Chapter 4

CHAPTER 4 Net Present Value Published by Lecturesheet.iiuc28a9.com

Chapter Outline ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

4.1 The One-Period Case: Future Value ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

4.1 The One-Period Case: Future Value ,[object Object],[object Object],[object Object],[object Object]

4.1 The One-Period Case: Present Value ,[object Object],The amount that a borrower would need to set aside today to to able to meet the promised payment of $10,000 in one year is call the Present Value ( PV ) of $10,000. Note that $10,000 = $9,523.81 ×(1.05).

4.1 The One-Period Case: Present Value ,[object Object],[object Object],[object Object]

4.1 The One-Period Case: Net Present Value ,[object Object],[object Object],Yes!

4.1 The One-Period Case: Net Present Value ,[object Object],[object Object],If we had not undertaken the positive NPV project considered on the last slide, and instead invested our $9,500 elsewhere at 5-percent, our FV would be less than the $10,000 the investment promised and we would be unambiguously worse off in FV terms as well: $9,500 ×(1.05) = $9,975 < $10,000.

4.2 The Multiperiod Case: Future Value ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

4.2 The Multiperiod Case: Future Value ,[object Object],[object Object],[object Object],[object Object]

Future Value and Compounding ,[object Object],[object Object],[object Object]

Future Value and Compounding 0 1 2 3 4 5

Present Value and Compounding ,[object Object],0 1 2 3 4 5 $20,000 PV

[object Object],How Long is the Wait?

[object Object],What Rate Is Enough? About 21.15%.

4.3 Compounding Periods ,[object Object],For example, if you invest $50 for 3 years at 12% compounded semi-annually, your investment will grow to

Effective Annual Interest Rates ,[object Object],The Effective Annual Interest Rate (EAR) is the annual rate that would give us the same end-of-investment wealth after 3 years:

Effective Annual Interest Rates (continued) ,[object Object]

Effective Annual Interest Rates (continued) ,[object Object],[object Object],[object Object]

EAR on a financial Calculator keys: display: description: 12 [gold] [P/YR] 12.00 Sets 12 P/YR. Hewlett Packard 10B 18 [gold] [NOM%] 18.00 Sets 18 APR. Texas Instruments BAII Plus keys: description: [2nd] [ICONV] Opens interest rate conversion menu [↓] [EFF=] [CPT] 19.56 [↓] [NOM=] 18 [ ENTER] Sets 18 APR. [↑] [C/Y=] 12 Sets 12 payments per year [gold] [EFF%] 19.56

Continuous Compounding (Advanced) ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

4.4 Simplifications ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Perpetuity ,[object Object],… The formula for the present value of a perpetuity is: 0 1 C 2 C 3 C

Perpetuity: Example ,[object Object],[object Object],… 0 1 £15 2 £15 3 £15

Growing Perpetuity ,[object Object],… The formula for the present value of a growing perpetuity is: 0 1 C 2 C ×(1+ g ) 3 C ×(1+ g ) 2

Growing Perpetuity: Example ,[object Object],[object Object],0 … 1 $1.30 2 $1.30 ×(1.05) 3 $1.30 ×(1.05) 2

Annuity ,[object Object],The formula for the present value of an annuity is: 0 1 C 2 C 3 C T C

Annuity Intuition ,[object Object],[object Object],[object Object],0 1 C 2 C 3 C T C

Annuity: Example ,[object Object],0 1 $400 2 $400 3 $400 36 $400

How to Value Annuities with a Calculator ,[object Object],PMT I/Y FV PV N – 400 7 0 12,954.59 36 PV Then enter what you know and solve for what you want.

What is the present value of a four-year annuity of $100 per year that makes its first payment two years from today if the discount rate is 9%? 0 1 2 3 4 5 $100 $100 $100 $100 $323.97 $297.22

How to Value “Lumpy” Cash Flows ,[object Object],[object Object],CF2 CF1 F2 F1 CF0 1 0 4 297.22 0 100 I NPV 9

Growing Annuity ,[object Object],The formula for the present value of a growing annuity: 0 1 C 2 C ×(1+ g ) 3 C ×(1+ g ) 2 T C ×(1+ g ) T -1

PV of Growing Annuity You are evaluating an income property that is providing increasing rents. Net rent is received at the end of each year. The first year's rent is expected to be $8,500 and rent is expected to increase 7% each year. Each payment occur at the end of the year. What is the present value of the estimated income stream over the first 5 years if the discount rate is 12%? $34,706.26 0 1 2 3 4 5

PV of Growing Annuity: Cash Flow Keys ,[object Object],[object Object],$34,706.26 I NPV 12 CF0 CF1 CF2 CF3 CF4 CF5 8,500.00 9,095.00 9,731.65 10,412.87 11,141.77 0

PV of Growing Annuity Using TVM Keys ,[object Object],PMT I/Y FV PV N 7,973.93 4.67 0 – 34,706.26 5 PV

Why it works ,[object Object],Since FV = 0, we can ignore the last term. We want to get to this equation:

We begin by substituting for PMT and for r becomes

We continue to simplify terms. Note that:

We continue to simplify terms. Finally, note that: (1 + r ) – (1 + g ) = r – g

The Result of our Algebrations: ,[object Object],PMT I/Y FV PV N 0 PV

Growing Annuity ,[object Object],0 1 $20,000 2 $20,000 ×(1.03) 40 $20,000 ×(1.03) 39

PV of Growing Annuity: BAII Plus PMT I/Y FV PV N 19,417.48 = 6.80 = 0 – 265,121.57 40 PV A defined-benefit retirement plan offers to pay $20,000 per year for 40 years and increase the annual payment by three-percent each year. What is the present value at retirement if the discount rate is 10 percent per annum? 20,000 1.03 1.10 1.03 – 1 × 100

PV of a delayed growing annuity ,[object Object],If investors demand a 10% return on investments of this risk level, what price will they be willing to pay? Year: 1 2 3 4 Dividends per share $1.50 $1.65 $1.82 5% growth thereafter

PV of a delayed growing annuity The first step is to draw a timeline. The second step is to decide on what we know and what it is we are trying to find. Year 0 1 2 3 Cash flow $1.50 $1.65 $1.82 4 $1.82×1.05 …

PV of a delayed growing annuity Year 0 1 2 3 Cash flow $1.50 $1.65 $1.82 dividend + P 3 PV of cash flow $32.81 = $1.82 + $38.22

4.5 What Is a Firm Worth? ,[object Object],[object Object]

4.6 Summary and Conclusions ,[object Object],[object Object],[object Object]

4.6 Summary and Conclusions (continued) ,[object Object]

How do you get to Carnegie Hall? ,[object Object],[object Object],[object Object],[object Object]

This is my calculator. This is my friend! ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

Problems ,[object Object],[object Object],[object Object],[object Object],[object Object],Hint: don’t even think about doing this: 15,000   30,000 × 7% 8,000 30,000 × 8%  7,000 30,000 × 15%

Problems ,[object Object],[object Object],PMT I/Y FV PV N PV 9.25 0 7 120 15,000 – 174.16 0 8 120 8,000 – 97.06 0 15 120 7,000 – 112.93 – 384.16 30,000 120 0 + + + + = =

Problems ,[object Object],CF2 CF1 F2 F1 CF0 9 $12,500 1 $14,662.65 0 0 I NPV 14 CF4 CF3 F4 F3 9 $18,000 1 15,000 CF4 F4 $22,000 1

Problems ,[object Object],– 495.03 PMT I/Y FV PV N PV 0 7 24 11,056 – 495.03 PMT I/Y FV PV N 25,000 7 36 11,056 – 495.03 PMT I/Y FV PV N PV 0 7 60 25,000

Problems ,[object Object],[object Object],[object Object]

Problems ,[object Object],[object Object],PMT I/Y FV PV N 100,000 5 5 127,628.16 0 PMT I/Y FV PV N 0 10 60 $25,525.63 = 0.20×$127,628.16 – 329.63

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