Chap 10

Keeping uncertainties separate

Keeping uncertainties separate

The way to do this is to keep the major uncertainties separate and to model their interaction and effect on the project’s value explicitly.

If the management have the flexibility to make this investment, it holds an option called a learning option.

Learning options – uncorrelated uncertainties

Two sources of uncertainty – product/market and technological.

Technological uncertainty is assumed to be independent of market conditions and to be diffuse now, but reduced through time by doing research.

Compound option with technological uncertainty

The Pharma Company is considering investment in a research and development project that has a basic research phase that costs $3 million and, based on experience, has only a 20 percent chance of succeeding into the development phase.

This second phase costs $60 million and has only a 15 percent chance of realizing a great product whose present value will be $600 million.

It also has a 25 percent chance of developing a mediocre product with a $40 million present value.

But there is a 60 percent chance of having no marketable product.

To go to market, a final investment of $40 million is required to build a factory.

The level perpetual cash flows start at the end of the plant construction phase (year 3) and are discounted at the weighted average cost of capital (10 percent).

The risk-free rate is 5 percent. This will be the correct rate for calculating the

cash flows because, given their independence of the market, their Capital Asset Pricing Model beta is zero.

600 400.15 40 0.25 40 0.60 40

1.05 1.053 0.2 60 (1.05)

1.05

00.8 60 1.05

1.05

NPV

)05.1()60(8.0)05.1(6005.1

23.552.03

3 1.41 45.71 50.12

6000.15 40 0.25(0) 0.60 0

1.05( ) 60

1.05NPV at node B

60 75.92 15.92

15.92( ) 3 0.2 0.8(0) 3 3.03 0.03

1.05NPV at node A

($0.03)

($ 50.12)

= ($50.15)

NPV with flexibility

NPV given precommitment

Value of flexibility

Compound rainbow option with two uncorrelated uncertainties

The second uncertainty is product / market uncertainty.

The expected value can go up or down by 20 percent each year.

0.15(417 40) 0.25(0) 0.60(0)( ) 55.17

1.025NPV at node D

Choosing node F to illustrate, the end-of-period payoffs are $82million in the up state, and $55 million in the down state.

The beginning-of-period value of the underlying is simply the expected technological outcomes from the research phase,

(i.e., 0.15($500)+0.25($33)=$83.25)

0.15($600) 0.25($40) 0.60($0) $100

0.15($417) 0.25($28) 0.60($0) $69.55

(1 ) 82

(1 ) 55

0.887, 6.54

0.887($83.25) $6.54 $67.30

f

f

uV

dV

muV r B

mdV r B

m B

Value mV B

Learning options - the quadranomial approach

Introduction to the quadranomial approach The quadranomial approach is a two-variable binomial t

ree.

1 2

1 2

1 2

1 2

1 2 0

1 2 0

1 2 0

1 2 0

(1 )

(1 )

(1 )

(1 )

f u u

f u d

f d u

f d d

mu u V r B C

mu d V r B C

md u V r B C

md d V r B C

1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2

0 (1 )u u u u u d u d d u d u d d d d

f

p C p C p C p CC

r

1 2 1 2

1 2 1 2

1 2 1 2

1 2 1 2

u u u u

u d u d

d u d u

d d d d

p p p

p p p

p p p

p p p

dut XXX ,

dut YYY , If the two uncertainties are independent, In this case the conditional probabilities for X

and Y are equal to their respective unconditional probabilities.

( )

( )u u uY

u u uX

p Y X p

p X Y p

( ) ( )( )( ) ( )

( ) ( )u u uu u

u uu u

p Y p X Yp Y Xp Y X Bayes Law

p X p X

VdzVdtdV

VdzV

CdtV

V

C

t

CV

V

CdC

222

2

2

1

ln( )C V1C

V V

2

2 2

1C

V V

0t

C

2

2dC dt dz

21

1ln( )2fd P r dt dz

21

1 2fg t r dt

22

2ln( )2fd Q r dt dz

22

2 2fg t r dt

21

1 1 2 1 2 1 1 2 1 2 1( )2f u u u d d u d dE g t r t p p u p p u

212121

212121

21 uppuppt ddudduuu

22

2 1 2 1 2 2 1 2 1 2 2( )2f u u d u u d d dE g t r t p p u p p u

222121

222121

22 uppuppt ddduuduu

21212121212112 )( uuppppt ddduuduu

121212121 ddduuduu pppp

1 2

1 2 2 1 1 2 12 1 2

1 24u u

u u u g t u g t tp

u u

1 2

1 2 2 1 1 2 12 1 2

1 24u d

u u u g t d g t tp

u u

1 2

1 2 2 1 1 2 12 1 2

1 24d u

u u d g t u g t tp

u u

1 2

1 2 2 1 1 2 12 1 2

1 24d d

u u d g t d g t tp

u u

tu 11 tu 22

Examples using the quadranomial approach

The two-period project lasts 6 months and provides cash flow of P×Q in revenues less $4,000 of fixed cash cost.

At the end of the second period, the cash flows become a constant perpetuity with a multiple of six.

The project can be sold to a competitor (i.e., abandoned) for $50,000 at any time.

Finally, the continuously compounded annual risk-free rate is 5 percent.

Currently expected to be 1,000 units per period but its volatility is believed to be 20 percent per year.

0.25 0.20(0.5) 0.10per quarter annual

fr

dQppuQQ

1

)1( 000

5378.09048.010517.1

9048.0125.011

du

drp f

0.32(68,398.9) 0.22(57,498.2) 0.27(50,924.8) 0.19(42,000.0)

1.012555572.3

BPV

We assume that the two uncertainties are correlated, having a positive 30 percent correlation coefficient ( ). 3.012

1.025.02.011 tu

1.011 ud

06.025.012.022 tu

2 2 0.06d u

2 21

1

0.20.05 0.25 0.0075

2 2fg t r dt

2 22

2

0.120.05 0.25 0.0107

2 2fg t r dt

1 2

0.1 0.06 0.06 0.0075 0.1 0.0107 0.3 0.2 0.12 0.250.39

4 0.1 0.06u up

1 2

0.1 0.06 0.06 0.0075 0.1 0.0107 0.3 0.2 0.12 0.250.15

4 0.1 0.06u dp

1 2

0.1 0.06 0.06 0.0075 0.1 0.0107 0.3 0.2 0.12 0.250.20

4 0.1 0.06d up

1 2

0.1 0.06 0.06 0.0075 0.1 0.0107 0.3 0.2 0.12 0.250.26

4 0.1 0.06d dp

66.059.0

39.0

)(

)()(

u

uuuu Pp

PQpPQp

( ) 0.15( ) 0.36

( ) 0.41u d

u dd

p Q Pp Q P

p P

1 2

0.1 0.06 0.06 0.0075 0.1 0.01070.32

4 0.1 0.06u up

1 2

0.1 0.06 0.06 0.0075 0.1 0.01070.22

4 0.1 0.06u dp

1 2

0.1 0.06 0.06 0.0075 0.1 0.01070.27

4 0.1 0.06d up

1 2

0.1 0.06 0.06 0.0075 0.1 0.01070.19

4 0.1 0.06d dp

無相關之機率為 :

We can see that with 30 percent correlation, the quantity is almost twice as likely to move up when the price moves up.

With no correlation the probability of the quantity of the quantity increasing is constant.

54.059.0

32.0

)(

)()(

u

uuuu Pp

PQpPQp

( ) 0.22( ) 0.54

( ) 0.41u d

u dd

p Q Pp Q P

p P

( ) [0.39(6839) 0.15(57498.2) 0.20(50294) 0.26(42000)] (0.0125)

55708.9

PV B

The present value of the project with flexibility and correlated factors is 52499.4, compared with a value of 44777.4 without flexibility – a difference of 7722.

Note that introduction of positive correlation between price and quality has resulted in an increase in the value of the flexibility.