JEM034 Corporate Finance Winter Semester 2017/2018 Lecture #5 Olga Bychkova
Topics Covered Today Risk and the Cost of Capital (chapter 9 in BMA) Understading Options (chapter 20 in BMA) Valuing Options (chapter 21 in BMA)
Risk and the Cost of Capital: Topics Covered Company and Project Costs of Capital Measuring the Cost of Equity Analyzing Project Risk Certainty Equivalents
Company Cost of Capital A firm s value can be stated as the sum of the value of its various assets: Firm value = PV (AB) = PV (A) + PV (B)
Company Cost of Capital where D Cost of capital = r assets = r debt V + r E equity V, D = Market Value of Debt, E = Market Value of Equity, V = D + E = Total Firm Value, r debt = r D = YTM on bonds, r equity = r E = r f + β(r m r f ).
Company Cost of Capital D After tax WACC = (1 T c )r D V + r E E V, where T c the marginal corporate tax rate.
Estimating Beta First estimate a raw beta using regression, then improve the estimates by using industry comparables and smoothing techniques: The measurement period for raw regression should include at least 60 data points (e.g., five years of monthly returns). Rolling betas should be graphed to search for any patterns or systematic changes in the stock s risk. Raw regression should be based on monthly returns. Using more frequent return periods, such as weekly or daily returns, leads to systematic biases. Company stock returns should be regressed against a value weighted, well diversified market portfolio, bearing in mind this portfolio s value may be distorted if measured during a market bubble.
Measuring Betas
Company Cost of Capital Company Cost of Capital is based on the average beta of the assets. The average beta of the assets is based on the % of funds in each asset. Assets = Debt + Equity β assets = β debt D V + β equity E V
Allowing for Possible Bad Outcomes Example: Project Z will produce just one cash flow, forecasted at $1 million at year 1. It is regarded as average risk, suitable for discounting at a 10% company cost of capital: PV = C 1 1 + r = $1, 000, 000 1.1 = $909, 100.
Allowing for Possible Bad Outcomes Example continued: But now you discover that the company s engineers are behind schedule in developing the technology required for the project. They are confident it will work, but they admit to a small chance that it will not. You still see the most likely outcome as $1 million, but you also see some chance that project Z will generate zero cash flow next year.
Allowing for Possible Bad Outcomes Example continued: This might describe the initial prospects of project Z. But if technological uncertainty introduces a 10% chance of a zero cash flow, the unbiased forecast could drop to $900,000. PV = $900, 000 1.1 = $818, 000.
Risk, DCF and CEQ PV = C t (1 + r) t = CEQ t (1 + r f ) t
Risk, DCF and CEQ Example: Project A is expected to produce CF = $100 million for each of three years. Given a risk free rate of 6%, a market premium of 8%, and beta of 0.75, what is the PV of the project? r = r f + β(r m r f ) = 6 + 0.75 8 = 12%
Risk, DCF and CEQ Example continued: Now assume that the cash flows change, but are RISK FREE. Since the 94.6 is risk free, we call it a Certainty Equivalent of the 100.
Risk, DCF and CEQ Example continued: The difference between the 100 and the certainty equivalent, 94.6, is 5.4% This % can be considered the annual premium on a risky cash flow. risky cash flow = certainty equivalent cash flow 1.054 $100 Year 1 : 1.054 = $94.6 Year 2 : Year 3 : $100 1.054 2 = $89.6 $100 1.054 3 = $84.8
Risk and the Cost of Capital: Problem 24, Chapter 9 of BMA Textbook An oil company executive is considering investing $10 million in one or both of two wells: well 1 is expected to produce oil worth $3 million a year for 10 years; well 2 is expected to produce $2 million for 15 years. These are real (inflation adjusted) cash flows. The beta for producing wells is 0.9. The market risk premium is 8%, the nominal risk free interest rate is 6%, and expected inflation is 4%. The two wells are intended to develop a previously developed oil field. Unfortunately, there is still a 20% chance of a dry hole in each case. A dry hole means zero cash flows and a complete loss of the $10 million investment. (Ignore taxes and make further assumptions as needed.)
Risk and the Cost of Capital: Problem 24, Chapter 9 of BMA Textbook (a) What is the correct discount rate for cash flows from the developed wells? Since the risk of a dry hole is unlikely to be market related, we can use the same discount rate as for producing wells. Thus, using the Security Market Line: We know that: r nominal = 0.06 + 0.9 0.08 = 0.132 or 13.2%. 1 + r nominal = (1 + r real ) (1 + r inflation ). Therefore: r real = 1.132 1.04 1 = 0.0885 or 8.85%.
Risk and the Cost of Capital: Problem 24, Chapter 9 of BMA Textbook (b) The oil company executive proposes to add 20 percentage points to the real discount rate to offset the risk of a dry hole. Calculate the NPV of each well with this adjusted discount rate. $3 million NPV 1 = $10 million + 0.2885 ( 1 ) 1 1.2885 10 = $425, 800 ( ) $2 million 1 NPV 2 = $10 million + 1 0.2885 1.2885 15 = $3, 222, 300
Risk and the Cost of Capital: Problem 24, Chapter 9 of BMA Textbook (c) What do you say the NPVs of the two wells are? Expected income from Well 1: 0.2 0 + 0.8 3 million = $2.4 million. Expected income from Well 2: 0.2 0 + 0.8 2 million = $1.6 million. Discounting at 8.85 percent gives: ( ) $2.4 million 1 NPV 1 = $10 million + 1 0.0885 1.0885 10 = $5, 504, 600 ( ) $1.6 million 1 NPV 2 = $10 million + 1 0.0885 1.0885 15 = $3, 012, 100
Risk and the Cost of Capital: Problem 24, Chapter 9 of BMA Textbook (d) Is there any single fudge factor that could be added to the discount rate for developed wells that would yield the correct NPV for both wells? Explain. For Well 1, one can certainly find a discount rate (and hence a fudge factor ) that, when applied to cash flows of $3 million per year for 10 years, will yield the correct NPV of $5,504,600. Similarly, for Well 2, one can find the appropriate discount rate. However, these two fudge factors will be different. Specifically, Well 2 will have a smaller fudge factor because its cash flows are more distant. With more distant cash flows, a smaller addition to the discount rate has a larger impact on present value.
Understading Options: Topics Covered Calls, Puts and Shares Financial Alchemy with Options What Determines Option Values?
Option Terminology Call option right to buy an asset at a specified exercise price on or before the exercise date. Put option right to sell an asset at a specified exercise price on or before the exercise date. Long Short Call option Right to buy asset Obligation to sell asset Put option Right to sell asset Obligation to buy asset
Terminology Derivatives any financial instrument that is derived from another (e.g. options, warrants, futures, swaps, etc.). Option premium the price paid for the option, above the price of the underlying security. Exercise price (strike price) the price at which you buy or sell the security. Intrinsic value difference between the strike price and the stock price. Time premium value of option above the intrinsic value. Expiration date the last date on which the option can be exercised.
Terminology American option can be exercised at any time prior to and including the expiration date. European option can be exercised only on the expiration date. All options usually act like European options because you make more money if you sell the option before expiration (vs. exercising it).
Option Value The value of an option at expiration is a function of the stock price and the exercise price: Value of call option at expiration = = max{market price of the share exercise price, 0}, Value of put option at expiration = = max{exercise price market price of the share, 0}. Example: Option values given an exercise price of $80. Stock Price $60 $70 $80 $90 $100 $110 Call Value 0 0 0 $10 $20 $30 Put Value $20 $10 0 0 0 0
Option Value Call option value (to buyer) given a $430 exercise price.
Option Value Put option value (to buyer) given a $430 exercise price.
Option Value Call option payoff (to seller) given a $430 exercise price.
Option Value Put option payoff (to seller) given a $430 exercise price.
Option Value Call buyer profit: assume strike of $430 and option price of $54.35.
Option Value Put seller profit: assume strike of $430 and option price of $48.55.
Financial Alchemy Masochists Strategy you lose if the stock price falls but do not gain if it rises: long stock and short call.
Financial Alchemy Protective Put you gain if the stock price rises but are protected on the downside: long stock and long put.
Financial Alchemy Straddle strategy for profiting from high volatility: long call and long put.
Financial Alchemy The solid black line shows the payoff from buying a call with an exercise price of $120. The dotted line shows the sale of a call with an exercise price of $160. The combined purchase and sale is shown by the colored line.
Put Call Parity A fundamental relationship for European options: Value of call + present value of exercise price = = value of put + share price. This relationship holds because the payoff of Buy call, invest present value of exercise price in safe asset is identical to the payoff from Buy put, buy share. This basic relationship among share price, call and put values, and the present value of the exercise price is called put call parity. If the stock makes a dividend payment before the final exercise date, you need to recognize that the investor who buys the call misses out on this dividend. In this case the relationship is Value of call + present value of exercise price = = value of put + share price present value of dividend.
Option Value The value of an option is bound, on the high end, by the value of the underlying stock. The lower bound is the value of exercising the option. In between, the major determinants are exercise price and stock price.
Option Value Point A: When the stock is worthless, the option is worthless. The value of an option increases as stock price increases, if the exercise price is held constant. Point B: When the stock price becomes large, the option price approaches the stock price less the present value of the exercise price. The value of an option increases with both the rate of interest and the time to maturity. Point C: The option price always exceeds its minimum value (except when stock price is zero). Thus the value of an option increases with both the volatility of the share price and the time to maturity.
Option Value The greater the distribution of possible outcomes, relative to the final price of the stock, the higher the value of the option. This is due to the greater potential for profit. Thus, Y will have a higher option price, ceribus paribus.
Time Decay Chart Option prices decline when the time to expiration declines (all other things equal).
What The Price of A Call Option Depends on: Sum Up
Option Value Which package of executive stock options would you choose? The package offered by Digital Organics is more valuable, because the volatility of that company s stock is higher.
Valuing Options: Topics Covered Simple Option Valuation Model A Binomial Model for Valuing Options Black Scholes Formula Black Scholes in Action Option Values at a Glance The Option Menagerie
Option Valuation Methods: Replicating Portfolio Google call options have an exercise price of $430 and the current stock price is also $430. There are only two states of the world in the future: Case 1 Case 2 Stock price falls to $322.5 Stock price rises to $573.33 Option value = $0 Option value = $143.33 Assume you buy 4 /7 of a Google share and borrow $181.57 from the bank (at 1.5%). Value of Call = value of 4 /7 shares $181.57 bank loan = = $430 4/7 $181.57 = $64.14.
Option Valuation Methods The number of shares needed to replicate one call is called the hedge ratio or option delta. Since the Google call option is equal to a leveraged position in 4 /7 shares, the option delta can be computed as follows: Delta of call option = spread of possible option prices spread of possible share prices = Delta of put option = = 143.33 0 573.33 322.5 = 4 7. spread of possible option prices spread of possible stock prices. The delta of a put option is always equal to the delta of a call option with the same exercise price minus one. The delta of a put option is always negative; that is, you need to sell delta shares of stock to replicate the put.
Option Valuation Methods: Alternative If we are risk neutral, the expected return on Google call options is 1.5%. We also know that Google stock can either rise by 33.3% to $573.33 or fall by 25% to $322.5. Accordingly, we can determine the probability of a rise in the stock price as follows: Expected Retrun = probability of rise 33.33 + + (1 probability of rise) ( 25) = 1.5 Probability of rise = 0.4543 or 45.43% The general formula for calculating the risk neutral probability of a rise in value is: p = interest rate downside change upside change downside change.
Option Valuation Methods: Alternative The Google option can then be valued based on the following method: Option value = probability of rise 143.33+(1 probability of rise) 0 = = 0.4543 143.33 + 0.5457 0 = $65.11 This is in 6 months, PV is 65.11 /1.015 = $64.14.
Binomial Model The price of an option, using the Binomial method, is significantly impacted by the time intervals selected. The Google example illustrates this fact.
How to Value An Option? Two ways to calculate the value of an option: 1. Find the combination of stock and loan that replicates an investment in the option. Since the two strategies give identical payoffs in the future, they must sell for the same price today. 2. Pretend that investors do not care about risk, so that the expected return on the stock is equal to the interest rate. Calculate the expected future value of the option in this hypothetical risk neutral world and discount it at the risk free interest rate.
Black Scholes Option Pricing Model where Value of call option = delta share price bank loan = = N(d 1 ) P N(d 2 ) PV (EX), d 1 = log (P /PV (EX)) σ t d 2 = d 1 σ t, + σ t 2, N(d) = cumulative normal probability density function, EX = exercise price of option; PV (EX) is calculated by discounting at the risk free interest rate r f, t = number of periods to exercise date (as % of year), P = price of stock now, σ = standard deviation per period of (continuously compounded) rate of return on stock.
Black Scholes Option Pricing Model Example: Google What is the price of a call option given the following? P = $430, EX = $430, r f = 3% per annum, or 1.5% for six months, σ = 0.4068, t = 180 days = 0.5 year. d 1 = log (P /PV (EX)) σ t + σ t 2 = = log (430 /(430/1.015)) 0.4068 + 0.4068 0.5 = 0.1956 N(d 1 ) = 0.5775. 0.5 2 d 2 = d 1 σ t = 0.1956 0.4068 0.5 = 0.0921 N(d 2 ) = 0.4633. Value of call option = N(d 1 ) P N(d 2 ) PV (EX) = = 0.5775 430 0.4633 430 1.015 = $52.05.