Real Options Approach to Capital Budgeting and capital structure

1
Real Options Approach to Capital Budgeting (and
capital structure)

• Traditional NPV does not take into account
  potential flexibility (i.e. possibility to react
  to changing conditions) hence, systematically
  underestimates project values
• Decision Trees approach accounts for flexibility
  but does not answer at what rate to discount cash
  flows
• ROA the way to properly account for managerial
  flexibility. Relying on the arbitrage argument
  (replicating portfolio) it properly finds
  Certainty Equivalents of future cash flows (or
  values) that are to be discounted at the riskless
  rate.

2
Types of real options

• Options of call type
• Increase payoff in the good state of nature
• Normally require large investment
• Option to defer investment
• Option to expand
• Option to enter (a new market)
• Options of put type
• Decrease loss in the bad state of nature
• Often require selling the asset or part of it for
  salvage value
• Option to contract
• Option to abandon for salvage value

3

• Compound options (options on options)
• Call option on the equity of a levered firm
  (simultaneous compound option)
• Phased investments (sequential compound options)
• RD project
• New product development
• Exploration and production

4
A bit more details

• ?????? ?? ??????? / ???????????? ??????????
  (option to defer)
• ?????? ????? ????????? ??????? ??????, ?? ???
  ??????? ??????? ??????
• ????????????? ???????, ??????? ?? ?????????
  ??????????? ??????????, ???????? ?? ??????????
  ???????
• ?????????? ??????, ?? ??????????? ??
  ????????????????, ?? ???????? ?????????????
  ??????? ??????
• ?????? ?? ?????????? ???????????? (option to
  expand)
• ??????????? ??????? ???????, ????? ?????
  ???????????? upside (?.?. ??? ?????????????
  ???????????)
• ???????? ?????????? ??? ???? ?????????? ???????
• ????????????? ???????, ??????? ?? ?????????
  status quo, ??????????

5

• ?????? ?? ?????????? ??????? (option to abandon
  for salvage value)
• ??????????? ??????, ????? ?????????? ????, ??? ??
  ?? ???????? ???????
• ????? ???????????? ???????? ?? ???????? ?
  ????????? ?????? ?? ?????????? ?????????
• ????????????? ???????, ??????? ?? ?????????
  ???????????, ??????????? ????????????
• ?????? ?? ?????????? ???????????? (option to
  contract)
• ????????? ??????? ???????, ????? ??????????????
  ??????
• ????? ??? ??????? ? ?????????? ??????????????
  ??????????
• ????????????? ???????, ??????? ?? ?????????
  status quo, ??????????

6
??? ???? ???????? ????????

• ????? ?????!
• ????????, ??????? ??????????????
• ????????????
• ????????????????
• ??????????????
• ????????? ????????
• ...
• ...

7
Option to expand
??????????? ???????? (?????? ????? ???, ????????
????????????, ????? ??? ???? ????? ? ?.?.)
8

• ? ??????? ????? ?? ?????? ? ??????? ?????
  ???????? ?? ?????
• ?? ????? ???????? ??????????? ? ???????????? 20
• ?????????? ???? ???????? ???????????
  ????????????? ?? ???? ??????
• ??????? ????? ?????????
• ????? ???????? ????????

9
Analogy between real and financial options

• Underlying asset the project (PV of future cash
  flows)
• Exercise price investment cost or
• Time to expiration time until opportunity
  disappears

10
Approaches to valuing real options

• Analytical
• Binomial model
• Black-Scholes formula
• Simulations

11
Binomial model
Vu2V
q
VuV
q

1-q
V-udV
V
q
1-q
V- dV
1-q
V--d2V

• V the gross value of the project (expected
  value of subsequent CF)
• d 1/u
• There exists a twin security that can be
  traded, which price S is perfectly correlated
  with V.
• If there is an option on the project, we use
  replicating portfolio technique (or risk
  neutral probabilities, which is the same) to
  determine its value

12
Previous lectures example (option to defer
investment)
V180, S36
u 1.8 d 0.6 r 8
q0.5
V100, S20I0104
0.5
V-60, S-12
I1112.321041.08

• E NS - (1r)B,
• E- NS- - (1r)B
• ? N (E - E-)/(S - S-), B (NS- E-)/(1r)
• ? the risk-neutral valuation
• E0 NS B (pE (1-p)E-)/(1r)
• where p ((1 r)S S-)/(S - S-)
  risk-neutral probability (in Copeland-Weston-Sha
  stri its the other way round risk neutral prob.
  is denoted q, and the actual one is denoted p)

13

• At the end p depends only on u, d and r
• p ((1 r)S S-)/(S - S-) (1 r d)/(u
  - d) (since S- dS, S uS)
• In fact, p can be found from the following
• S (puS (1-p)dS)/(1r),
• i.e. p must be such that the risk-neutral
  valuation of the twin security yields its
  actual price.
• Thus
• p does not depend on the actual probability of
  going up q. Reason q is already incorporated in
  the price S.
• Given the tree, p does not depend on the
  particular option (in particular on where we are
  in the tree)
• For our tree (u 1.8, d 0.6) and r 8 p
  0.4

14
Example Option to abandon for salvage value or
switch use
Current project. Values of V
Alternative use. Values of V
191
324
127.5
180
102
108
85
100
68
60
54.4
36

• We should switch at such points (If the option
  is to switch any time we want, we switch the
  first time we get to such a node)

15
What is the value of the option to switch in year
1?
E180
E0?
E-68
E0 (pE (1-p)E-)/(1r) - I0 0.44 (we can
use the same probabilities p as before)
If we had no option to switch, the project would
have NPV -4 (previous lecture) Hence, the value
of the option is 4.44
16
Black-Scholes Pricing Formula(no dividend case,
call option)

• C0 the value of a European option at time t
  0
• r the risk-free interest rate
• S the price of the underlying asset (or twin
  security)
• N(.) cumulative standard normal distribution
  function

17
Adjusting for dividends (i.e. if the project
generates cash flows before the option
expiration date)
Assume a constant dividend yield (i.e. constant
cash flow) every year. Then
18
Some caveats the real options approach

• Black-Scholes formula presumes a diffusion Wiener
  process for underlying (twin) security
• Is it always the case?
• Can we always find a twin security? If not,
  people do market asset disclamer assumption
  the project itself is a twin security as if it
  could be traded.

19
Analogy between the Black-Scholes and binomial
models

• At the limit, as the time period length in the
  binomial model goes to zero, the binomial process
  converges to the corresponding Wiener process.
  Thus, the Black-Scholes formula is nothing else
  but a binomial risk-neutral pricing formula (or
  riskless hedge formula) but in continuous time
  (for comparison see e.g. Copeland-Weston, pp. 264
  - 269)
• An example of two techniques yielding close
  results even when a two-period binomial
  approximation is used Copeland-Weston, pp. 269
  273.

20
Example venture project
???????????? ??????
300
8
6
5
7
????????? ?????????? ??????
????????? ?????????? ??????
(1500)
21
Venture project (continued)

• ?????? ??????????????? (????????? ????????) 20
• ????????? ?????? ???????? ? ???????NPV - 56???
• ?? ????????? ????????? ???????????? ??????
  (????????????? ?????? ? ??????? ????????)
• Ex ante (? ??????? ???????), ???????????? ??????
  ???? ? ??????? ???????? NPV -356/1.24 -81
  ???
• ?????? ???????????????? ???????????? ????????
  ??????? ?? ????????????? ??????? ?????????? ?
  ?????? 4, ? ????? ?? ????? ????????? ?????
  ????????
• ????????? ?????? ???????, ??? ??? ???????????
  ??????????? ???????????? ?????? ????? ?????. ???
  ???????? ????

22
Venure project (continued)

• ?????????? ????????? ??????? ?????? ?? ??????????
  ????????????? ???????
• ???????????????? ???????????? ?????????? ???????
  ? ?????? ? 5 ?? 8 ??. ?????????? ????? ??????????
  ?????????? ???????
• ?????? ????? ??????? ? ???????? ????????????,
  ?????? ???? ?? ????? ???????
• ??? ??????????? ?????? (?????????? ????? 4 ????,
  ???? ?????????? 1500 ???, ?? ????? ? ?????????
  ?????? 300, 600, 900, 300)
• ??????????? ????????????? ? 0.35, ???????????
  ?????? ???????? r0.10
• ????? ?? ??????? ?????-?????? ?????? ????? 71
  ???
• ????? ???????, ????????? ?????? ???????
• ?????? 56 ???
• ?? ???? ?????? ?????????? 71 ???
• ????? NPV15 ???

23
Equity as a Call Option on the Firm

• The equity in a firm is a residual claim, i.e.,
  equity holders lay claim to all cash flows left
  over after other financial claim-holders (debt,
  preferred stock etc.) have been satisfied.
• If a firm is liquidated, the same principle
  applies, with equity investors receiving whatever
  is left over in the firm after all outstanding
  debts and other financial claims are paid off.
• The principle of limited liability, however,
  protects equity investors in publicly traded
  firms if the value of the firm is less than the
  value of the outstanding debt, and they cannot
  lose more than their investment in the firm.

24
Equity as a call option

• The payoff to equity investors, on liquidation,
  can therefore be written as
• Payoff to equity on liquidation V - D if V gt
  D
• 0 if V ? D,
• where
• V Value of the firm
• D Face Value of the outstanding debt
• A call option, with a strike price of K, on an
  asset with a current value of S, has the
  following payoffs
• Payoff on exercise S - K if S gt K
• 0 if S ? K

25
Application to valuation A simple example

• Assume that you have a firm whose assets are
  currently valued at 100 million and that the
  standard deviation in this asset value is 40.
• Further, assume that the face value of debt is
  80 million (It is zero coupon debt with 10 years
  left to maturity).
• If the ten-year treasury bond rate is 10,
• how much is the equity worth?
• how much is the debt worth?

26
Model Parameters

• Value of the underlying asset S Value of the
  firm 100 million
• Exercise price K Face Value of outstanding
  debt 80 million
• Life of the option t Life of zero-coupon debt
  10 years
• Variance in the value of the underlying asset
  s2 Variance in firm value 0.16
• Riskless rate r Treasury bond rate
  corresponding to option life 10

27
Valuing Equity as a Call Option

• Based upon these inputs, the Black-Scholes model
  provides the following
• d1 1.5994 N(d1) 0.9451
• d2 0.3345 N(d2) 0.6310
• Value of the call Value of equity
• 100 (0.9451) - 80 exp(0.1010) (0.6310)
  75.94 million
• Value of the outstanding debt 100 - 75.94
  24.06 million