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week 4

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Week 4 – Discussion

In this week’s discussion, prepare a synopsis of the material discussed in the chapter readings. In your post, share any questions you may have regarding the managerial finance concepts presented in the textbook. This synopsis should be 450+ words

for 2 students

Please, address each of the questions below, in 100-150 words (per question). Include any relevant examples and links to your sources.

1. 1. How can the opportunity cost of an investment be defined?

2. 2. What is the difference between an ordinary annuity and an annuity due?

3. 3. What is the definition of a perpetuity?

Time Value of Money

CHAPTER 4

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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
Or
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
Or
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?

The Impact of Compounding (Answer)
LARGER!
If compounding is more frequent than once a year–for example, semiannually, quarterly, or daily–interest is earned on interest more often.

FV Formula with Different Compounding Periods

$100 at a 12% nominal rate with semiannual compounding for 5 years

FV of $100 at a 12% nominal rate for 5 years with different compounding
FV(Ann.) = $100(1.12)5 = $176.23
FV(Semi.) = $100(1.06)10 = $179.08
FV(Quar.) = $100(1.03)20 = $180.61
FV(Mon.) = $100(1.01)60 = $181.67
FV(Daily) = $100(1+(0.12/365))(5×365) = $182.19

Effective Annual Rate
The effective annual rate (EAR or EFF%) is the annual rate that causes PV to grow to the same FV as under multi-period compounding.

Effective Annual Rate Example
Example: Invest $1 for one year at 12%, semiannual:

EFF% = 12.36%, because $1 invested for one year at 12% semiannual compounding would grow to the same value as $1 invested for one year at 12.36% annual compounding.

Comparing Rates
An investment with monthly payments is different from one with quarterly payments. Must put on EFF% basis to compare rates of return. Use EFF% only for comparisons.
Banks say “interest paid daily.” Same as compounded daily.

EFF% for a nominal rate of 12%, compounded semiannually

Finding EFF with HP10BII
Type in nominal rate, then Orange Shift key, then NOM% key (in orange).
Type in number of periods, then Orange Shift key, then P/YR key (in orange).
To find effective rate, hit Orange Shift key, then EFF% key (in orange).

EAR (or EFF%) for a Nominal Rate of of 12%

Can the effective rate ever be equal to the nominal rate?
Yes, but only if annual compounding is used, i.e., if M = 1.
If M > 1, EFF% will always be greater than the nominal rate.

When is each rate used? (1 of 3)
INOM: Written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

When is each rate used? (2 of 3)
IPER: Used in calculations, shown on time lines.
If INOM has annual compounding,
then IPER = INOM/1 = INOM.

When is each rate used? (3 of 3)
EAR (or EFF%): Used to compare returns on investments with different payments per year.
Used for calculations if and only if dealing with annuities where payments don’t match interest compounding periods.

Amortization (1 of 2)
Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.

Step 1: Find the required payments.

Step 2: Find interest charge for Year 1.
INTt = Beg balt (I)
INT1 = $1,000(0.10) = $100

Step 3: Find repayment of principal in Year 1.

Step 4: Find ending balance after Year 1.

Repeat these steps for Years 2 and 3
to complete the amortization table.

Amortization Table
YEAR BEG BAL PMT INT PRIN PMT END BAL
1 $1,000 $402 $100 $302 $698
2 698 402 70 332 366
3 366 402 37 366 0
TOT 1,206.34 206.34 1,000

Interest declines because outstanding balance declines.

Amortization (2 of 2)
Amortization tables are widely used–for home mortgages, auto loans, business loans, retirement plans, and more. They are very important!
Financial calculators (and spreadsheets) are great for setting up amortization tables.

Fractional Time Periods
On January 1 you deposit $100 in an account that pays a nominal interest rate of 11.33463%, with daily compounding (365 days).
How much will you have on October 1, or after 9 months (273 days)? (Days given.)

Convert interest to daily rate

Find FV

Calculator Solution (1 of 2)

Non-matching rates and periods (1 of 2)
What’s the value at the end of Year 3 of the following CF stream if the quoted interest rate is 10%, compounded semiannually?

Time line for non-matching rates and periods

Non-matching rates and periods (2 of 2)
Payments occur annually, but compounding occurs each 6 months.
So we can’t use normal annuity valuation techniques.

1st Method: Compound Each CF

2nd Method: Treat as an annuity, use financial calculator
Find the EFF% (EAR) for the quoted rate:

Use EAR = 10.25% as the annual rate in calculator.

What’s the PV of this stream?

Comparing Investments
You are offered a note that pays $1,000 in 15 months (or 456 days) for $850. You have $850 in a bank that pays a 6.76649% nominal rate, with 365 daily compounding, which is a daily rate of 0.018538% and an EAR of 7.0%. You plan to leave the money in the bank if you don’t buy the note. The note is riskless.
Should you buy it?

Daily time line

Three solution methods
1. Greatest future wealth: FV
2. Greatest wealth today: PV
3. Highest rate of return: EFF%

1. Greatest Future Wealth
Find FV of $850 left in bank for
15 months and compare with
note’s FV = $1,000.
FVBank = $850(1.00018538)456
= $924.97 in bank.
Buy the note: $1,000 > $924.97.

Calculator Solution to FV

2. Greatest Present Wealth
Find PV of note, and compare
with its $850 cost:
PV = $1,000/(1.00018538)456
= $918.95
Buy the note: $918.95 > $850

Financial Calculator Solution
PV of note is greater than its $850 cost, so buy the note. Raises your wealth.

3. Rate of Return
Find the EFF% on note and compare with 7.0% bank pays, which is your opportunity cost of capital:
FVN = PV(1 + I)N
$1,000 = $850(1 + I)456
Now we must solve for I.

Calculator Solution (2 of 2)
Convert % to decimal:
Decimal = 0.035646/100 = 0.00035646.
EAR = EFF% = (1.00035646)365 – 1
= 13.89%.

Using interest conversion
P/YR = 365
NOM% = 0.035646(365) = 13.01
EFF% = 13.89
Since 13.89% > 7.0% opportunity cost,
buy the note.

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