Lattice Modeling for Equity Solutions David C. Dufendach, CPA/ABV
Transcript of Lattice Modeling for Equity Solutions David C. Dufendach, CPA/ABV
Lattice Modeling for Equity Solutions
© 2013 Business Valuation Resources, LLC
© Grant Thornton LLP. All rights reserved.
Lattice Modeling for Equity Solutions
David C. Dufendach, CPA/ABV, ASA
Candice M. Bassell, CPA/ABV/CFF
August 15, 2013
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David C. Dufendach, CPA/ABV, ASA
David C. Dufendach, CPA/ABV, ASA, is a Partner in Grant Thornton LLP's Advisory Services
– Valuations Group. He specializes in the valuation of businesses and business segments,
intellectual property and intangible assets, financial instruments, derivatives, and related
matters for purposes of financial statement reporting, corporate planning, and other purposes.
Dave has published articles in Valuation Strategies and Business Valuation Review. He is an
adjunct at Seattle University's Albers School of Business and Economics and a guest lecturer
in both the University of Washington's MBA program and Law School. Dave is a past
member of the AICPA Business Valuation Committee and currently serves on the AICPA
IPR&D Task Force. He is a member of the American Society of Appraisers, the American
Institute of CPAs, and the Washington Society of CPAs.
Dave holds an MBA from the Wharton School, University of Pennsylvania, and a BA in
Business Administration from the University of Washington.
Lattice Modeling for Equity Solutions
© 2013 Business Valuation Resources, LLC
© Grant Thornton LLP. All rights reserved.
Candice M. Bassell, CPA/ABV/CFF
Candice M. Bassell, CPA/ABV/CFF, is a senior manager in Grant Thornton LLP's Advisory
Services – Valuations Group. She specializes in the valuation of closely held business
interests and financial instruments for purposes of financial statement reporting, litigation
support (marriage dissolutions and shareholder suits), and estate planning and taxation.
Candice serves on the technical advisory board for the AICPA's FVS Consulting Digest. She
is a member of the American Society of Appraisers, the American Institute of CPAs, and the
Washington Society of CPAs.
Candice holds an MBA in finance and a BA in English from the University of Kansas.
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Agenda
• Review of option valuation fundamentals
• Types of lattice models
• How to build a simple equity lattice model
• Equity solutions using lattice models
– Equity valuation
– Equity allocation
– Options on equity
• Advanced applications
Lattice Modeling for Equity Solutions
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Option Characteristics
Stock options have the following features that impact their valuation:
– Contractual features: exercise price, maximum term, and possibly a
market condition
– Market features: stock price (if publicly traded), risk-free interest rate
– Estimated features: stock price (if closely-held), dividends, volatility,
expected term
Highlighted features represent the six inputs that are employed to value
employee stock options (ESOs) granted by publicly-traded companies
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Option Pricing Fundamentals
Conceptually, option pricing is an expected present value
technique:
1. Potential future (maturity date) stock prices are estimated.
2. The value of the option at each potential future stock price is
determined.
3. The maturity-date option value is probability-weighted and
discounted to the present value.
All commonly used option pricing models perform this 3-step
process.
Lattice Modeling for Equity Solutions
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Option Pricing Models
There are three basic types of models. Two are widely used:
1. Closed-form models: the Black-Scholes model and its variants,
such as the Black-Scholes-Merton model.
2. Lattice models: binomial, trinomial and others.
A third type of model (Monte Carlo simulation) is employed for
more exotic options.
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Black-Scholes-Merton Model
• This widely-used closed-form model requires six inputs: exercise
price, stock price, risk-free interest rate, dividends, volatility,
expected term.
• It was developed to value publicly-traded short-term call options.
• Its advantages are ease of use and broad acceptance.
• Its primary disadvantage is its inflexibility; only one amount can
be selected for each input.
Lattice Modeling for Equity Solutions
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Black-Scholes-Merton Example
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Discussion of Inputs
You just saw traditional six inputs
common to all widely used option
pricing models.
The impact on option value of an
increase in each, while the others
are held constant, is summarized
here.
Lattice Modeling for Equity Solutions
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Lattice Models
• There are many variants of the lattice model; two of the most
common are the binomial and trinomial models.
• Unlike the closed-form models, lattice models visually display the
option valuation process.
• The term "lattice" derives from the appearance of the binomial
"tree" used to solve the option’s value.
• The valuation process is best illustrated by separating the model
into two separate trees, as shown in the following slides.
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The "Underlying Asset Tree"
• As mentioned, the first step in valuing a call option is to estimate potential future prices of the underlying stock.
• The range of potential future values depends on three parameters:
1. Today’s stock price
2. The expected future volatility of the stock price
3. The term of the option
• In a binomial model, these factors are modeled in a 'lattice' (the underlying asset tree):
– It calculates the expected evolution of the underlying stock price at various future time periods leading up to the option expiry date.
Lattice Modeling for Equity Solutions
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Asset Tree Example
Period 0 1 2 3 4 5 6 7 8 9 10 11 12
141.31 122.31
105.87 105.87 91.64 91.64
79.33 79.33 79.33 68.66 68.66 68.66
59.44 59.44 59.44 59.44 51.45 51.45 51.45 51.45
44.53 44.53 44.53 44.53 44.53 38.55 38.55 38.55 38.55 38.55
33.37 33.37 33.37 33.37 33.37 33.37 28.88 28.88 28.88 28.88 28.88 28.88
Value of
underlying
equity 25.00 25.00 25.00 25.00 25.00 25.00 25.00
21.64 21.64 21.64 21.64 21.64 21.64 18.73 18.73 18.73 18.73 18.73 18.73
16.21 16.21 16.21 16.21 16.21 14.03 14.03 14.03 14.03 14.03
12.15 12.15 12.15 12.15 10.52 10.52 10.52 10.52
9.10 9.10 9.10 7.88 7.88 7.88
6.82 6.82 5.90 5.90
5.11 4.42
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Maturity Date Values
• At each possible 'end node' on the asset tree (period 12), the option either:
– Has positive value (if the stock price exceeds the contractual exercise price), or
– Expires worthless.
• In our example, all asset tree end nodes that exceed the $25 exercise price are 'in-the-money' and represent values at which the option will be exercised.
• At each 'in-the-money' node, the option will therefore have an intrinsic value = (stock price – exercise price).
Lattice Modeling for Equity Solutions
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The Option Solution Tree
• Once the 'in-the-money' end nodes are identified and valued, a second
'lattice' (the solution tree) is then employed.
• Unlike the asset tree, which shows the evolution in time from left-to-right,
the solution tree operates from right-to-left, performing two operations.
• Each end-value is probability-weighted and then discounted back one
period, i.e., the value in period 11 is calculated based on its two related
period 12 values.
• This weighting and discounting process is repeated in what is referred to
as a 'backward-solving' process until the period 0 (valuation date) is
reached.
• Note the consistency of the binomial result with the Black-Scholes value
calculated earlier.
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Solution Tree Example
Period 0 1 2 3 4 5 6 7 8 9 10 11 12
116.31 97.38
81.00 80.87 66.83 66.71
54.58 54.45 54.33 43.98 43.85 43.73
34.81 34.68 34.56 34.44 26.95 26.76 26.63 26.51
20.36 20.04 19.78 19.66 19.53 14.98 14.54 14.11 13.73 13.61
10.74 10.23 9.69 9.10 8.49 8.37 7.52 7.00 6.42 5.77 4.98 3.94 Value of
Option 5.14 4.66 4.13 3.53 2.81 1.86 0.00 3.03 2.59 2.10 1.54 0.88 0.00
1.59 1.22 0.83 0.41 0.00 0.00 0.70 0.44 0.19 0.00 0.00
0.23 0.09 0.00 0.00 0.00 0.04 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00
0.00 0.00 0.00
0.00
Lattice Modeling for Equity Solutions
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Consistency of the Binomial Result with the Black-
Scholes Value
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Comparison of Models
• As the previous examples show, both the B-S-M and binomial models
produce results that are substantially similar.
– If the number of periods used in the binomial model is increased,
the binomial model’s value converges with the B-S-M if the same
inputs are used in each model.
• The advantage goes to the binomial model if flexibility of inputs is
important;
– i.e., the binomial model can accommodate assumptions that change
over time; the B-S-M cannot.
Lattice Modeling for Equity Solutions
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Types of Lattice Models
• Equity
• Debt/Interest rate
• Binomial, trinomial
• Single v. multiple risk resolution
• Lognormal vs. mean-reverting
The focus of this presentation will be on equity
solutions using lognormally distributed binomial
lattice models.
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Lattice Models in Debt Valuations
• When debt has a call or put feature, the debt has a variable
rather than a contractual life as it can be called/put, and can
therefore terminate before maturity.
• The value of the call/put option is driven by the shape of the
yield curve and the terms of the embedded call/put feature.
• One model to value callable/putable debt is the Black-
Derman-Toy (BDT) model.
• The BDT model assumes that interest rates follow a
binomial process and uses current observed yields and
yield volatility to estimate future outcomes for the yields (by
means of a binomial tree of future interest rates).
Lattice Modeling for Equity Solutions
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Lattice Models in Valuations of Hybrid Securities
• Can be used to value convertible debt.
• Holder may convert to equity depending on stock
price.
• At each time interval, model compares
– Value of conversion
– Value of holding conversion option to next
period
• Optimal decision is selected
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Lattice Models in Equity Valuations
• Can be used to value enterprises and business
segments.
• Can be used to allocate value between different
classes of equity in "cheap stock" valuations
performed for 409(A) tax purposes.
• Can be used to value stock options for financial
reporting purposes.
• Other uses?
Lattice Modeling for Equity Solutions
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Building a Lattice Model
Valuation of a Risky Business Opportunity
Market Risk Only
Pre-revenue stage
• Value of opportunity today: $1.0 million
• Annual volatility of opportunity: 50 percent
• Estimated time to commercialization: 6 months
• Cost of commercialization/product rollout: $0.8 million
• Risk-free rate: 3.0 percent
What is the value of this opportunity?
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Mapping Opportunity onto the Black-Scholes Model
• Value of underlying asset = $1.0 million
• Strike Price = $0.8 million
• Time to expiry = 6 months
• Risk-free rate (annual) = 3.0 percent
• Volatility (annual) = 50 percent
• Value = $259,000
Lattice Modeling for Equity Solutions
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Interpretation of Black-Scholes Results
• Intrinsic value of opportunity = $200,000
– Underlying asset: $1,000,000
– "Exercise price": $800,000
• Due to volatility and ability to defer investment in
commercialization, value is higher
What if there is technology risk as well?
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Mapping the Opportunity onto a Binomial Model
• Same parameters as Black-Scholes
• Since binomial model is "discrete", we must
choose an appropriate number of incremental time
periods – we chose six one-month periods
• Necessitates calculating all annual parameters in
terms of one-month equivalents
Lattice Modeling for Equity Solutions
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Calculation of Volatility
• Volatility varies with the square root of time
• Annual volatility = monthly volatility times square
root of 12 (12 monthly periods)
• Therefore, monthly volatility = annual volatility
divided by square root of 12
• 50%/(sq rt 12) = 14.43 percent per month
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Calculation of Binomial "Jumps"
• Up movements (lognormal) are equal to the
exponential of the volatility
• Exp(0.1443) = 1.1553
• Down movements are the multiplicative inverse of
the up movements
• 1/1.1553 = 0.8656
Lattice Modeling for Equity Solutions
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Probabilities of Up and Down Jumps
• These are often referred to as "risk-neutral
probabilities"
• They are a function of risk-free rates and the
magnitude of up and down moves
• Formula: [exp(Rf) – down]/[up – down]
• In our case: [1.0025 – 0.8656]/[1.1553-0.8656] =
0.4725 is probability of an up move
• Down move = 1 – up = 0.5275
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The Underlying Asset Value Tree
• Value today is $1.0 million
• Value in one month, if "market" is up, is $1,155,300 ($1.0
million x 1.1553) and has a 47.25 percent "probability"
• Value in one month, if "market" is down, is $865,600 ($1.0
million x 0.8656) and has a 52.75 percent "probability"
• Therefore, value in six months may range from $2,377,000
(six consecutive up moves) to $421,000 (six consecutive
down moves)
What is the probability of six consecutive up moves?
Lattice Modeling for Equity Solutions
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Annual volatility 50% Rf/year 3.00%
Per period 0.1443 Rf/period 1.0025
Up movement 1.1553 p 0.4726
Down movement 0.8656 1-p 0.5274
Period 0 1 2 3 4 5 6
Value of opportunity 1,000 1,155 1,335 1,542 1,781 2,058 2,377
866 1,000 1,155 1,335 1,542 1,781
749 866 1,000 1,155 1,335
649 749 866 1,000
561 649 749
486 561
421
The Underlying Asset
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Our Rollout Decision
• We will commercialize only if value exceeds $0.8
million cost
• Four of our seven outcomes are "in-the-money"
and provide us with positive values net of exercise
cost
• Bringing these values back to the present by
applying risk-neutral probabilities and discounting
at the monthly risk-free rate, we get a value of
$260,000
Lattice Modeling for Equity Solutions
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Strike price 800
Value of current equity 260 384 547 748 985 1,260 1,577
150 240 369 539 744 981
71 126 218 357 535
21 44 94 200
0 0 0
0 0
0
Value of the Opportunity
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Example Calculation – Binomial Model
• Time period 6 value with six up moves: $1,000 x
(1.1553)6 = $2,377 – 800 exercise cost = $1,577
• Time period 6 value with five up and one down
move: $1,000 x (1.1553)5 x 0.8656 = $1,781 – 800
= $981
• Time period 5 value of $1,260 (working
backwards):
[($1,577 x 0.4725) + ($981 x 0.5275)]/1.0025 =
$1,260
Lattice Modeling for Equity Solutions
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Modified Binomial Model
• Why did we go through the effort on the binomial model if
the Black-Scholes model gives us the same result?
– Because the binomial model lends itself to customization
• A simple example – we add a second risk and a second
investment decision:
– Technology risk: 30 percent chance technology fails at
the end of six months
– Cost of R&D efforts to prove out technology: $100,000,
which must be invested today
• The value ($82,000) is calculated directly by this modified
binomial model
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Expanding the Lattice Model
Multiple Risks, Multiple Investment Decisions
• Value of opportunity if technology is successful: $1.0 million
• Annual volatility of opportunity – market risk only: 50
percent
• Chance of achieving technological success: 70 percent
• Cost of R&D to prove technology: $0.1 million
• Estimated time to completion of R&D: 6 months
• Cost of commercialization/product rollout: $0.8 million
Lattice Modeling for Equity Solutions
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Example Calculation – Modified Binomial Model
• Time period 6 value with six up moves: $1,577 x 70% =
$1,104
• Time period 6 value with five up and one down move: $981
x 70% = $687
• Time period 5 value of $882 (working backwards):
[($1,104 x 0.4725) + ($687x 0.5275)]/1.0025 = $882
Lattice Modeling for Equity Solutions
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Lattice Models in Equity Allocation
• Three methodologies for allocating value between
different classes of equity.
– Current value method
– Option pricing method
• Black-Scholes
• Lattice
– Probability-weighted expected return method
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When should a lattice model be used?
Can be helpful in allocations when:
• Allocation changes completely above a certain equity value (e.g., auto-
conversion upon IPO).
• Classes of equity receive different payouts based upon the
achievement of certain metrics and/or liquidation value.
– A class of preferred stock may be pari passu with the other
preferred at certain exit values, but first in order of preference at
other exit values to guarantee them a certain return.
– Management may receive a "carveout" – a certain percentage of the
exit value when exit values fall within a certain range.
Lattice Modeling for Equity Solutions
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Equity Allocation - Example
• Series A, B, C and D preferred and common stock outstanding
• Term: 2 years
• Risk-free rate: 0.27 percent
• Volatility: 60 percent
• Management has stated that if the equity value of the Company is $145
million, they will pursue an IPO rather than a sale. In an IPO all
preferred stock automatically converts to common stock.
• The preferred stock has liquidation preferences totaling $82 million.
The preferred does not participate, but is convertible into common at a
1 to 1 ratio. All preferred converts when the equity value reaches $232
million.
• The current equity value of the Company has been determined to be
$90 million.
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Equity Allocation – Example Asset Tree
Period 0 1 2 3 4 5 6 7 8 9 10 11 12
Value of underlying equity 90.0 115.0 146.9 187.7 239.8 306.3 391.3 499.9 638.7 815.9 1,042.4 1,331.7 1,701.4
70.4 90.0 115.0 146.9 187.7 239.8 306.3 391.3 499.9 638.7 815.9 1,042.4
55.1 70.4 90.0 115.0 146.9 187.7 239.8 306.3 391.3 499.9 638.7
43.2 55.1 70.4 90.0 115.0 146.9 187.7 239.8 306.3 391.3
33.8 43.2 55.1 70.4 90.0 115.0 146.9 187.7 239.8
26.4 33.8 43.2 55.1 70.4 90.0 115.0 146.9
20.7 26.4 33.8 43.2 55.1 70.4 90.0
16.2 20.7 26.4 33.8 43.2 55.1
12.7 16.2 20.7 26.4 33.8
9.9 12.7 16.2 20.7
7.8 9.9 12.7
6.1 7.8
4.8
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Equity Allocation - Example
• The preferred would not fully convert to common stock until $232 million
under a sale.
• Because management will pursue an IPO at lower values, there are
nodes where preferred stock's optimal outcome would be to receive its
liquidation preference, but it does not.
• This results in a higher value to the common stock.
• For the highlighted (IPO) nodes, common receives the asset value
times their pro rata interest. (12.1 percent in this example)
• For nodes below that, common's value is determined based on the
equity breakpoints, which consider preferred liquidation preferences
and conversion rights.
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Equity Allocation – Example Common Stock Tree
Period 0 1 2 3 4 5 6 7 8 9 10 11 12
Value of common stock 7.09 10.3 14.7 20.4 27.5 36.3 47.0 60.3 77.1 98.5 125.8 160.7 205.3
4.5 6.9 10.3 14.8 20.6 27.9 36.6 47.2 60.3 77.1 98.5 125.8
2.7 4.3 6.7 10.2 14.9 21.0 28.3 37.0 47.2 60.3 77.1
1.4 2.4 4.0 6.5 10.2 15.2 21.6 28.9 37.0 47.2
0.6 1.1 2.0 3.6 6.2 10.2 15.9 22.6 28.9
0.2 0.4 0.8 1.6 3.1 5.8 10.5 17.7
0.0 0.1 0.2 0.4 0.9 2.1 4.9
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0
0.0 0.0 0.0
0.0 0.0
0.0
Lattice Modeling for Equity Solutions
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Lattice Models to Value Stock Options
• ASC Topic 718 requires entities to use
– Fair-value-based measurement method
• To estimate value of employee awards
• Entities required to apply requirements of fair-
value-based measurement method in determining
award value
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Explicitly Excluded Characteristics
• Service conditions
• Performance conditions that affect vesting or
exercisability
• Vesting period restrictions
• Reload features
• Certain contingent features (e.g., clawbacks)
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Hull-White Lattice
• Lattice model to value employee stock options
• Accounts for possibility of early exercise and termination
• Example:
– Stock price: $100
– Strike price: $100
– Volatility: 35 percent
– Risk-free rate: 5 percent
– Term: 6 years
– Vesting period: 3 years
– Early exercise assumed when stock price is twice the strike price.
– The annual probability the employee will terminate is 3.0 percent.
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Hull-White Example
• Used a 24-period lattice model (4 periods per year)
• The up movement is calculated at 1.1912 = EXP(0.35/SQRT(4))
• The down movement is calculated at 0.8395 = (1/1.1912)
• The up probability is calculated at 49.12% = (EXP(0.05)-
0.8395)/(1.1912-0.8395)
• The down probability is calculated at 50.88% = 1-0.4912
• First step is to prepare an asset tree for underlying stock price
Lattice Modeling for Equity Solutions
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Hull-White Example – Asset Tree (Periods 7 through 20 not shown)
Period 0 1 2 3 4 5 6 21 22 23 24
Value of underlying equity 100 119 142 169 201 240 286 3,945 4,699 5,598 6,669
84 100 119 142 169 201 2,780 3,312 3,945 4,699
70 84 100 119 142 1,959 2,334 2,780 3,312
59 70 84 100 1,380 1,644 1,959 2,334
50 59 70 973 1,159 1,380 1,644
42 50 686 817 973 1,159
35 483 575 686 817
340 406 483 575
240 286 340 406
169 201 240 286
119 142 169 201
84 100 119 142
59 70 84 100
42 50 59 70
29 35 42 50
21 25 29 35
15 17 21 25
10 12 15 17
7 9 10 12
5 6 7 9
4 4 5 6
3 3 4 4
2 3 3
2 2
1
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Hull-White Example
• Next, a similar lattice can be developed to calculate the
value of the underlying option.
• First, consider the value of the option at maturity (period
24).
• The option will have value at stock prices greater than the
$100 strike price.
• Calculate value in period 24 as MAX(stock price – strike
price, $0)
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Hull-White Example
• Then, consider the value of the option in nodes prior to maturity (1-23).
• Early exercise and termination will be factors in these nodes.
– IF(stock price > 2x strike after vesting, stock price – strike),
accounts for voluntary early exercise (also referred to as suboptimal
exercise and the 2x multiple is referred to as the S.O.E.F.)
– IF(stock price < 2x strike)
• (1-.0075) x [(0.4912 x $6,569) + (0.5088 x $4,599)]/1.012 hold
• + 0.0075 x MAX(stock price – strike, $0) exercise if terminated
(also referred to as post-vesting termination factor)
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Hull-White Example – Solution Tree
Period 0 1 2 3 4 5 6 21 22 23 24
Value of option 35.65 48 64 85 111 144 186 3,845 4,599 5,498 6,569
25 34 47 63 83 110 2,680 3,212 3,845 4,599
17 24 33 45 61 1,859 2,234 2,680 3,212
11 16 23 32 1,280 1,544 1,859 2,234
7 10 15 873 1,059 1,280 1,544
4 6 586 717 873 1,059
2 383 475 586 717
240 306 383 475
140 186 240 306
71 101 140 186
26 44 70 101
5 10 20 42
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0
0 0
0
Lattice Modeling for Equity Solutions
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Lattice Models in Valuation/Real Options
• Lattice models can be used to capture need for
future financing and possible future dilution
• Discussed in new AICPA Accounting & Valuation
Guide Valuation of Privately-Held-Company Equity
Securities Issued as Compensation
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Using a Lattice Model to Capture Future Milestones
• Can adjust a lattice model to account for future
rounds of financing prior to a liquidity event.
• This is modeled as a sequence of options or
"compound option"
• Can address spikes in value due to success/failure
of financing rounds and technological/market risk
resolution
Lattice Modeling for Equity Solutions
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Using a Lattice Model to Capture Future Milestones - Example
• Value of company (ignoring need for future
financing): $2.0 million
• Current capital structure is Series A preferred and
common stock.
• Volatility: 80 percent
• Risk-free rate: 3 percent
• Company requires financing of $800,000 in one
year (Series B) and $1.5 million in three years
(Series C) to achieve an exit.
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Future Milestones - Asset Tree
Period 0 1 2 3 4 5 6 7 8 9 10 11 12
Value of opportunity 2,000 2,984 4,451 6,640 9,906 14,778 22,046 32,889 49,065 73,196 109,196 162,902 243,021
1,341 2,000 2,984 4,451 6,640 9,906 14,778 22,046 32,889 49,065 73,196 109,196
899 1,341 2,000 2,984 4,451 6,640 9,906 14,778 22,046 32,889 49,065
602 899 1,341 2,000 2,984 4,451 6,640 9,906 14,778 22,046
404 602 899 1,341 2,000 2,984 4,451 6,640 9,906
271 404 602 899 1,341 2,000 2,984 4,451
181 271 404 602 899 1,341 2,000
122 181 271 404 602 899
82 122 181 271 404
55 82 122 181
37 55 82
25 37
16
Lattice Modeling for Equity Solutions
© 2013 Business Valuation Resources, LLC
© Grant Thornton LLP. All rights reserved.
Future Milestones – First and Second Option
• Determine when the second option (opportunity to
invest $1.5 million in Series C) would be exercised.
– This would happen at equity values above $1.5
million in period 12.
• Determine when the first option (opportunity to
invest $0.8 million in Series B) would be exercised.
– This would happen at equity values above $0.8
million in period 4.
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Future Milestones – Tree with First and Second Option
Value of current equity 693.31 1,351 2,548 4,594 7,754 13,369 20,612 31,445 47,609 71,730 107,719 161,413 241,521
244 535 1,156 2,453 5,312 8,496 13,333 20,591 31,423 47,587 71,708 107,696
45 109 268 1,861 3,162 5,237 8,450 13,311 20,569 31,401 47,565
0 0 530 978 1,758 3,066 5,174 8,428 13,289 20,546
0 111 225 448 869 1,637 2,973 5,151 8,406
15 33 73 160 345 728 1,495 2,951
2 6 14 34 83 204 500
0 0 0 0 0 0
0 0 0 0 0
0 0 0 0
0 0 0
0 0
0
Lattice Modeling for Equity Solutions
© 2013 Business Valuation Resources, LLC
© Grant Thornton LLP. All rights reserved.
Future Milestones – Accounting for Dilution
• At valuation date, only Series A and common outstanding.
• At period 12, the Series C has been removed, but the
dilution from Series B still exists.
• How can we determine the dilution from the Series B
financing that exists in the period 12 nodes?
– Determine the extent to which the Series B investors
dilute the holdings of the Series A and common at the
end of year one (assuming a Series B financing occurs).
– Determine how this financing affects the year three exit
values.
– Remove the Series B dilution from the exit date values.
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Summary
• Review of option valuation fundamentals
• Types of lattice models
• How to build a simple equity lattice model
• Equity solutions using lattice models
– Equity valuation
– Equity allocation
– Options on equity
• Advanced applications
Lattice Modeling for Equity Solutions
© 2013 Business Valuation Resources, LLC
© Grant Thornton LLP. All rights reserved.
Questions?