econ 659: real options and investment i. 1.traditional ...minsley/ec659f13/i. introduction to real...

Econ 659, I Introduction

Econ 659: Real options and investment I. Introduction

1. Traditional investment theory versus the options approach - traditional approach: determine whether the expected net present

value exceeds zero, ENPV>0

- This rule is the basis for neoclassical investment theory


o Two equivalent approaches of neoclassical investment theory

(i) Compare the MP of capital with an equivalent per period rental cost or user cost that can be computed from the purchase price of capital, interest, depreciation rates, and applicable taxes

(ii) Compare the expected present value of future revenue

with the PV of investment cost

This second approach can be put in the context of Tobin (1969) – compare the capitalized value of marginal investment to the purchase cost

The capitalized value can be found directly if the ownership of the investment can be traded in a market; otherwise the


value can be imputed from the expected present value of the future stream of profits

Tobin’s q – ratio of capitalized value to purchase price -

should undertake the investment if Tobin’s q exceeds 1.


- The traditional ENPV approach implicitly assumes o an investment is reversible in that expenditures can be

recovered if the market turns out to be worse than expected, or o if the expenditures are irreversible, it cannot be delayed – if not

undertaken now it cannot be undertaken in the future

- Many investments can be delayed, and this can affect investment decision

- There are also other options available in many investment projects which are often ignored by traditional analysis – management has the flexibility to expand, contract, shut down etc – as time unfolds

- Ex. Decision to drill in an Arctic wildlife refuge o We are considering whether to develop oil reserves in an area

that has previously been protected because of sensitive herds of caribou and other wildlife.


o There is a non-zero probability that great damage will be done

to the wildlife.

o Suppose the ENPV>0 – i.e. (Expected PV oil revenues – costs of production – Expected PV of damages to caribou herd) >0

o Does that mean you should begin drilling right away? What are some reasons you might want to delay? Get more info about caribou through a study or just

observing information over time – could avoid the worst case scenario completely

Wait to see trends in oil demand and development of energy substitutes.


o Suppose over the next five years time you can do a study that will tell you whether the caribou herd will be harmed or not. We have an option to delay development to wait for more information

o Your ENPV might be higher if you delay your decision and undertake this study

o If you were correctly valuing this investment you would take

into account the fact that the investment can be delayed, or even started and halted part way through


- Extending the example: The option to shut down the investment

midstream

o Consider again our Arctic drilling example, and suppose the ENPV (naively calculated) is negative.

o Suppose the ENPV of oil development (not including

environmental damage) is $200 million o You estimate the expected value of damages to the caribou as

follows. Note that all amounts are present values.


EV(damages) = 0.05(-5000 million)+0.5(-3 million) = -$251.5 million ENPV of oil development including environmental damage= $200 million - $251.5 million = -$51.5 million Conclusion: Do not undertake the project

5 %

50 %

45 %

-$5 billion

-$3 million

0


Now suppose you recognize that if the worst case scenario appears to be happening to the caribou, you could suspend development in time for the herd to recover. A correct analysis of the project would take this into account. Suppose if your biologists notice that the herd is being damaged and you suspend development the damages will only be $500 million instead of $5 billion. To correctly calculate the option to suspend production you create the following decision tree for caribou damage:


o The EV of damages is now: 0.05 Max[-5 bil, -500 mil]+ 0.5(-$3 million)=-$26.5 million

- We also need to recalculate the ENPV of the oil production because of the possibility that production will be shut down.

5 %

50 %

45 %

-$5 billion

-$3 million

0

-$500 million

continue

suspend


- Suppose if production is shut down the ENPV from the oil development alone will be -$10 million (an assumption)

- ENPV of oil production = 0.95($200 million) + 0.05 (-10) = $189.5 million

- ENPV of oil development including EV of damages is: $189.5 million - $26.5 million = $163.0 million.

- Conclusion: We should go ahead with the project

o The simplest ENPV approach ignores these issues o The real options approach focuses on correctly valuing all

important embedded options in a project – like the option to delay or the option to shut down.


- A firm with an investment opportunity is holding an option like a financial call option

- If you treat an investment opportunity like a call option, then you explicitly deal with the optimal timing of investment and optimal decisions regarding other embedded options.


2. Introduction to Financial options A derivative is a financial instrument whose value depends on the value of other underlying variables – usually other traded assets. Futures and forward contracts and options are all derivatives. Also credit derivatives, electricity derivatives, weather derivatives, insurance derivatives Traded on derivatives exchanges or over-the-counter market See Hull ch 1 for a good introduction to derivatives


2.1 Call options

A financial call option: gives you the right, but not the obligation, to buy a financial asset at some future date.

o Example: An investor buys European call option to purchase 100 shares of a particular stock. strike price (or exercise price) = $100 current price of the stock is $98 options expires in 4 months price of an option to purchase one share is $5 The investor spends $5 X 100 on options = $500

A European option can only be exercised at the expiry date. (An American option can be exercised at any time up to the expiry date.)


o If in four month’s time the price of the stock is less than $100, the investor will not exercise the option and loses $500 (initial investment)

o If in four months time the stock price exceeds $100, he will exercise the option.

o Suppose stock price in 4 months time is $120. By exercising the option investor can buy the stock at $100 and sell immediately for a gain of $20 per share = $2000.

o Deducting initial investment of $500, the net profit is $1500.

o Draw a diagram showing the profit from buying this call option versus the stock price at the expiry of the option. (the hockey stick diagram)


o Note that holder of a call option benefits if the stock price rises

o Profit per share = max[stock price – exercise price, 0 ] – option price


2.2 Put options o Purchaser of a put option benefits if the price of the asset falls

o A put option gives the right but not the obligation to sell an

asset at a given price at a date specified by the option contract.

o Consider an investor who buys a European put option to sell 100 shares of a particular stock for $70 per share (strike price or exercise price). The current price of the stock is $65 and the option expires in three months time.

o Price of an option to sell one share is $7. o Initial investment:

- Profit per share = max[E-S,0] – op price


o Suppose the price in three months time is $50. o The investor’s gain is: o Draw a diagram showing the profit from buying this European

put option versus the stock price at the expiry date.


2.3 Valuing financial options For derivatives that are traded in markets, we can observe their market price For new types of derivatives, analysts will need to estimate their fair market value. Ex. An analyst may want to determine whether a new derivative offered by an insurance company is being priced fairly. There is a very large literature on the pricing of financial derivatives – can get very complex- due to the proliferation of exotic options – as opposed to vanilla options - Bermudan options, chooser options, knock-out options


2.4 A firm’s opportunity to acquire a real asset There is an analogy between a firm’s investment opportunity and a call option.

A firm with an investment opportunity has the right but not the obligation to buy an asset at some future time of its choosing.

- Consider the Arctic drilling example again:


Analogy between financial and real options Financial option Real option Call option on a stock Opportunity to develop an oil field Exercise price Cost of the investment – drilling the

wells etc Option price – sometimes observable in the market – other times must be calculated

Value of the right to drill and extract oil from the reservoir – what you would need to pay to acquire that right – sometimes observable in the market, other times must be calculated

Expiration date of option May be infinite, or have a fixed time American or European types

Most real options are American-type options

Option price depends on the price of the underlying asset – the price of the stock

Value of the investment opportunity depends on the price of underlying uncertain variables – the price of oil, the caribou herds


Irreversibility: Once you have exercised the option, you don’t have the right to reverse your decision.

Irreversibility: Once you begin spending on the oil field a certain portion of your investment cannot be recovered – a sunk cost.

For American style options (both financial and real) we are interested in finding the optimal time to exercise the option – a critical threshold of the underlying asset at which it is optimal to exercise the option.

Makes American style options more complex than European style.


When we exercise an option, we lose opportunity to wait for a time when the investment might be more profitable – we “kill the option”. This lost option value should be included as an investment cost when determining the optimal time to invest. Modify the traditional investment rule –

- From o Invest if the expected present value of the net revenue from

capital ≥ purchase cost

- To o Invest if the expected present value of revenue from capital ≥

purchase cost + value of keeping investment option alive

- The value of keeping the option alive may also be considered the opportunity cost of exercising the option.


- Investment decisions can be very wrong if this opportunity cost is ignored

- Optimal investment rules for irreversible investment under

uncertainty can be obtained using methods that have been developed for pricing options in financial markets – contingent claims approach

- Dixit and Pindyck also present an equivalent approach – dynamic

programming – an approach from the theory of optimal sequential decisions under uncertainty

- The problem with the dynamic programming approach is an

inadequate treatment of the discount rate for problems involving embedded options and a finite time horizon.


- In this course we will focus on the methodologies from finance, but we will also discuss the dynamic programming approach and why it is problematic


3. A two period example of valuing an investment under uncertainty This example is a variation on one in Dixit and Pindyck, chapter 2. A firm is considering whether to invest in a widget factory. An irreversible investment Investment can be built instantly for a cost of I=$1600 and will produce one widget per year forever with zero operating costs. Price of widget today, P0=$200. Price next year and forever after:

P1= $340 with probability q = 0.5 P1= $100 with probability (1-q) = 0.5


Discount rate = 10% (Assume for the moment that risk in investing in widgets is fully diversifiable – i.e. it can be completely eliminated in a diversified portfolio. Hence we are able to use the risk-free interest rate, which we assume is 10%.) Should the firm invest now or would it be better to wait until next year to see whether the price of widgets goes up or down? What is the value of this investment opportunity? There are two ways to answer these questions. In the calculations we assume revenues and costs occur at the beginning of each period.


3.1 Method 1: Compare the ENPV of investing immediately, versus waiting until next year. Show that the ENPV of investing immediately is $800 while the ENPV of investing a year from now is $973. EPNV(of immediate investment) =

1 $801.1 1 220

1600 200 ( ) 1600 2000.1 1.

01 0.1

E P


What if our choice was to invest today or never? How much is it worth to have the flexibility to make the investment decision next year, rather than having to invest either now or never? Answer: $173 How high would the cost of the flexible investment have to be to make us indifferent between the flexible versus the now-or-never investment? Answer: $1980


3.2 Method 2: Using option pricing methods – creating a hedging portfolio 3.2.1 An aside on short sales Short selling or shorting an asset: selling an asset that is not owned. Short selling is possible for some, but not all, investment assets. Eg. (see Hull page 100) An investor tells his broker to short 500 shares of RBC. The broker borrows these shares from another client and sells them in the market and gives the revenue to the investor. The investor can maintain the short position as long as the broker has shares that he can borrow. The investor must pay the broker any dividends or income that accrue to the shares during the time the shares are shorted. When the investor wants to close out his position he will purchase 500 shares of RBC and return them to the broker. The investor profits if the stock has fallen in price in the meantime.


Suppose the original price of the stock is $200 and when the short position is closed out the price has fallen to $150. 500 shares are shorted in April and the position is closed out in July.

Cash flow from short sale of shares: April Borrow 500 shares and sell at $200 per share $100,000 May Pay dividend, $2 per share -$1000 July Buy 500 shares for $150 per share -$75,000

Net profit $24,000


The cash flows from a short sale are the mirror image of the cash flow from purchasing the shares in April and selling in July. Cash flow from purchase of shares: April Purchase 500 shares at $200 per share -$100,000 May Receive dividend, $2 per share $1000 July Sell 500 shares for $150 per share $75,000 Net profit -$24,000


3.2.2 Creating a hedging portfolio In finance, options are typically valued by creating a portfolio of the option and a sufficient number of another related asset so that the risk from holding the option is eliminated – i.e. the portfolio is riskfree By setting the portfolio’s return equal to the riskfee rate we can back out the value of the option. We can use this same approach for valuing an investment opportunity.


Let: F0 : the value today of the investment opportunity – what we would be willing to pay for the opportunity to build a widget factory – This is what we want to determine. F1 : the value of the investment opportunity next year, a random variable which depends on the price of widgets

If P1=$340, 1

(1 )max 340 1600,0 $2140

rF

r

If P1=$100, 1

(1 )max 100 1600,0 0

rF

r

The value of the investment opportunity depends on the price of widgets. We can eliminate the risk of our investment opportunity by shorting an appropriate number of widgets


We create a portfolio – hold the investment opportunity and sell short n widgets. We choose the number of widgets so that the portfolio is riskfree – and therefore must earn a return equal to the riskfree rate.


The value of this portfolio today (Φ0) is 0 0F nP .

The value of this portfolio next year is:

1 1 1

1

1 1 1

2140 340

0 100

F nP n

F nP n

(1.1)

We want to choose n (the number of widgets to sell short) so that the value of the portfolio next year is the same whether the price of widgets rises or falls. Show that n=8.9167 If price rises our portfolio in period 1 is worth:


If price falls our portfolio in period 1 is worth: What is the return from holding this portfolio? The return will be capital gains less any payments needed to hold the short position. Capital gain is: 1 0 0891.67 ( 8.9167 200)F

Payments to hold the short position: assumed to be zero (contrary to the Dixit and Pindyck example) Return from holding the portfolio over the year is therefore.


1 0 0891.67 F (1.2)

The return must equal the riskfree rate. So we can calculate F0 as follows.

0 0891.67r F

Substituting for 0 and solving for F0 we get F0=$972.7.


The value of the investment opportunity calculated in this manner is the same as using the previous approach. Using this latter approach how do we decide whether or not to exercise the option to invest? The payoff from investing immediately is $800, found earlier. But if we invest immediately we “kill the option” and lose a value of $973. Hence it is better to delay.


Another way to look at this: The full cost of investing today is the $1600 capital cost plus the $973 lost value of the option to invest next period. Full cost = $2573 We invest today only if the PV of revenues from investing today exceeds the full cost of investment. Expected PV revenues = $200+$220/.1=$2400. Since $2400<$2573, we delay our investment. The rule is to invest immediately if 0 0V I F where V0 refers to the

present value of revenues from widgets. Note that this approach requires that we can trade widgets – hold long or short positions. If this is not possible, we could look for


another asset, or combination of assets, the price of which is perfectly correlated with widgets. Alternately we could use Method 1 if all price risk is diversifiable and we can use the riskfree rate. If this does not hold, then we have to find a way to determine an appropriate risk premium. This will be discussed further in future classes.


3.2.3 Hedging strategy for previous example At beginning of year:

- sell 8.9166 widgets short and receive 8.9166X$200=$1783.32 - use proceeds to purchase the investment opportunity for $972.72 (=F0) - this leaves $1783-$973 = $810, which can be put in a riskfree bank account and earn 10%

At end of the year: - How much do we have in the bank? $810 X (1.10)=$891 - If 1 100P our investment opportunity ( 1F ) is worth zero.

- To pay back the short position requires 100 X 8.9166 = $891.66, which is what we have in our bank account


- If 1 340P , 1 $2140F

- To close out the short position we have to buy shares for a total cost of 340 X 8.9166 = $3031.7 - This can be financed by selling our investment opportunity for $2140 and using the bank account of $892 for a total of $3031. - Hence whether the price of widgets rises or falls, we end up with zero dollars. - Since I put up zero dollars of my own money initially, I must just break even in the end. Otherwise there is an arbitrage opportunity.


3.2.4 An aside on the no-arbitrage assumption and risk

The no-arbitrage assumption

A key assumption in economics and finance.

An arbitrage opportunity refers to an opportunity to make an instantaneous riskless profit.

The assumption of no arbitrage opportunities means that there is “no free lunch”.

More correctly, it is assumed that arbitrage opportunities cannot persist for long as changes in prices will eliminate them.


Finance theory assumes the existence of risk-free investments that give a guaranteed return with no chance of default. One approximation would be a Canadian government short term bond or a bank deposit in a sound bank.

Under the no-arbitrage assumption the greatest risk-free return you could make on a portfolio of assets is the riskless return one would get on a bank deposit.

To see why, suppose you can purchase a portfolio of assets for $V and that these assets earn a guaranteed return in the form of a dividend, $D per year. Suppose that the dividend yield (D/V) is 10% while the risk-free rate on a bank account is only 5%.


No one would hold the bank account and everyone would try to buy the portfolio of assets. This would push up the price of the portfolio. V would rise until the return on V equalled the risk-free rate of 5%.

Anyone who wants a return greater than the risk-free rate must accept some risk.

An implication is that if a portfolio requires no investment and is riskless, then its terminal value must be zero. We demonstrated this in the previous example of buying an asset using the proceeds of the short sale of widgets.


A note on risk

Risk is commonly classified as either specific or non-specific. (Non-specific risk is also sometimes called systematic risk, or market risk.)

Specific risk is due to the specific characteristics of a particular asset. For example, if a company operates in an area prone to earth quakes, its stock is subject to that specific risk.

A non-specific risk refers to risks that affect all assets, such as a change in interest rates or inflation.

It is possible to diversify away specific risk by having a portfolio of assets from different sectors of the economy. A diversified portfolio implies that your wealth is not significantly affected by the specific risk faced by any individual asset.


In theory, an individual will only be rewarded with extra return in the market by taking on non-specific risk.

One popular definition of risk is the variance of return. There are other definitions as well.


3.2.5 Impact of changing investment cost, I Recall our equilibrium condition that the return to our portfolio must equal the riskfree rate.

1 0 0r

State F0 as a function of I for our example. Sketch a graph of F0 and the value of investing right away, V0-I, versus I. Show the critical I* at which one is indifferent between investing now or delaying for a year. Note: I*=1283, given V0=2400-I


3.2.6 Changing the initial price P0 Specify F0 as a function of P0. Sketch a graph of F0 and V0-I as functions of P0. Observe that there are three regions in this graph.

*

0 0

* **

0 0 0

**

0 0

Never invest

Delay to period 1 and invest if price goes up.

Invest immediately

P P

P P P

P P

Note we can write the expected value of net revenues from investing immediately in terms of P0 as follows: V0=12P0

The upper and lower critical values are: P*=$85.56 and P**=$249.35


3.2.7 Increasing uncertainty over price Suppose we keep P0 fixed but increase the size of upward and downward movements in period 1. How is the value of F0 changed? What about the critical price P* above which you will delay investing until period 1? Try an example on your own.


4. Extending the example to more periods We could extend the previous example to three periods so that the price could rise or fall in period 1 as well as period 2, and then remain constant thereafter. This would increase the complexity of the problem in at least two ways.

(i) The number of investment strategies to choose from depending on P0 will increase to 5:


T=0 T=1 T=2 P0 P1+ P1- P2+ P2-

Never invest - - - - Delay delay never invest - Delay invest never continue - Delay invest delay invest or continue don’t invest Invest continue continue continue continue

(ii) Also the number of widgets to sell short would need to change in period 2 when the price of widgets changes. A dynamic hedging strategy is required. If the price of widgets is assumed to be able to change to any value and is described as a continuous time stochastic process, then F0 will be a smooth, continuous function of P0.


5. Other common types of real options The ability to delay investment is the most common type of option considered in the literature. Other important options (See Table 1.1 of Trigeorgis, pages 2-3, for references.) (i) option to alter operating scale – expand or contract; shut down and

restart (ii) time to build options (staged investment) (iii) option to abandon (iv) option to switch outputs or inputs – eg switch energy source of power

plant (v) growth options – an early investment is a prerequisite for future growth

opportunities – eg R&D or industries with multiple product generations

(vi) multiple interacting options – collection of various options – combined value may differ from the sum of separate values

econ 659: real options and investment i. 1.traditional ...minsley/ec659f13/i. introduction to real...

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