3b.1 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson...

3b.1 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.

Chapter 3 - Chapter 3 - SupportSupport

The Time Value The Time Value of Moneyof Money

The Time Value The Time Value of Moneyof Money


Assume that you deposit $1,000$1,000 at a compound interest rate of 7% for

2 years2 years.

Remember?Remember? Future Value Future ValueSingle Deposit (Graphic)Single Deposit (Graphic)Remember?Remember? Future Value Future ValueSingle Deposit (Graphic)Single Deposit (Graphic)

0 1 22

$1,000$1,000

FVFV22

7%


Future Value Excel FormulaFuture Value Excel Formula[Calculates a single value in the future based on current [Calculates a single value in the future based on current expectations]expectations]

• Excel function is =FV(rate,nper,pmt,pv,type)• rate: the interest rate per period

• nper: the total number of compounding periods

• pmt: the payment made each period and cannot change over the life of the annuity ($0 in a single cash flow)

• pv: the present value you begin with

• type: is the number 0 (normal period-end) or 1 (beginning of the period) and indicates when payments are due/occur


Future Value ExampleFuture Value Example

Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value

problems on the “Future Value” tab. Students may find this much easier!

B C D E F

2 Explanations

3 7.00% Compound 7% per year

4 2 2 periods in the problem

5 -$ $0 per year invested

6 (1,000.00)$ Start with $1,000 today

7 0 Not relevant in single flow

8

9 $1,144.90 =FV(D3,D4,D5,D6,D7)

OutputsFuture Value (FV):

Inputsrate:

nper:

pmt:

pv:

type:


Story Problem Story Problem RevisitedRevisitedStory Problem Story Problem RevisitedRevisited

Julie Miller wants to know how large her deposit of $10,000$10,000 today will become at a compound annual interest rate of 10% for 5 years5 years.

0 1 2 3 4 55

$10,000$10,000

FVFV55

10%


Future Value ExampleFuture Value Example

Refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) on the “Future Value” tab. This is the identical answer as our other methods in

one formula shown in the boxed area above!

B C D E F

2 Explanations



5 -$ No payment as single flow

6 (10,000.00)$ Invests $10,000 today


8

9 $16,105.10 =FV(D3,D4,D5,D6,D7)

pv:type:


Inputsrate:

nper: pmt:


AnotherAnother Future Future Value Story ProblemValue Story Problem

• John and Mary are saving for retirement and currently have $127,833.56 as a nest egg.

• John indicates that they plan to retire 25 years from today while Mary expects that a 6% rate of return is appropriate for their risk level given historical returns.

• Calculate how large the account is expected to grow.


““New” FV Story ProblemNew” FV Story Problem

Refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) on the “Future Value” tab.

B C D E F

2 Explanations




6 (127,833.56)$ Invests $127,833.56 today


8

9 $548,645.11 =FV(D3,D4,D5,D6,D7)

pv:type:


Inputsrate:

nper: pmt:


““New” FV Story ProblemNew” FV Story Problem

• John and Mary will have their $ 127,833.56 investment grow to $ 548,645.11 in 25 years if they earn exactly 6% each year.

• Note that the Excel answer is a ‘positive’ amount. This indicates that John and Mary DEPOSITED $ 127,833.56 (the negative amount as they have less cash) to receive the positive $548,645.11 (when they receive cash at retirement when they WITHDRAW the funds).


Remember?Remember? Present Value Present Value Single Deposit (Graphic)Single Deposit (Graphic)Remember?Remember? Present Value Present Value Single Deposit (Graphic)Single Deposit (Graphic)

Assume that you need $1,000$1,000 in 2 years.2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually.

0 1 22

$1,000$1,000

7%

PV1PVPV00


Present Value Excel FormulaPresent Value Excel Formula[Calculates a single current value based on future [Calculates a single current value based on future expectations]expectations]

• Excel function is =PV(rate,nper,pmt,fv,type)• rate: the interest rate per period

• nper: the total number of discounted periods

• pmt: the payment made each period and cannot change over the life of the annuity ($0 in a single cash flow)

• fv: the future value you expect to attain



Present Value ExamplePresent Value Example


problems on the “Present Value” tab. Students may find this much easier!

B C D E F

2 Explanations

3 7.00% Discount 7% per period



6 1,000$ Want $1,000 in future


8

9 ($873.44) =PV(D3,D4,D5,D6,D7)

type:

nper:

Inputs

Outputs

rate:

present value (pv)

pmt:fv:


Story Problem Story Problem RevisitedRevisitedStory Problem Story Problem RevisitedRevisited

Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000$10,000 in 5 years5 years at a discount rate of 10%.

0 1 2 3 4 55

$10,000$10,000PVPV00

10%


Present Value ExamplePresent Value Example


problems on the “Present Value” tab. Students may find this much easier!

B C D E F

2 Explanations




6 10,000$ Want $10,000 in future


8

9 ($6,209.21) =PV(D3,D4,D5,D6,D7)

type:

nper:

Inputs

Outputs

rate:

present value (pv)

pmt:fv:


AnotherAnother Present Value Present Value Story ProblemStory Problem

• John and Mary are expecting to build a $100,000 nest egg to use to travel the world upon retirement. They would like to know how much they need to set aside today to reach this goal.

• John indicates that they will retire 20 years from today while Mary thinks that a 6% rate of return is appropriate for their risk level. Calculate how much they need to set aside today.


B C D E F

2 Explanations




6 100,000$ Want $100,000 in future


8

9 ($31,180.47) =PV(D3,D4,D5,D6,D7)

type:

nper:

Inputs

Outputs

rate:

present value (pv)

pmt:fv:

““New” PV Story ProblemNew” PV Story Problem

Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xls) to use in developing their own solutions to other present value

problems on the “Present Value” tab.


““New” PV Story ProblemNew” PV Story Problem

• John and Mary need to set aside $31,180.47 today if they earn exactly 6% each year for the next 20 years to reach their goal.

• Note that the Excel answer is a ‘negative’ amount. This indicates that John and Mary will need to DEPOSIT this amount of money (they have less cash) to receive the positive $100,000 (when they receive cash) they would WITHDRAW monies.


We will use the ““Rule-of-72Rule-of-72””..

Remember?Remember? Double Double Your Money!!!Your Money!!!Remember?Remember? Double Double Your Money!!!Your Money!!!

Quick! How long does it take to double $5,000 at a compound rate

of 12% per year (approx.)?


Output: The Periods!Output: The Periods!

Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other interest rate problems on the “Periods” tab. Students may find this much easier!

NOTE: The same answer! Solved in only one cell in Excel!!

B C D E F

2 Explanations

3 12% Rate is 12% per period (e.g., year)


5 (5,000)$ Start with $5,000 invested (negative)

6 10,000$ Want $10,000 in future (positive)

7 0 Best guess - 10% assumed if ignore

8

9 6.12 =NPER(D3,D4,D5,D6,D7)

type

OutputsPeriods (nper)

Inputsrate

pmt:pv:fv:


Remember?Remember? Double Double Your Money!!!Your Money!!!Remember?Remember? Double Double Your Money!!!Your Money!!!

What if you were given the periods (years) and wanted to solve for the

interest rate? Wouldn’t it be the same concept for the rule of 72?

Yes! We simply need to solve using a different function in Excel called “rate”.


Output: The Interest Rate!Output: The Interest Rate!

Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other interest rate

problems on the “Rate” tab.

Note that we “guessed 12% as 12 x 6 = 72. The answer was very close!

B C D E F

2 Explanations



5 (5,000)$ Start with $5,000 invested (negative)

6 10,000$ Want $10,000 in future (positive)

0 "0" as no other cash flows

7 12.00% Best guess - 10% assumed if ignore

8

9 12.25% =RATE(D3,D4,D5,D6,D7,D8)

Outputsinterest rate (rate)

type:

Inputsnper: pmt:

pv:fv:

guess:


PVAPVA33 = $1,000/(1.07)1 + $1,000/(1.07)2 +

$1,000/(1.07)3

= $934.58 + $873.44 + $816.30 = $2,624.32$2,624.32

Remember? Remember? Example of an Example of an Ordinary Annuity -- PVAOrdinary Annuity -- PVARemember? Remember? Example of an Example of an Ordinary Annuity -- PVAOrdinary Annuity -- PVA

$1,000 $1,000 $1,000

0 1 2 3 3 4

$2,624.32 = PVA$2,624.32 = PVA33

7%

$934.58$873.44 $816.30

Cash flows occur at the end of the period


Present Value of an AnnuityPresent Value of an Annuity [Calculates a single current value based on future expectations][Calculates a single current value based on future expectations]

• Excel function is =PV(rate,nper,pmt,fv,type)• rate: the interest rate per period

• nper: the total number of payments or periods

• pmt: the payment that is made/received each period and cannot change over the life of the annuity

• fv: a single future value you expect to receive (can be $0)



PV Annuity ExamplePV Annuity Example(Ordinary Annuity)(Ordinary Annuity)

B C D E F

2 Explanations



5 1,000$ $10,000 invested per yr

6 -$ No future amount in additon

7 0 Ordinary Annuity

8

9 ($2,624.32) =PV(D3,D4,D5,D6,D7)

fv:type:

Outputspresent value (pv)

Inputsrate:

nper: pmt:

Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “PV Annuity” tab. Students will find this much easier!


““New” PV Annuity Example New” PV Annuity Example (Ordinary Annuity)(Ordinary Annuity)

• John and Mary are trying to build a nest egg to use in the future. They would like to know how much they need to set aside in a single lump sum today to be equivalent to investing $10,000 each year starting one year from today to reach this goal.

• John indicates that they will use the money 20 years from today while Mary thinks that a 6% rate of return is appropriate for their risk level.

• Calculate the equivalent present value of this ordinary annuity stream.



B C D E F

2 Explanations



5 (10,000)$ $10,000 invested per yr



8

9 $127,833.56 =PV(D3,D4,D5,D6,D7)

fv:type:


Inputsrate:

nper: pmt:

Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “PV Annuity” tab. Students will find this much easier!



• John and Mary need to set aside $127,833.56 today to be equivalent to setting aside $10,000 per year at exactly 6% each year for the next 25 years.

• In this case, John and Mary need to decide which is their preference. This sum will grow to exactly the same as the future value of an ordinary annuity (see slides 7 to 9).


Remember? Remember? Example of an Example of an Ordinary Annuity -- FVAOrdinary Annuity -- FVARemember? Remember? Example of an Example of an Ordinary Annuity -- FVAOrdinary Annuity -- FVA

FVAFVA33 = $1,000(1.07)2 + $1,000(1.07)1 + $1,000(1.07)0

= $1,145 + $1,070 + $1,000 = $3,215$3,215

$1,000 $1,000 $1,000

0 1 2 3 3 4

$3,215 = FVA$3,215 = FVA33

7%

$1,070

$1,145

Cash flows occur at the end of the period


• Excel function is = FV(rate,nper,pmt,pv,type)• rate: the interest rate per period

• nper: the total number of payments or periods

• pmt: the payment that is made/received each period and cannot change over the life of the annuity

• pv: a single present amount you begin with (can be $0)


Future Value of an AnnuityFuture Value of an Annuity[Calculates a single current value based on future expectations][Calculates a single current value based on future expectations]


Future Value Annuity Future Value Annuity Example (Ordinary Annuity)Example (Ordinary Annuity)


problems on the “FV Annuity” tab. The same answer as before is arrived at in a single step in Excel - one formula ‘=FV(.07,3,-1000,0,0)’!

B C D E F

2 Explanations



5 (1,000)$ $1,000 per year invested (negative)

6 Start with $0 today in account


8

9 $3,214.90 =FV(D3,D4,D5,D6,D7)


Inputsrate:

nper:

pmt:

pv:

type:


““New” Future Value Annuity New” Future Value Annuity Example (Ordinary Annuity)Example (Ordinary Annuity)

• John and Mary are trying to build a nest egg to use in the future. They would like to know how much they need to set aside in a single lump sum today to be equivalent to investing $10,000 each year starting one year from today to reach this goal. (See slides 21 to 23 and also 7 to 9)


• Calculate the equivalent present value of this ordinary annuity stream.



B C D E F

2 Explanations



5 (10,000)$ $10,000 per year invested

6 -$ No additional monies today


8

9 $548,645.12 =FV(D3,D4,D5,D6,D7)

pv:

type:


Inputsrate:

nper:

pmt:

Students may refer to the supporting Excel file for Chapter 3 (VW13E-03.xlsx) to use in developing their own solutions to other present value problems on the “FV

Annuity” tab. Students will find this is the exact same answer as those derived using PV of an annuity and then Future Value!



• John and Mary will accumulate nearly $550,000 by investing $10,000 per year at exactly 6% each year for the next 25 years.

• In this case, note that this result is equivalent to the future value of a single sum where John and Mary needed to set aside over $127,000 to generate this sum.


Remember?Remember? Example of Example of an Annuity Due – PVADan Annuity Due – PVADRemember?Remember? Example of Example of an Annuity Due – PVADan Annuity Due – PVAD

PVADPVADnn = $1,000/(1.07)0 + $1,000/(1.07)1 + $1,000/(1.07)2 = $2,808.02$2,808.02

$1,000.00 $1,000 $1,000

0 1 2 33 4

$2,808.02 $2,808.02 = PVADPVADnn

7%

$ 934.58$ 873.44

Cash flows occur at the beginning of the period


PV Annuity PV Annuity Example (Annuity Due)Example (Annuity Due)

Students may refer to the supporting Excel file for Chapter 3 to use in developing their own solutions to other present value

annuity problems.

B C D E F

2 Explanations



5 (1,000)$ $1,000 invested per yr (negative)

6 No future amount in additon

7 1 Annuity DUE!!!!

8

9 $2,808.02 =PV(D3,D4,D5,D6,D7)

fv:type:


Inputsrate:

nper: pmt:


““New” PV Annuity New” PV Annuity Example (Annuity Due)Example (Annuity Due)

• John and Mary are trying to build a nest egg to use in the future. They would like to know how much they need to set aside in a single lump sum today to be equivalent to investing $10,000 each year starting today to reach this goal.


• Calculate the equivalent present value of this annuity due stream.



Students may refer to the supporting Excel file for Chapter 3 to use in developing their own solutions to other present value

annuity problems.

B C D E F

2 Explanations



5 (10,000)$ $10,000 invested per yr



8

9 $135,503.58 =PV(D3,D4,D5,D6,D7)

fv:type:


Inputsrate:

nper: pmt:



• John and Mary need to set aside $135,503.58 today to be equivalent to setting aside $10,000 per year at exactly 6% each year for the next 25 years.

• In this case, John and Mary need to decide which is their preference. This sum will grow to exactly the same as the future value of an ordinary annuity (see slides 24 to 26) plus EXTRA interest.

• EXTRA INTEREST: So the amount is one years interest higher for each payment or $127,833.56*6% = $7,670.01 higher PV!


Annuity DueAnnuity Due

• An annuity due is used when the cash flow occurs at the beginning of the period

• As before, you see the value is higher by an amount equal to i% times the ordinary annuity value.

• Present values of annuities will be larger because each cash flow is “discounted” one less period. See previous examples.

• Future values of annuities will be larger because each cash flow gets compounded one “extra” period. No examples shown here.


Mixed Flows ExampleMixed Flows ExampleMixed Flows ExampleMixed Flows Example

Julie Miller will receive the set of cash flows below. What is the Present Value Present Value at a discount rate of 10%10%.

0 1 2 3 4 55

$600 $600 $400 $400 $100$600 $600 $400 $400 $100

PVPV00

10%10%


Solve a “mixed flows” problem Solve a “mixed flows” problem using the NPV functionusing the NPV function

Students may refer to the supporting Excel file for Chapter 3 (VW13E-

03.xlsx) to use in developing their own

solutions to other present value problems on the “Mixed Flows”

tab.

Simply type in the cash flows in the green

column and the correct interest rate in yellow

and you get your answer!

Period Cash Flows1 600.00$

Interest Rate:2 600.00$ (discount rate)3 400.00$

4 400.00$ Present Value*: $1,677.155 100.00$

=NPV(F3,C3:C22)678

* Do NOT include cash 9flows in period 0. Please10add or subtract these flows11from the final answer above.12(Cell F6)13

14151617181920

10%


Solve a “mixed flows” problem Solve a “mixed flows” problem using the NPV functionusing the NPV function

Students can also solve this in one formula step

again!

=npv(.10, 600, 600, 400, 400, 100)

This will generate the same and correct output. The model on the right is

designed to be slightly more flexible in solving

these types of problems.

Remember the NPV function, because we will

use it later!

Period Cash Flows1 600.00$

Interest Rate:2 600.00$ (discount rate)3 400.00$

4 400.00$ Present Value*: $1,677.155 100.00$

=NPV(F3,C3:C22)678

* Do NOT include cash 9flows in period 0. Please10add or subtract these flows11from the final answer above.12(Cell F6)13

14151617181920

10%


Remember? Remember? BWs Effective BWs Effective

Annual Interest RateAnnual Interest Rate

Remember? Remember? BWs Effective BWs Effective

Annual Interest RateAnnual Interest RateBasket Wonders (BW) has a $1,000 CD at the bank. The interest rate is

6% compounded quarterly for 1 year. What is the Effective Annual

Interest Rate (EAREAR)?

EAREAR = ( 1 + 6% / 4 )4 - 1 = 1.0614 - 1 = .0614 or

6.14%!6.14%!


BWs Effective BWs Effective Annual Interest RateAnnual Interest RateBWs Effective BWs Effective Annual Interest RateAnnual Interest Rate

We can use a single function in Excel again to solve for the Effective Annual Interest Rate (EAREAR)

Nominal Periods Effective Formula Used6% 1 6.0000% =EFFECT(B3,C3)6% 2 6.0900% =EFFECT(B4,C4)6% 4 6.1364% =EFFECT(B5,C5)6% 6 6.1520% =EFFECT(B6,C6)6% 12 6.1678% =EFFECT(B7,C7)6% 365 6.1831% =EFFECT(B8,C8)6% 1,000,000 6.1837% =EFFECT(B9,C9)

So it is very simple to solve for the effective rate using Excel. Note that a large number of periods per year can be used to

approximate continuous compounding!


Julie Miller is borrowing $10,000 $10,000 at a compound annual interest rate of 12%.

Amortize the loan if annual payments are made for 5 years.

Step 1: Payment

PVPV00 = R (PVIFA i%,n)

$10,000 $10,000 = R (PVIFA 12%,5)

$10,000$10,000 = R (3.605)

RR = $10,000$10,000 / 3.605 = $2,774$2,774

Remember?Remember? Amortizing Amortizing a Loan Examplea Loan ExampleRemember?Remember? Amortizing Amortizing a Loan Examplea Loan Example


1.1. The first step is to use the “PMT” function to determine the The first step is to use the “PMT” function to determine the yearly (in this case) payment on the loanyearly (in this case) payment on the loan

2.2. Now you can use Excel to easily create the table you see below! Now you can use Excel to easily create the table you see below! Refer to Refer to ‘VW13E-03.xlsx’‘VW13E-03.xlsx’ on the ‘Effect and Loan’ tab. on the ‘Effect and Loan’ tab.

Remember?Remember? Amortizing Amortizing a Loan Examplea Loan ExampleRemember?Remember? Amortizing Amortizing a Loan Examplea Loan Example

Amortizing a loanStep 1: Calculating the loan amountRate: 12.00% Interest rate per period (year in this case)nper: 5 Number of periods (5 years in this case)pv 10,000.00$ Beginning loan balance today (positive)fv -$ Ending loan balance at end of periodspayment: ($2,774.10) Payment needed (negative)

Step 2: Create a tablePeriod Beginning Bal Payment Interest in Period Principal in Period Ending Balance

0 10,000.00$ 1 10,000.00$ $2,774.10 1,200.00$ $1,574.10 8,425.90$ 2 8,425.90$ $2,774.10 1,011.11$ $1,762.99 6,662.91$ 3 6,662.91$ $2,774.10 799.55$ $1,974.55 4,688.37$ 4 4,688.37$ $2,774.10 562.60$ $2,211.49 2,476.87$ 5 2,476.87$ $2,774.10 297.22$ $2,476.87 0.00$

3b.1 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson...

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Transcript of 3b.1 Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson...