Chapter 5, Problem 41SP

a.

Summary Introduction

To calculate: Future value of $1,000 after 5 years at 10% annual interest rate.

Introduction:

Time Value of Money: It is a vital concept to the investors, as it suggests them the money they are having today is worth more than the value promised in the future.

Explanation of Solution

Calculation in spreadsheet by “FV” formula,

Table (1)

Steps required to calculate present value by using “FV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “FV” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $1,610.50.

Conclusion

Future value of $1,000 is $1,610.51.

b.

Summary Introduction

To calculate: Investments future value at 0%,5% and 20% rate after 0,1,2,3,4 and 5 years.

Introduction:

Time Value of Money: It is a vital concept to the investors, as it suggests them the money they are having today is worth more than the value promised in the future.

Explanation of Solution

Calculation spreadsheet by “FV” formula,

Table (2)

Steps required to calculate present value by using “FV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “FV” and then press OK.
• A window will pop up.
• Input data in required field.

Conclusion

Investment future values are different for the different years with 0%, 5% and 20% interest rate.

c.

Summary Introduction

To calculate: Present value due of $1,000 in 5 years at the discount rate of 10%.

Introduction:

Time Value of Money: It is a vital concept to the investors, as it suggests them the money they are having today is worth more than the value promised in the future.

Explanation of Solution

Calculation in spreadsheet by “PV” formula,

Table (3)

Steps required to calculate present value by using “PV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “PV” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $620.92.

Conclusion

Present value of $1,000 is $620.92 at 10 % discount rate.

d.

Summary Introduction

To calculate: Rate of return of security $1,000 and returns as $2,000 after 5 years.

Introduction:

Time Value of Money: It is a vital concept to the investors, as it suggests them the money they are having today is worth more than the value promised in the future.

Explanation of Solution

Calculationin spreadsheet by “RATE” formula,

Table (4)

Steps required to calculate present value by using “RATE” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “RATE” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is 14.87%

Conclusion

The rate of return is14.87%.

e.

Summary Introduction

To calculate: Time taken by 36.5 million populations to double with annual growth rate of 2%

Introduction:

Time Value of Money: It is a vital concept to the investors, as it suggests them the money they are having today is worth more than the value promised in the future.

Explanation of Solution

Calculation is solved in spreadsheet by “NPER” formula

Table (5)

Steps required to calculate present value by using “NPER” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “NPER” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is 35 years.

Conclusion

Conclusion:

It will take 35 years to double the population from 36.5 million to 73 million.

f.

Summary Introduction

To calculate: Present and future value of annuity that pays $1,000 each of the next 5 years with interest rate of 15%.

Introduction:

Time Value of Money: It is a vital concept to the investors, as it suggests them the money they are having today is worth more than the value promised in the future.

Explanation of Solution

Calculation in spreadsheet by “PV” formula,

Table (6)

Steps required to calculate present value by using “PV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “PV” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $3,352.16.

So, the present value is $3,352.16.

Calculation of future value of annuity in spreadsheet by using “FV” formula,

Table (7)

Steps required to calculate present value by using “FV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “FV” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $6,742.38.

So the future value is $6,742.38.

Conclusion

Present value is $3,352.16 and future value is $6,742.38 of annuity that pays $1,000 each of the next 5 years with interest rate of 15%.

g.

Summary Introduction

To calculate: Present and future value of part ‘f’ if the annuity is due.

Introduction:

Time Value of Money: It is a vital concept to the investors, as it suggests them the money they are having today is worth more than the value promised in the future.

Explanation of Solution

Calculation in spreadsheet by “PV” function,

Table (8)

Steps required to calculate present value by using “PV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “PV” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $3,351.66.

Present value of annuity due is $3,351.66.

Future value of annuity in spreadsheet by “FV” function,

Table (9)

Steps required to calculate present value by using “FV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “FV” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $7,753.74

Future value of annuity due is $7,753.74.

Conclusion

Present value of annuity due is $3,351.66 while future value is $7,753.74.

h.

Summary Introduction

To calculate: Present and future value for $1,000, due in 5 years with 10% semiannual compounding.

Introduction:

Time Value of Money: It is a vital concept to the investors, as it suggests them the money they are having today is worth more than the value promised in the future.

Explanation of Solution

Calculation in spreadsheet by “PV” formula,

Table (10)

Steps required to calculate present value by using “PV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “PV” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $613.91.

Present value is $613.91.

Calculation in spreadsheet by “FV” formula,

Table (11)

Steps required to calculate present value by using “FV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “FV” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $1,628.89.

Future value of annuity is $1,628.89.

Conclusion

Present andfuture value for $1,000, due in 5 years with 10% semiannual compoundingwill be $613.91 and $1,628.89, respectively.

i.

Summary Introduction

To calculate: Annual payments for an ordinary annuity for 10 years with PV of $1,000, interest rate 8% and payments of annuity due.

Introduction:

Time Value of Money: It is a vital concept to the investors, as it suggests them the money they are having today is worth more than the value promised in the future.

Explanation of Solution

Calculation in spreadsheet by “PMT” formula,

Table (12)

Steps required to calculate present value by using “PMT” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “PMT” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $1,628.89.

Payment of ordinary annuity is $149.03.

Calculation in spreadsheet by “PMT” formula,

Table (13)

Steps required to calculate present value by using “PMT” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “PMT” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $1,628.89.

Payment of ordinary annuity due is $137.99.

Conclusion

Annual payments are$149.03 for an ordinary annuity for 10 years with PV of $1,000, interest rate 8% and payments of annuity due is $137.99.

j.

Summary Introduction

To calculate: Present value and future value of an investment that pays 8% annually and makes the year end payments of $100, $200,$300.

Explanation of Solution

Calculation in spreadsheet by “NPV” formula,

Table (14)

Steps required to calculate present value by using “NPV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “NPV” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $581.59.

Present value is $581.59.

Calculation in spreadsheet by “FV” formula,

Table (15)

Steps required to calculate present value by using “FV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “FV” and then press OK.
• A window will pop up.
• Input data in the required field.
• Final answer will be shown by the formula that is $732.54.

Future value is $732.54.

Conclusion

Present value is $581.89 while future value is $732.54.

k.1.

Summary Introduction

To calculate: Effective annual rate each bank pays and the future value of $5,000 at the end of 1 and 2 year.

Explanation of Solution

Given for Bank A,

Nominal interest rate is 5%.

Compounding is annual.

Formula to calculate effective annual rate is,

$\left(\text{EFF\%}\right)={\left(1+\frac{{\text{I}}_{\text{NOM}}}{\text{M}}\right)}^{\text{M}}-1.0$

Where,

• EFF is the effective annual rate.
• I_NOM is the nominal interest rate.
• M is the compounding period.

Substitute 5% for I_NOM and 1 for M.

$\begin{array}{c}\left(\text{EFF\%}\right)={\left(1+\frac{0.05}{1}\right)}^1-1.0\\ =0.05\\ =5\%\end{array}$

So, effective annual rate for Bank A is 5%.

Given for Bank B,

Nominal interest rate is 5%.

Compounding is semiannual.

Formula to calculate effective annual rate is,

$\left(\text{EFF\%}\right)={\left(1+\frac{{\text{I}}_{\text{NOM}}}{\text{M}}\right)}^{\text{M}}-1.0$

Where,

• EFF is the effective annual rate.
• I_NOM is the nominal interest rate.
• M is the compounding period.

Substitute 5% for I_NOM and 2 for M.

$\begin{array}{c}\left(\text{EFF\%}\right)={\left(1+\frac{0.05}{2}\right)}^2-1.0\\ =0.0506\\ =5.06\%\end{array}$

So,effective annual rate for Bank B is 5.06%.

Given for Bank C,

Nominal interest rate is 5%.

Compounding is quarterly.

Formula to calculate effective annual rate is,

$\left(\text{EFF\%}\right)={\left(1+\frac{{\text{I}}_{\text{NOM}}}{\text{M}}\right)}^{\text{M}}-1.0$

Where,

• EFF is the effective annual rate.
• I_NOM is the nominal interest rate.
• M is the compounding period.

Substitute 5% for I_NOM and 4 for M.

$\begin{array}{c}\left(\text{EFF\%}\right)={\left(1+\frac{0.05}{4}\right)}^4-1.0\\ =0.0509\\ =5.09\%\end{array}$

So, effective annual rate for Bank C is 5.09%.

Given for Bank D,

Nominal interest rate is 5%.

Compounding ismonthly.

Formula to calculate effective annual rate is,

$\left(\text{EFF\%}\right)={\left(1+\frac{{\text{I}}_{\text{NOM}}}{\text{M}}\right)}^{\text{M}}-1.0$

Where,

• EFF is the effective annual rate.
• I_NOM is the nominal interest rate.
• M is the compounding period.

Substitute 5% for I_NOM and 12 for M.

$\begin{array}{c}\left(\text{EFF\%}\right)={\left(1+\frac{0.05}{12}\right)}^{12}-1.0\\ =0.0511\\ =5.11\%\end{array}$

So, effective annual rate for Bank D is 5.11%.

Given for Bank E,

Nominal interest rate is 5%.

Compounding is daily.

Formula to calculate effective annual rate is,

$\left(\text{EFF\%}\right)={\left(1+\frac{{\text{I}}_{\text{NOM}}}{\text{M}}\right)}^{\text{M}}-1.0$

Where,

• EFF is the effective annual rate.
• I_NOM is the nominal interest rate.
• M is the compounding period.

Substitute 5% for I_NOM and 365 for M.

$\begin{array}{c}\left(\text{EFF\%}\right)={\left(1+\frac{0.05}{365}\right)}^{365}-1.0\\ =0.0512\\ =5.12\%\end{array}$

So, effective annual rate for Bank E is 5.12%.

Calculation of future value in spreadsheet by “FV” formula,

Table (16)

Steps required to calculate present value by using “FV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “FV” and then press OK.
• A window will pop up.
• Input data in the required field.

So, the future value at different effective rate after a year are $5, 250, $5,253,$5,254.50, $5,255.50 and $5,256.

Calculation of future value in spreadsheet by “FV” formula,

Table (17)

Steps required to calculate present value by using “FV” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “FV” and then press OK.
• A window will pop up.
• Input data in the required field.

So, the future value at different effective rate after a year are $5, 512, $5,518.80, $5,521.95, $5,524.06 and $5,525.11.

Conclusion

Each bank pays different effective rate as there compounding is different, the rates are5% for bank A, 5.06% Bank B, 5.09% Bank C, 5.11% Bank D, 5.12% Bank E, also the future values also change on the basis of their number of periods.

2.

Summary Introduction

To explain: If banks are insured by the government and are equally risky, will they be equally able to attract funds and at what nominal rate all banks provide equal effective rate as Bank A.

Answer to Problem 41SP

No, it is not possible for the banks to equally attract funds.

The nominal rate which causes same effective rate for all banks are,

Particulars	A	B	C	D	E
Nominalrate	5%	5.06%	5.09%	5.11%	5.12%

Table (18)

Explanation of Solution

• Bank will not be equally able to attract funds because of compounding, as people prefer to invest in that bank which have more frequent compounding in comparison to the bank which have lesser frequent compounding.
• Nominal rate is opposite of the effective rate which is calculated in part ‘1’. The nominal rate indicated in above table will cause same effective rate for all banks as it is for A bank.

Conclusion

Due to frequent compounding, banks will not be able to equally attract funds and the nominal rate will be 5% for Bank A, 5.06% Bank B, 5.09% Bank C, 5.11% Bank D, 5.12% Bank E.

3.

Summary Introduction

To calculate: Present value of amount to get $5,000 after 1 year.

Explanation of Solution

Calculation of payment to be made in spreadsheet by “PMT” formula,

Table (19)

Steps required to calculate present value by using “PMT” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “PMT” and then press OK.
• A window will pop up.
• Input data in the required field.

Conclusion

So, the amounts to be paid are$4,761.90,$2498.10,$1,102.40,$296.96 and $15.83.

4.

Summary Introduction

To explain: If all banks are providing a same effective interest rate would rational investor be indifferent between the banks.

Answer to Problem 41SP

Yes, a rational investor would be indifferent between the banks.

Explanation of Solution

Rational investor chooses the bank which will provide him better return so he would be indifferent, if all the banks are giving same effective rate because he chooses the bank which will have more frequent compounding than others.

Conclusion

A bank offers frequent compounding is able to attract more number of customers than others.

Summary Introduction

To prepare: Amortization schedule to show annual payments, interest payments, principal payments, and beginning and ending loan balances.

Amortization:

Amotization means to write off or pay the debt over the priod of time it can be for loan or intangible assets. Its main purpose is to get cost recovery. Example of amortization is ,an automobile company that spent $20 million dollars on a design patent with a useful life of 20 years. The amortization value for that company will be $1 million each year.

Explanation of Solution

Calculation of annual installment is done by using “PMT” formula in spreadsheet at the amortization schedule.

Amortization schedule is prepared below,

Table (20)

Steps required to calculate present value by using “PMT” function in excel are given,

• Select ‘Formulas’ option from Menu Bar of Excel sheet.
• Select insert Function that is (fx).
• Choose category of Financial.
• Then select “PMT” and then press OK.
• A window will pop up.
• Input data in the required field.

Conclusion

Amortization schedule represents annual payments, interest payments, principal payments, and beginning and ending loan balances.