirement plan of Mrs Jackson is shown as,

  

Consider the functions.

  $y\left(t\right)=5000{\left(1+\frac{0.065}{4}\right)}^{4t}\mathrm{...}\left(1\right)$

  $y\left(t\right)=5000{\left(1+\frac{0.042}{12}\right)}^{12t}+5000{\left(1+\frac{0.023}{52}\right)}^{52t}\mathrm{...}\left(2\right)$

As the equation (1) has higher interest rate per dollar of investment than equation (2).

Therefore, the option A is better than the option B.