September 27, 2022

Assume that you are nearing graduation of your MBA program and have applied for a job with a local bank. As part of the bank’s evaluation process, you have been asked to take an exam

Assume that you are nearing graduation of your MBA program and have applied for a job with a local bank. As part of the bank's evaluation process, you have been asked to take an examination that covers several financial analysis techniques. The first section of the test addresses time value of money analysis. See how you would do by answering the following questions.

A customer of the bank, Raj Kami, wants to deposit $100,000 in a savings account that pays a nominal rate of 8%.

1. If the bank compounds interest annually, how much will the customer have in his account 3 years from now?

2. What would the balance be in 3 years from now if the bank used quarterly compounding rather than annual compounding?

3. If Raj Kami deposited the $100,000 in 4 equal payments of $25,000 each at the end of years 1, 2, 3, and 4. How much would he have in the savings account at the end of year 4, based on 8% annual compounding?

4. Raj Kami wants to know how long it will take his sum of money to double if the growth rate per year is 8%

5. Raj Kami wants to buy a car, and a local bank will lend him $20,000. The loan would be fully amortized over 5 years (60 months), and the nominal interest rate would be 12%, with interest paid monthly. What is the monthly loan payment? What is the loan’s effective (or equivalent) rate EFF?

6. What is the present value of $100,000 to be received in 4 years if the appropriate interest rate is 5%?

7. Jackson Corporation’s bonds have 10 years remaining to maturity. Interest is paid annually, the bonds have a $1,000 par value, and the coupon interest rate is 9%. The bonds have a yield to maturity of 10%. What is the current market price of these bonds?

8. Renfro Rentals has issued bonds that have a 10% coupon rate, payable semiannually. The bonds mature in 10 years, have a face value of $1,000, and a yield to maturity of 9%. What is the price of the bonds?

9. Wilson Wonders’ bonds have 10 years remaining to maturity. Interest is paid annually, the bonds have a $1,000 par value, and the coupon interest rate is 10%. The bonds sell at a price of $900. What is their yield to maturity?

10. What is the present value of a perpetuity that pays $1,000 per year if the appropriate interest rate is 5%?

Submit your answers in a Word document.

Chapter4-TimeValueofMoney..pptx

Time Value of Money

CHAPTER 4

© 2020 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part, except for use as permitted in a license distributed with a certain product or service or otherwise on a password-protected website for classroom use.

Time Value Topics

Future value

Present value

Rates of return

Amortization

Determinants of Intrinsic Value: The Present Value Equation

Time lines show timing of cash flows.

Tick marks at ends of periods, so Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2.

Time line for a $100 lump sum due at the end of Year 2.

Time line for an ordinary annuity of $100 for 3 years

Time line for uneven CFs

FV of an initial $100 after 3 years (I = 10%)

Finding FVs (moving to the right

on a time line) is called compounding.

After 1 year

FV1 = PV + INT1 = PV + PV (I)

= PV(1 + I)

= $100(1.10)

= $110.00

After 2 years

FV2 = FV1(1+I)

= $110(1.10)

= $121.00

FV2 = PV(1 + I)(1+I)

= PV(1+I)2

= $100(1.10)2

= $121.00

After 3 years

FV3 = FV2(1+I)

= $121.00(1.10)

= $133.10

FV3 = FV2(1+I)=PV(1 + I)2(1+I)

= PV(1+I)3

= $100(1.10)3

= $133.10

Using the General Formula: FVN= PV(1 + I)N

Generalizing the approach from previous slides:

FVN = PV(1 + I)N

FV3 = $100(1.10)3

= $133.10

Four Ways to Find FVs

Step-by-step approach using time line (as shown in previous slides).

Solve the equation with a regular calculator (formula approach).

Use a financial calculator.

Use a spreadsheet.

Financial calculator: HP 10bII+ (1 of 4)

Adjust display brightness: hold down ON and push + or –.

Set number of decimal places to display: Orange Shift key, then DISP key (in orange), then desired decimal places (e.g., 3).

To temporarily show all digits, hit Orange Shift key, then DISP, then =.

Financial calculator: HP 10bII+ (2 of 4)

To permanently show all digits, hit ORANGE shift, then DISP, then . (period key).

Set decimal mode: Hit ORANGE shift, then ./, key. Note: many non-US countries reverse the US use of decimals and commas when writing a number.

HP 10bII+: Set Time Value Parameters

To set END (for cash flows occurring at the end of the year), hit ORANGE shift key, then BEG/END.

To set 1 payment per period, hit 1, then ORANGE shift key, then P/YR.

Financial calculator: BAII+

Set number of decimal places to display: 2nd Format; use the up and down arrows to display DEC=; press 9; press ENTER

Set AOS calculation; 2nd Format; down arrow 4 times until you see Chn (if you see AOS then just stop and hit CE/C, you are done); 2nd SET (AOS should display); CE/C you are done.

BAII +: Set Time Value Parameters

To set END (for cash flows occurring at the end of the year), hit 2nd BGN; 2nd SET will toggle between cash flows at the beginning of the year (BGN) and end of the year (END). Leave it as END.

To set 1 payment per period, hit 2nd P/Y 1 ENTER.

BAII+

To reset TVM calculations; 2nd CLR TVM.

To reset cash flow register; CF; 2nd CLR Work.

Financial Calculator Solution (1 of 3)

Financial calculators solve this equation:

FVN + PV (1+I)N = 0.

There are 4 variables. If 3 are known, the calculator will solve for the 4th.

Here’s the setup to find FV

Clearing automatically sets everything to 0, but for safety enter PMT = 0.

Set: P/YR = 1, END.

Spreadsheet Solution (1 of 6)

Use the FV function: see spreadsheet in Ch04 Mini Case.xls

FVN = FV(I, N, PMT, PV)

FVN = FV(0.10, 3, 0, -100) = 133.10

What’s the PV of $100 due in 3 years if I/YR = 10%?

Finding PVs is discounting, and it’s the reverse of compounding.

Formula Solution

FVN = PV(1 + I)N

Solve for PV:

Financial Calculator Solution (1 of 2)

Either PV or FV must be negative. Here

PV = -75.13. Put in $75.13 today, take

out $100 after 3 years.

Spreadsheet Solution (2 of 6)

Use the PV function: see spreadsheet in Ch04 Mini Case.xls

PV = PV(I, N, PMT, FV)

PV = PV(0.10, 3, 0, 100) = -75.13

Finding the Time to Double (1 of 2)

FVN = PV(1 + I)N

Continued on next slide

Finding the Time to Double (2 of 2)

Financial Calculator Solution (2 of 2)

Spreadsheet Solution (3 of 6)

Use the NPER function: see spreadsheet in Ch04 Mini Case.xls

N = NPER(I, PMT, PV, FV)

N = NPER(0.10, 0, -1, 2) = 3.8

Finding the Interest Rate

Financial Calculator

Spreadsheet Solution (4 of 6)

Use the RATE function: see spreadsheet in Ch04 Mini Case.xls

I = RATE(N, PMT, PV, FV)

I = RATE(3, 0, -1, 2) = 0.2599

Ordinary Annuity vs. Annuity Due

What’s the FV of a 3-year ordinary annuity of $100 at 10%?

FV Annuity Formula

The future value of an annuity with N periods and an interest rate of I can be found with the following formula:

Financial Calculator Formula for Annuities

Financial calculators solve this equation:

There are 5 variables. If 4 are known, the calculator will solve for the 5th.

Financial Calculator Solution (2 of 3)

Have payments but no lump sum PV, so enter 0 for present value.

Spreadsheet Solution (5 of 6)

Use the FV function: see spreadsheet in Ch04 Mini Case.xls.

FVN = FV(I, N, PMT, PV)

FVN = FV(0.10, 3, -100, 0) = 331.00

What’s the PV of this ordinary annuity?

PV Annuity Formula

The present value of an annuity with N periods and an interest rate of I can be found with the following formula:

Financial Calculator Solution (3 of 3)

Have payments but no lump sum FV, so enter 0 for future value.

Spreadsheet Solution (6 of 6)

Use the PV function: see spreadsheet in Ch04 Mini Case.xls.

PV = PV(I, N, PMT, FV)

PV = PV(0.10, 3, 100, 0) = -248.69

Find the FV and PV if the annuity were an annuity due.

PV and FV of Annuity Due vs. Ordinary Annuity

PV of annuity due:

= (PV of ordinary annuity) (1+I)

= ($248.69) (1+ 0.10) = $273.56

FV of annuity due:

= (FV of ordinary annuity) (1+I)

= ($331.00) (1+ 0.10) = $364.10

PV of Annuity Due: Switch from “End” to “Begin”

FV of Annuity Due: Switch from “End” to “Begin”

Excel Function for Annuities Due

Change the formula to:

=PV(0.10,3,-100,0,1)

The fourth term (0) tells the function there are no other cash flows. The fifth term (1) tells the function that it is an annuity due. A similar function gives the future value of an annuity due:

=FV(0.10,3,-100,0,1)

What is the PV of this uneven cash flow stream?

Financial calculator: HP 10bII+ (3 of 4)

Clear all: Orange Shift key, then C All key (in orange).

Enter number, then hit the CFj key.

Repeat for all cash flows, in order.

To find NPV: Enter interest rate (I/YR). Then Orange Shift key, then NPV key (in orange).

Financial calculator: HP 10bII+ (4 of 4)

To see current cash flow in list, hit RCL CFj CFj

To see previous CF, hit RCL CFj –

To see subsequent CF, hit RCL CFj +

To see CF 0-9, hit RCL CFj 1 (to see CF 1). To see CF 10-14, hit RCL CFj . (period) 1 (to see CF 11).

Financial calculator: BAII + (1 of 4)

Clear all cash flows: CF; 2nd CLR Work.

CF0 displayed. Enter number, then hit the ENTER key.

Hit the down arrow to display C01. Enter number, hit ENTER.

F01 displayed. Usually just hit 1 ENTER. If you have several cash flows that are all the same, then use F01 to say how many you have.

Financial calculator: BAII + (2 of 4)

Repeat for all cash flows, in order.

To find NPV: Hit NPV; I = will display. Enter interest rate (as a percentage, so enter 10 for 10%) ENTER; Down Arrow; Displays NPV =; hit CPT and the NPV will display.

Financial calculator: BAII + (3 of 4)

To see current cash flow in list, hit CF

Scroll up or down using the up and down arrows.

Financial calculator: BAII + (4 of 4)

Input in “CFLO” register:

CF0 = 0

CF1 = 100

CF2 = 300

CF3 = 300

CF4 = -50

Enter I/YR = 10, then press NPV button to get NPV = 530.09. (Here NPV = PV.)

Excel Formula in cell A3: = NPV(10%,B2:E2)

Nominal rate (INOM)

Stated in contracts, and quoted by banks and brokers.

Not used in calculations or shown on time lines

Periods per year (M) must be given.

Examples:

8%; Quarterly

8%, Daily interest (365 days)

Periodic rate (IPER )

IPER = INOM/M, where M is number of compounding periods per year. M = 4 for quarterly, 12 for monthly, and 360 or 365 for daily compounding.

Used in calculations, shown on time lines.

Examples:

8% quarterly: IPER = 8%/4 = 2%.

8% daily (365): IPER = 8%/365 = 0.021918%.

The Impact of Compounding

Will the FV of a lump sum be larger or smaller if we compound more often, holding the stated I% constant?

Why?

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Assume that you are nearing graduation of your MBA program and have applied for a job with a local bank. As part of the bank’s evaluation process, you have been asked to take an exam

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