Chapter 3.1, Problem 77E

a.

To determine

Calculate the equation of the model of the growth of each investment and use a graph to draw the model over a $20$ - years period.

Answer to Problem 77E

The equation of the three model of investment are $A=500{(1.07)}^t$ , $A=500{(1+\frac{0.07}{4})}^{4t}$ and $A=500e^{0.07t}$ .

Explanation of Solution

Given:

It is given in the question that there are three options for investing $\$500$ ,the first earns $7\%$ compounded annually ,the second earns $7\%$ compounded quarterly ,and the third earns $7\%$ compounded continuously.

Concept Used:

In this ,use the concept of compound interest and the formula used is $A=P{(1+\frac{r}{n})}^{nt}$ and $A=P{e}^{rt}$ .

Calculation:

Since,Initial amount, $P=500$

  $7\%$ rate is $r=0.07$

For compounded annually, use $A=P{(1+\frac{r}{n})}^{nt}withn=1$ ,

  $\begin{array}{}$ A=500{(1+\frac{0.07}{1})}^{1\times t}$\end{array}$

  $A=500{(1+0.07)}^t$

  $A=500{(1.07)}^t$

For quarterly compounding , use the same formula with $n=4$ ,

  $\begin{array}{}$ A=500{(1+\frac{0.07}{4})}^{4\times t}$\end{array}$

  $A=500{(1+\frac{0.07}{4})}^{4t}$

For continous compounding ,use the formula of $A=P{e}^{rt}$ ,

  $A=500e^{0.07t}$ .

Now,the graph of the three model in the same viewing window is :

  

The red curve shows the equation of compounded annually.

The green curve shows the equation of compounded continouslly.

The blue curve shows the equation of compounded quarterly.

Conclusion:

The equation of the three model of investment are $A=500{(1.07)}^t$ , $A=500{(1+\frac{0.07}{4})}^{4t}$ and $A=500e^{0.07t}$ .

b.

To determine

Find the investment that yields the highest return after $20$ years. Also ,find the differences in earning among the three investment.

Answer to Problem 77E

The investment in continuous compounding yields the highest return.The difference is $93.1$ more than annual.

Explanation of Solution

Given:

It is given in the question that the three equation is $A=500{(1.07)}^t$ , $A=500{(1+\frac{0.07}{4})}^{4t}$ and $A=500e^{0.07t}$ .

Concept Used:

In this ,use the concept of compound interest and the formula used is $A=P{(1+\frac{r}{n})}^{nt}$ and $A=P{e}^{rt}$ .Also,put the value and calculation will be more accurate.

Calculation:

From the graph obtained in part (a) , the continuous one yield the highest return ,as it will always do for the same interest rate.

But it has to be , put $t=20$ in all the equation to check it.

  $\begin{array}{}$ A=500{(1.07)}^t$\end{array}$

  $A=500{(1.07)}^{20}$

  $\begin{array}{l}A=500\times 3.869\\ A=1934.5\end{array}$

  $\begin{array}{}$ A=500{(1+\frac{0.07}{4})}^{4t}$\end{array}$

  $A=500{(1.0175)}^{4\times 20}$

  $A=500{(1.0175)}^{80}$

  $\begin{array}{l}A=500\times 4.006\\ A=2003\end{array}$

  $\begin{array}{l}A=500e^{0.07t}\\ A=500e^{0.07\times 20}\\ A=500e^{1.4}\\ A=500\times 4.055=2027.60\end{array}$

The quarterly yields $2003-1934.5=68.5$ more than the annual.

The continuous yields $2027.60-2003=24.6$ more than the quarterly and

  $2027.60-1934.5=93.1$ more than the annual.

Conclusion:

The investment in continuous compounding yields the highest return.The difference is $93.1$ more than annual.