Chapter 5, Problem 5.1STP

Learning Goals 2, 5

ST5-1 Future values for various compounding frequencies Delia Martin has $10,000 that she can deposit in any of three savings accounts for a 3-year period. Bank A compounds interest on an annual basis, bank B compounds interest twice each year, and bank C compounds interest each quarter. All three banks have a stated annual interest rate of 4%.

1. a. What amount would Ms. Martin have after 3 years, leaving all interest paid on deposit, in each bank?
2. b. What effective annual rate (EAR) would she earn in each of the banks?
3. c. On the basis of your findings in parts a and b, which bank should Ms. Martin deal with? Why?
4. d. If a fourth bank (bank D), also with a 4% stated interest rate, compounds interest continuously, how much would Ms. Martin have after 3 years? Does this alternative change your recommendation in part c? Explain why or why not.

Summary Introduction

To calculate: Future value.

Introduction:

Future value (FV): The future value refers the value of present amount at a future date.

Answer to Problem 5.1STP

Bank A= $11,250

Bank B= $11,260

Bank C= $11,270

Explanation of Solution

Given:

Present value, $10,000, interest rate 4%, time period 3.

Calculation

The general formula for calculating future values is shown below.

$\text{Future value=Present value}{\left(1+\text{interest rate}\right)}^{\text{time period}}$ (1)

Substitute the values in equation (1) to calculate future values of $10,000.

$\begin{array}{c}\text{FV}=\$10,000{\left(1+0.04\right)}^3\\ =\$10,000\times 1.125\\ =\$11,250\end{array}$

FV is $11,250.

By using the same equation (1), the future value of deposit $10,000 in each bank is shown below.

Table 1 shows the future value.

Table 1

Bank	Present value	Years to compound	Interest	Future value
A	$10,000	3 years	4%	$11,250
B	$10,000	Two times in a year	4%	$11,260
C	$10,000	Quarterly (4)	4%	$11,270

Summary Introduction

To calculate: Effective annual rate.

Introduction:

Effective annual rate (EAR): Effective annual rate refers, it is the annual rate of interest which is actually paid or earned.

Answer to Problem 5.1STP

Bank A= 4%

Bank B= 4.04%

Bank C= 4.06%

Explanation of Solution

Given:

Interest rate 4%, time period 3.

Calculation

The general formula for calculating the effective annual rate is shown below.

$\text{EAR=}{\left(1+\frac{\text{r}}{\text{m}}\right)}^{\text{n}}-1$ (2)

Substitute the values in equation (2) to calculate the EAR for deposit $10,000 in bank A.

$\begin{array}{c}\text{EAR=}{\left(1+\frac{0.04}{1}\right)}^1-1\\ ={\left(1+0.04\right)}^1-1\\ =1.04-1\\ =0.04\end{array}$

EAR is 4%.

By using the same equation (1), the EAR for each case is shown below.

Table 2 shows the EAR.

Table 2

Bank	Years to compound	Interest	EAR
A	3 years	4%	4%
B	Two times in a year	4%	4.04%
C	Quarterly (4)	4%	4.06%

Summary Introduction

To discuss: Which bank the investor will choose.

Answer to Problem 5.1STP

Bank C

Explanation of Solution

The investor will choose bank C, because, the compounding frequency is 4 (quarterly). Higher the compounding frequency increases the future value of deposit. Thus, bank C is best.

Summary Introduction

To calculate: Future value.

Introduction:

Future value (FV): The future value refers the value of present amount at a future date.

Answer to Problem 5.1STP

$11,274.97

Explanation of Solution

Given:

Present value, $10,000, interest rate 4%, time period 3.

Calculation

In bank D, the interest is compounded continuously. Calculation of future values of $10,000 with continues compounding is shown below.

${\text{Future value=Present value}}_0\times {\text{e}}^{\text{interest rate}\times \text{time period}}$

The value of “e” is approximately “2.7183”.

$\begin{array}{c}\text{FV=\$10,000}\times {2.7183}^{0.04\times 3}\\ =\$10,000\times 1.127497\\ =\$11,274.97\end{array}$

FV is $11,274.97.

Comparing Bank D with C, bank D is better than C, because, the compounding frequency and future value obtained in bank D is higher than bank C. thus, bank D is a good alternative.