lecture 8

Lecture 8 – “ATH Lecture 8 – “ATH Microtechnologies & Project Microtechnologies & Project

Valuation and ROI - I” Valuation and ROI - I”

Steve MontgomerySteve Montgomery

04/07/23 2

Introduction To The Business Introduction To The Business EnvironmentEnvironment

● What to do when someone’s not telling you what to doWhat to do when someone’s not telling you what to do● Topics covered:Topics covered:

• Course introduction and Overview of corporate structure (1 Course introduction and Overview of corporate structure (1 session)session)

• Fundamentals of business strategy (2 sessions)Fundamentals of business strategy (2 sessions)

• Introduction to Marketing (2 sessions)Introduction to Marketing (2 sessions)

• Overview of Accounting and Finance (2 sessions)Overview of Accounting and Finance (2 sessions)

• Project Valuation and ROI (2 sessions)Project Valuation and ROI (2 sessions)

• Course review & strategy illustration (1 session)Course review & strategy illustration (1 session)

04/07/23 3

Accounting/FinanceAccounting/Finance● Accounting: Two typesAccounting: Two types

• Bookkeeping and managerialBookkeeping and managerial– BookkeepingBookkeeping: Your company’s overall financial health: Your company’s overall financial health

• Balance sheets income statements and how to read themBalance sheets income statements and how to read them

– ManagerialManagerial: Measurable metrics that drive decision making: Measurable metrics that drive decision making• What incentives are you creating, and can you measure the right things? What incentives are you creating, and can you measure the right things?

● Finance: The art and science of cash managementFinance: The art and science of cash management• Cash is KingCash is King

• Treat it like it’s yoursTreat it like it’s yours

• Time = moneyTime = money

End goal: End goal: Understanding your business, how you Understanding your business, how you

reach customers, reach customers, and your firm’s business and your firm’s business conditioncondition allow you to pick smart projects allow you to pick smart projects

and sell them to managementand sell them to management

End goal: End goal: Understanding your business, how you Understanding your business, how you

reach customers, reach customers, and your firm’s business and your firm’s business conditioncondition allow you to pick smart projects allow you to pick smart projects


04/07/23 4

Project Valuation and ROIProject Valuation and ROI

● The project decision: Is this project a good The project decision: Is this project a good use of company resources?use of company resources?• How can we tell? What should we measure?How can we tell? What should we measure?

• What are we not doing by starting this project?What are we not doing by starting this project?

● Techniques for estimating Return on Techniques for estimating Return on InvestmentInvestment

End goal: End goal: Understanding your business, how you Understanding your business, how you reach customers, your firm’s business reach customers, your firm’s business

condition and the condition and the potential impact your of potential impact your of your workyour work allow you to pick smart projects allow you to pick smart projects


End goal: End goal: Understanding your business, how you Understanding your business, how you reach customers, your firm’s business reach customers, your firm’s business

condition and the condition and the potential impact your of potential impact your of your workyour work allow you to pick smart projects allow you to pick smart projects


04/07/23 5

Lecture 8: Project Valuation and Lecture 8: Project Valuation and ROI IROI I

● ATH Microtechnologies Case ATH Microtechnologies Case

● Finance and the time value of moneyFinance and the time value of money

● Present/future value problemsPresent/future value problems

04/07/23 6

ATH MicrotechnologiesATH Microtechnologies

● Founding Period:Founding Period:• Scepter’s bid:Scepter’s bid:

– $90M initial payment$90M initial payment

– $30M if new product approved by $30M if new product approved by FDAFDA

– $35M if independent study proved $35M if independent study proved product superiorityproduct superiority

– Up to $120M in cash for Up to $120M in cash for sales/profit goalssales/profit goals

● What are some features of What are some features of this payment system?this payment system?

● Is there anything missing Is there anything missing here?here?

Growth

Profit Control

04/07/23 7

ATH Microtechnologies, cont.ATH Microtechnologies, cont.

● Did the earn out structure focus on the right Did the earn out structure focus on the right performance goals?performance goals?

● Were there adequate controls in place?Were there adequate controls in place?

● What are some good metrics to measure What are some good metrics to measure ATH’s performance by?ATH’s performance by?

04/07/23 8


● Growth Phase:Growth Phase:• Autonomy granted to ATH Autonomy granted to ATH

(why?)(why?)

• Goal: Get market share Goal: Get market share through new product through new product development and marketingdevelopment and marketing– Were bonuses linked to Were bonuses linked to

market share growth?market share growth?

• First earn out paid as FDA First earn out paid as FDA approved product, but approved product, but European tech proved European tech proved superior in test, so no second superior in test, so no second phase earn outphase earn out

Growth

Profit Control

04/07/23 9


● How did they do during the growth period?How did they do during the growth period?

● Did they achieve their strategic objectives?Did they achieve their strategic objectives?

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● Push to profitability:Push to profitability:• 20% bonus and trip 20% bonus and trip

for two to Hawaii if for two to Hawaii if we’re profitable!we’re profitable!–What’s not to like?What’s not to like?

• Would you design an Would you design an incentive program incentive program this way?this way?

Growth

Profit Control

04/07/23 11


● Was ATH successful in becoming profitable?Was ATH successful in becoming profitable?● How did they do it?How did they do it?● What effects did this have on the rest of the business?What effects did this have on the rest of the business?

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● Refocus on Refocus on ProcessProcess• Issued vision Issued vision

statement statement • Restructured bonus Restructured bonus

programprogram

• Launched education Launched education initiativeinitiative

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● How did they do?How did they do?● Were the issues contained Were the issues contained

at this point?at this point?● Final earn out paid at this Final earn out paid at this

pointpoint

Growth

Profit Control

04/07/23 14


● New Management:New Management:• Declining salesDeclining sales

• Founders left the companyFounders left the company

• New European tech eating at New European tech eating at business + other competitors business + other competitors entering marketentering market

• New spending focused on New spending focused on new product development new product development and tech leadershipand tech leadership

• Newest products withdrawn Newest products withdrawn from market; corners cut to from market; corners cut to make deadlinesmake deadlines

Growth

Profit Control

04/07/23 15


● How well is ATH positioned for the future?How well is ATH positioned for the future?

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ATH SummaryATH Summary

● People respond to incentives:People respond to incentives:• You get what you measure!You get what you measure!

• Consequently, measures need to be balanced: Reward one thing, Consequently, measures need to be balanced: Reward one thing, get one thingget one thing

● Strategy is everything!Strategy is everything!• In this case, did ATH focus on the short or the long term?In this case, did ATH focus on the short or the long term?

● It’s difficult to balance short vs. long term goals, but:It’s difficult to balance short vs. long term goals, but:• Design incentives that try to do bothDesign incentives that try to do both

– How could that have been done here?How could that have been done here?

• Keep tension in the system between growth, profit and controlKeep tension in the system between growth, profit and control– If one is sacrificed completely, negative consequences can appear If one is sacrificed completely, negative consequences can appear

later!later!

04/07/23 17

Time…Is MoneyTime…Is Money

● The key to understanding financing decisions is The key to understanding financing decisions is to note that the value of money decreases over to note that the value of money decreases over timetime

● Two factors contribute:Two factors contribute:• Inflation. It takes more dollars to buy something Inflation. It takes more dollars to buy something

tomorrow that it does todaytomorrow that it does today

• Opportunity cost. By doing x instead of y, you lost out Opportunity cost. By doing x instead of y, you lost out on potential revenue. on potential revenue.

● How do we assign a value to these things?How do we assign a value to these things?

04/07/23 18

Time Is Money, cont.Time Is Money, cont.Effect Of Compounding Interest Over Time

$-

$2.00

$4.00

$6.00

$8.00

$10.00

$12.00

$14.00

$16.00

$18.00

$20.00

0 5 10 15 20 25 30 35Compounding Period

Val

ue,

[$]

With Rate 1

With Rate 2

With Rate 3

nnj

jj

iValueValue

or

iValueValue

1

1

0,

1

10%

5%

3%

04/07/23 19

Future Value:Future Value:

● Future value: ExampleFuture value: Example• You take $100 and buy a CD paying 4% annual You take $100 and buy a CD paying 4% annual

interest rate:interest rate:

• ValueValueYear 1Year 1 = $100(1+4%) = $104.00 = $100(1+4%) = $104.00

• ValueValueYear 2 Year 2 = $104(1+4%) = $108.16= $104(1+4%) = $108.16

● Therefore, your $100 has a future value of Therefore, your $100 has a future value of $108.16 after 2 years.$108.16 after 2 years.

04/07/23 20

Future Value, cont.Future Value, cont.

● CFCFjj = Cash flow value at some j = Cash flow value at some jthth compounding period compounding period

● j = The jj = The jth th compounding periodcompounding period

● FF00 = The initial investment = The initial investment

● i = The interest rate i = The interest rate (Also called the discount rate)(Also called the discount rate)● Note that this is the value of some Note that this is the value of some singlesingle investment at some investment at some futurefuture

datedate● The factor (1+i)The factor (1+i)jj is called the “Future Value interest factor” is called the “Future Value interest factor”

• Sometimes tabulatedSometimes tabulated

• Handy to use in spreadsheets when doing discounted cash flow analysisHandy to use in spreadsheets when doing discounted cash flow analysis

jj iFCF 10

04/07/23 21

Using Future Value to Derive Using Future Value to Derive Present ValuePresent Value

● Sometimes it’s desirable to know what the Sometimes it’s desirable to know what the present value of a future payment will bepresent value of a future payment will be

● Drives decision making: If you know what Drives decision making: If you know what something is worth in today’s dollars, is this a something is worth in today’s dollars, is this a good investment or not?good investment or not?

● The value of money isn’t static:The value of money isn’t static:• Inflation degrades its valueInflation degrades its value

• Other opportunities that Other opportunities that weren’tweren’t pursued should be pursued should be thought of as thought of as costscosts

04/07/23 22

Using Future Value to Derive Using Future Value to Derive Present Value, cont.Present Value, cont.

● Revisit the equation Revisit the equation ● Recast slightly:Recast slightly:

• Note that the “initial Note that the “initial investment” is the same investment” is the same as the value of that as the value of that investment today, or is investment today, or is also the “Present Value”also the “Present Value”

jj iFCF 10

–Becomes–

jj iPVCF 1

Where PV = “Present Value”

04/07/23 23

Using Future Value to Derive Using Future Value to Derive Present Value, cont.Present Value, cont.

● Solving for present value:Solving for present value:

● Same formula, we’re just thinking about it differently.Same formula, we’re just thinking about it differently.● The factor 1/(1+i)The factor 1/(1+i)jj is called the “Present Value interest is called the “Present Value interest

factor” (PVIF)factor” (PVIF)• Sometimes tabulatedSometimes tabulated

• Handy to use in spreadsheets when doing discounted cash flow Handy to use in spreadsheets when doing discounted cash flow analysisanalysis

jj

i

CFPV

1

04/07/23 24

Present Value Example:Present Value Example:

● A friend of yours owes you A friend of yours owes you $1,000. He can’t pay you back $1,000. He can’t pay you back today, but promises that he will today, but promises that he will pay you back one year from pay you back one year from now.now.

● Suppose that inflation is 3%. Suppose that inflation is 3%. What is the present value of What is the present value of the $1,000 payment (that the $1,000 payment (that happens 1 year from now)?happens 1 year from now)?

jj

i

CFPV

1

87.970$

03.01

1000$1

PV

(You lost $30 on that deal)

04/07/23 25

Present Value Example, cont.:Present Value Example, cont.:

● A friend of yours owes you A friend of yours owes you $1,000. He can’t pay you back $1,000. He can’t pay you back today, but promises that he will today, but promises that he will pay you back one year from pay you back one year from now.now.

● Suppose you could have Suppose you could have invested some money at 7% invested some money at 7% interest. How much would you interest. How much would you have had to set aside this year have had to set aside this year to get $1000 next year?to get $1000 next year?

jj

i

CFPV

1

58.934$

07.01

1000$1

PV

(This time your friend cost you

~$65)

04/07/23 26

What About A Series Of Cash What About A Series Of Cash Flows?Flows?

● Suppose you invest $4,000 a year in an Suppose you invest $4,000 a year in an IRA, and that you expect the IRA to earn IRA, and that you expect the IRA to earn 5.5% on average for the next 20 years.5.5% on average for the next 20 years.

● How much will you have 20 years from How much will you have 20 years from now?now?

● This is a sum of future years’ problem: Add This is a sum of future years’ problem: Add the future value of each of the next 20 the future value of each of the next 20 years together:years together:

18

192020

055.014000$

055.014000$055.014000$CF

04/07/23 27

Stream Of Cash Flows, cont.Stream Of Cash Flows, cont.Calculations for Application #1: Personal savings

Number of years of annual savings = 20Expected annual rate of return = 5.50%

Years from now

Amount invested FVIF Future value

0 4000 2.9178 11671.031 4000 2.7656 11062.592 4000 2.6215 10485.873 4000 2.4848 9939.214 4000 2.3553 9421.055 4000 2.2325 8929.916 4000 2.1161 8464.377 4000 2.0058 8023.108 4000 1.9012 7604.839 4000 1.8021 7208.37

10 4000 1.7081 6832.5811 4000 1.6191 6476.3812 4000 1.5347 6138.7513 4000 1.4547 5818.7214 4000 1.3788 5515.3715 4000 1.3070 5227.8416 4000 1.2388 4955.3017 4000 1.1742 4696.9718 4000 1.1130 4452.1019 4000 1.0550 4220.00

Sum of all invested amounts = 147144.30

Source: J. Karpoff, UW.

04/07/23 28

Visualizing Cash FlowsVisualizing Cash Flows

● Sometimes it’s helpful to consider a picture:Sometimes it’s helpful to consider a picture:

● Each of these cash flows is listed in their value at Each of these cash flows is listed in their value at that particular time period. We then use that particular time period. We then use discounting discounting (more on this in a minute) to relate (more on this in a minute) to relate them all back to today.them all back to today.

Period n

Period 1

Period 4

Period 3

Period 2

Period 5

Period 0 (Today)

CF1 CF2 CF3 CF4 CF5 CFn

04/07/23 29

Present Value of a Present Value of a StreamStream Of Of Cash Flows:Cash Flows:

● Note that CFNote that CFnn = PV(1+k) = PV(1+k)n n

• Cash flow (or future value) at some year (or Cash flow (or future value) at some year (or compounding period) equals the initial investment compounding period) equals the initial investment adjusted by the interest rate and number of periods)adjusted by the interest rate and number of periods)

nn

n

nn

nn

nn

nn

ikPVCF

iPViPViPViPVCFCFCFCF

iPVCF

iPVCF

iPVCF

iPVCF

1by sidesboth Divide 1

1...111...

1

1

1

1

00

20210

22

1

00

n

nn

n

i

CFPV

0 1 (Total)

k = interest raten = number of periods

04/07/23 30

Present Value of a Present Value of a StreamStream Of Of Cash Flows, cont.Cash Flows, cont.● Considering inflation, the value of a dollar degrades Considering inflation, the value of a dollar degrades

over timeover time• Example: What’s the value of $1 ten years from now in today’s Example: What’s the value of $1 ten years from now in today’s

dollars? (Assume inflation rate of 3%)dollars? (Assume inflation rate of 3%)

• Ans: Ans:

● Note that there’s no summation here, since there’s only Note that there’s no summation here, since there’s only 1 cash flow in this case (or think of it as a series of zero 1 cash flow in this case (or think of it as a series of zero cash flows)cash flows)

74.0$

03.01

1$

11 101010

0

k

CF

k

CFPV

n

nn

n

04/07/23 31

Present Value of a Present Value of a StreamStream Of Of Cash Flows, cont.Cash Flows, cont.● What if we have multiple cash flows?What if we have multiple cash flows?

• Example: Ichiro Suzuki signs an endorsement deal with Mizuno Example: Ichiro Suzuki signs an endorsement deal with Mizuno paying him $2M/year for the next 4 years (payment starts 1 year paying him $2M/year for the next 4 years (payment starts 1 year from now). What’s the present value of this contract? (Assume from now). What’s the present value of this contract? (Assume inflation is 3%)inflation is 3%)

● In other words, inflation knocks ~$565k off the value of In other words, inflation knocks ~$565k off the value of his contract.his contract.

M

k

CF

k

CF

k

CF

k

CF

k

CF

k

CFPV

n

nn

n

434.7$%31

000,000,2$

%31

000,000,2$

%31

000,000,2$

%31

000,000,2$

%31

0

111111

43210

44

33

22

11

00

0

04/07/23 32

Visualizing Cash Flows, againVisualizing Cash Flows, again

● To visualize what happens when we perform the discounting, multiply To visualize what happens when we perform the discounting, multiply each CF by the PVIF of that period and see…each CF by the PVIF of that period and see…

Period n

Period 1

Period 4

Period 3

Period 2

Period 5

Period 0 (Today)

CF1 CF2 CF3 CF4 CF5 CFn

*PVIF1 *PVIF2 *PVIF3 *PVIF4 *PVIF5 *PVIFn

Period n

Period 1

Period 4

Period 3

Period 2

Period 5

Period 0 (Today)

PV, CF1 PV, CF2 PV, CF3 PV, CF4 PV, CF5

PV, CFn

04/07/23 33

PV, cont.PV, cont.

● A good way to format this on spreadsheets is the A good way to format this on spreadsheets is the following:following:

● What if Ichiro took this class and wanted $8M in What if Ichiro took this class and wanted $8M in today’s dollars? How would we figure out what today’s dollars? How would we figure out what his payments should be?his payments should be?

Ichiro and Spresheet-friendly formatting:Interest rate: 3%

Year 0 1 2 3 4Payout 0 2,000,000$ 2,000,000$ 2,000,000$ 2,000,000$

* * * * *PVIF 1 0.970873786 0.942595909 0.915141659 0.888487048

PV, each pay: -$ 1,941,747.57$ 1,885,191.82$ 1,830,283.32$ 1,776,974.10$

Total value: (sum of PV's): 7,434,196.81$

04/07/23 34

Back to IchiroBack to Ichiro

● Notice the power of inflation – to get $8M in Notice the power of inflation – to get $8M in today’s dollars, he needs to be paid $618k more today’s dollars, he needs to be paid $618k more over the course of 4 yearsover the course of 4 years

● Why do the payments get bigger further out in Why do the payments get bigger further out in time?time?

What if Ichiro wanted $8M in today's dollars?

Interest rate 3%Year New payout PVIF PV

0 1 -$ 1 2,060,000$ 0.970874 2,000,000.00$ 2 2,121,800$ 0.942596 2,000,000.00$ 3 2,185,454$ 0.915142 2,000,000.00$ 4 2,251,018$ 0.888487 2,000,000.00$

TOTAL 8,618,272$ 8,000,000.00$ Total PV

618,272$ Difference

04/07/23 35

A Bond ExampleA Bond Example

● You want to invest in something, yet you’re leery You want to invest in something, yet you’re leery of the stock market. How about buying bonds?of the stock market. How about buying bonds?

● A bond is a promissory note from an entity A bond is a promissory note from an entity obligating the payee to pay out interest over the obligating the payee to pay out interest over the life of the bond, and refund the principal at some life of the bond, and refund the principal at some later datelater date• Corporate bondsCorporate bonds

• Municipal bonds (tax-free!)Municipal bonds (tax-free!)

04/07/23 36

A Bond Example, cont.A Bond Example, cont.

● Bonds are present value problems: You get the Bonds are present value problems: You get the interest (Called coupon payments) + the amount interest (Called coupon payments) + the amount of the bond (Called the face value) back when the of the bond (Called the face value) back when the bond expiresbond expires

● Bonds are issued in lots of $1000 (Face) with Bonds are issued in lots of $1000 (Face) with some pre-determined interest rate (Coupon)some pre-determined interest rate (Coupon)• Typically paid annually or semi-annuallyTypically paid annually or semi-annually

● Let’s buy a bond and figure out how much our Let’s buy a bond and figure out how much our investment is worthinvestment is worth

04/07/23 37

Bond Example, cont.Bond Example, cont.

● Suppose you want to buy a bond for $1000 Suppose you want to buy a bond for $1000 face value and a coupon rate of 10% The face value and a coupon rate of 10% The market interest rate is 12% (The market market interest rate is 12% (The market expects to earn a 12% return on this bond).expects to earn a 12% return on this bond).• Bond makes payments annuallyBond makes payments annually

• Say there are 10 years to maturity, or 10 Say there are 10 years to maturity, or 10 compounding periodscompounding periods

● What is this bond worth today?What is this bond worth today?

04/07/23 38


● First, how much is each interest payment?First, how much is each interest payment?• Each payment = (10%)(1000)= $100Each payment = (10%)(1000)= $100

● Next step is to find the present value of each of the interest Next step is to find the present value of each of the interest payments, then add those up:payments, then add those up:

● Next, find the present value of $1000 at a rate of 12% in 10 years Next, find the present value of $1000 at a rate of 12% in 10 years ($321.97)($321.97)

● The sum is the price of the bond: $287.48 + $565.02 = $887.The sum is the price of the bond: $287.48 + $565.02 = $887.

02.565$

%121

100$

%121

100$

%121

100$

%121

100$

%121

0

111111

43210

44

33

22

11

00

0

k

CF

k

CF

k

CF

k

CF

k

CF

k

CFPV

n

nn

n

04/07/23 39


Bond Example

Interest rate 12.00%

Compounding Period

Interest Payment Principal PVIF CF

0 $0.00 0 1 $0.00

1 $100.00 0 0.892857143 $89.29

2 $100.00 0 0.797193878 $79.72

3 $100.00 0 0.711780248 $71.18

4 $100.00 0 0.635518078 $63.55

5 $100.00 0 0.567426856 $56.74

6 $100.00 0 0.506631121 $50.66

7 $100.00 0 0.452349215 $45.23

8 $100.00 0 0.403883228 $40.39

9 $100.00 0 0.360610025 $36.06

10 $100.00 1000 0.321973237 $354.17

5.650223028 $887.00 Total

04/07/23 40

Bonds, cont.Bonds, cont.

● When a bond is first issued, you can pay When a bond is first issued, you can pay the face value and realize payments at the the face value and realize payments at the coupon ratecoupon rate• But, the market will value/devalue bonds just But, the market will value/devalue bonds just

like stocks! This affects the mathlike stocks! This affects the math

• Also, there’s the matter of getting in on a bond Also, there’s the matter of getting in on a bond at the beginning of its term or somewhere in at the beginning of its term or somewhere in the middlethe middle

● Let’s go buy a real bondLet’s go buy a real bond

04/07/23 41

Bond Example, Cont.Bond Example, Cont.

● Emerson Electric:Emerson Electric:• 5% coupon, semi-annual5% coupon, semi-annual

• $1000 face$1000 face

• ~10 payments remaining of (5%)($1000)/2 = $25.~10 payments remaining of (5%)($1000)/2 = $25.

● What’s the price of this bond?What’s the price of this bond?

04/07/23 42

Bond Example, cont.Bond Example, cont.● The market’s expected rate of return is called the “Yield The market’s expected rate of return is called the “Yield

To Maturity” (2.456% in this case)To Maturity” (2.456% in this case)● So the interest rate to use is (2.456/2)%:So the interest rate to use is (2.456/2)%:

● The actual price of this bond is $1120.31 – why the The actual price of this bond is $1120.31 – why the difference?difference?

Bond Example

Interest rate 1.228%Compounding

PeriodInterest

Payment Principal PVIF CF

0 $0.00 0 1 $0.00

1 $25.00 0 0.987868969 $24.702 $25.00 0 0.9758851 $24.403 $25.00 0 0.964046608 $24.104 $25.00 0 0.952351728 $23.815 $25.00 0 0.94079872 $23.526 $25.00 0 0.929385862 $23.237 $25.00 0 0.918111453 $22.958 $25.00 0 0.906973815 $22.679 $25.00 0 0.895971287 $22.40

10 $25.00 $1,000 0.885102232 $907.23$1,119.01 Total

04/07/23 43

Lump Sum or Monthly Payments?Lump Sum or Monthly Payments?

● You being a hot commodity thanks to some class You being a hot commodity thanks to some class you took at UW has landed you a new job. Upon you took at UW has landed you a new job. Upon leaving your old company, you have the option to leaving your old company, you have the option to take your retirement plan savings as either:take your retirement plan savings as either:• Lump sum of $50,000 nowLump sum of $50,000 now

• 18 monthly payments of $3,000 (for a total of $54k)18 monthly payments of $3,000 (for a total of $54k)

● In either case, you can invest your money at In either case, you can invest your money at 6%/year (0.5%/month)6%/year (0.5%/month)

*Example adapted from J. Karpoff, UW

04/07/23 44

Take The Monthly PaymentsTake The Monthly Payments

● With the monthly With the monthly payments, at this payments, at this interest rate you came interest rate you came our ahead by our ahead by $1,518.30.$1,518.30.

● What would change What would change this situation?this situation?

New Job Example

Interest rate 0.500%

Compounding Period

Monthly Payment PVIF CF

0 $0.00 1 $0.00

1 $3,000.00 0.995024876 $2,985.07

2 $3,000.00 0.990074503 $2,970.22

3 $3,000.00 0.985148759 $2,955.45

4 $3,000.00 0.980247522 $2,940.74

5 $3,000.00 0.975370668 $2,926.11

6 $3,000.00 0.970518078 $2,911.55

7 $3,000.00 0.96568963 $2,897.07

8 $3,000.00 0.960885204 $2,882.66

9 $3,000.00 0.95610468 $2,868.31

10 $3,000.00 0.951347941 $2,854.04

11 $3,000.00 0.946614866 $2,839.84

12 $3,000.00 0.94190534 $2,825.72

13 $3,000.00 0.937219243 $2,811.66

14 $3,000.00 0.932556461 $2,797.67

15 $3,000.00 0.927916877 $2,783.75

16 $3,000.00 0.923300375 $2,769.90

17 $3,000.00 0.918706841 $2,756.12

18 $3,000.00 0.91413616 $2,742.41

$51,518.30 Total

04/07/23 45

Present/Future Value Problem Present/Future Value Problem Summary:Summary:

● PV/FV techniques are useful for a variety of PV/FV techniques are useful for a variety of different calculations – different calculations – • Investment decisionsInvestment decisions

• Retirement savings Retirement savings

• Bond buyingBond buying

• ……and much moreand much more

● They’re also useful for evaluating projects. The They’re also useful for evaluating projects. The only thing we do differently is change the only thing we do differently is change the discount rate and the cash flowsdiscount rate and the cash flows

lecture 8

Documents

firms business

visualizing

cash flows

initial investment

discount rate

product development

reach customers

interest rate