Finance 1
Finance 1 Core Finance Course: Lecture 1 Week 1, 2006 Alexander Guembel & Han Ozsoylev
What is Finance? • Finance is concerned with several problems: • Which investments should the firm make? – Business entry/exit decisions – Resource allocation
• How should the firm finance its decisions? – Debt, Equity, Convertible bonds, etc
• How should investors select their investments? – Choice of asset classes – Appropriate price
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Selecting Investments • When capital is scarce, where should it be allocated? • How does one value a new investment? – Does it have the appropriate risk/return characteristics?
• How should one evaluate project risk? • What sort of return should we demand for a given level of risk?
Financing Decisions • Does it matter where we get our funds? • What are the incentive consequences of different funding decisions? • Can financing decisions send useful signals to investors? • What is the best governance structure for the firm? • Can risk management add value to the corporation?
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Course Structure Finance 1 (Michaelmas Term) • Valuing Financial Assets Finance 2 (Core Elective, Hilary Term) • Capital Structure and Financial Management • Text: Principles of Corporate Finance, R.A. Brealey, S.C. Myers and F. Allen: McGraw-Hill
Our Lectures 1. 2. 3. 4. 5. 6. 7. 8.
Introduction to Finance Valuing Safe Projects Portfolio Theory Capital Asset Pricing Model Valuing Risky Projects Arbitrage, Market Efficiency and Risk Management Option Valuation More Options and Review
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Other Administrative Information • Attempt to resolve confusion in your study groups first • Don’t use email until you have tried the study group • We will try to be in the Common Room for coffee every day • Office hours are: – Alexander: Thursday 12:15pm – 1:15pm – Han: Thursday 12:15pm – 1:15pm
• The e-mail address is: [email protected] [email protected]
Class Format • Starts on time – Please be there then. Late arrival can disrupt the class
• Please make sure that you have your name plate – This will affect cold-calling incentives…
• No laptops in class, please • Expect to participate
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External Sources of Capital Debt
Equity
• The debt-equity split selected by an organisation is known as its capital structure • Two key perspectives on capital structure in the table: – A way of splitting up the cash flows generated by the company – A way of splitting up control of the company’s assets
Corporations vs. Other Organisations • Corporations have debt and equity (see previous slide) • Ownership is separated from control • The manager’s job is to maximise the wealth of the shareholders – Is this the same as maximising profits? – How can we be sure that managers actually do this?
• Alternatives: – Sole proprietor business – Partnership
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Time Value of Money • I prefer £100 now to £100 in a year – Inflation, opportunity cost, risk, …
• Suppose interest rates are 5% • I can deposit £100 now and get £105 in a year • Equivalently, to get £100 in a year I could deposit the following amount now:
100 1 = = 0.9524 105 1.05
Discount Factors • We say that £95.24 is the present value of £100 in one year • The multiplier 1/1.05 is the discount factor (D.F.) • Interest rates (equivalently, DFs) tell me how to compare money received on different dates / DF 1 year
Now
time
£100
£95.24
x DF
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What are Capital Markets for? • Capital markets are the places where debt and equity claims are traded • Capital markets are where discount factors are established • Capital markets allow the transfer of wealth (and hence consumption) between different dates – MBA students transfer wealth from the future to today – Salaried executives transfer wealth from today to retirement
• Companies pay for investments by using the capital markets to transfer anticipated profits from the future to today
Example: Bonds • Bond is a tradeable debt contract • Its coupon is the amount of interest paid • Maturity date is the time when the face value is repaid • Example: – Bond A: 1 year maturity, zero coupon, £100 face value, currently trades at £96 – What is its rate of return (yield)? – Bond B: identical to A, but pays £6 coupon – What is the price of bond B?
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Discount Factors for Longer Intervals • Suppose that we deposit £100 at 5% for two years: x1.05
x1.05
/ DF
/ DF
100
105
110.25
Now
1 yr
2 yrs
time xDF2 2 • Two year DF: 1 1.05 = 0.907029 • Three year DF: 1 1.053 = 0.863838 , and so on.
Yield Curves • A yield curve is a plot of discount rates against maturities
Rate (%)
US Treasury Yields, 5th August 2005 4.80% 4.60% 4.40% 4.20% 4.00% 3.80% 3.60% 3.40% 3.20% 3.00% 0
5
10
15
20
Years
• We will mostly assume that the yield curve is flat
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Compounding Frequency • Suppose that interest is paid two times per year • Annual interest rate is 6%: what is 1 yr DF? 6 mths
Today x 1.03
1
12 mths x 1.03
1.03
1.0609
DF = 1/1.032=1/1.0609 = 0.9426
• In general:
D.F. =
1 Ann. Rate ⎞ ⎛ ⎟ ⎜1 + ⎝ Comp. Freq ⎠
yrs×comp freq
Continuous Compounding • Interest rates are always quoted for one year • If deposit at R% for t years with compounding frequency f, tf tf − tf R R R Return = ⎜⎛1 + ⎟⎞ ; D.F. = 1 ⎜⎛1 + ⎞⎟ = ⎜⎛1 + ⎞⎟ ⎝ f⎠ ⎝ f⎠ ⎝ f⎠ • As f gets very large (f →∝), return and DF approach: Return = e Rt ; D.F. = e -Rt
• Continuously compounded interest rates are useful for theory (especially derivatives) but otherwise seldom employed
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Example of Discounting • Your firm has an investment opportunity: – Invest £25 mio in a new machine – The machine will generate £5 mio per year after costs for 5 years – Sell the machine for £5 mio to the scrap market in 5 years
• All cash flows are certain • The interest rate is 5.5% • Should the firm make the investment?
Cash Flow Table Date (Years)
Qty
D.F.
Present Value (P.V.)
0
-25
1.0000
-25.00
1
5
2
5
3
5
4
5
5
10
(1 1.05 5) = 0.9479 5 × 0.9479 = 4.74 (1 1.05 5)2 = 0.8985 5 × 0.8985 = 4.49 (1 1.05 5)3 = 0.8516 5 × 0.8516 = 4.26
(1 1.05 5)4 = 0.8072 5 × 0.8072 = 4.04 (1 1.05 5)5 = 0.7651 10 × 0.7651 = 7.65 Net Present Value (NPV):
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Investment Decision • Is it OK just to add up the present values? – Yes: each is a sum of money today. We can add those up
• So … the project is equivalent to an immediate cash gift of £180,000. Accept it. • In general: ∞
NPV = ∑ t =0
Investment rule:
CFt (1 + r )t
Accept positive NPV investments
• What happens when cash flows are risky?
Unanimous Decisions • Suppose that some of the shareholders want to spend money now, while others prefer to wait • Will they all agree on this investment rule? • Why?
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Useful Formulae (1): Perpetuities • What’s the value of a payment of £100 per year forever (first payment is one year from now)? – We call infinite constant streams perpetuities
• Suppose interest rate is 7% 100 100 0.07 = , so PV = = 1428.57 PV 0.07 Alternatively, Value =
100 100 100 100 + + +K = = 1428.57 2 3 1.07 1.07 1.07 0.07
Perpetuitity Value =
Cash Flow Interest Rate
Useful Formulae (2): Annuities • What about £100 for the next 20 years – A constant finite stream of regular payments is an annuity
• 20 year annuity = (perpetuity) – (perpetuity starting in 20 yrs) 100 0.07 1 100 Perpetuity starting in 20 years : value = × 20 1.07 0.07 1 ⎛ 1 ⎞ Annuity value = 100 × ⎜ − = 1059.4 20 ⎟ ⎝ 0.07 0.07 ×1.07 ⎠
Perpetuity starting today :
value =
Annuity factor
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Useful Formulae (2): Annuities • In general: The present value of a payment CF for T periods when the per period discount rate is r can be calculated as
⎛1 1 PV = CF ⎜⎜ − T ⎝ r r (1 + r )
⎞ ⎟⎟ ⎠
Annuity factor
Example: Mortgage • You need a £200,000 housing loan, which you want to repay in equal monthly instalments (mortgage) • The (monthly compounded) interest rate is 6% • Suppose you want to repay the loan over 10 years. What monthly payment can you expect?
• How about if you want to repay over 20 years?
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Useful Formulae (3): Growing Perpetuities • What about a perpetuity starting at £100 and growing at 3% per year forever? • Value of growing perpetuity is
PV =
CF r-g
where CF = first payment (one year from now) g = growth rate of the cash payment r = discount rate
• So value is
100 = £2 ,500 0.07 - 0.03
Share Valuation • Valuation is conceptually simple: • A share is the right to a stream of dividends • The value of the share is the NPV of the expected future dividends • Now all you need to do is establish the dividend stream and the appropriate interest rate
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Trivial Example • Smiths holdings makes pins. Its cost of equity capital is 14%, its next dividend (in 1 year) will be 10p and the dividend is expected to grow at 5% every year. What is the value of a share? r = 14%; g = 5%; C1 = 0.1 So share price is
C1 0.1 = = £1.11 r - g 0.09
• Where do r and g come from?
Estimating Dividend Growth Rates • Suppose that I re-invest 40% of my profits in the business and that my return on equity investment is 15%: Div = 0 .4 eps EPS Return on Equity (ROE) = = 0.15 Book equity per share
Ploughback ratio = 1 - payout ratio = 1 -
• Then my earnings and dividends grow at 40%x15%: g = Ploughback Ratio × ROE = 0.4 × 0.15 = 6%
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Estimating Cost of Equity Capital • Cost of capital for risky firms usually computed using the Capital Asset Pricing Model (weeks 3 and 4) • But we can also turn the growth formula around: P0 = so r =
Div1 , r−g
Div1 + g = div yield + growth rate P0 INCOME CAPITAL APPRECIATION
Examples: Last week of July, 2002 Company
Equity Price
Barclays Bank Northern Foods Bellway Scottish Power
460.50 159.00 409.50 329.25
EPS (Analyst Dividend estimate in Yield 1 year) 40.04 4.01 13.32 5.73 71.23 3.98 25.29 9.23
Return on Ploughback Equity Ratio 24.21 24.03 18.62 5.94
0.54 0.32 0.77 -0.20
g 13% 8% 14% -1%
r 17% 13% 18% 8%
• What do the numbers suggest about investment at Scottish Power? • What are we assuming about future growth rates? Is this reasonable?
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Growth Stocks • What is the difference between growth and income stocks? • Consider a firm whose share price P and earnings per share (eps) you know. • How much of the firm’s value does the market attribute to its growth potential? – Figure out how much the firm would be worth if it did not grow. – What would its ploughback ratio be in that case? – What would its fair value P* be?
P* =
eps r
Growth Opportunities “PVGO” • In general, eps + Value of Future Growth Opportunities P0 = r
• Under what circumstances does the discount rate r equal the earnings/price ratio? Example Calculation: Barclays Bank • P0=460.50; eps=40.04; r=17%. 40.04 PVGO = 460.50 − = 225.72 0.17
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Growth & P/E ratios • Share value includes current and expected future projects • A growth stock is one whose value includes a large component due to future growth •50
P/E ratio of S&P 500 (10yr earnings), 1881-2004
•40 •30 •20 •10 •0 •1860 •1880 •1900 •1920 •1940 •1960 •1980 •2000 •2020
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Valuing Safe Projects Finance Week 2, 2006 Alexander Guembel & Han Ozsoylev
Lecture Plan How before why… 1. NPV valuation of real projects 1. What should you discount 2. Dealing with depreciation 3. Dealing with taxation
2. Is NPV appropriate? 1. Maximising shareholder wealth 2. Alternatives to NPV 3. Attempting to fool the market
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Investment Appraisal: Contrived Example • Monster plc is contemplating a new project to build fluffy dinosaurs – Requires a new factory: cost £500,000 – It will be necessary to hire another sales person: her salary will be £50,000 per annum and she will spend half of her time on this project – Dinosaurs will be sold for £23 each in today’s money – Market will be 4,000 in the first year and will grow at 30% for two years and 10% for one year, before stabilising – Working capital requirement will be £50,000 – Allocated overheads in the warehouse will be £12,000 per anum – The factory will last for 10 years, after which it must be scrapped: the cost of doing so is £150,000 (environmental clean-up costs) – Costs will be straight-line depreciated over ten years – Corporate tax rate is 40%
Evaluating the Investment
• … Should we attempt to compute the expected profit?
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Accounting numbers & cash flows • • • •
Only cash earns interest – not accounting profits Use Free Cash Flow (FCF) in present value calculations It is cash available for distribution to providers of capital (equityholders and creditors) FCF = Revenue – Cost – CAPEX – ∆WC – Tax – CAPEX = capital expenditure – ∆WC = changes in working capital
• •
In accounts you often find Earnings before Interest and after Tax (EBIAT) (after tax ‘profit’) FCF = EBIAT – CAPEX – ∆WC + DEP – DEP = depreciation
We only care about the consequences of our decision •
We only care about cash flow generated by the new project, i.e., incremental cash flows
•
This means you have to include – All changes in cash flow that result from your decision to invest in a particular project – But not cash flows that you would receive / pay regardless of whether or not you choose to invest.
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First Attempt at Determining Cash Flows (it is NOT the solution) Year 0 1 2 3 4 5 6 7 8 9 10
Factory Costs Depreciation -500,000 -50,000 -50,000 -50,000 -50,000 -50,000 -50,000 -50,000 -50,000 -50,000 -150,000 -50,000
Sales Dinosaur Revenue Staff Sales from toys -25,000 -25,000 -25,000 -25,000 -25,000 -25,000 -25,000 -25,000 -25,000 -25,000
4,000 5,200 6,760 7,436 7,436 7,436 7,436 7,436 7,436 7,436
92,000 119,600 155,480 171,028 171,028 171,028 171,028 171,028 171,028 171,028
Allocated ∆ WC Overheads -50,000 -12,000 -12,000 -12,000 -12,000 -12,000 -12,000 -12,000 -12,000 -12,000 -12,000 50,000 -12,000
Issues?
Identifying Cash Flows • How do we deal with depreciation?
• Where does the allocated overhead come from?
• Only half of the sales person’s salary is attributed to the project: why? Is this correct?
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Price Level • The numbers in the first table are all expressed in real terms. – What does this mean?
• • • •
Are interest rates in real terms? Are received cash flows in real terms? How should we express prices? In practise, wage inflation could be different to cuddly dinosaur inflation, but we will assume that all prices increase at a rate of 3%
Price Level • Usually cash flow projections and rates of return (i.e., the rate at which you discount) are quoted as nominal numbers • In practise you need to check that this is true (e.g., ask your marketing manager whether her projections are real or nominal) • Then just discount nominal numbers at nominal rates • Suppose someone gives you projections in real cash flows – Apply inflation rate to them to make them nominal. Then discount at nominal rate. – You can see in the cash flow table later on how this is done
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Working Capital • Working capital is the difference between short term assets and liabilities – Short term assets: inventory, money in a cash till or a non-interest bearing account, bills that someone needs to pay to you (‘receivables’) – Short term liabilities: bills that you need to pay to someone (‘payables’)
• For Monster plc’s project, it is the £50,000 of soft toys and soft toy manufacturing equipment which it holds • Incorporate changes in working capital into the calculation: in this case, £50,000 in at the start and then the same amount retrieved at the end of the project
Allow for Tax Effects • Tax is a real cash flow and hence should be included in the calculation • It is 40% of taxable profit and is paid a year in arrears • Which cash flows are needed to determine this? – The taxman does not work off real cash flows – Capital expenditure is regarded as occurring in stages, as depreciation filters through – This is the only reason we incorporate depreciation into our calculations
• Taxable profit = Revenue – Cost – DEP
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Correct Cash Flow Table Year 0 1 2 3 4 5 6 7 8 9 10 11
• • • •
Price Level CAPEX Deprcn 1.000 -500,000 1.030 -50,000 1.061 -50,000 1.093 -50,000 1.126 -50,000 1.159 -50,000 1.194 -50,000 1.230 -50,000 1.267 -50,000 1.305 -50,000 1.344 -50,000 1.384
Wage Revenue cost Sales from toys -51,500 -53,045 -54,636 -56,275 -57,964 -59,703 -61,494 -63,339 -65,239 -67,196
4,000 5,200 6,760 7,436 7,436 7,436 7,436 7,436 7,436 7,436
Other costs
94,760 126,884 169,897 192,494 198,268 204,216 210,343 216,653 223,153 229,847 -201,587
∆WC -50,000
67,196
Taxable Profit -6,740 23,839 65,261 86,218 90,305 94,514 98,849 103,315 107,914 -88,936
Tax
2,696 -9,535 -26,104 -34,487 -36,122 -37,806 -39,540 -41,326 -43,166 35,574
How is the price level determined? Which columns are used to determine taxable profit? What is being assumed in the tax column? Which columns go into the final appraisal of the project?
Project Value • Assume that cash flows are discounted at 7% (nominal rate) Year 0 1 2 3 4 5 6 7 8 9 10 11
Discount Factor 1.0000 0.9346 0.8734 0.8163 0.7629 0.7130 0.6663 0.6227 0.5820 0.5439 0.5083 0.4751
Net C/Flow -550,000 43,260 76,535 105,725 110,114 105,817 108,392 111,044 113,775 116,588 -14,906 35,574 NPV:
PV -550,000 40,430 66,848 86,303 84,005 75,446 72,226 69,152 66,218 63,416 -7,577 16,901 83,370
• Positive NPV, so accept
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Choosing Between Different Investments •
You produce widgets. Each widget is sold at £3.50 and there is no inflation. Your widget bending machine is broken and you need to buy a new one. Two machines are on the market: Machine 1 Machine 2 Useful life (years): 4 3 Fixed Cost: 10,000 3,000 Widget Production Cost: 2.5 3
• •
You sell 3,250 widgets per year. Discount rate = 4%. Which machine do you buy?
Widget Machine NPV Calculation Yr 0 1 2 3 4
Cap Ex D.F. Turnover 1 2 1.0000 -10,000 -3,000 0.9615 11,375 0.9246 11,375 0.8890 11,375 0.8548 11,375
Prodn Costs 1 2 8,125 8,125 8,125 8,125
Net Cash Flow 1 2 -10,000 -3,000 9,750 3,250 1,625 9,750 3,250 1,625 9,750 3,250 1,625 3,250 NPV:
1,797
1,510
• Is this the full story?
•
Calculations for this section on ChoiceBetweenProjs.xls on lecturer’s drive
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Machine Replacement • Suppose we keep replacing the machines for twelve years Yr 0 1 2 3 4 5 6 7 8 9 10 11 12
Cap Ex D.F. Turnover 1 2 1.0000 -10,000 -3,000 0.9615 11,375 0.9246 11,375 0.8890 11,375 -3,000 0.8548 11,375 -10,000 0.8219 11,375 0.7903 11,375 -3,000 0.7599 11,375 0.7307 11,375 -10,000 0.7026 11,375 -3,000 0.6756 11,375 0.6496 11,375 0.6246 11,375
Prodn Costs 1 2 8,125 8,125 8,125 8,125 8,125 8,125 8,125 8,125 8,125 8,125 8,125 8,125
9,750 9,750 9,750 9,750 9,750 9,750 9,750 9,750 9,750 9,750 9,750 9,750 NPV:
Net Cash Flow 1 2 -10,000 -3,000 3,250 1,625 3,250 1,625 3,250 -1,375 -6,750 1,625 3,250 1,625 3,250 -1,375 3,250 1,625 -6,750 1,625 3,250 -1,375 3,250 1,625 3,250 1,625 3,250 1,625 4,647
5,105
A General Approach • The original calculation yielded a present value • What is the equivalent income per year of the machine’s life? NPV = Equivalent Annual Income × Annuity Factor So Income = NPV Annuity Factor 1 1 − = 2.7751 0.04 0.04 ×1.043 1 1 4 year annuity factor = − = 3.6299 0.04 0.04 ×1.04 4
3 year annuity factor =
Equiv. annual income from machine 1 = 1,797 3.6299 = 495 Equiv. annual income from machine 2 = 1,510 2.7751 = 544
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Perturbations • How would the results be affected if the number of widgets sold per year was different? • Exercise: download the spreadsheet and see. • What do we have to assume to use this method to choose between machines 1 and 2?
Questioning the NPV Approach 1. Should the firm be attempting to maximise shareholder wealth? 2. Is NPV the best way to maximise shareholder wealth? 3. Does the market really value shares using the NPV rule?
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Should Firms Maximise Shareholder Value? • Legally, yes in the UK/US: the shareholders are the residual claimants and they own the firm • The UK and US corporate governance systems are based upon the incentives that the shareholders provide to managers. • A powerful mechanism is disciplining effect of takeovers (or simply firing) for non maximising managers • If firms have other objectives this system may come unravelled
Corporate Social Responsibility? • There is an increasing trend towards corporate social responsibility (CSR) • A recent European Commission green paper states: Being socially responsible means not only fulfilling legal expectations, but also going beyond compliance and investing ‘more’ into human capital, the environment and the relations with stakeholders. (see http://europa.eu.int/comm/employment_social/soc-dial/csr/greenpaper_en.pdf)
• “Going beyond compliance”: managers selecting social objectives on behalf of shareholders? • Will this encourage investment?`
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Alternatives to NPV •
If we can find a discount rate, NPV measures the value of a new project to the providers of capital
•
But some financial managers persist in using other approaches: 1. Payback periods 2. Internal Rate of Return
Payback Period • Payback period is the amount of time it takes for the accumulated cash flow to turn positive – i.e., at the beginning you invest and have big cash outflow: payback tells you how long it takes until the project has paid for itself
• More sophisticated version of payback looks at period until present value of cash flows turns positive – i.e., take into account the opportunity cost of funds, so that project has to pay for itself in present value terms
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Payback Period • Examine cumulative NPV for our example project: Year 0 1 2 3 4 5 6 7 8 9 10 11
Discount Factor 1.0000 0.9346 0.8734 0.8163 0.7629 0.7130 0.6663 0.6227 0.5820 0.5439 0.5083 0.4751
Net C/Flow -550,000 43,260 76,535 105,725 110,114 105,817 108,392 111,044 113,775 116,588 -14,906 35,574 NPV:
PV -550,000 40,430 66,848 86,303 84,005 75,446 72,226 69,152 66,218 63,416 -7,577 16,901 83,370
Cumulative PV -550,000 -509,570 -442,722 -356,418 -272,413 -196,967 -124,741 -55,588 10,630 74,046 66,469 83,370
Payback Period • The project takes eight years to break even. Is it better than a project which takes nine years? • Some managers examine payback based on undiscounted cash flows. What do you think about this?
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Internal Rate of Return • When the NPV=0 the discount rate is equal to the opportunity cost of capital • The IRR of a project is the discount rate which makes its value equal to zero • Calculation is via trial and error: for a long lived project no formula gives the answer (Excel can help!) – Guess a discount rate. Adjust until value is zero.
• In general, IRR is a reasonable measure for rate of return – IRR>CoC Îinvest – IRR sB, manager A performed better
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σ
8
Portfolio Holdings • Empirically, the Sharpe ratio of actively managed funds (net of expenses) does not on average beat the market • So everyone holds the market portfolio – Risk averse people hold very little and put the rest in saving accounts – Risk lovers borrow money so that they can buy more shares
• The part of an asset’s risk which remains in the market portfolio after diversification is measured by its beta – See last lecture
Pricing (1) Suppose I hold the market portfolio and invest a further small amount x in stock i, with βi. How much does risk change? •
it increases by x βi σM2 .
(2) Suppose instead I invest x βi in the market portfolio and x(1- βi) in the riskfree. How much does risk change? •
it increases by x βi σM2
→ (1) and (2) must have the same price.
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Pricing • •
If prices are the same, expected returns are also the same What is expected return of a ‘portfolio’ with βi in the market and (1-βi) in the risk free asset? •
•
So the expected return on stock i is: •
•
(1-βi )rf + βi E(rM)
E(ri) = (1-βi )rf + βi E(rM)
This can be written as E(ri) = rf + βi (E(rM) – rf )
(CAPM)
Which stock is more risky? • Two stocks with the same expected payoff: which has the highest discount rate? Probability
Stock A
Stock B Stocks’ payoff 70
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80
90
100
110
120
130
140
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Risk decomposition • From CAPM formula we can calculate overall risk: ri = rf + βi (rM – rf ) + εi,, where E(εi,) = 0, Cov(εi,,rM)=0 →
σi2 = βi2σM2
+
σε2
Market, or Specific, idiosyncratic, systemic, risk or diversifiable risk
How Risk is Priced: Intuition • •
After diversification only market or systemic risk remains So you are rewarded only for bearing market risk
• •
Another way of thinking about it: If an asset’s expected return included diversifiable risk you could: 1. Buy the asset and take the reward for diversifiable risk 2. Diversify your holdings • You would be left with a reward (for diversifiable risk) and no risk! This reward would quickly be snapped up by traders and would vanish
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Pricing Risk • The Cost of Capital has two components: 1. rf: the risk free rate, for bearing no risk 2. E(rM)-rf: the risk premium, for bearing market risk • •
The amount of market risk in an asset is measured by its β So the Cost of Capital is given by:
E (ri ) = rf + β i (E (rM ) − rf ) •
This is often referred to as the Security Market Line
The Security Market Line E(ri) E (ri ) = r f + β i (E (rM ) − rf )
E(rM) rf 0
1
βi= risk
• What about shares with negative beta?
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Examples • Suppose rF=6% and E[rM]=10%. If stock beta is 1.4, then its equilibrium expected return is: E[r ] = 0.06 + 1.4 × (0.10 − 0.06 ) = 11.6%
•
A project promises payoffs in one year with expected value $1,500. It has a beta of 0.9. Present value? 1. Compute expected return E[r ] = 0.06 + 0.9 × 0.04 = 9.6% 2. Discount expected payoff 1,500 P.V. = $ = $1,368.61 1.096
Examples Stock i
ρiM
σi
E(rM) = 12%
A
0.5
0.25
rf = 5%
B
0.3
0.30
σΜ2 = 0.01
• Portfolio P: 50% A, 50% B. Expected return? β A = ρ AM σ A σ M = 0.5 × 0.25 β B = ρ BM σ B σ M = 0.9
0.01 = 1.25
rA = rF + β A (rM − rF ) = 0.05 + 1.25 × 0.07 = 13.75% rB = 0.05 + 0.9 × 0.07 = 11.3% rP = 0.5 × 13.75% + 0.5 × 11.3% = 12.525%
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Examples • Alternative way to calculate this • Portfolio beta = (value-weighted) average beta
βP = 0.5 βA + 0.5 βB = 1.075 • Then apply CAPM using βP rP = 5% + 1.075 (12% - 5%) = 12.525 %
Does CAPM hold? 1. Can the Market Portfolio be Measured? • In theory, M contains everything: shares in every country, bonds, illiquid investments, human capital • Hard to observe: typically use a proxy in tests (FTSE 100, S&P index etc) 2. Is the Market Index Mean-Variance Efficient? • Tests (Black, Jensen & Scholes (1972), Fama & MacBeth (1974) etc) use a proxy for the market portfolio – – – –
Estimate stock betas Form portfolios of similar beta stocks and estimate portfolio betas Compare betas either to historical or realised payoffs Expect linear relationship, crossing market portfolio where β=1
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Valuing Risky Projects Finance: Week 5 Michaelmas Term, 2006 Alexander Guembel & Han Ozsoylev
So Far… 1. Interest rates and Discount Factors 2. Picking the right cash flows for project evaluation 3. Defining and measuring risk
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Important Insights • • • • • • • •
Only cash flows matter There is no such thing as “corporate risk appetite” Companies should worry about shareholders’ risk appetite Shareholders tend to hold widely diversified portfolios They only care about contribution to portfolio risk The CAPM argues that they hold the market portfolio So the relevant risk measure is market beta In practice, this is probably a good proxy for the shareholder’s portfolio
Valuing Real Projects We have a model (CAPM) relating risk to return:
r = rf + β(E{rM } − rf ) • Does the company CoC reflect investor’s required return and hence give the hurdle rate for new projects? • How should we estimate β? • How is share β related to company CoC? • Is it OK to use the one-period CAPM for projects lasting several years?
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Company Cost of Capital • In July 2001 the CoC for Pfizer was 9.2% (B&M p. 196) • What does this mean? • If Pfizer contemplates a new project, should it discount the cash flows at 9.2%? • Value additivity principle: you can sum present values: Firm Value = PV(A+B) = PV(A) + PV(B) • A project’s value is independent of the portfolio it sits in • The cost of capital depends upon the way capital is used • So why mess about computing company CoC?
Measuring Equity Beta • Equities contribute to market risk and so have a β Equity Return (%)
β 1 Market Return (%)
r = rf + β(E{rM } − rf ) • Estimate β by finding the best line through realised returns • We do this by regression
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What goes into the regression? • We can calculate stock and index returns during a given interval (say 1 month) • How do we do that?
• We do this for many months (say for 5 years) • Each months has two data points: a stock return & an index return • Run regression (e.g., Excel) with stock returns on the left and index returns on the right
Example Regression: Dell Computer Dell return (%)
Prices from Aug 88 to Jan 95 Regression parameters: β = 1.62 (standard error 0.52) R2 = 0.11
Market return (%)
• Which market proxy is being used?
Source: Brealey and Myers, 7th Edn
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Interpreting Regression Parameters Remember that
σ i2 = βi2σ M2
σ ε2
+
Market risk
i
Specific risk
R2 is the proportion of total variance due to market movements • Dell stocks had total variance 5,750 over this period, so: – Market risk = 0.11 × 5750 = 632.50 – Specific risk = 0.89 × 5750 = 5,117.34
• The β estimate is drawn from a distribution with SD 0.52 – With 95% confidence, 1.62 - (1.96 × 0.52) < β < 1.62 + (1.96 × 0.52)
Accuracy of β Estimates • The confidence interval for individual stock βs is wide • Fortunately, we can achieve better estimates by examining groups of similar stocks: Beta Standard. Error Burlington Northern .64 CSX Transportation .46
.20 .24
Norfolk Southern Union Pacific
.52 .40
.26 .21
Industry Portfolio
.50
.17
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Another Dell Regression Dell return (%)
Prices from Feb 95 to Jul 01 Regression parameters: β = 2.02 (standard error 0.38) R2 = 0.27
Market return (%)
• Equity β is not stable Source: Brealey and Myers, 7th Edn
International Investment • • • •
Typically, betas are computed relative to a local market index Is this consistent with the CAPM? Is this appropriate for value maximising managers? B&M note that, although the Egyptian market has a much higher SD than the US one, it has a beta of 0.56 relative to the S&P index • So should US managers be rewarded for diversifying into the Egyptian market?
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β Estimation and Judgement • In practice, we usually want the CoC for a project • Projects are typically not priced and are not traded • How should we react? • First observation: remember the difference between cash flow analysis and risk analysis – Do not increase the discount factor simply because there is a danger that the projected cash flows will not materialise
β Estimation: Cyclicality • • • •
Is the provision of catastrophe insurance risky? Why? What is the relationship between cyclicality and beta? How can we estimate cyclicality? – Cash flow betas and accounting betas
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β Estimation: Operating Leverage • Does the fixed cost base make a difference? • Why?
Company Cost of Capital • We can work out debt β in the same way (for bonds, at least) • The debt/equity split just divides up the cash flows (lecture 1) – If you own all debt and all equity you receive all of the cash flows
• Example. You own all the shares and debt of this company: Company Value V = 110 Debt value D = 65 Equity value E = 45
• The debt returns 8% in a year and the equity returns 15% Total Income = (65 × 0.08) + (45 × 0.15) = 11.95 Percentage Return = 11.95 ÷ 110 = 10.86%
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Company CoC (Ignoring Tax) • The company CoC is a weighted average of debt and equity costs • In general (without taxes), Remember that E D rfirm ≡ rassets ≡ WACC = rdebt + requity D, E and V are market values V V • Think about rdebt as effective cost. This could include tax savings (see Finance 2). • What happens to the firm’s CoC if we increase the debt level from 59% to 70%? • What about the required return on debt and on equity?
Company CoC Increasing Debt Level • • • • •
Total cash flows have not changed so nor has value V … … Aggregate risk has not changed so nor has CoC Debt is now more risky. Debt holders will demand more Suppose the required return goes from 8% to 9% The required return on equity is given by E D rfirm ≡ rassets ≡ WACC = rdebt + requity V V • So D V requity = ⎛⎜ rassets − rdebt ⎞⎟ × = (0.1086 − 0.7 × 0.09) 0.3 V ⎝ ⎠ E = 15.2%
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Company β • The capital structure divides cash flows and risk • But the total risk is not affected: D E βfirm ≡ βassets = βdebt + βequity V V • Earlier example: βdebt = 0.2 and βequity = 0.9 before refinancing
⎛ 65 ⎞ ⎛ 45 ⎞ βfirm = ⎜ × 0.2 ⎟ + ⎜ × 0.9 ⎟ = 0.486 ⎝ 110 ⎠ ⎝ 110 ⎠ • What will happen to βdebt, βequity and βassets when the debt level is increased to 70%?
Financial and Asset Risk • • •
The capital structure divides the cash flows between debt and equity holders It determines the riskiness of their claims, but not of the firm So we can distinguish between:
1. Financial risk: comes from the nature of the cash flow claim 2. Asset risk: comes from the nature of the assets which generate the cash flows
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Example •
Company A: – Half debt, half equity – Debt has yield 5%; equity β is 1.2 – Risk free rate is 3% and the expected market return is 7%
•
What is Company A’s Cost of Capital?
Company A’s CoC • Cost of Debt: rD=5% • Equity: rE = rf + β E (rM − rf ) = 0.03 + 1.2 × 0.04 = 7.8% • So: CoC1 = WACC1 = 0.5 × 0.078 + 0.5 × 0.05 = 6.4%
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Another Example •
1. 2. 3. 4.
ABC Corporation has no debt and its equity β is 0.85. The finance director is going to buy back 30% of the equity and will finance her purchase with debt, yielding 6% What is ABC’s CoC before the equity buy back? What is the company CoC after the buy-back? Find the post buy-back cost of equity and cost of debt. Find equity and debt betas after the buy-back Useful data: rf = 2.5%; E{rM} = 8%
ABC’s Equity Buy-Back 1. Before repurchase: CoC = rf + β(E{rM } − rf ) = 2.5% + 0.85 × (8% − 2.5% ) = 7.175% 2. The company CoC would remain 7.175% 3. Post Buy-Back: Debt CoC = 6% 0.07175 = (0.3 × 0.06 ) + (0.7 × rE ) so rE = (0.07175 − 0.3 × 0.06 ) 0.7 = 7.6786%
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Financial Risk After Buy-Back 4. The security market line implies β = (CoC − rf ) (E{rM } − rf ) So we immediately get: β E = (0.076786 − 0.025) (0.08 − 0.025) = 0.9416 β D = (0.06 − 0.025) (0.08 − 0.025) = 0.6364
The tax shield of debt • Interest payments can be deducted from corporate tax – dividend payments cannot • How do we value the tax shield of debt? • Adjusted Present Value APV = NPV + PV(tax shield) • At what rate should we discount the tax savings?
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Example • A company has a WACC = 10%, rD = 7% and a long-term debt of D=£50,000,000 • The corporate tax rate is 35% • Suppose next year’s after tax cash flows (not counting interest tax shield) are expected to be £8,000,000, which grow forever at the inflation rate (2%) • What is the value of the firm if its debt level remains constant forever?
Pharmaceutical Start-Up Woozy Drugs plc’s new R&D project has two phases Initial research. £1.5 mio cost, lasts 1 year. Outcomes: 1. Highly effective. Probability 10% 2. Moderately effective. Probability 10% 3. No Drug. Probability 80%
Development Phase. £ 5 mio cost, lasts 1 year. Outcomes: 1. Passes the regulatory safety tests. Probability 40% 2. Fails the regulatory safety tests. Probability 60%
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Marketing the Drug • • •
The drug can be brought to market only if it passes the regulatory safety tests Marketing requires a new factory, costing £130 mio Operating costs of drug production are assumed to be zero
1. Highly effective drug: yields perpetual income £19 mio 2. Moderately effective drug: perpetual income £14 mio
Decision Data • • • •
Cost of Capital is 10% There are no taxes There is no inflation Should Woozy plc embark upon the new project?
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Project Evolution Highly effective 1 0.
Initial Research: Cost £ 1.5 mio
0.1
0.4
0.6
Testing: Cost £ 5mio 0.4 Moderately 0.6 effective
Pass
Factory Costs £130 mio Income £19 mio pa
Fail Pass
Factory Costs £130 mio Income £14 mio pa
Fail
8 0.
No drug
Standard NPV Valuation
Year DF 0 1 1 0.909 2 0.826 Note:
Cash out -1,500 -5,000 -130,000
Prob 1 0.2 0.08
High
190,000
190,000= 19,000/0.1
Finance 1, SBS, Michaelmas Term 2006
Cash in Prob Moderate Prob
0.04
140,000
0.04
E(CF) -1,500 -1,000 2,800 NPV=
PV -1,500 -909 2,314 -95
16
Identifying Exit Options 0.1
Highly effective 0.4 Value £ 16.82 mio 0.6 0.4 × 60 1.1 − 5 Testing: Cost £ 5mio
Initial Research: -£ 1.5 mio
0.1
Value £ 0.03 mio 0.1×16.82 1.1 − 1.5
0.4 0.6
So DO NOT PERFORM TESTING 0.8
ACCEPT THE PROJECT
Moderately effective Value -£ 1.36 mio 0.4 ×10 1.1 − 5
No drug
Pass Fail
Pass Fail
Factory Costs £130 mio Income £19 mio pa Value £60 mio Value £0
Factory Costs £130 mio Income £14 mio pa Value £10 mio Value £0
Value £0
Finding Options • Projects often include options to expand and to abandon • Decision trees help you to find them • These are real options, as opposed to the financial options which we will price in lecture 7 • Do real options always increase the value of a project? • What are the possible limitations of this type of analysis?
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Market efficiency & arbitrage Finance 1 Week 6, MT 2006 Alexander Guembel & Han Ozsoylev
Market efficiency • So far we have assumed that prices in financial markets reflect fundamentals. • What exactly does this mean?
• Is it a reasonable assumption to make?
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Arbitrage • Suppose the fundamental value of an asset is V and its price is P • If V > P what would happen? • If V < P what would happen?
Informational efficiency • Not everybody may be in an equal position to estimate V – Different people have different expectations on V. – Jack is a dentist and reads the FT. He has expectation VJack – Should he trade if P ≠ VJack ?
• The answer depends on – How well informed Jack is – How much information is reflected in prices
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Strong form efficiency • Lawrence is the CEO of DyeHard Corp. – He learns that secret tests for a new textile dye are promising. His estimate of firm value changes to VCEO > P. – Should he trade? (Assume for the sake of the argument that insider trading is legal)
• If an insider has no incentive to trade, we call prices strong form efficient • In this case prices already reflect insiders’ information
Semi-strong form efficiency • Camilla has an MBA and reads the FT regularly – Based on her reading she believes that VCamilla ≠ P – Should she trade?
• If trade based on publicly available information is not profitable, prices are semi-strong form efficient
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Weak form efficiency • Tom has a degree in Engineering and does not read the FT • He carefully examines past stock price patterns – He discovered that when the current stock price moves below its 200 day average, prices tend to increase afterwards
• Suppose prices are semi-strong form efficient. Should he trade? • If profitable trade based on past data is impossible, prices are weak form efficient
Efficient Markets There is agreement on very little in economics, but the following have been tested to death: 1. Price changes are random. It seems to be impossible to make consistent profits based upon past price histories. Markets are weak form efficient. 2. Prices reflect new information very quickly. You can’t make much money from publicly available data. Markets are semi-strong form efficient. 3. Insiders could typically make money from trading on their information. Prices are not strong form efficient.
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Efficient markets • Going back to Jack – the FT reading dentist – Should he trade?
Efficient markets • The ‘arbitrage’ opportunity implied by V ≠ P implies taking a risky position – The realization of future cash flows is uncertain – Trading when V ≠ P may yield profits on average, but it also requires risk taking
• Efficient markets really mean that you cannot make excess risk adjusted returns • How do we adjust returns for risk? – This requires a model of the price of risk, e.g., CAPM – Testing for efficient markets is therefore always testing the joint hypothesis “markets are efficient and CAPM holds”
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Risk free arbitrage • If arbitrage involves risk we cannot be sure that someone will emerge to trade, unless we know how risk averse people are. • If arbitrage is risk free, someone will definitely want to trade: doing so means making money for sure • Example: Siemens (ADR) trades at $81.30 in New York and at $82.10 in Frankfurt. – What trades do you execute?
Risk free arbitrage • Risk free arbitrage allows – Making money without taking risk – And without committing any of your own capital: the trade is selffinancing
• It is like picking up money from the street – For how long will a £10 note lie on a busy street?
• Risk free arbitrage opportunities are unlikely to exist for long • R.f.a. provides a potentially powerful tool for pricing of assets: knowing price of one asset implies price of another asset • This will be important when we price derivatives
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Derivatives • Financial contracts whose payoff depends on the value of another asset. • Forward and Futures contracts – Agreement to buy (sell) an asset at a specific future date at a determined price. This price is called the forward (futures) price
• Example – I agree to buy 1000 gallons of heating oil on the 15 March 2007 at a per gallon price F. – What is my payoff on the 15 March? – At what price F will I find someone who will agree (now) to sell me the oil on the 15 March?
Example: Payoff diagram • Suppose spot price is P – I buy at F and sell at P – Profit is P-F
Payoff
• I am forced to buy, even if P < F! F
Finance 1, SBS, Michaelmas Term 2006
Spot price P on 15/3/07
7
Hedging • I can use the forward (futures) market to hedge risk • Example – – – –
I need to heat a production facility during the winter Suppose this requires 1000 gallons of oil What is my payoff as a function of the spot price in the future? How does this change if I enter the forward contract?
Hedged & unhedged payoffs Payoff Forward payoff
F
Spot price P
Hedged payoff Unhedged payoff
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Forward prices • How do we determine the forward price F? • Instead of buying the oil in the future, I could buy it now at price S0. – My payoff on the 15 March is then P
• Suppose I put X into a bank account so that I have just enough on the 15 March 2007 to pay F – My payoff on the 15 March is then also P – I need X = F/(1+rf)t/12 , where t is number of months until 15 March
Forward Prices • If F is such that X < S0 (X > S0) ? – Someone could make money buy short selling (buying) oil now and entering forward contract to buy (sell) oil at F in the future
• Only if S0 = F/(1+rf)t/12 there is no arbitrage opportunity • We can determine the forward price using the “No-Arbitrage” principle • We find that
Finance 1, SBS, Michaelmas Term 2006
F = S0(1+rf )t/12
9
Example: Currency hedging • BA buys $100m worth of aircraft from Boeing, Payoff delivered & payable one year from now • Spot exchange rate £/$ = 0.5 (i.e. £50m for planes) • If next year – £/$ = 1 ⇒ planes cost £100m!! – £/$ = 1/3 ⇒ planes cost only £33m
Forward payoff
F
Exchange rate £/$
Hedged payoff Unhedged payoff
Forwards & Futures • Forwards are customized contracts – Payment requires counter party to be solvent – There is counter-party risk
• Futures are standardized and exchange traded contracts – Exchange acts as intermediary and guarantees payments – No counter-party risk – Exchange requires cash as ‘collateral’ so as to insure against default – Customers have ‘margin’ accounts where positions are ‘marked-tomarket’: if my liabilities increase, I have to pay money into the account
• See Fixed Income and Derivatives elective for more!
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Options • An option contract gives you the right to perform a transaction on a given date • Two standard, or “vanilla” types: – Call options: give you the right to buy something – Put options: give you the right to sell something
• If you use your rights, you are exercising the option • The price you can trade at is the exercise price • The date when you have the rights is the expiry date – American options: can exercise early – European options: can exercise only on the expiry date
Payoff diagrams • These are helpful to understand the behaviour of options • Suppose you have a $10 call expiring on 1 March – i.e.. You have the right to buy a share for $10 on 1 March
• A payoff diagram is a graph showing what the option is worth to you on its expiry date at every share price • Value if expiry share price is $5? • If it is $10? • If it is $14?
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Payout on 1 March Payout
$10
Share Price
Option Premium • • • •
This contract will never lose money! You have to pay something for a position like this one… The cost of an option is referred to as its premium It compensates the option seller for the possibility that she may lose a lot of money • What does the payoff diagram for a $10 strike put option look like?
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Put Option Payoff Diagram Payout • What does the pay off diagram look like for an option seller?
$10
Share Price
Adding Together Payoffs • What happens when I combine options? • For example, I buy a $10 call and sell a $12 call: Option Payoff
$10
Finance 1, SBS, Michaelmas Term 2006
$12
Share Price
13
Combined Payoff Option Payoff
$12
$10
Share Price
• This type of structure is sometimes referred to as a “call spread”, or a “bull call spread.” • Why would anyone choose to execute this trade?
Another Example • What happens when I buy a call option with $10 strike and at the same time sell a put option with $10 strike? Option Payoff
$10
Finance 1, SBS, Michaelmas Term 2006
Share Price
14
Combined Payoff Option Payoff
$10
Share Price
• This is the same as the payoff from a promise to buy at $10 on the expiry date, with no choice: this is a forward
CALL OPTION – PUT OPTION = FORWARD POSITION • This relationship is called put-call parity • In general, positions with the same payout have the same values
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Financial Options Finance 1 Week 7, 2006 Alexander Guembel & Han Ozsoylev
Financial Options • What are options? • Where do we find them in practice? • What is the fair value of an option?
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Option Pricing • • • •
Suppose a non-dividend paying share has price 10 In one year the price will be 21 or 5 with equal probability One year risk free interest rate is 10% Value of a one year call with strike of 10?
½
21
Call Value = 11
½
5
Call Value = 0
Call Value = C 10
The Big Idea: “No arbitrage” • We can trade the share • We can also trade a risk free bond. – Value 1 in a year – Value 1/1.1 today
• Try buying ∆ shares and B risk free bonds • If we can replicate option payoff with ∆ shares and B risk free bonds then by “no-arbitrage” principle, we can pin down option price
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Portfolio Behaviour To ensure replication
10∆ +
21∆ + B
= 11
5∆ + B
=0
B 1.1
Solve for ∆ and B 21∆ + B = 11 5∆ + B = 0
Subtract
5∆ + B = 55 16 + B = 0
16∆ = 11
∆ = 11 16
B = − 55 16
• This portfolio has the same payoffs as the option • So it must have the same value today (why?) C = Option Value = 10∆ +
Finance 1, SBS, Michaelmas Term 2006
B 110 55 = − = 3.75 1.1 16 16 ×1.1
3
Observations 1. Where did we use the probability of an upward or downward movement in the share price? 2. What do you have to assume about risk preferences? 3. How do you interpret the minus sign in the bond position?
Trading • • • • •
The pricing approach tells us how to trade… A customer wants to buy this 10 strike call option What does the market maker do to cover her risks? What does this cost? How does the market maker decide what to charge?
• Why does anyone bother buying the option from the market maker?
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Trading: Two stages • Suppose that the stock price moved once every six months: 21
11
15 10
U
Pu C
11
7 D
1 Pd
5
0
Hedging With Two Stages • If position U is reached, faced with these trees: ∆u = 1; Bu = -10 11 (EXERCISE) 21 Pu 15 Pu = 15∆u+Bu/1.05 = 5.48 1 11 • Similarly, from position D: 7
11 5
Pd
1 0
(EXERCISE)
∆d = 1/6; Bd = -5/6 Pd = 7∆d+Bd/1.05 = 0.37
• To hedge you will need 5.48 at U and 0.37 at D • You can ensure this by holding another replicating portfolio • It transpires that (exercise): – ∆ = 0.6379, B = -4.0923, C = 10∆+B/1.05 = 2.4816
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General Hedging • The delta changes every time the share price moves • So hedging is required every time the price moves • This is quite hard work and there are economies of scale. Delegation to a market maker makes economic sense • At every stage, the maturing portfolio exactly covers the cost of the replacement portfolio. The trader never adds extra money: the strategy is sometimes referred to as selffinancing
Outstanding Issues • What should we assume about the movement of the share at each stage? • Surely there is a simpler way to do this…
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Risk Neutral Probabilities • Do all of this again • Assume share prices move by percentages u or d • Do it all algebraically. Int. rate = r%; expiry t years
Su
Call Value = Cu
Call Value = C S Sd Call Value = Cd
General Hedge Ratios Su∆ + B
C = S∆ +
To ensure replication = Cu
B 1 + rt Sd∆ + B
= Cd
• Solving is an (easy) but optional exercise • We obtain: C − Cd C d − Cd u ; B=- u ∆= u Su − Sd u−d
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General Valuation Formula C=
1 ⎡ ⎛ 1 + rt − d ⎞ 1 ⎛ u − (1 + rt ) ⎞⎤ [qC u + (1 − q )C d ] Cu ⎜ ⎟⎥ = ⎟ + Cd ⎜ ⎢ 1 + rt ⎣ ⎝ u − d ⎠ ⎝ u − d ⎠⎦ 1 + rt q = “Risk Neutral Probability” of up movement
• q is not the real world probability of an up movement • But if we pretend it is and act like risk neutral traders we get the right price: hence the name q Su S 1-q
Sd
Risk Neutrality •
We can value options (or any other security) by:
1. Computing percentage up and down movements u and d 2. Computing the risk neutral probability q 3. Valuing the option as if we are risk neutral and q is the probability of up movement (which it is not) •
This is completely equivalent to assuming that we can hedge our position in the way we have been describing
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Standard Assumptions: u and d • We assume per period percentage share returns are constant: i.e., u and d the same throughout the tree • This is called Geometric Brownian Motion • What property of share price movements do u and d capture? • Let σ be annualised volatility – Means that SD of percentage price movements in t years is σ t
• We always assume that, with time interval t years, u = eσ t and d = e −σ t
Two Step Pricing: Example • • • •
Non-dividend paying stock has an initial price 1.25 Riskless interest rates are 10% at monthly compounding Stock volatility is 22% What would you charge for a two month 1.20 call option?
• Use a two step tree:
⎧ r = 0.10 ⎫ ⎪ u = eσ t = 1.0656 ⎪ ⎪ σ = 0.22 ⎬ ⇒ ⎨ d = 1 u = 0.9385 t = 1 12 ⎪⎭ ⎪q = 1 + rt − d = 0.5497 ⎪⎩ u−d
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Stock Price Evolution • We use u and d to work out where S will go: 1.3320 x u = 1.4193 1.25 x u = 1.3320 1.2500
1.3320 x d = 1.2500 1.25 x d = 1.1731 1.1731 x d = 1.1009
• The risk neutral probability of an up movement is q – Remember this is not the real probability of an up movement
Call Option Price Tree
1.4193 - 1.2000 = 0.2193 P1 1.2500 - 1.2000 = 0.0500
0.0000
Expected expiry price using q: q × 0.2193 + (1 − q ) × 0.0500 = 0.1431 Discount at risk free interest rate: 0.1431 P1 = = 0.1419 1+ r ×t
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Call Option Price Tree (Continued)
1.4193 - 1.2000 = 0.2193 0.1419 1.2500 - 1.2000 = 0.0500 P2 0.0000
P2 =
q × 0.05 + (1 − q ) × 0 = 0.0273 1+ r ×t
Call Option Value
1.4193 - 1.2000 = 0.2193 0.1419 P3
1.2500 - 1.2000 = 0.0500 0.0273 0.0000
Finally, the correct value for the option: P3 =
q × 0.1419 + (1 − q ) × 0.0273 = 0.0895 1+ r × t
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The Black Scholes Model • Now suppose you implement this method with very low t – Spot price moves with greater frequency – Hedging is performed more often
• In the limiting case where t=0: – Prices move continuously – Hedging is performed continuously
• The limiting case is the Black Scholes model • It is the standard model for trading options on anything except interest rates and is frequently used even for these
The Black Scholes Model • Black Scholes formula : C = SN (d1 ) − Xe −rT N (d 2 )
Long share position ∆=N(d1)
Short bond position (borrow) B=XN(d2)
• X is the strike, T is time to maturity in years and the riskless rate r is expressed on a continuous basis d1 =
ln (S X ) + (r + σ 2 2 )T and d 2 = d 1 − σ T σ T
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Problems with Black-Scholes • • •
To use this, we need to be able to hedge We need to do this continuously, as the share price moves around So we need:
1. Liquid share markets 2. Share markets which do not move around violently •
What happens during a market crash?
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What is Finance? • Finance is concerned with several problems: • Which investments should the firm make? – Business entry/exit decisions – Resource allocation
• How should the firm finance its decisions? – Debt, Equity, Convertible bonds, etc
• How should investors select their investments? – Choice of asset classes – Appropriate price
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Selecting Investments • When capital is scarce, where should it be allocated? • How does one value a new investment? – Does it have the appropriate risk/return characteristics?
• How should one evaluate project risk? • What sort of return should we demand for a given level of risk?
Financing Decisions • Does it matter where we get our funds? • What are the incentive consequences of different funding decisions? • Can financing decisions send useful signals to investors? • What is the best governance structure for the firm? • Can risk management add value to the corporation?
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Course Structure Finance 1 (Michaelmas Term) • Valuing Financial Assets Finance 2 (Core Elective, Hilary Term) • Capital Structure and Financial Management • Text: Principles of Corporate Finance, R.A. Brealey, S.C. Myers and F. Allen: McGraw-Hill
Our Lectures 1. 2. 3. 4. 5. 6. 7. 8.
Introduction to Finance Valuing Safe Projects Portfolio Theory Capital Asset Pricing Model Valuing Risky Projects Arbitrage, Market Efficiency and Risk Management Option Valuation More Options and Review
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Other Administrative Information • Attempt to resolve confusion in your study groups first • Don’t use email until you have tried the study group • We will try to be in the Common Room for coffee every day • Office hours are: – Alexander: Thursday 12:15pm – 1:15pm – Han: Thursday 12:15pm – 1:15pm
• The e-mail address is: [email protected] [email protected]
Class Format • Starts on time – Please be there then. Late arrival can disrupt the class
• Please make sure that you have your name plate – This will affect cold-calling incentives…
• No laptops in class, please • Expect to participate
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External Sources of Capital Debt
Equity
• The debt-equity split selected by an organisation is known as its capital structure • Two key perspectives on capital structure in the table: – A way of splitting up the cash flows generated by the company – A way of splitting up control of the company’s assets
Corporations vs. Other Organisations • Corporations have debt and equity (see previous slide) • Ownership is separated from control • The manager’s job is to maximise the wealth of the shareholders – Is this the same as maximising profits? – How can we be sure that managers actually do this?
• Alternatives: – Sole proprietor business – Partnership
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Time Value of Money • I prefer £100 now to £100 in a year – Inflation, opportunity cost, risk, …
• Suppose interest rates are 5% • I can deposit £100 now and get £105 in a year • Equivalently, to get £100 in a year I could deposit the following amount now:
100 1 = = 0.9524 105 1.05
Discount Factors • We say that £95.24 is the present value of £100 in one year • The multiplier 1/1.05 is the discount factor (D.F.) • Interest rates (equivalently, DFs) tell me how to compare money received on different dates / DF 1 year
Now
time
£100
£95.24
x DF
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What are Capital Markets for? • Capital markets are the places where debt and equity claims are traded • Capital markets are where discount factors are established • Capital markets allow the transfer of wealth (and hence consumption) between different dates – MBA students transfer wealth from the future to today – Salaried executives transfer wealth from today to retirement
• Companies pay for investments by using the capital markets to transfer anticipated profits from the future to today
Example: Bonds • Bond is a tradeable debt contract • Its coupon is the amount of interest paid • Maturity date is the time when the face value is repaid • Example: – Bond A: 1 year maturity, zero coupon, £100 face value, currently trades at £96 – What is its rate of return (yield)? – Bond B: identical to A, but pays £6 coupon – What is the price of bond B?
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Discount Factors for Longer Intervals • Suppose that we deposit £100 at 5% for two years: x1.05
x1.05
/ DF
/ DF
100
105
110.25
Now
1 yr
2 yrs
time xDF2 2 • Two year DF: 1 1.05 = 0.907029 • Three year DF: 1 1.053 = 0.863838 , and so on.
Yield Curves • A yield curve is a plot of discount rates against maturities
Rate (%)
US Treasury Yields, 5th August 2005 4.80% 4.60% 4.40% 4.20% 4.00% 3.80% 3.60% 3.40% 3.20% 3.00% 0
5
10
15
20
Years
• We will mostly assume that the yield curve is flat
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Compounding Frequency • Suppose that interest is paid two times per year • Annual interest rate is 6%: what is 1 yr DF? 6 mths
Today x 1.03
1
12 mths x 1.03
1.03
1.0609
DF = 1/1.032=1/1.0609 = 0.9426
• In general:
D.F. =
1 Ann. Rate ⎞ ⎛ ⎟ ⎜1 + ⎝ Comp. Freq ⎠
yrs×comp freq
Continuous Compounding • Interest rates are always quoted for one year • If deposit at R% for t years with compounding frequency f, tf tf − tf R R R Return = ⎜⎛1 + ⎟⎞ ; D.F. = 1 ⎜⎛1 + ⎞⎟ = ⎜⎛1 + ⎞⎟ ⎝ f⎠ ⎝ f⎠ ⎝ f⎠ • As f gets very large (f →∝), return and DF approach: Return = e Rt ; D.F. = e -Rt
• Continuously compounded interest rates are useful for theory (especially derivatives) but otherwise seldom employed
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Example of Discounting • Your firm has an investment opportunity: – Invest £25 mio in a new machine – The machine will generate £5 mio per year after costs for 5 years – Sell the machine for £5 mio to the scrap market in 5 years
• All cash flows are certain • The interest rate is 5.5% • Should the firm make the investment?
Cash Flow Table Date (Years)
Qty
D.F.
Present Value (P.V.)
0
-25
1.0000
-25.00
1
5
2
5
3
5
4
5
5
10
(1 1.05 5) = 0.9479 5 × 0.9479 = 4.74 (1 1.05 5)2 = 0.8985 5 × 0.8985 = 4.49 (1 1.05 5)3 = 0.8516 5 × 0.8516 = 4.26
(1 1.05 5)4 = 0.8072 5 × 0.8072 = 4.04 (1 1.05 5)5 = 0.7651 10 × 0.7651 = 7.65 Net Present Value (NPV):
Finance 1, SBS, Michaelmas Term 2006
0.18
10
Investment Decision • Is it OK just to add up the present values? – Yes: each is a sum of money today. We can add those up
• So … the project is equivalent to an immediate cash gift of £180,000. Accept it. • In general: ∞
NPV = ∑ t =0
Investment rule:
CFt (1 + r )t
Accept positive NPV investments
• What happens when cash flows are risky?
Unanimous Decisions • Suppose that some of the shareholders want to spend money now, while others prefer to wait • Will they all agree on this investment rule? • Why?
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Useful Formulae (1): Perpetuities • What’s the value of a payment of £100 per year forever (first payment is one year from now)? – We call infinite constant streams perpetuities
• Suppose interest rate is 7% 100 100 0.07 = , so PV = = 1428.57 PV 0.07 Alternatively, Value =
100 100 100 100 + + +K = = 1428.57 2 3 1.07 1.07 1.07 0.07
Perpetuitity Value =
Cash Flow Interest Rate
Useful Formulae (2): Annuities • What about £100 for the next 20 years – A constant finite stream of regular payments is an annuity
• 20 year annuity = (perpetuity) – (perpetuity starting in 20 yrs) 100 0.07 1 100 Perpetuity starting in 20 years : value = × 20 1.07 0.07 1 ⎛ 1 ⎞ Annuity value = 100 × ⎜ − = 1059.4 20 ⎟ ⎝ 0.07 0.07 ×1.07 ⎠
Perpetuity starting today :
value =
Annuity factor
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Useful Formulae (2): Annuities • In general: The present value of a payment CF for T periods when the per period discount rate is r can be calculated as
⎛1 1 PV = CF ⎜⎜ − T ⎝ r r (1 + r )
⎞ ⎟⎟ ⎠
Annuity factor
Example: Mortgage • You need a £200,000 housing loan, which you want to repay in equal monthly instalments (mortgage) • The (monthly compounded) interest rate is 6% • Suppose you want to repay the loan over 10 years. What monthly payment can you expect?
• How about if you want to repay over 20 years?
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Useful Formulae (3): Growing Perpetuities • What about a perpetuity starting at £100 and growing at 3% per year forever? • Value of growing perpetuity is
PV =
CF r-g
where CF = first payment (one year from now) g = growth rate of the cash payment r = discount rate
• So value is
100 = £2 ,500 0.07 - 0.03
Share Valuation • Valuation is conceptually simple: • A share is the right to a stream of dividends • The value of the share is the NPV of the expected future dividends • Now all you need to do is establish the dividend stream and the appropriate interest rate
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Trivial Example • Smiths holdings makes pins. Its cost of equity capital is 14%, its next dividend (in 1 year) will be 10p and the dividend is expected to grow at 5% every year. What is the value of a share? r = 14%; g = 5%; C1 = 0.1 So share price is
C1 0.1 = = £1.11 r - g 0.09
• Where do r and g come from?
Estimating Dividend Growth Rates • Suppose that I re-invest 40% of my profits in the business and that my return on equity investment is 15%: Div = 0 .4 eps EPS Return on Equity (ROE) = = 0.15 Book equity per share
Ploughback ratio = 1 - payout ratio = 1 -
• Then my earnings and dividends grow at 40%x15%: g = Ploughback Ratio × ROE = 0.4 × 0.15 = 6%
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Estimating Cost of Equity Capital • Cost of capital for risky firms usually computed using the Capital Asset Pricing Model (weeks 3 and 4) • But we can also turn the growth formula around: P0 = so r =
Div1 , r−g
Div1 + g = div yield + growth rate P0 INCOME CAPITAL APPRECIATION
Examples: Last week of July, 2002 Company
Equity Price
Barclays Bank Northern Foods Bellway Scottish Power
460.50 159.00 409.50 329.25
EPS (Analyst Dividend estimate in Yield 1 year) 40.04 4.01 13.32 5.73 71.23 3.98 25.29 9.23
Return on Ploughback Equity Ratio 24.21 24.03 18.62 5.94
0.54 0.32 0.77 -0.20
g 13% 8% 14% -1%
r 17% 13% 18% 8%
• What do the numbers suggest about investment at Scottish Power? • What are we assuming about future growth rates? Is this reasonable?
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Growth Stocks • What is the difference between growth and income stocks? • Consider a firm whose share price P and earnings per share (eps) you know. • How much of the firm’s value does the market attribute to its growth potential? – Figure out how much the firm would be worth if it did not grow. – What would its ploughback ratio be in that case? – What would its fair value P* be?
P* =
eps r
Growth Opportunities “PVGO” • In general, eps + Value of Future Growth Opportunities P0 = r
• Under what circumstances does the discount rate r equal the earnings/price ratio? Example Calculation: Barclays Bank • P0=460.50; eps=40.04; r=17%. 40.04 PVGO = 460.50 − = 225.72 0.17
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Growth & P/E ratios • Share value includes current and expected future projects • A growth stock is one whose value includes a large component due to future growth •50
P/E ratio of S&P 500 (10yr earnings), 1881-2004
•40 •30 •20 •10 •0 •1860 •1880 •1900 •1920 •1940 •1960 •1980 •2000 •2020
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Valuing Safe Projects Finance Week 2, 2006 Alexander Guembel & Han Ozsoylev
Lecture Plan How before why… 1. NPV valuation of real projects 1. What should you discount 2. Dealing with depreciation 3. Dealing with taxation
2. Is NPV appropriate? 1. Maximising shareholder wealth 2. Alternatives to NPV 3. Attempting to fool the market
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Investment Appraisal: Contrived Example • Monster plc is contemplating a new project to build fluffy dinosaurs – Requires a new factory: cost £500,000 – It will be necessary to hire another sales person: her salary will be £50,000 per annum and she will spend half of her time on this project – Dinosaurs will be sold for £23 each in today’s money – Market will be 4,000 in the first year and will grow at 30% for two years and 10% for one year, before stabilising – Working capital requirement will be £50,000 – Allocated overheads in the warehouse will be £12,000 per anum – The factory will last for 10 years, after which it must be scrapped: the cost of doing so is £150,000 (environmental clean-up costs) – Costs will be straight-line depreciated over ten years – Corporate tax rate is 40%
Evaluating the Investment
• … Should we attempt to compute the expected profit?
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Accounting numbers & cash flows • • • •
Only cash earns interest – not accounting profits Use Free Cash Flow (FCF) in present value calculations It is cash available for distribution to providers of capital (equityholders and creditors) FCF = Revenue – Cost – CAPEX – ∆WC – Tax – CAPEX = capital expenditure – ∆WC = changes in working capital
• •
In accounts you often find Earnings before Interest and after Tax (EBIAT) (after tax ‘profit’) FCF = EBIAT – CAPEX – ∆WC + DEP – DEP = depreciation
We only care about the consequences of our decision •
We only care about cash flow generated by the new project, i.e., incremental cash flows
•
This means you have to include – All changes in cash flow that result from your decision to invest in a particular project – But not cash flows that you would receive / pay regardless of whether or not you choose to invest.
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First Attempt at Determining Cash Flows (it is NOT the solution) Year 0 1 2 3 4 5 6 7 8 9 10
Factory Costs Depreciation -500,000 -50,000 -50,000 -50,000 -50,000 -50,000 -50,000 -50,000 -50,000 -50,000 -150,000 -50,000
Sales Dinosaur Revenue Staff Sales from toys -25,000 -25,000 -25,000 -25,000 -25,000 -25,000 -25,000 -25,000 -25,000 -25,000
4,000 5,200 6,760 7,436 7,436 7,436 7,436 7,436 7,436 7,436
92,000 119,600 155,480 171,028 171,028 171,028 171,028 171,028 171,028 171,028
Allocated ∆ WC Overheads -50,000 -12,000 -12,000 -12,000 -12,000 -12,000 -12,000 -12,000 -12,000 -12,000 -12,000 50,000 -12,000
Issues?
Identifying Cash Flows • How do we deal with depreciation?
• Where does the allocated overhead come from?
• Only half of the sales person’s salary is attributed to the project: why? Is this correct?
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Price Level • The numbers in the first table are all expressed in real terms. – What does this mean?
• • • •
Are interest rates in real terms? Are received cash flows in real terms? How should we express prices? In practise, wage inflation could be different to cuddly dinosaur inflation, but we will assume that all prices increase at a rate of 3%
Price Level • Usually cash flow projections and rates of return (i.e., the rate at which you discount) are quoted as nominal numbers • In practise you need to check that this is true (e.g., ask your marketing manager whether her projections are real or nominal) • Then just discount nominal numbers at nominal rates • Suppose someone gives you projections in real cash flows – Apply inflation rate to them to make them nominal. Then discount at nominal rate. – You can see in the cash flow table later on how this is done
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Working Capital • Working capital is the difference between short term assets and liabilities – Short term assets: inventory, money in a cash till or a non-interest bearing account, bills that someone needs to pay to you (‘receivables’) – Short term liabilities: bills that you need to pay to someone (‘payables’)
• For Monster plc’s project, it is the £50,000 of soft toys and soft toy manufacturing equipment which it holds • Incorporate changes in working capital into the calculation: in this case, £50,000 in at the start and then the same amount retrieved at the end of the project
Allow for Tax Effects • Tax is a real cash flow and hence should be included in the calculation • It is 40% of taxable profit and is paid a year in arrears • Which cash flows are needed to determine this? – The taxman does not work off real cash flows – Capital expenditure is regarded as occurring in stages, as depreciation filters through – This is the only reason we incorporate depreciation into our calculations
• Taxable profit = Revenue – Cost – DEP
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Correct Cash Flow Table Year 0 1 2 3 4 5 6 7 8 9 10 11
• • • •
Price Level CAPEX Deprcn 1.000 -500,000 1.030 -50,000 1.061 -50,000 1.093 -50,000 1.126 -50,000 1.159 -50,000 1.194 -50,000 1.230 -50,000 1.267 -50,000 1.305 -50,000 1.344 -50,000 1.384
Wage Revenue cost Sales from toys -51,500 -53,045 -54,636 -56,275 -57,964 -59,703 -61,494 -63,339 -65,239 -67,196
4,000 5,200 6,760 7,436 7,436 7,436 7,436 7,436 7,436 7,436
Other costs
94,760 126,884 169,897 192,494 198,268 204,216 210,343 216,653 223,153 229,847 -201,587
∆WC -50,000
67,196
Taxable Profit -6,740 23,839 65,261 86,218 90,305 94,514 98,849 103,315 107,914 -88,936
Tax
2,696 -9,535 -26,104 -34,487 -36,122 -37,806 -39,540 -41,326 -43,166 35,574
How is the price level determined? Which columns are used to determine taxable profit? What is being assumed in the tax column? Which columns go into the final appraisal of the project?
Project Value • Assume that cash flows are discounted at 7% (nominal rate) Year 0 1 2 3 4 5 6 7 8 9 10 11
Discount Factor 1.0000 0.9346 0.8734 0.8163 0.7629 0.7130 0.6663 0.6227 0.5820 0.5439 0.5083 0.4751
Net C/Flow -550,000 43,260 76,535 105,725 110,114 105,817 108,392 111,044 113,775 116,588 -14,906 35,574 NPV:
PV -550,000 40,430 66,848 86,303 84,005 75,446 72,226 69,152 66,218 63,416 -7,577 16,901 83,370
• Positive NPV, so accept
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Choosing Between Different Investments •
You produce widgets. Each widget is sold at £3.50 and there is no inflation. Your widget bending machine is broken and you need to buy a new one. Two machines are on the market: Machine 1 Machine 2 Useful life (years): 4 3 Fixed Cost: 10,000 3,000 Widget Production Cost: 2.5 3
• •
You sell 3,250 widgets per year. Discount rate = 4%. Which machine do you buy?
Widget Machine NPV Calculation Yr 0 1 2 3 4
Cap Ex D.F. Turnover 1 2 1.0000 -10,000 -3,000 0.9615 11,375 0.9246 11,375 0.8890 11,375 0.8548 11,375
Prodn Costs 1 2 8,125 8,125 8,125 8,125
Net Cash Flow 1 2 -10,000 -3,000 9,750 3,250 1,625 9,750 3,250 1,625 9,750 3,250 1,625 3,250 NPV:
1,797
1,510
• Is this the full story?
•
Calculations for this section on ChoiceBetweenProjs.xls on lecturer’s drive
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Machine Replacement • Suppose we keep replacing the machines for twelve years Yr 0 1 2 3 4 5 6 7 8 9 10 11 12
Cap Ex D.F. Turnover 1 2 1.0000 -10,000 -3,000 0.9615 11,375 0.9246 11,375 0.8890 11,375 -3,000 0.8548 11,375 -10,000 0.8219 11,375 0.7903 11,375 -3,000 0.7599 11,375 0.7307 11,375 -10,000 0.7026 11,375 -3,000 0.6756 11,375 0.6496 11,375 0.6246 11,375
Prodn Costs 1 2 8,125 8,125 8,125 8,125 8,125 8,125 8,125 8,125 8,125 8,125 8,125 8,125
9,750 9,750 9,750 9,750 9,750 9,750 9,750 9,750 9,750 9,750 9,750 9,750 NPV:
Net Cash Flow 1 2 -10,000 -3,000 3,250 1,625 3,250 1,625 3,250 -1,375 -6,750 1,625 3,250 1,625 3,250 -1,375 3,250 1,625 -6,750 1,625 3,250 -1,375 3,250 1,625 3,250 1,625 3,250 1,625 4,647
5,105
A General Approach • The original calculation yielded a present value • What is the equivalent income per year of the machine’s life? NPV = Equivalent Annual Income × Annuity Factor So Income = NPV Annuity Factor 1 1 − = 2.7751 0.04 0.04 ×1.043 1 1 4 year annuity factor = − = 3.6299 0.04 0.04 ×1.04 4
3 year annuity factor =
Equiv. annual income from machine 1 = 1,797 3.6299 = 495 Equiv. annual income from machine 2 = 1,510 2.7751 = 544
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Perturbations • How would the results be affected if the number of widgets sold per year was different? • Exercise: download the spreadsheet and see. • What do we have to assume to use this method to choose between machines 1 and 2?
Questioning the NPV Approach 1. Should the firm be attempting to maximise shareholder wealth? 2. Is NPV the best way to maximise shareholder wealth? 3. Does the market really value shares using the NPV rule?
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Should Firms Maximise Shareholder Value? • Legally, yes in the UK/US: the shareholders are the residual claimants and they own the firm • The UK and US corporate governance systems are based upon the incentives that the shareholders provide to managers. • A powerful mechanism is disciplining effect of takeovers (or simply firing) for non maximising managers • If firms have other objectives this system may come unravelled
Corporate Social Responsibility? • There is an increasing trend towards corporate social responsibility (CSR) • A recent European Commission green paper states: Being socially responsible means not only fulfilling legal expectations, but also going beyond compliance and investing ‘more’ into human capital, the environment and the relations with stakeholders. (see http://europa.eu.int/comm/employment_social/soc-dial/csr/greenpaper_en.pdf)
• “Going beyond compliance”: managers selecting social objectives on behalf of shareholders? • Will this encourage investment?`
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Alternatives to NPV •
If we can find a discount rate, NPV measures the value of a new project to the providers of capital
•
But some financial managers persist in using other approaches: 1. Payback periods 2. Internal Rate of Return
Payback Period • Payback period is the amount of time it takes for the accumulated cash flow to turn positive – i.e., at the beginning you invest and have big cash outflow: payback tells you how long it takes until the project has paid for itself
• More sophisticated version of payback looks at period until present value of cash flows turns positive – i.e., take into account the opportunity cost of funds, so that project has to pay for itself in present value terms
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Payback Period • Examine cumulative NPV for our example project: Year 0 1 2 3 4 5 6 7 8 9 10 11
Discount Factor 1.0000 0.9346 0.8734 0.8163 0.7629 0.7130 0.6663 0.6227 0.5820 0.5439 0.5083 0.4751
Net C/Flow -550,000 43,260 76,535 105,725 110,114 105,817 108,392 111,044 113,775 116,588 -14,906 35,574 NPV:
PV -550,000 40,430 66,848 86,303 84,005 75,446 72,226 69,152 66,218 63,416 -7,577 16,901 83,370
Cumulative PV -550,000 -509,570 -442,722 -356,418 -272,413 -196,967 -124,741 -55,588 10,630 74,046 66,469 83,370
Payback Period • The project takes eight years to break even. Is it better than a project which takes nine years? • Some managers examine payback based on undiscounted cash flows. What do you think about this?
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Internal Rate of Return • When the NPV=0 the discount rate is equal to the opportunity cost of capital • The IRR of a project is the discount rate which makes its value equal to zero • Calculation is via trial and error: for a long lived project no formula gives the answer (Excel can help!) – Guess a discount rate. Adjust until value is zero.
• In general, IRR is a reasonable measure for rate of return – IRR>CoC Îinvest – IRR sB, manager A performed better
Finance 1, SBS, Michaelmas Term 2006
σ
8
Portfolio Holdings • Empirically, the Sharpe ratio of actively managed funds (net of expenses) does not on average beat the market • So everyone holds the market portfolio – Risk averse people hold very little and put the rest in saving accounts – Risk lovers borrow money so that they can buy more shares
• The part of an asset’s risk which remains in the market portfolio after diversification is measured by its beta – See last lecture
Pricing (1) Suppose I hold the market portfolio and invest a further small amount x in stock i, with βi. How much does risk change? •
it increases by x βi σM2 .
(2) Suppose instead I invest x βi in the market portfolio and x(1- βi) in the riskfree. How much does risk change? •
it increases by x βi σM2
→ (1) and (2) must have the same price.
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Pricing • •
If prices are the same, expected returns are also the same What is expected return of a ‘portfolio’ with βi in the market and (1-βi) in the risk free asset? •
•
So the expected return on stock i is: •
•
(1-βi )rf + βi E(rM)
E(ri) = (1-βi )rf + βi E(rM)
This can be written as E(ri) = rf + βi (E(rM) – rf )
(CAPM)
Which stock is more risky? • Two stocks with the same expected payoff: which has the highest discount rate? Probability
Stock A
Stock B Stocks’ payoff 70
Finance 1, SBS, Michaelmas Term 2006
80
90
100
110
120
130
140
10
Risk decomposition • From CAPM formula we can calculate overall risk: ri = rf + βi (rM – rf ) + εi,, where E(εi,) = 0, Cov(εi,,rM)=0 →
σi2 = βi2σM2
+
σε2
Market, or Specific, idiosyncratic, systemic, risk or diversifiable risk
How Risk is Priced: Intuition • •
After diversification only market or systemic risk remains So you are rewarded only for bearing market risk
• •
Another way of thinking about it: If an asset’s expected return included diversifiable risk you could: 1. Buy the asset and take the reward for diversifiable risk 2. Diversify your holdings • You would be left with a reward (for diversifiable risk) and no risk! This reward would quickly be snapped up by traders and would vanish
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Pricing Risk • The Cost of Capital has two components: 1. rf: the risk free rate, for bearing no risk 2. E(rM)-rf: the risk premium, for bearing market risk • •
The amount of market risk in an asset is measured by its β So the Cost of Capital is given by:
E (ri ) = rf + β i (E (rM ) − rf ) •
This is often referred to as the Security Market Line
The Security Market Line E(ri) E (ri ) = r f + β i (E (rM ) − rf )
E(rM) rf 0
1
βi= risk
• What about shares with negative beta?
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Examples • Suppose rF=6% and E[rM]=10%. If stock beta is 1.4, then its equilibrium expected return is: E[r ] = 0.06 + 1.4 × (0.10 − 0.06 ) = 11.6%
•
A project promises payoffs in one year with expected value $1,500. It has a beta of 0.9. Present value? 1. Compute expected return E[r ] = 0.06 + 0.9 × 0.04 = 9.6% 2. Discount expected payoff 1,500 P.V. = $ = $1,368.61 1.096
Examples Stock i
ρiM
σi
E(rM) = 12%
A
0.5
0.25
rf = 5%
B
0.3
0.30
σΜ2 = 0.01
• Portfolio P: 50% A, 50% B. Expected return? β A = ρ AM σ A σ M = 0.5 × 0.25 β B = ρ BM σ B σ M = 0.9
0.01 = 1.25
rA = rF + β A (rM − rF ) = 0.05 + 1.25 × 0.07 = 13.75% rB = 0.05 + 0.9 × 0.07 = 11.3% rP = 0.5 × 13.75% + 0.5 × 11.3% = 12.525%
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Examples • Alternative way to calculate this • Portfolio beta = (value-weighted) average beta
βP = 0.5 βA + 0.5 βB = 1.075 • Then apply CAPM using βP rP = 5% + 1.075 (12% - 5%) = 12.525 %
Does CAPM hold? 1. Can the Market Portfolio be Measured? • In theory, M contains everything: shares in every country, bonds, illiquid investments, human capital • Hard to observe: typically use a proxy in tests (FTSE 100, S&P index etc) 2. Is the Market Index Mean-Variance Efficient? • Tests (Black, Jensen & Scholes (1972), Fama & MacBeth (1974) etc) use a proxy for the market portfolio – – – –
Estimate stock betas Form portfolios of similar beta stocks and estimate portfolio betas Compare betas either to historical or realised payoffs Expect linear relationship, crossing market portfolio where β=1
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Valuing Risky Projects Finance: Week 5 Michaelmas Term, 2006 Alexander Guembel & Han Ozsoylev
So Far… 1. Interest rates and Discount Factors 2. Picking the right cash flows for project evaluation 3. Defining and measuring risk
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Important Insights • • • • • • • •
Only cash flows matter There is no such thing as “corporate risk appetite” Companies should worry about shareholders’ risk appetite Shareholders tend to hold widely diversified portfolios They only care about contribution to portfolio risk The CAPM argues that they hold the market portfolio So the relevant risk measure is market beta In practice, this is probably a good proxy for the shareholder’s portfolio
Valuing Real Projects We have a model (CAPM) relating risk to return:
r = rf + β(E{rM } − rf ) • Does the company CoC reflect investor’s required return and hence give the hurdle rate for new projects? • How should we estimate β? • How is share β related to company CoC? • Is it OK to use the one-period CAPM for projects lasting several years?
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Company Cost of Capital • In July 2001 the CoC for Pfizer was 9.2% (B&M p. 196) • What does this mean? • If Pfizer contemplates a new project, should it discount the cash flows at 9.2%? • Value additivity principle: you can sum present values: Firm Value = PV(A+B) = PV(A) + PV(B) • A project’s value is independent of the portfolio it sits in • The cost of capital depends upon the way capital is used • So why mess about computing company CoC?
Measuring Equity Beta • Equities contribute to market risk and so have a β Equity Return (%)
β 1 Market Return (%)
r = rf + β(E{rM } − rf ) • Estimate β by finding the best line through realised returns • We do this by regression
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What goes into the regression? • We can calculate stock and index returns during a given interval (say 1 month) • How do we do that?
• We do this for many months (say for 5 years) • Each months has two data points: a stock return & an index return • Run regression (e.g., Excel) with stock returns on the left and index returns on the right
Example Regression: Dell Computer Dell return (%)
Prices from Aug 88 to Jan 95 Regression parameters: β = 1.62 (standard error 0.52) R2 = 0.11
Market return (%)
• Which market proxy is being used?
Source: Brealey and Myers, 7th Edn
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Interpreting Regression Parameters Remember that
σ i2 = βi2σ M2
σ ε2
+
Market risk
i
Specific risk
R2 is the proportion of total variance due to market movements • Dell stocks had total variance 5,750 over this period, so: – Market risk = 0.11 × 5750 = 632.50 – Specific risk = 0.89 × 5750 = 5,117.34
• The β estimate is drawn from a distribution with SD 0.52 – With 95% confidence, 1.62 - (1.96 × 0.52) < β < 1.62 + (1.96 × 0.52)
Accuracy of β Estimates • The confidence interval for individual stock βs is wide • Fortunately, we can achieve better estimates by examining groups of similar stocks: Beta Standard. Error Burlington Northern .64 CSX Transportation .46
.20 .24
Norfolk Southern Union Pacific
.52 .40
.26 .21
Industry Portfolio
.50
.17
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Another Dell Regression Dell return (%)
Prices from Feb 95 to Jul 01 Regression parameters: β = 2.02 (standard error 0.38) R2 = 0.27
Market return (%)
• Equity β is not stable Source: Brealey and Myers, 7th Edn
International Investment • • • •
Typically, betas are computed relative to a local market index Is this consistent with the CAPM? Is this appropriate for value maximising managers? B&M note that, although the Egyptian market has a much higher SD than the US one, it has a beta of 0.56 relative to the S&P index • So should US managers be rewarded for diversifying into the Egyptian market?
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β Estimation and Judgement • In practice, we usually want the CoC for a project • Projects are typically not priced and are not traded • How should we react? • First observation: remember the difference between cash flow analysis and risk analysis – Do not increase the discount factor simply because there is a danger that the projected cash flows will not materialise
β Estimation: Cyclicality • • • •
Is the provision of catastrophe insurance risky? Why? What is the relationship between cyclicality and beta? How can we estimate cyclicality? – Cash flow betas and accounting betas
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β Estimation: Operating Leverage • Does the fixed cost base make a difference? • Why?
Company Cost of Capital • We can work out debt β in the same way (for bonds, at least) • The debt/equity split just divides up the cash flows (lecture 1) – If you own all debt and all equity you receive all of the cash flows
• Example. You own all the shares and debt of this company: Company Value V = 110 Debt value D = 65 Equity value E = 45
• The debt returns 8% in a year and the equity returns 15% Total Income = (65 × 0.08) + (45 × 0.15) = 11.95 Percentage Return = 11.95 ÷ 110 = 10.86%
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Company CoC (Ignoring Tax) • The company CoC is a weighted average of debt and equity costs • In general (without taxes), Remember that E D rfirm ≡ rassets ≡ WACC = rdebt + requity D, E and V are market values V V • Think about rdebt as effective cost. This could include tax savings (see Finance 2). • What happens to the firm’s CoC if we increase the debt level from 59% to 70%? • What about the required return on debt and on equity?
Company CoC Increasing Debt Level • • • • •
Total cash flows have not changed so nor has value V … … Aggregate risk has not changed so nor has CoC Debt is now more risky. Debt holders will demand more Suppose the required return goes from 8% to 9% The required return on equity is given by E D rfirm ≡ rassets ≡ WACC = rdebt + requity V V • So D V requity = ⎛⎜ rassets − rdebt ⎞⎟ × = (0.1086 − 0.7 × 0.09) 0.3 V ⎝ ⎠ E = 15.2%
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Company β • The capital structure divides cash flows and risk • But the total risk is not affected: D E βfirm ≡ βassets = βdebt + βequity V V • Earlier example: βdebt = 0.2 and βequity = 0.9 before refinancing
⎛ 65 ⎞ ⎛ 45 ⎞ βfirm = ⎜ × 0.2 ⎟ + ⎜ × 0.9 ⎟ = 0.486 ⎝ 110 ⎠ ⎝ 110 ⎠ • What will happen to βdebt, βequity and βassets when the debt level is increased to 70%?
Financial and Asset Risk • • •
The capital structure divides the cash flows between debt and equity holders It determines the riskiness of their claims, but not of the firm So we can distinguish between:
1. Financial risk: comes from the nature of the cash flow claim 2. Asset risk: comes from the nature of the assets which generate the cash flows
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Example •
Company A: – Half debt, half equity – Debt has yield 5%; equity β is 1.2 – Risk free rate is 3% and the expected market return is 7%
•
What is Company A’s Cost of Capital?
Company A’s CoC • Cost of Debt: rD=5% • Equity: rE = rf + β E (rM − rf ) = 0.03 + 1.2 × 0.04 = 7.8% • So: CoC1 = WACC1 = 0.5 × 0.078 + 0.5 × 0.05 = 6.4%
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Another Example •
1. 2. 3. 4.
ABC Corporation has no debt and its equity β is 0.85. The finance director is going to buy back 30% of the equity and will finance her purchase with debt, yielding 6% What is ABC’s CoC before the equity buy back? What is the company CoC after the buy-back? Find the post buy-back cost of equity and cost of debt. Find equity and debt betas after the buy-back Useful data: rf = 2.5%; E{rM} = 8%
ABC’s Equity Buy-Back 1. Before repurchase: CoC = rf + β(E{rM } − rf ) = 2.5% + 0.85 × (8% − 2.5% ) = 7.175% 2. The company CoC would remain 7.175% 3. Post Buy-Back: Debt CoC = 6% 0.07175 = (0.3 × 0.06 ) + (0.7 × rE ) so rE = (0.07175 − 0.3 × 0.06 ) 0.7 = 7.6786%
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Financial Risk After Buy-Back 4. The security market line implies β = (CoC − rf ) (E{rM } − rf ) So we immediately get: β E = (0.076786 − 0.025) (0.08 − 0.025) = 0.9416 β D = (0.06 − 0.025) (0.08 − 0.025) = 0.6364
The tax shield of debt • Interest payments can be deducted from corporate tax – dividend payments cannot • How do we value the tax shield of debt? • Adjusted Present Value APV = NPV + PV(tax shield) • At what rate should we discount the tax savings?
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Example • A company has a WACC = 10%, rD = 7% and a long-term debt of D=£50,000,000 • The corporate tax rate is 35% • Suppose next year’s after tax cash flows (not counting interest tax shield) are expected to be £8,000,000, which grow forever at the inflation rate (2%) • What is the value of the firm if its debt level remains constant forever?
Pharmaceutical Start-Up Woozy Drugs plc’s new R&D project has two phases Initial research. £1.5 mio cost, lasts 1 year. Outcomes: 1. Highly effective. Probability 10% 2. Moderately effective. Probability 10% 3. No Drug. Probability 80%
Development Phase. £ 5 mio cost, lasts 1 year. Outcomes: 1. Passes the regulatory safety tests. Probability 40% 2. Fails the regulatory safety tests. Probability 60%
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Marketing the Drug • • •
The drug can be brought to market only if it passes the regulatory safety tests Marketing requires a new factory, costing £130 mio Operating costs of drug production are assumed to be zero
1. Highly effective drug: yields perpetual income £19 mio 2. Moderately effective drug: perpetual income £14 mio
Decision Data • • • •
Cost of Capital is 10% There are no taxes There is no inflation Should Woozy plc embark upon the new project?
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Project Evolution Highly effective 1 0.
Initial Research: Cost £ 1.5 mio
0.1
0.4
0.6
Testing: Cost £ 5mio 0.4 Moderately 0.6 effective
Pass
Factory Costs £130 mio Income £19 mio pa
Fail Pass
Factory Costs £130 mio Income £14 mio pa
Fail
8 0.
No drug
Standard NPV Valuation
Year DF 0 1 1 0.909 2 0.826 Note:
Cash out -1,500 -5,000 -130,000
Prob 1 0.2 0.08
High
190,000
190,000= 19,000/0.1
Finance 1, SBS, Michaelmas Term 2006
Cash in Prob Moderate Prob
0.04
140,000
0.04
E(CF) -1,500 -1,000 2,800 NPV=
PV -1,500 -909 2,314 -95
16
Identifying Exit Options 0.1
Highly effective 0.4 Value £ 16.82 mio 0.6 0.4 × 60 1.1 − 5 Testing: Cost £ 5mio
Initial Research: -£ 1.5 mio
0.1
Value £ 0.03 mio 0.1×16.82 1.1 − 1.5
0.4 0.6
So DO NOT PERFORM TESTING 0.8
ACCEPT THE PROJECT
Moderately effective Value -£ 1.36 mio 0.4 ×10 1.1 − 5
No drug
Pass Fail
Pass Fail
Factory Costs £130 mio Income £19 mio pa Value £60 mio Value £0
Factory Costs £130 mio Income £14 mio pa Value £10 mio Value £0
Value £0
Finding Options • Projects often include options to expand and to abandon • Decision trees help you to find them • These are real options, as opposed to the financial options which we will price in lecture 7 • Do real options always increase the value of a project? • What are the possible limitations of this type of analysis?
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Market efficiency & arbitrage Finance 1 Week 6, MT 2006 Alexander Guembel & Han Ozsoylev
Market efficiency • So far we have assumed that prices in financial markets reflect fundamentals. • What exactly does this mean?
• Is it a reasonable assumption to make?
Finance 1, SBS, Michaelmas Term 2006
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Arbitrage • Suppose the fundamental value of an asset is V and its price is P • If V > P what would happen? • If V < P what would happen?
Informational efficiency • Not everybody may be in an equal position to estimate V – Different people have different expectations on V. – Jack is a dentist and reads the FT. He has expectation VJack – Should he trade if P ≠ VJack ?
• The answer depends on – How well informed Jack is – How much information is reflected in prices
Finance 1, SBS, Michaelmas Term 2006
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Strong form efficiency • Lawrence is the CEO of DyeHard Corp. – He learns that secret tests for a new textile dye are promising. His estimate of firm value changes to VCEO > P. – Should he trade? (Assume for the sake of the argument that insider trading is legal)
• If an insider has no incentive to trade, we call prices strong form efficient • In this case prices already reflect insiders’ information
Semi-strong form efficiency • Camilla has an MBA and reads the FT regularly – Based on her reading she believes that VCamilla ≠ P – Should she trade?
• If trade based on publicly available information is not profitable, prices are semi-strong form efficient
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Weak form efficiency • Tom has a degree in Engineering and does not read the FT • He carefully examines past stock price patterns – He discovered that when the current stock price moves below its 200 day average, prices tend to increase afterwards
• Suppose prices are semi-strong form efficient. Should he trade? • If profitable trade based on past data is impossible, prices are weak form efficient
Efficient Markets There is agreement on very little in economics, but the following have been tested to death: 1. Price changes are random. It seems to be impossible to make consistent profits based upon past price histories. Markets are weak form efficient. 2. Prices reflect new information very quickly. You can’t make much money from publicly available data. Markets are semi-strong form efficient. 3. Insiders could typically make money from trading on their information. Prices are not strong form efficient.
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Efficient markets • Going back to Jack – the FT reading dentist – Should he trade?
Efficient markets • The ‘arbitrage’ opportunity implied by V ≠ P implies taking a risky position – The realization of future cash flows is uncertain – Trading when V ≠ P may yield profits on average, but it also requires risk taking
• Efficient markets really mean that you cannot make excess risk adjusted returns • How do we adjust returns for risk? – This requires a model of the price of risk, e.g., CAPM – Testing for efficient markets is therefore always testing the joint hypothesis “markets are efficient and CAPM holds”
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Risk free arbitrage • If arbitrage involves risk we cannot be sure that someone will emerge to trade, unless we know how risk averse people are. • If arbitrage is risk free, someone will definitely want to trade: doing so means making money for sure • Example: Siemens (ADR) trades at $81.30 in New York and at $82.10 in Frankfurt. – What trades do you execute?
Risk free arbitrage • Risk free arbitrage allows – Making money without taking risk – And without committing any of your own capital: the trade is selffinancing
• It is like picking up money from the street – For how long will a £10 note lie on a busy street?
• Risk free arbitrage opportunities are unlikely to exist for long • R.f.a. provides a potentially powerful tool for pricing of assets: knowing price of one asset implies price of another asset • This will be important when we price derivatives
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Derivatives • Financial contracts whose payoff depends on the value of another asset. • Forward and Futures contracts – Agreement to buy (sell) an asset at a specific future date at a determined price. This price is called the forward (futures) price
• Example – I agree to buy 1000 gallons of heating oil on the 15 March 2007 at a per gallon price F. – What is my payoff on the 15 March? – At what price F will I find someone who will agree (now) to sell me the oil on the 15 March?
Example: Payoff diagram • Suppose spot price is P – I buy at F and sell at P – Profit is P-F
Payoff
• I am forced to buy, even if P < F! F
Finance 1, SBS, Michaelmas Term 2006
Spot price P on 15/3/07
7
Hedging • I can use the forward (futures) market to hedge risk • Example – – – –
I need to heat a production facility during the winter Suppose this requires 1000 gallons of oil What is my payoff as a function of the spot price in the future? How does this change if I enter the forward contract?
Hedged & unhedged payoffs Payoff Forward payoff
F
Spot price P
Hedged payoff Unhedged payoff
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Forward prices • How do we determine the forward price F? • Instead of buying the oil in the future, I could buy it now at price S0. – My payoff on the 15 March is then P
• Suppose I put X into a bank account so that I have just enough on the 15 March 2007 to pay F – My payoff on the 15 March is then also P – I need X = F/(1+rf)t/12 , where t is number of months until 15 March
Forward Prices • If F is such that X < S0 (X > S0) ? – Someone could make money buy short selling (buying) oil now and entering forward contract to buy (sell) oil at F in the future
• Only if S0 = F/(1+rf)t/12 there is no arbitrage opportunity • We can determine the forward price using the “No-Arbitrage” principle • We find that
Finance 1, SBS, Michaelmas Term 2006
F = S0(1+rf )t/12
9
Example: Currency hedging • BA buys $100m worth of aircraft from Boeing, Payoff delivered & payable one year from now • Spot exchange rate £/$ = 0.5 (i.e. £50m for planes) • If next year – £/$ = 1 ⇒ planes cost £100m!! – £/$ = 1/3 ⇒ planes cost only £33m
Forward payoff
F
Exchange rate £/$
Hedged payoff Unhedged payoff
Forwards & Futures • Forwards are customized contracts – Payment requires counter party to be solvent – There is counter-party risk
• Futures are standardized and exchange traded contracts – Exchange acts as intermediary and guarantees payments – No counter-party risk – Exchange requires cash as ‘collateral’ so as to insure against default – Customers have ‘margin’ accounts where positions are ‘marked-tomarket’: if my liabilities increase, I have to pay money into the account
• See Fixed Income and Derivatives elective for more!
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Options • An option contract gives you the right to perform a transaction on a given date • Two standard, or “vanilla” types: – Call options: give you the right to buy something – Put options: give you the right to sell something
• If you use your rights, you are exercising the option • The price you can trade at is the exercise price • The date when you have the rights is the expiry date – American options: can exercise early – European options: can exercise only on the expiry date
Payoff diagrams • These are helpful to understand the behaviour of options • Suppose you have a $10 call expiring on 1 March – i.e.. You have the right to buy a share for $10 on 1 March
• A payoff diagram is a graph showing what the option is worth to you on its expiry date at every share price • Value if expiry share price is $5? • If it is $10? • If it is $14?
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Payout on 1 March Payout
$10
Share Price
Option Premium • • • •
This contract will never lose money! You have to pay something for a position like this one… The cost of an option is referred to as its premium It compensates the option seller for the possibility that she may lose a lot of money • What does the payoff diagram for a $10 strike put option look like?
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Put Option Payoff Diagram Payout • What does the pay off diagram look like for an option seller?
$10
Share Price
Adding Together Payoffs • What happens when I combine options? • For example, I buy a $10 call and sell a $12 call: Option Payoff
$10
Finance 1, SBS, Michaelmas Term 2006
$12
Share Price
13
Combined Payoff Option Payoff
$12
$10
Share Price
• This type of structure is sometimes referred to as a “call spread”, or a “bull call spread.” • Why would anyone choose to execute this trade?
Another Example • What happens when I buy a call option with $10 strike and at the same time sell a put option with $10 strike? Option Payoff
$10
Finance 1, SBS, Michaelmas Term 2006
Share Price
14
Combined Payoff Option Payoff
$10
Share Price
• This is the same as the payoff from a promise to buy at $10 on the expiry date, with no choice: this is a forward
CALL OPTION – PUT OPTION = FORWARD POSITION • This relationship is called put-call parity • In general, positions with the same payout have the same values
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Financial Options Finance 1 Week 7, 2006 Alexander Guembel & Han Ozsoylev
Financial Options • What are options? • Where do we find them in practice? • What is the fair value of an option?
Finance 1, SBS, Michaelmas Term 2006
1
Option Pricing • • • •
Suppose a non-dividend paying share has price 10 In one year the price will be 21 or 5 with equal probability One year risk free interest rate is 10% Value of a one year call with strike of 10?
½
21
Call Value = 11
½
5
Call Value = 0
Call Value = C 10
The Big Idea: “No arbitrage” • We can trade the share • We can also trade a risk free bond. – Value 1 in a year – Value 1/1.1 today
• Try buying ∆ shares and B risk free bonds • If we can replicate option payoff with ∆ shares and B risk free bonds then by “no-arbitrage” principle, we can pin down option price
Finance 1, SBS, Michaelmas Term 2006
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Portfolio Behaviour To ensure replication
10∆ +
21∆ + B
= 11
5∆ + B
=0
B 1.1
Solve for ∆ and B 21∆ + B = 11 5∆ + B = 0
Subtract
5∆ + B = 55 16 + B = 0
16∆ = 11
∆ = 11 16
B = − 55 16
• This portfolio has the same payoffs as the option • So it must have the same value today (why?) C = Option Value = 10∆ +
Finance 1, SBS, Michaelmas Term 2006
B 110 55 = − = 3.75 1.1 16 16 ×1.1
3
Observations 1. Where did we use the probability of an upward or downward movement in the share price? 2. What do you have to assume about risk preferences? 3. How do you interpret the minus sign in the bond position?
Trading • • • • •
The pricing approach tells us how to trade… A customer wants to buy this 10 strike call option What does the market maker do to cover her risks? What does this cost? How does the market maker decide what to charge?
• Why does anyone bother buying the option from the market maker?
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Trading: Two stages • Suppose that the stock price moved once every six months: 21
11
15 10
U
Pu C
11
7 D
1 Pd
5
0
Hedging With Two Stages • If position U is reached, faced with these trees: ∆u = 1; Bu = -10 11 (EXERCISE) 21 Pu 15 Pu = 15∆u+Bu/1.05 = 5.48 1 11 • Similarly, from position D: 7
11 5
Pd
1 0
(EXERCISE)
∆d = 1/6; Bd = -5/6 Pd = 7∆d+Bd/1.05 = 0.37
• To hedge you will need 5.48 at U and 0.37 at D • You can ensure this by holding another replicating portfolio • It transpires that (exercise): – ∆ = 0.6379, B = -4.0923, C = 10∆+B/1.05 = 2.4816
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General Hedging • The delta changes every time the share price moves • So hedging is required every time the price moves • This is quite hard work and there are economies of scale. Delegation to a market maker makes economic sense • At every stage, the maturing portfolio exactly covers the cost of the replacement portfolio. The trader never adds extra money: the strategy is sometimes referred to as selffinancing
Outstanding Issues • What should we assume about the movement of the share at each stage? • Surely there is a simpler way to do this…
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Risk Neutral Probabilities • Do all of this again • Assume share prices move by percentages u or d • Do it all algebraically. Int. rate = r%; expiry t years
Su
Call Value = Cu
Call Value = C S Sd Call Value = Cd
General Hedge Ratios Su∆ + B
C = S∆ +
To ensure replication = Cu
B 1 + rt Sd∆ + B
= Cd
• Solving is an (easy) but optional exercise • We obtain: C − Cd C d − Cd u ; B=- u ∆= u Su − Sd u−d
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General Valuation Formula C=
1 ⎡ ⎛ 1 + rt − d ⎞ 1 ⎛ u − (1 + rt ) ⎞⎤ [qC u + (1 − q )C d ] Cu ⎜ ⎟⎥ = ⎟ + Cd ⎜ ⎢ 1 + rt ⎣ ⎝ u − d ⎠ ⎝ u − d ⎠⎦ 1 + rt q = “Risk Neutral Probability” of up movement
• q is not the real world probability of an up movement • But if we pretend it is and act like risk neutral traders we get the right price: hence the name q Su S 1-q
Sd
Risk Neutrality •
We can value options (or any other security) by:
1. Computing percentage up and down movements u and d 2. Computing the risk neutral probability q 3. Valuing the option as if we are risk neutral and q is the probability of up movement (which it is not) •
This is completely equivalent to assuming that we can hedge our position in the way we have been describing
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Standard Assumptions: u and d • We assume per period percentage share returns are constant: i.e., u and d the same throughout the tree • This is called Geometric Brownian Motion • What property of share price movements do u and d capture? • Let σ be annualised volatility – Means that SD of percentage price movements in t years is σ t
• We always assume that, with time interval t years, u = eσ t and d = e −σ t
Two Step Pricing: Example • • • •
Non-dividend paying stock has an initial price 1.25 Riskless interest rates are 10% at monthly compounding Stock volatility is 22% What would you charge for a two month 1.20 call option?
• Use a two step tree:
⎧ r = 0.10 ⎫ ⎪ u = eσ t = 1.0656 ⎪ ⎪ σ = 0.22 ⎬ ⇒ ⎨ d = 1 u = 0.9385 t = 1 12 ⎪⎭ ⎪q = 1 + rt − d = 0.5497 ⎪⎩ u−d
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Stock Price Evolution • We use u and d to work out where S will go: 1.3320 x u = 1.4193 1.25 x u = 1.3320 1.2500
1.3320 x d = 1.2500 1.25 x d = 1.1731 1.1731 x d = 1.1009
• The risk neutral probability of an up movement is q – Remember this is not the real probability of an up movement
Call Option Price Tree
1.4193 - 1.2000 = 0.2193 P1 1.2500 - 1.2000 = 0.0500
0.0000
Expected expiry price using q: q × 0.2193 + (1 − q ) × 0.0500 = 0.1431 Discount at risk free interest rate: 0.1431 P1 = = 0.1419 1+ r ×t
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Call Option Price Tree (Continued)
1.4193 - 1.2000 = 0.2193 0.1419 1.2500 - 1.2000 = 0.0500 P2 0.0000
P2 =
q × 0.05 + (1 − q ) × 0 = 0.0273 1+ r ×t
Call Option Value
1.4193 - 1.2000 = 0.2193 0.1419 P3
1.2500 - 1.2000 = 0.0500 0.0273 0.0000
Finally, the correct value for the option: P3 =
q × 0.1419 + (1 − q ) × 0.0273 = 0.0895 1+ r × t
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The Black Scholes Model • Now suppose you implement this method with very low t – Spot price moves with greater frequency – Hedging is performed more often
• In the limiting case where t=0: – Prices move continuously – Hedging is performed continuously
• The limiting case is the Black Scholes model • It is the standard model for trading options on anything except interest rates and is frequently used even for these
The Black Scholes Model • Black Scholes formula : C = SN (d1 ) − Xe −rT N (d 2 )
Long share position ∆=N(d1)
Short bond position (borrow) B=XN(d2)
• X is the strike, T is time to maturity in years and the riskless rate r is expressed on a continuous basis d1 =
ln (S X ) + (r + σ 2 2 )T and d 2 = d 1 − σ T σ T
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Problems with Black-Scholes • • •
To use this, we need to be able to hedge We need to do this continuously, as the share price moves around So we need:
1. Liquid share markets 2. Share markets which do not move around violently •
What happens during a market crash?
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