“Valuation and Financial Forensics – Educate, Communicate, Preserve”
“Valuation and Financial Forensics – Educate, Communicate, Preserve”
NACVA and IBA’s Annual Consultants’ ConferenceNACVA and IBA’s Annual Consultants’ Conference
Discount for Lack of Marketability (Liquidity) Models: A Comparative Analysis
Discount for Lack of Marketability (Liquidity) Models: A Comparative Analysis
Mainstream Track
and
Ashok Abbott, PhD present
Mainstream Track
and
Ashok Abbott, PhD present
May 27-30, 2009Boston, MA
May 27-30, 2009Boston, MA
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Please set pagers, cell phones, etc. on Please set pagers, cell phones, etc. on vibrate modevibrate mode or turn them off. or turn them off.
Please complete and hand in the Please complete and hand in the Presenter EvaluationPresenter Evaluation forms. forms.
Don’t forget to complete your CPE Don’t forget to complete your CPE Attestation Form.Attestation Form. Add your NACVA# Add your NACVA# (located on your name badge) and sign it. (located on your name badge) and sign it. Keep the white copy for your records. Turn Keep the white copy for your records. Turn in the yellow copy at the NACVA in the yellow copy at the NACVA Registration Desk.Registration Desk.
Discount for Lack of Marketability (Liquidity) Models: A Comparative Analysis
Discount for Lack of Marketability (Liquidity) Models: A Comparative Analysis
Ashok Abbott, PhDAshok Abbott, PhDAshok Abbott, PhDAshok Abbott, PhD
4
Comparative Analysis of Liquidity and Marketability Discount Models
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Core Concepts in Discounting
Marketability Liquidity Holding period Liquidation period Price pressure Price risk/volatility
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Distinction between Marketability and Liquidity Marketability and Liquidity are aligned but
distinct concepts. Marketability -The capability and ease of
transfer or salability of an asset, business, business ownership interest or security.
Liquidity-The ability to readily convert an asset, business, business ownership interest or security into cash without significant loss of principal.
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Marketability versus liquidity: ASA definitions adopted July 2004
Marketability: The capability and ease of transfer or salability of an asset, business, business ownership interest or security
Liquidity: The ability to readily convert an asset, business, business ownership interest or security into cash without significant loss of principal
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Degrees of Marketability
Registered stock in an Exchange Listed Publicly Traded firm
Registered stock in an Exchange Listed Publicly Traded firm subject to Reg 144 restrictions
Unregistered stock in an Exchange Listed Publicly Traded firm
Unregistered stock in a closely held unlisted large firm (potential to go public)
Unregistered stock in a closely held unlisted small firm
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Model Classes for DLOM QMDM- Quantitative Model of Discount for
Marketability, proposed by Z. Christopher Mercer(1997) Time Value Model proposed by John J.Stockdale
(2006) CAPM based approach to calculating illiquidity
discounts proposed by David I. Tabak which deals with lack of
diversification Meulbroek model for cost of lack of diversification Proposed by Lisa K. Meulbroek(2002) Time Volatility Models
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Models considering Marketability and assuming Liquidity
Restricted Stock Discounts Registered vs. Unregistered Stock IPO cost studies
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Silber Model LN (RPRS) = 4.33 + 0.036 LN (REV) - 0.142 LN (RBRT) + 0.174 DERN - 0.332 DCUST
LN (RPRS) is natural logarithm of the relative price of restricted stock expressed in percentage terms [(p*/p) • 100].
LN (REV), the natural logarithm of the firm's revenues (in millions);
LN (RBRT), the natural logarithm of the restricted block relative to total common stock (in per cent);
DERN, a dummy variable equal to one if the firm's earnings are positive and equal to zero otherwise; and
DCUST, a dummy variable equal to one if there is a customer relationship between the investor and the firm issuing the restricted stock and zero otherwise.
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Bajaj ( 2001)
Discount = a + 0.40 x Fraction of Shares Issued -0.08 x Z-Score + 3.13 x Standard Deviation of Returns + b 4 x Registration Indicator.
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Abbott (2004)
DLOM (delisting change in value) = - 0.22220 +0.39571XCumexret +1.146XCap90X10^-5 +0.02491XTurnover
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Models Considering lack of diversification but assuming Marketability and Liquidity
Tabak Meulbroek
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Tabak
DLOM= 1-exponent ( σs2 / σm
2)XRPXT Discount for lack of diversification RP is the equity risk premium T is the time to liquidation
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Meulbroek DLOM= 1-(1/(1+R)n)
Where R is the product of the market risk premium multiplied by the difference between the asset’s beta and the ratio of standard deviation of returns for the asset and the market, a measure of incremental risk.
((σs / σm)- β)XRP
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Models considering Delayed Liquidation as lack of Marketability
QMDM Stockdale
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DCF Models QMDM /Stockdale
QMDM Framework The Expected Holding Period (HP) Expected Distribution Yield (D%) Expected Growth in Distributions (GD
%) Projected Terminal Value Stockdale Enhancements
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Stockdale explicitly accommodates inherent
uncertainty in the estimated liquidation period.
Assumes a linear liquidation probability, the model is flexible enough to accommodate any selected probability distribution.
Allows for the starting point for the period of liquidation to be any time in future rather than the present time period
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Time Volatility (Option) Models
Black Scholes Put (BSP) (Chaffee 1993) Average Price Asian Put (AAP) (Finnerty 2002) Look Back Put (LBP) (Longstaff 1995) (Abbott 2007)
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BSP
Black Scholes Put (BSP) is a simple contract. It provides protection against any realized loss in value at maturity of the contract. (LOSS I) The minimum value any asset can reach is zero. Therefore, the maximum value payable under a BSP contract is the exercise price for the put.
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BSP
P(T) = e-rT N(-d2)- N (-d1) Where d1= [(r+σ2/ 2) T]/ σT And d2 = d1- σT
The estimated BSP discount for lack of liquidity then becomes
P(T)/[1+P(T)]
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BSP Basics
σ is the standard deviation for the returns computed for the same
d1= [Lognormal(S/K) + (r+σ2/ 2) T]/ σT
d2 = d1- σT And N(-d1) and N(-d2) are the Normal
cumulative distribution probabilities Setting S=K=1
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Finnerty model :Asian Average Put
D(T) = V[ e rt N(r/ T +T / 2) –N (r/ T -T / 2)]
and 2 = σ2 T + Ln[2( e σ2 T - σ2 T -1)]
-2 Ln [e σ2 T -1]
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Finnerty Model Discount
Once again setting V to 1, D(T) becomes
[ e rt N(r/ T +T / 2) –N (r/ T -T / 2)] and the corresponding discount for
lack of liquidity becomes D(T)/ 1+ D(T)
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Look Back Put
F ( V,T) = V(2+ σ2 T/ 2) N(σ2T /2)
+ V ( σ2 T/2) e (-σ2T
/8) -V
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Look back Put DISCOUNT
set V to 1, the LBP option premium becomes
F(T) = (2+ σ2 T/ 2) N( σ2 T /2) + (σ2 T/2) e (- σ2 T /8) -1
the corresponding LBP discount for lack of liquidity becomes
F(T)/ (2+ σ2 T/ 2) N( σ2 T /2) + (σ2 T/2) e (- σ2 T /8)
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Estimated DLOL
Low Volatility (Annual σ 0.10-0.30),
Low Risk free rate (3%),
short duration (1 year)
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Discounts ComparisonAnnual Standard Deviation BSP LBP AAP
0.1 2.56% 7.61% 4.00%
0.11 2.92% 8.33% 4.19%
0.12 3.27% 9.04% 4.38%
0.13 3.63% 9.75% 4.58%
0.14 3.98% 10.45% 4.77%
0.15 4.33% 11.14% 4.97%
0.16 4.68% 11.83% 5.17%
0.17 5.03% 12.51% 5.37%
0.18 5.38% 13.19% 5.56%
0.19 5.72% 13.86% 5.76%
0.2 6.07% 14.52% 5.96%
0.21 6.41% 15.17% 6.16%
0.22 6.74% 15.82% 6.35%
0.23 7.08% 16.47% 6.55%
0.24 7.41% 17.10% 6.75%
0.25 7.74% 17.74% 6.94%
0.26 8.07% 18.36% 7.14%
0.27 8.40% 18.98% 7.33%
0.28 8.72% 19.60% 7.52%
0.29 9.04% 20.20% 7.71%
0.3 9.36% 20.81% 7.90%
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Estimated DLOL
Mid range Volatility (Annual σ . 0.40-0.60),
Medium Risk free rate (6%), Medium duration (5 year)
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Annual Standard Deviation BSP LBP AAP
0.4 15.72% 48.38% 33.79%
0.41 16.18% 49.16% 34.22%
0.42 16.64% 49.92% 34.64%
0.43 17.09% 50.68% 35.06%
0.44 17.53% 51.41% 35.46%
0.45 17.97% 52.14% 35.86%
0.46 18.40% 52.85% 36.25%
0.47 18.83% 53.55% 36.63%
0.48 19.25% 54.23% 37.00%
0.49 19.66% 54.90% 37.36%
0.5 20.07% 55.56% 37.70%
0.51 20.47% 56.21% 38.04%
0.52 20.87% 56.84% 38.37%
0.53 21.26% 57.47% 38.69%
0.54 21.64% 58.08% 39.00%
0.55 22.02% 58.68% 39.30%
0.56 22.39% 59.27% 39.58%
0.57 22.76% 59.85% 39.86%
0.58 23.12% 60.42% 40.13%
0.59 23.47% 60.97% 40.39%
0.6 23.82% 61.52% 40.64%
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Estimated DLOL
for High Volatility (Annual σ . 0.70-0.90),
High Risk free rate (9%), Long duration (10 year)
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Annual Standard Deviation BSP LBP AAP
0.7 19.56% 76.91% 59.57%
0.71 19.81% 77.30% 59.57%
0.72 20.07% 77.69% 59.58%
0.73 20.31% 78.06% 59.59%
0.74 20.55% 78.43% 59.59%
0.75 20.79% 78.79% 59.60%
0.76 21.02% 79.14% 59.60%
0.77 21.24% 79.48% 59.60%
0.78 21.46% 79.81% 59.61%
0.79 21.68% 80.14% 59.61%
0.8 21.89% 80.46% 59.61%
0.81 22.09% 80.78% 59.62%
0.82 22.29% 81.08% 59.62%
0.83 22.49% 81.38% 59.62%
0.84 22.68% 81.68% 59.62%
0.85 22.86% 81.97% 59.62%
0.86 23.04% 82.25% 59.63%
0.87 23.22% 82.52% 59.63%
0.88 23.39% 82.79% 59.63%
0.89 23.56% 83.06% 59.63%
0.9 23.72% 83.32% 59.63%
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