Chapter 4, Problem 22PS

a

Summary Introduction

To compute: The annual rate of return when the plan of investment is for 2 years.

Introduction:

Rate of return on Investment: Normally, when an investor invests his funds, his intention will be to earn profits. A rate of return on investment can be described as the net gain or loss incurred on investing the funds.

Explanation of Solution

Given Information:

Expense ratio = 0.5%

Mutual Fund = 4% load

Bank CD interest = 6%

There is another option for investing i.e., investment in Bank CD (Certificate of Deposit). So, better option for investment is has to be analyze- whether investing in mutual funds is better or in Bank CD. In case option mutual funds is chosen, then it should be ensured that it provides rate of return more than any bank CD. Normally a mutual fund requires front-end load and this adversely reduces the net value investment amount. Further the investor has to shell down the expense ratio also on this investment. So, to balance the rate of return of bank CD and the mutual fund, the following calculation will be helpfully.

  $\begin{array}{}$ {(1+R_{CD})}^t=(1-Front-endload)\times {(1+R_{{}_{MutualFund}}-ExpenseRatio)}^t$\end{array}$

where r= rate of return

t=number of period

Now depict the values in the equation and proceed with the calculation:

  $\begin{array}{l}{\left(1+6\%\right)}^2=\left(1-4\%\right)\times (1+R_{{}_{MutualFund}}-{0.5\%)}^2\\ {\left(1+\frac{6}{100}\right)}^2={\left(\left(1-\frac{4}{100}\right)\times \left(1+R_{{}_{MutualFund}}-\frac{0.5}{100}\right)\right)}^2\\ {\left(1+0.06\right)}^2=\left(1-0.04\right)\times {\left(1+R_{{}_{MutualFund}}-0.005\right)}^2\\ {\left(1.06\right)}^2=0.96\times {\left(\left(1-0.005\right)+R_{{}_{MutualFund}}\right)}^2\\ \left(1.06\times 1.06\right)=0.96\times {\left(0.995+R_{{}_{MutualFund}}\right)}^2\\ 1.1236=0.96\times {\left(0.995+R_{{}_{MutualFund}}\right)}^2\\ \frac{1.1236}{0.96}={\left(0.995+R_{{}_{MutualFund}}\right)}^2\\ 1.170417={\left(0.995+R_{{}_{MutualFund}}\right)}^2\end{array}$

We can rewrite the equation as :

  $\begin{array}{l}\left(0.995+R_{{}_{MutualFund}}\right)={\left(1.170417\right)}^{\frac{1}{2}}\\ \left(0.995+R_{{}_{MutualFund}}\right)=1.0819\end{array}$

So, $R_{{}_{MutualFund}}=1.0819-0.995$

  $R_{{}_{MutualFund}}=0.869$

Converting the above value into percentages

= 8.69%

Therefore, the mutual fund portfolio will be better than Bank CD if it earns more than 8.69% per annum.

Conclusion

The mutual fund portfolio will be better than Bank CD if it earns more than 8.69% per annum.

b

Summary Introduction

To compute: The annual rate of return when the plan of investment is for 6 years.

Introduction:

Investment: When an individual deposit or does an act of investing his money for profit, it is called investment. Investments can be for short term and long term.

Explanation of Solution

Given Information:

Expense ratio = 0.5%

Mutual Fund = 4% load

Bank CD interest = 6%

To calculate the rate of return to balance the return earned on investment in mutual fund with than of Bank CD when the investment is made for 6 years. The formula to be used for this computation is as follows:

  ${(1+R_{CD})}^t=(1-Front-endload)\times {(1+R_{{}_{MutualFund}}-ExpenseRatio)}^t$

Where ‘r’ = rate of return

‘t’ = number of period

Now let us depict the values in the equation and proceed with the calculation:

  $\begin{array}{l}{\left(1+6\%\right)}^6=\left(1-4\%\right)\times (1+R_{{}_{MutualFund}}-{0.5\%)}^6\\ {\left(1+\frac{6}{100}\right)}^6={\left(\left(1-\frac{4}{100}\right)\times \left(1+R_{{}_{MutualFund}}-\frac{0.5}{100}\right)\right)}^6\\ {\left(1+0.06\right)}^2=\left(1-0.04\right)\times {\left(1+R_{{}_{MutualFund}}-0.005\right)}^6\\ {\left(1.06\right)}^6=0.96\times {\left(\left(1-0.005\right)+R_{{}_{MutualFund}}\right)}^6\\ \left(1.06\times 1.06\times 1.06\times 1.06\times 1.06\times 1.06\right)=0.96\times {\left(0.995+R_{{}_{MutualFund}}\right)}^6\\ 1.418519=0.96\times {\left(0.995+R_{{}_{MutualFund}}\right)}^6\\ \frac{1.418519}{0.96}={\left(0.995+R_{{}_{MutualFund}}\right)}^6\\ 1.477624={\left(0.995+R_{{}_{MutualFund}}\right)}^6\end{array}$

The equation can be written as :

  $\begin{array}{l}\left(0.995+R_{{}_{MutualFund}}\right)={\left(1.477624\right)}^{\frac{1}{6}}\\ \left(0.995+R_{{}_{MutualFund}}\right)=1.0672\end{array}$

So, $R_{{}_{MutualFund}}=1.0672-0.995$

  $R_{{}_{MutualFund}}=0.0722$

Or 7.22%.

Therefore, the mutual fund portfolio will be better than Bank CD if it earns more than 7.22% per annum.

Conclusion

Incase, the investor decides to hold the investment in mutual fund for 6 years, then the mutual fund portfolio will be better than Bank CD if it earns more than 7.22% per annum.

c

Summary Introduction

To compute: The annual rate of return of the Mutual Fund portfolio in case there is no front-end load and 12b-1 fee of 0.75% per year is utilized.

Introduction:

Mutual Funds: It is a type of investment which can be done even by individual investors and can be managed at a low price. It consists of portfolio of stocks, bonds and other types of securities.

Explanation of Solution

Given Information:

Expense ratio = 0.5%

12b-1 fee= 0.75%

Bank CD interest = 6%

Like in previous cases, here we are not considering (1-Front-end load) calculation as there is no front-end load and instead we will use 12b-1 fee. So the formula will be as follows:

  ${(1+R_{CD})}^t={(1+R_{{}_{MutualFund}}-\left(12b-1fee\right)-ExpenseRatio)}^t$

where r= rate of return

t=number of period

Now let us depict the values in the equation and proceed with the calculation when the investment is made for 2 years.

  $\begin{array}{l}{\left(1+6\%\right)}^2=(1+R_{{}_{MutualFund}}\left(0.75\%\right)-{0.5\%)}^2\\ {\left(1+\frac{6}{100}\right)}^2={\left(\left(1-\frac{4}{100}\right)\times \left(1+R_{{}_{MutualFund}}-\frac{0.75}{100}-\frac{0.5}{100}\right)\right)}^6\\ {\left(1+0.06\right)}^2={\left(1+R_{{}_{MutualFund}}-\left(0.075\right)-0.005\right)}^2\\ {\left(1.06\right)}^2={\left(1++R_{{}_{MutualFund}}-\left(0.0125\right)\right)}^2\\ \left(1.06\times 1.06\right)={\left(\left(1-0.0125\right)+R_{{}_{MutualFund}}\right)}^2\\ 1.1236={\left(0.9875+R_{{}_{MutualFund}}\right)}^2\end{array}$

We can rewrite the equation as :

  $\begin{array}{l}\left(0.9875+R_{{}_{MutualFund}}\right)={\left(1.1236\right)}^{\frac{1}{2}}\\ \left(0.9875+R_{{}_{MutualFund}}\right)=1.06\end{array}$

So, $R_{{}_{MutualFund}}=1.06-0.9875$

  $R_{{}_{MutualFund}}=0.0725$

When we convert the above value into percentages we get 7.25%

Therefore, the mutual fund portfolio will be better than Bank CD if it earns more than 7.25% per annum.

The same calculation has to be done when the investment is made for 6 years.

  ${(1+R_{CD})}^t={(1+R_{{}_{MutualFund}}-\left(12b-1fee\right)-ExpenseRatio)}^t$

where r= rate of return

t=number of period

Now substitute the values in the formula

  $\begin{array}{l}{\left(1+6\%\right)}^6=(1+R_{{}_{MutualFund}}-\left(0.75\%\right)-{0.5\%)}^6\\ {\left(1+\frac{6}{100}\right)}^6={\left(\left(1+R_{{}_{MutualFund}}-\left(\frac{0.75}{100}\right)-\frac{0.5}{100}\right)\right)}^6\\ {\left(1+0.06\right)}^6={\left(1+R_{{}_{MutualFund}}-\left(0.075\right)-0.005\right)}^6\\ {\left(1.06\right)}^6={\left(1+R_{{}_{MutualFund}}-\left(0.0125\right)\right)}^6\\ \left(1.06\times 1.06\times 1.06\times 1.06\times 1.06\times 1.06\right)={\left(\left(1-0.0125)\right)+R_{{}_{MutualFund}}\right)}^6\\ 1.418519={\left(0.9875+R_{{}_{MutualFund}}\right)}^6\end{array}$

We can rewrite the equation as :

  $\begin{array}{l}\left(0.9875+R_{{}_{MutualFund}}\right)={\left(1.418519\right)}^{\frac{1}{6}}\\ \left(0.9875+R_{{}_{MutualFund}}\right)=1.06\end{array}$

So, $R_{{}_{MutualFund}}=1.06-09875$

  $R_{{}_{MutualFund}}=0.0725$

When we convert the above value into percentages we get 7.25%

Therefore, the mutual fund portfolio will be better than Bank CD if it earns more than 7.25% per annum.

Conclusion

Given the situation where investment is made for 2 years and 6 years, the mutual fund portfolio will be better than Bank CD if it earns more than 7.25% per annum in the both cases.