## Annex 1: BPM6-Recommended Valuation Methods

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### Purpose and scope
- Compilers’ choice of valuation method and estimation technique can have a substantial effect on official FDI figures and a country’s international investment position (IIP).
- The paper investigates whether the BPM6-shortlisted methods reduce valuation issues and produce reliable market value equivalents and robust results (robustness = producing similar estimates when different estimation techniques are applied).
- Empirical illustration uses Danish IIP data; limitations:
  - Only a subset of recommended methods tested due to data availability and general applicability concerns.
  - Test limited to one detailed Nordic dataset.

### Key empirical findings (selected)
- Applying different estimation techniques to the price-to-earnings method generated a difference corresponding to 63% of the total liabilities in the official Danish IIP.
- Total market value estimates of unlisted direct investment equity (selected models) vary from EUR 130 billion to EUR 181 billion; discrepancy corresponds to 11% of the total liabilities in the official Danish IIP.
- P/E models produced estimates ranging from EUR 54 billion to EUR 341 billion in one sensitivity exercise; the largest estimate among P/E models is 530% higher than the smallest.
- Aggregate impacts on Danish IIP using P/B regression model approximations (DKK billion; percentage change in brackets):
  - OFBV (official figures):
    - Direct investment equity: Assets 589, Liabilities 496, Net assets 93
    - All other financial instruments: Assets 2785, Liabilities 2895, Net assets -110
    - Total: Assets 3374, Liabilities 3391, Net assets -17
  - Market value (P/B regression model approximations):
    - Direct investment equity: Assets 1439 (144%), Liabilities 1212 (144%), Net assets 227
    - All other financial instruments: Assets 2785 (0%), Liabilities 2895 (0%), Net assets -110
    - Total: Assets 4224 (25%), Liabilities 4107 (21%), Net assets 117
  - DKK/EUR exchange rate at end-2006: 7.4560

### Main conclusion on methods
- No single BPM6 method dominates; all seven recommended methods have strengths and weaknesses and different applicability depending on data availability.
- Among tested methods (selected for public-data practicality and general applicability):
  - Method C2: Price to earnings (relative)
  - Method D: Price to book value (relative)
  - Method E: Own funds at book value (absolute)
- Empirical preference in this study: P/B (Method D) models are more robust and produce more reliable market value estimates than P/E (Method C2) models; OFBV (Method E) is precise and symmetric but likely understates market values because accounting book values capture intangibles to a limited extent.

### Practical recommendation for compilers (high level)
- Encourage adoption of harmonized valuation principles across compilers to reduce bilateral asymmetries.
- Use multiple BPM6-recommended methods and different datasets where possible to cross-check estimates.
- Prefer estimation techniques that:
  - Use publicly available data where feasible.
  - Produce robust central-tendency measures (prefer central tendency measures for relative valuation methods over level-based regression when scale effects and multicollinearity are concerns).
- When robust multiples with low dispersion are observed, consider regression approaches (with careful diagnostics) to allow inclusion of liquidity and size variables.

---

### Equity valuation theory (absolute vs relative)
- Absolute valuation models
  - Core idea: value = discounted future cash flows using investors’ required risk-adjusted rate of return.
  - Example: Gordon Growth Model (GGM): V0 = (D0 (1 + g)) / (r − g) for r > g.
  - Practical issues: sensitivity to assumptions on growth g and required return r; risk premium estimation via CAPM or APT; DCF models rarely used in practice for unlisted equity due to uncertainty in inputs.
- Relative valuation models
  - Based on the law of one price (LOOP); value inferred from prices/multiples of comparable companies.
  - Synthetic arbitrage: V_it = (F_it / F_jt) V_jt when exact peers unavailable.
  - Common multiples: P/E, P/B; EV-based multiples (EV/EBIT, EV/S) preferred when adjusting for leverage.
  - Multi-factor form: V_it = Σ_{n=1}^x a_nt (F_intτ / F_jntτ), often estimated by regression.

### Theoretical practicalities and sensitivities
- Relative models are simpler and rely on available market prices but depend heavily on choice of comparables, accounting practices, timing τ versus valuation time t, and treatment of leverage and liquidity.
- Valuation multiples are undefined when denominators are non-positive; multi-company industry sums can sometimes allow inclusion of such companies.
- Liquidity, value of control, and negative equity values are key complications for unlisted equity valuation.

---

### BOX 1 — Including future earnings in relative models (summary)
- PEG concept: P/E divided by expected earnings growth to account for differing growth expectations; PEG still inherits forecast uncertainty.
- BPM6 market value principle: use closing market prices; if instruments not traded or infrequently traded, estimate a market equivalent (fair value) defined as “the amount for which an asset could be exchanged, or a liability settled, between knowledgeable, willing parties in an arm’s-length transaction” (BPM6, paragraph 3.88).
- Study focus: methods C2 (P/E), D (P/B), E (OFBV) selected because they rely on publicly available information.
- Three complications specific to unlisted equity highlighted:
  - Liquidity: unlisted equity lower liquidity, negative effect on prices if significant.
  - Value of control: control premiums vary by country; may be reflected in listed comparables or require additional variables (e.g., largest owner’s share).
  - Negative equity values: BPM6 allows inclusion of negative direct investment equity positions; country practices vary.
- Estimation approaches for compilers:
  - Regression (OLS) on market value or EV.
  - Central tendency measures (total summation, positive summation, arithmetic mean, weighted mean, median) for valuation multiples with practical trimming rules (e.g., exclude denominators ≤ 0, trim top/bottom 5%).

### BOX 2 — Model evaluation and dataset notes (selected)
- R2 limitations: R2 biased upward under scale effects; R2 comparisons across samples/time are unreliable without plausibility checks of parameter signs.
- Multicollinearity: borderline correlation between OFBV and earnings near 0.7 threshold; including both may distort parameter estimates.
- Dataset (ODIN Database; Nordic countries):
  - 1,027 listed companies in database; final 2006 dataset contains 682 companies where MVE and OFBV are available.
  - Descriptive statistics for 2006 (selected exact figures):
    - MVE: Observations 682; Mean 1,042.7; Std. deviation 4,052.3; Minimum 0.3; Maximum 63,388.1
    - OFBV: Observations 682; Mean 348.8; Std. deviation 1,196.0; Minimum -3.3; Maximum 11,124.1
    - EARNS: Observations 681; Mean 72.0; Std. deviation 422.8; Minimum -637.0; Maximum 7,404.0
    - EV: Observations 623; Mean 1,259.3; Std. deviation 4,904.0; Minimum -0.1; Maximum 70,326.1
    - VOL: Observations 654; Mean 15,872.7; Std. deviation 101,718.7; Minimum 0.0; Maximum 2,172,892.0
    - HOLD: Observations 682; Mean 28.4; Std. deviation 19.9; Minimum 0.1; Maximum 90.1
    - AGE: Observations 680; Mean 26.4; Std. deviation 25.7; Minimum 0.0; Maximum 110.0
  - Country composition (2006 sample):
    - Denmark 103 (15.1 percent); Finland 124 (18.2 percent); Norway 113 (16.6 percent); Sweden 342 (50.2 percent)

### BOX 3 — Practical data considerations and empirical diagnostics (selected)
- Timing: principle that all stock variables should be recorded at the same point in time; practical approximation used for account-closing dates.
- Flow variables prorated to 12 months when accounting periods differ.
- Dual share classes: consolidated into one observation using most traded share class.
- Robustness and reliability criteria:
  - Robustness: different typical estimation techniques should generate similar market value estimates.
  - Reliability: method must produce reliable market value estimates.
- Estimation caveats:
  - Level-based regression models subject to scale effects and multicollinearity; results vary considerably across industry groups.
  - Central tendency measures preferred for relative valuation methods in general.
- Empirical central-tendency statistics (All industries; exact figures reported):
  - P/E ratios:
    - Total summation 14.5
    - Positive summation 12.8
    - Arithmetic mean 40.5
    - Weighted mean 29.5
    - Median 20.7
    - Dispersion 216%
  - P/B ratios:
    - Total summation 3.0
    - Positive summation 3.0
    - Arithmetic mean 3.6
    - Weighted mean 4.2
    - Median 2.7
    - Dispersion 56%
  - EV/EBIT ratios:
    - Total summation 55.9
    - Positive summation 31.0
    - Arithmetic mean 78.6
    - Weighted mean 80.7
    - Median 33.6
    - Dispersion 160%
- Observed patterns:
  - OFBV is a better indicator for market value of equity than earnings (lower dispersion).
  - EV/EBIT ratios are slightly more stable across estimation techniques than P/E ratios.
  - Companies with OFBV below EUR 200 million have higher and more dispersed P/B ratios than larger companies.
- Sensitivity for Danish IIP:
  - Total market value estimates of inward unlisted direct investment equity vary widely by method; treatment of negative positions materially affects P/E-based aggregates (excluding negatives yields estimates ~50% larger than including negatives).

### Annex 6 — Practical advice and institutional recommendation (selected)
- To enhance cross-country consistency and reduce bilateral asymmetries:
  - Reduce interpretation diversity within each valuation method over time.
  - Carry out similar country studies to provide guidance on remaining methods; multiple methods may be necessary by country circumstances.
  - Establish a set-up allowing IIP compilers to share experiences and valuation models.
    - If each country develops models for valuing inward direct investment equity and shares them, workload and bilateral asymmetries would be reduced and quality improved.
- IMF recommendation: use OFBV as the valuation principle for unlisted equity in the CDIS to promote symmetry in reported bilateral direct investment positions.
  - This recommendation aligns with study conclusion that P/B ratios (OFBV-based relative models) should be used to value unlisted direct investment equity where feasible.

### Annex 2 — Issues specific to unlisted equity (selected)
- Illiquidity discounts
  - Empirical estimates vary; studies cite typical illiquidity discounts (examples):
    - Koeplin, Sarin and Shapiro (2000): average illiquidity discount of 20-30% when comparing public companies to private acquisition targets.
  - Practical approaches:
    - Include liquidity variables in multi-factor regressions or apply average illiquidity discounts; debate about using a marketability dummy versus treating liquidity as a continuum.
- Value of control
  - Acquisition-premium literature often reports approximately 25% premiums, though pure control premium may be considerably smaller.
  - Nenova (2003): control-block premiums vary by country; as high as 48% in low minority protection countries and less than 1% in Denmark, Finland, and Sweden.
  - Compilers may include largest-owner share as an explanatory variable to capture control effects.
- Treatment of negative positions
  - BPM6 allows inclusion of negative direct investment equity positions; country practices differ (some set negative to zero).
  - Differences in treatment can significantly impact IIP figures and bilateral asymmetries.
  - Recommendation: for symmetry, follow BPM6 and accept negative values under direct investment equity, or prefer valuation indicators that rarely take on negative values.

### Annex 3 — Model assumptions and transferability (selected)
- Four essential assumptions for relative methods (P/E and P/B):
  - LOOP (law of one price)
  - Existence of comparables
  - Transferability of model from listed to unlisted firms
  - Validity of projections outside input-data range
- For the Nordic dataset used in this study:
  - No obvious violations of the four assumptions.
  - Listed/unlisted accounting differences are not expected to systematically bias results when unconsolidated local GAAP data are used.
  - Illiquidity discounts can be included directly in models.
  - Dataset includes small firms (market capitalization as low as EUR 0.3 million) supporting transferability to smaller unlisted firms.

*Source: Annex 1 and related boxes and annexes from the supplied IMF working paper content unit (_wp09242).*

### Annex 1: BPM6-Recommended Valuation Methods.............................................................................

### Annex 1: BPM6-Recommended Valuation Methods

### Introduction
- Compilers’ choice of valuation method and estimation technique can have a substantial effect on official foreign direct investment (FDI) figures and a country’s international investment position (IIP).
- Market prices are the preferred principle for valuation of financial instruments; valuation of unlisted equity is complicated because market prices cannot be directly observed and must be estimated.
- Differences in country practices for valuing unlisted equity hinder international comparability and bilateral symmetry.
- The BPM6 contains a shortlist of recommended methods to estimate market value equivalents for unlisted direct investment equity and guidance on method suitability depending on country circumstances.
- Purpose of paper: investigate to what extent the BPM6-shortlisted methods reduce valuation issues and whether they produce reliable market value equivalents and robust results (robustness defined as producing similar estimates when different estimation techniques are applied).
- Empirical illustration uses Danish IIP data; finding example: applying different estimation techniques to the price-to-earnings method generated a difference corresponding to 63% of the total liabilities in the official Danish IIP.
- Empirical test limitations noted:
  - Only a subset of recommended methods tested (due to data availability and limited general applicability of some methods).
  - Test limited to one detailed dataset for Nordic companies.
- Recommendation emphasis: steps should be taken to increase cross-country harmonization in valuation principles for unlisted direct investment equity.

### Equity Valuation Theory — Overview
- Valuation models split into two groups:
  - Absolute valuation models (company-specific fundamentals; typically DCF-based).
  - Relative valuation models (value inferred from prices of similar companies; e.g., price-to-earnings (P/E)).
- Note: valuation theory applies to both listed and unlisted equity; issues specific to unlisted equity discussed elsewhere.

### Absolute Valuation Models
- Core idea: fundamental value equals discounted future cash flows using investors’ required risk-adjusted rate of return.
- Simple model presented: Gordon Growth Model (GGM)
  - Equation (2.1): V0 = (D0 (1 + g)) / (r − g) for r > g
  - Variables: V0 (equity value at time 0), D0 (dividend in period 0), g (expected dividend growth rate), r (required rate of return).
- Intuition from GGM:
  - Higher dividend growth (g) increases equity value.
  - Lower risk-free interest rate and/or lower risk premium increase equity value.
  - Equity value particularly sensitive to changes in expected dividend growth because it affects numerator and denominator.
- Practical considerations:
  - GGM assumptions: even relationship between dividends and earnings, constant dividend growth, constant and estimable risk premium.
  - Risk premium often estimated using CAPM or APT.
  - DCF models may replace dividends with earnings for unlisted equity where dividends are irregular.
  - Despite theoretical appeal, absolute (DCF) models are not widely used in practice due to uncertainty in risk premium and future earnings estimates and high sensitivity of valuations to small changes in inputs.

### Relative Valuation Models
- Based on the law of one price (LOOP): similar assets should trade at similar prices to prevent arbitrage.
- Relative models compare a company’s price to that of peers rather than deriving fundamental value directly.
- Simple equality under identical characteristics:
  - Equation (2.2): V_it = V_jt if D_i0 = D_j0, r_i = r_j, g_i = g_j.
- Synthetic arbitrage when exact peers unavailable:
  - Equation (2.3): V_it = (F_it / F_jt) V_jt, where F is the factor allowed to vary (e.g., earnings, sales); τ denotes factor measurement time.
  - Preferred that τ = t but often not possible; assumption that relationships are stable across times used.
- Valuation multiples:
  - Multiples typically price per share divided by factor per share; for whole-company valuation, notation with market value V used equivalently to price P since share counts cancel.
  - Common multiples: P/E, P/B; EV-based multiples used when accounting for differences in financial leverage.
- Enterprise Value (EV) concept:
  - EV = market value of debt + common equity + preferred equity − cash and investments.
  - EV ratios (e.g., EV/EBIT) are less sensitive to leverage differences; EBIT used as pre-interest flow to all providers of capital.
  - EV/S ratios preferred over P/S for firms with varying leverage, since sales flow to all providers of capital.
- Multi-factor models:
  - General form (2.4): V_it = Σ_{n=1}^x a_nt (F_intτ / F_jntτ), often estimated by regression with market value of equity or EV as dependent variable and factors as independent variables.
  - Weights a_nt sum to 1 for all t: Σ_{n=1}^x a_nt = 1.
- Practical strengths and weaknesses:
  - Strengths: simplicity, data availability, use of reference companies reduces need for strong forecasts.
  - Weakness: reliance on accounting data; despite IFRS and local GAAP, accounting interpretation differences can produce variation in multiples unrelated to economic differences.
  - Relative models can incorporate estimates of future earnings to connect to theoretical foundations.

### Practical Implications for Unlisted Equity Valuation (as drawn from theory and BPM6 context)
- Applying different recommended methods and estimation techniques can yield materially different market value estimates for unlisted equity, impacting IIP levels and bilateral asymmetries.
- Robustness of a valuation method is judged by the extent to which different estimation techniques produce similar market value estimates.
- To improve comparability and reduce bilateral asymmetries:
  - Encourage adoption of harmonized valuation principles across compilers.
  - Use multiple recommended BPM6 methods and different datasets where possible to cross-check estimates.
  - Be attentive to practical estimation considerations (data timing τ versus valuation time t, choice of factors, handling of leverage via EV, treatment of accounting differences).

### Key empirical note from the paper’s case study (Denmark)
- Example finding: varying estimation techniques for the P/E method generated a valuation difference equivalent to 63% of total liabilities in the official Danish IIP.

_Annex 1: BPM6-Recommended Valuation Methods — content excerpt from the source document._

### BOX 1: INCLUDING FUTURE EARNINGS IN RELATIVE VALUATION MODELS

### BOX 1: INCLUDING FUTURE EARNINGS IN RELATIVE VALUATION MODELS

### A. Including future earnings: PEG concept and limitation
- Main weakness of most relative valuation models: based on historical data and do not take future earnings into account; therefore comparing P/E ratios can be misleading if peer companies do not share the same expected future earnings.
- The concept of price-earnings-to-growth (PEG) ratios addresses differing expected growth: PEG is a special case of the multi-factor model where P/E ratios are divided by the expected earnings growth rate to take future differentials into account.
- Adjusted expression from the source:
  - ()
    ()
    11−−
    =
    ijjijtit
    ggEEVV
    ττ
    , where E represents earnings, and g denotes the earnings growth rate.
- Key insight: A high P/E ratio is justified if earnings growth is expected to be high.
- Limitation: Forecasts of earnings growth are fundamentally associated with a high degree of uncertainty; PEG is not a panacea and shares this uncertainty with absolute valuation models.

### B. Valuation in international macroeconomic statistical manuals (BPM6)
- Market value is the general valuation principle in the BPM6 and other international macroeconomic statistical manuals.
- For position data, "market value" is defined as the value of assets or liabilities using the closing market prices on the balance sheet reporting date.
- If financial instruments are not traded or traded infrequently, a market equivalent value (fair value) should be estimated; fair value is defined as “the amount for which an asset could be exchanged, or a liability settled, between knowledgeable, willing parties in an arm’s-length transaction” (BPM6, paragraph 3.88).
- Exceptions: positions in loans, deposits, and accounts receivable/payable are recorded at nominal value (BPM6, paragraph 3.86) for pragmatic concerns and legal/insolvency reasons.

- BPM6 (paragraph 7.16) includes a list of seven methods recommended for estimating market values of shareholder equity in unlisted direct investment enterprises; the list includes both absolute and relative valuation methods. All methods have individual strengths and weaknesses and different applicability depending on data availability.

- The study focuses on three methods (selected for public-data practicality and general applicability):
  - Method C2: Price to earnings (relative method)
  - Method D: Price to book value (relative method)
  - Method E: Own funds at book value (absolute method)
- Rationale for selection:
  - Preference for methods based on publicly available information to limit subjective assumptions and reduce bilateral asymmetries.
  - Excluded methods: C1 (Present value of earnings) and B (Net asset value) require subjective or inside information; A (Recent transaction price) and F (Apportioning global value) are not generally applicable to most unlisted equity.

### C. Issues specific to valuation of unlisted equity
- Three main complications that affect both absolute and relative methods:
  - (i) Liquidity:
    - Unlisted equity typically has lower liquidity than listed equity; lower liquidity tends to have a negative effect on prices and should be taken into account if significant.
  - (ii) Value of control:
    - Unlisted companies often have very few owners; a control premium is frequently included when investors obtain controlling stakes, but it can be argued that the premium should be offered to all shareholders and thus all shares should be valued uniformly.
    - Exception: in countries with low protection of minority shareholders controlling investors may reap private benefits.
  - (iii) Negative equity values:
    - Valuation methods may generate negative equity positions. Arguments:
      - Negative equity positions in direct investment enterprises might not be recorded in the IIP if direct investors would not be liable for losses exceeding invested capital.
      - Counterarguments include quasicorporations (branches, notional units) where investors may be liable, recapitalization by direct investors, explicit guarantees, or liability for damages.
    - BPM6 allows inclusion of negative direct investment equity positions in the IIP, but practices vary by country.
    - Practical implication: preferable to use valuation indicators that rarely take on negative values or only small negatives.

- The empirical results in Sections 4 and 5 quantify effects of including a liquidity variable and potential impact of negative values; Annex 2 discusses these three issues in detail.

### D. Study design and data — Selection of valuation methods (summary)
- Empirical study purpose: test reliability and robustness of BPM6-recommended valuation methods.
- Study limited to three methods (C2, D, E) because:
  - They are based on common, publicly available data: price, earnings, book values.
  - They are general in applicability across companies.
- Note: Limiting to three methods is a pragmatic choice for empirical testing, not a statement of inferiority of excluded methods (e.g., recent transaction price often gives the best market equivalents when available).

### E. Estimation techniques — regression models and central tendency measures
- Two typical practical approaches for compilers:
  - Regression approach (OLS on market value or EV)
  - Central tendency measures for valuation multiples (P/E, P/B)

- Regression models on market value
  - Idea: estimate a simple ordinary least squares (OLS) model applicable at company level.
  - Advantages over central tendency measures:
    - Multi-factor regression models do not require predetermined weights as in Equation 2.4.
    - Regression models are not restricted to positive observations in independent variables.
  - Basic estimation framework (Equation 3.1):
    - Y = Xβ + ε , where:
      - Y is a column vector of market values or EV.
      - X is the design matrix including a column of ones (constant) and quantitative/qualitative variables related to future earnings and risk.
      - β is the parameter column vector; ε is the error term column vector.
  - Interaction terms are included to capture effects that vary with company scale or other attributes.
  - Time is not explicitly modeled; model can be estimated at any point in time. Pooled OLS may be used to increase observations or to produce a multi-period model less sensitive to unusual market events.
  - Main weaknesses:
    - Scale effects: large companies can dominate regression, distorting parameter estimates and inflating R-squared.
    - Multicollinearity: high correlation between independent variables (caution if correlation > 0.7).
    - Heteroscedasticity: non-constant variance may lead to underestimated standard errors; corrected via White (1980) adjustment.

- Central tendency measures for valuation multiples
  - Used if OLS parameter estimates are affected by scale effects or multicollinearity.
  - For single-factor multiples (P/E, P/B) a range of central tendency measures can be calculated (Table 2):
    - Total summation
    - Positive summation (exclude listed companies where Xj ≤ 0)
    - Arithmetic mean (excluding listed companies where Xj ≤ 0 plus top 5% and bottom 5% of multiples)
    - Weighted mean (excluding listed companies where Xj ≤ 0 plus top 5% and bottom 5% of multiples)
    - Median (excluding listed companies where Xj ≤ 0)
  - Practical notes:
    - Valuation multiples are only defined when denominator is positive, but industry-level sums may allow inclusion of companies with non-positive denominators.
    - Applying multiples to companies with negative earnings/book values can yield negative market value estimates.
    - Liquidity cannot be included directly in single-factor multiples, but an average illiquidity discount from liquidity studies (Annex 2) can be applied to estimated multiples.
    - Central tendency measures require predetermined weights for multi-factor models; they are less dominated by a few large companies than OLS and are generally more stable (less multicollinearity).
  - Guidance on measure choice:
    - No universal rule; depends on distribution of multiples, proportion and size of negative values, and expected distribution for unlisted equity.
    - If robust multiples with low dispersion are observed, compilers can exploit regression advantages by constructing company-specific multiples and carrying out multivariate regression after exclusion of top 5% and bottom 5% (5% exclusion threshold used in the study; appropriate threshold depends on dataset).

- Notes on transformations and scale handling:
  - Log transformations can address scale effects but exclude companies with negative earnings/book values; study estimates regressions on non-transformed data and monitors signs of scale effects.
  - Pooled OLS uses all observations in a panel without time dimension; fixed/random effects are inappropriate because model must be transferable to other companies and require observed independent variables directly.

*Source: BOX 1: INCLUDING FUTURE EARNINGS IN RELATIVE VALUATION MODELS (excerpt).*

### BOX 2: MODEL EVALUATION

### BOX 2: MODEL EVALUATION

### Model-evaluation principles and diagnostics
- R2 as a goodness-of-fit measure:
  - R2 will be heavily affected by extreme observations in the presence of scale effects.
  - Brown, Lo, and Lys (1999) show that R2 is upward biased when scale effects are present.
  - R2 values should not be used to make comparisons over time and across samples.
  - R2 values of OLS models with and without a constant term cannot be compared directly.
- Multicollinearity and scale effects:
  - Neither scale effects nor multicollinearity can be directly detected in regression diagnostics; plausibility checks of parameter estimates are important.
  - Example: significantly negative parameter estimates on earnings or book value likely indicate multicollinearity.
- Robustness checks for valuation multiples:
  - Evaluate distribution of multiples: lower standard deviation and skewness imply more robust central-tendency measures for valuation.
  - Compare different central-tendency measures; low dispersion indicates consistency in multiples.

### Recommendations for application to unlisted direct investment equity
- Practical data-collection considerations:
  - IIP compilers usually collect data on unlisted direct investment equity directly from companies; response burden must be considered.
  - Prefer variables already collected for IIP or balance of payments (BOP) purposes or within companies’ own business needs.
  - If additional variables materially improve valuation models, assess reporter cost to supply information.
- Use and publication of model results:
  - Even if R2 is imperfect or the standard deviation of multiples is significant, a model may still be useful for macroeconomic statistics.
  - If no systematic bias exists between unlisted companies and their peer group, aggregation will eliminate or significantly reduce random company-level estimation errors.
  - Recommendation: publish results as aggregates and avoid overly detailed breakdowns.

### Data choice and dataset characteristics
- Choice rationale:
  - Empirical valuation models estimated on listed-company data, then applied to unlisted equity.
  - Traded-unlisted transaction data was not used due to: (1) transaction prices may be representative only for a short period and recent transactions for unlisted companies rarely exist; (2) limited number of observations from such transactions in BOP data due to limited trading activity.
- Dataset used:
  - All estimations are based on data from Bureau van Dijk’s ODIN Database covering public and private limited companies in Denmark, Finland, Norway, and Sweden.
  - Accounting data for last ten years for all companies except Denmark (only last five years available).
  - Total of 1,027 listed companies in the database.
  - Results for 2006 are presented; data for 2002-05 used to check consistency.
  - Dataset is unbalanced due to listing/delisting dynamics.
  - Final dataset for 2006 contains 682 companies where MVE and OFBV are available; number of observations varies across models when data are missing.

### Quantitative variables in final dataset (Table 3)
- MVE: Total market value of equity at sample year-end; Unit: EUR million
- OFBV: Own funds at book value at sample year-end; Unit: EUR million
- EARNS: P/L before taxes in sample year; Unit: EUR million
- EV: Enterprise value at sample year-end; Unit: EUR million
- EBIT: Earnings before interest and taxes in sample year; Unit: EUR million
- REVENUE: Revenue in sample year; Unit: EUR million
- VOL: Equity trading volume in December of sample year; Unit: EUR million
- HOLD: Largest owner’s share of total equity at sample year-end; Unit: Per cent
- SUBS: Number of recorded subsidiaries at sample year-end; Unit: Raw number
- AGE: Time from incorporation to sample year; Unit: Raw number

### Descriptive statistics for 2006 dataset (Table 4)
- MVE: Observations 682; Mean 1,042.7; Std. deviation 4,052.3; Minimum 0.3; Maximum 63,388.1
- OFBV: Observations 682; Mean 348.8; Std. deviation 1,196.0; Minimum -3.3; Maximum 11,124.1
- EARNS: Observations 681; Mean 72.0; Std. deviation 422.8; Minimum -637.0; Maximum 7,404.0
- EV: Observations 623; Mean 1,259.3; Std. deviation 4,904.0; Minimum -0.1; Maximum 70,326.1
- EBIT: Observations 674; Mean 23.0; Std. deviation 171.1; Minimum -132.1; Maximum 2,937.0
- REVENUE: Observations 619; Mean 204.4; Std. deviation 1,599.3; Minimum 0.0; Maximum 32,651.0
- VOL: Observations 654; Mean 15,872.7; Std. deviation 101,718.7; Minimum 0.0; Maximum 2,172,892.0
- HOLD: Observations 682; Mean 28.4; Std. deviation 19.9; Minimum 0.1; Maximum 90.1
- SUBS: Observations 682; Mean 26.7; Std. deviation 50.5; Minimum 0.0; Maximum 421.0
- AGE: Observations 680; Mean 26.4; Std. deviation 25.7; Minimum 0.0; Maximum 110.0

### Correlations, multicollinearity, and variable selection
- All independent variables in equity valuation models are positively correlated with market capitalization; same pattern holds in EV models.
- Correlations align with theoretical expectation (higher independent-variable values increase company value).
- Multicollinearity concerns:
  - Correlation between OFBV and earnings measures is at the borderline of the 0.7 rule-of-thumb threshold for multicollinearity.
  - Including both OFBV and earnings measures may distort parameter estimates in multi-factor models.

### Qualitative variables and industry grouping (Table 5)
- Qualitative variables in final dataset:
  - Country: Denmark (DK:1) and other Nordic countries; frequency counts shown.
  - Index inclusion: 1 = Included in main stock market index; 0 = Not included.
  - Industry: NACE rev. 1.1 classification, initially using an 11-industry-group breakdown from the European Test Exercise; combined into six groups due to small-sample concerns.
- Frequency and percent (excerpted from Table 5):
  - Country: Denmark 103 (15.1 percent); Finland 124 (18.2 percent); Norway 113 (16.6 percent); Sweden 342 (50.2 percent)
  - Index inclusion: Included 66 (9.7 percent); Not included 616 (90.3 percent)
  - Industry group counts and percents (selected):
    - Information and communication technology (ICT) activities: 103 (15.1 percent)
    - Mining/energy: 18 (2.6 percent)
    - Manufacturing (non-ICT): 119 (17.5 percent)
    - Construction: 7 (1.0 percent)
    - Trade: 51 (7.5 percent)
    - Hotel/restaurants/transports/communication (non-ICT): 36 (5.3 percent)
    - Financial intermediation: 46 (6.7 percent)
    - Insurance: 0 (0.0 percent)
    - Financial and insurance auxiliaries: 6 (0.9 percent)
    - Real estate/non-financial services/non-ICT/non-holdings/others: 157 (23.0 percent)
    - Holdings: 139 (20.4 percent)
- Treatment in estimation:
  - Country variable: dummy with Denmark = 1, others = 0.
  - Stock index inclusion: dummy variable.
  - Industry groups combined to six groups to avoid very small-sample industry estimates.

*Source: Calculations based on data from Bureau van Dijk’s ODIN Database (excerpted from BOX 2: MODEL EVALUATION).*

### BOX 3: PRACTICAL DATA CONSIDERATIONS

### BOX 3: PRACTICAL DATA CONSIDERATIONS

### Account closing dates and timing of stock variables
- Approximately 9% of the companies have a different account date than the end of the Gregorian calendar year.
- Principle: all stock variables should be recorded at the exact same point in time.
- Practical approach used: assume stock variables do not change until the end of the Gregorian calendar year rather than performing linear interpolation across accounting dates (interpolation is time-consuming and would affect a limited number of observations).
- Exception: if a significant company-specific event such as bankruptcy occurs, the compiler should take this into account.

### Accounting year length and flow variables
- Most companies follow the Gregorian calendar year, so flow variables (earnings, revenue) are measured for a 12-month period.
- Accounting periods may be longer or shorter than 12 months; data are prorated so all flow variables represent a 12-month period to avoid biased estimates.

### Companies with dual share classes
- Dual share classes are included as one observation in the dataset using the most traded share class as indicator for market price.
- Rationale: small pricing differences in the Nordic countries documented by Nenova (2003).

### Robustness and reliability criteria for valuation methods
- Two basic criteria for method selection:
  - Robustness: different but typical estimation techniques should generate similar market value estimates.
  - Reliability: method must produce reliable market value estimates.
- Robustness testing is a necessary but not sufficient condition for reliability.

### Regression estimation considerations and findings
- Regression approach (level-based) is natural starting point but subject to scale effects and multicollinearity.
- Models estimated jointly and by six industry groups with different combinations of independent variables (earnings, OFBV, liquidity variables, INDEX), and EV as dependent variable in one specification.
- Estimation results show parameter estimates vary considerably across industry groups, indicating need for industry-group-specific models.
- Examples of implausible parameter estimates and multicollinearity concerns:
  - Earnings parameter estimate for holding companies (Industry group 6) in the earnings model on market value of equity is 0.37 — economically implausible (interpreted as entire investment returned within 5 months under strict assumptions).
  - Correlation between independent variables exceeds 90% at the industry group level in some models.
  - Some OFBV parameter estimates below 1 for two industry groups, implying negative future earnings or overvalued assets—both considered unlikely.
- Multicollinearity effects in multi-factor models:
  - Inclusion of both earnings and OFBV sometimes led to negative parameter estimates; likely caused by multicollinearity or omitted variables.
  - Sample sizes for industry group models are often small, reducing robustness.
- Recommendation: compilers should use level-based regression models with utmost caution; if used, estimate on a large number of observations and assess plausibility of parameter estimates before application.

### Central tendency measures and valuation multiples
- Central tendency measures for P/E, P/B, and EV/EBIT ratios were calculated by six industry groups.
- General tendencies:
  - Arithmetic mean is higher than the median for P/E and EV/EBIT, indicating right-skewed distributions.
  - OFBV is a better indicator for market value of equity than earnings (lower dispersion).
  - EV/EBIT ratios are slightly more stable across estimation techniques than P/E ratios.
- Empirical statistics (selected, reported exactly as in source):
  - P/E ratios (All industries):
    - Total summation 14.5
    - Positive summation 12.8
    - Arithmetic mean 40.5
    - Weighted mean 29.5
    - Median 20.7
    - Dispersion 216%
  - P/B ratios (All industries):
    - Total summation 3.0
    - Positive summation 3.0
    - Arithmetic mean 3.6
    - Weighted mean 4.2
    - Median 2.7
    - Dispersion 56%
  - EV/EBIT ratios (All industries):
    - Total summation 55.9
    - Positive summation 31.0
    - Arithmetic mean 78.6
    - Weighted mean 80.7
    - Median 33.6
    - Dispersion 160%
- Observed pattern: companies with OFBV below EUR 200 million have higher and more dispersed P/B ratios than larger companies.
- For P/B ratios, total and positive summation measures yield identical results when reported with one decimal due to few and small companies with negative OFBV.

### Regression model for P/B ratios (selected coefficients and diagnostics)
- Model estimated after excluding companies with negative OFBV and trimming top and bottom 5% P/B ratios.
- Independent variables include trading volume (VOL), main stock market index dummy (Index), country dummy (DK), OFBV<200, and industry group dummies (D_IND1–D_IND5), with Industry group 6 as reference.
- Reported coefficients and statistics (exactly as in source):
  - Intercept 2.86; Std. Err. 0.33; t-value 8.73; P>t 0.00; [95% Conf. Interval] 2.21 3.50
  - VOL 3.69E-07; Std. Err. 9.84E-07; t-value 0.38; P>t 0.71; [95% Conf. Interval] -1.56E-06 2.30E-06
  - Index 1.04; Std. Err. 0.42; t-value 2.48; P>t 0.01; [95% Conf. Interval] 0.22 1.86
  - DK -0.10; Std. Err. 0.27; t-value -0.38; P>t 0.71; [95% Conf. Interval] -0.63 0.42
  - OFBV<200 0.77; Std. Err. 0.29; t-value 2.62; P>t 0.01; [95% Conf. Interval] 0.19 1.34
  - D_IND1 0.08; Std. Err. 0.34; t-value 0.25; P>t 0.80; [95% Conf. Interval] -0.57 0.74
  - D_IND2 -0.07; Std. Err. 0.30; t-value -0.22; P>t 0.82; [95% Conf. Interval] -0.65 0.52
  - D_IND3 -0.26; Std. Err. 0.34; t-value -0.74; P>t 0.46; [95% Conf. Interval] -0.93 0.42
  - D_IND4 -0.88; Std. Err. 0.42; t-value -2.09; P>t 0.04; [95% Conf. Interval] -1.71 -0.05
  - D_IND5 -0.10; Std. Err. 0.30; t-value -0.33; P>t 0.74; [95% Conf. Interval] -0.69 0.49
  - Adjusted R2 0.01
- Interpretation:
  - Trading volume contribution is positive but insignificant; INDEX dummy is positive and significant.
  - OFBV<200 dummy is positive and significant (companies with OFBV less than EUR 200 million have higher P/B ratios).
  - Industry group differences are generally small; D_IND4 (financial intermediation) is significant in a one-sided test.
  - Model has a low coefficient of determination (Adjusted R2 0.01), indicating much unexplained variation.

### Construction and use of P/B ratios for unlisted equity
- If trading volumes are assumed zero, twelve different P/B ratios can be constructed: two ratios for each of six industry groups depending on OFBV size (< EUR 200 million or ≥ EUR 200 million).
- Caveat: the model's low R2 implies significant unexplained variation; however, on an aggregate level the law of large numbers may yield reliable approximations if there is no systematic bias between listed and unlisted equity not already included in the model.

### Comparative evaluation and recommended estimation technique
- Level-based regression approaches on non-deflated accounting data are not recommended generally due to multicollinearity and scale effects causing large volatility.
- Preferred estimation technique: central tendency measures for relative valuation methods (P/E, P/B, EV/EBIT).
- Empirical support:
  - OFBV-based models are more robust than earnings- or EBIT-based models because earnings are volatile (flow variable).
  - 37% of all companies in the sample display negative before-tax earnings while market value is positive, illustrating volatility of earnings.
  - P/B multiples show lower variation (standard deviation as proportion of mean) than P/E and EV/EBIT, making P/B more robust and produces more reliable market value estimates than earnings multiples.

### Application to the Danish IIP — sensitivity and magnitudes
- Applying estimated P/E central tendency measures to inward unlisted direct investment equity in the Danish IIP yields highly sensitive results to estimation technique and treatment of negative positions.
- Reported aggregate estimates (exactly as in source):
  - Total market value estimates vary from EUR 54 billion to EUR 341 billion.
  - The largest estimate among P/E models is 530% higher than the smallest.
  - The difference corresponds to 63% of the total liabilities in the official Danish IIP.
- Treatment of negative positions:
  - Excluding negative positions leads to aggregate estimates approximately 50% larger than estimates including negative positions.
  - BPM6 allows inclusion of negative direct investment equity positions.
- Comparative robustness: variation across estimation techniques is significantly lower for P/B models than for P/E models, illustrating greater robustness and predictive reliability for P/B.

*Source: BOX 3: PRACTICAL DATA CONSIDERATIONS, _wp09242 - BOX 3: PRACTICAL DATA CONSIDERATIONS*

### Section 4C. Total market value estimates of unlisted direct investment equity vary from EUR

### _wp09242 - Section 4C. Total market value estimates of unlisted direct investment equity vary from EUR

### Comparison of valuation estimates and model behavior
- Total market value estimates of unlisted direct investment equity vary from EUR 130 billion to EUR 181 billion; the discrepancy corresponds to 11% of the total liabilities in the official Danish IIP.
- The difference of 40% between the highest and lowest direct investment estimates is considerably smaller than the difference of 530% observed for the P/E models.
- Unadjusted OFBV generates significantly lower estimates than the relative valuation models, most likely because accounting standards only capture intangibles to a limited extent; OFBV may consequently underestimate market values.
- OFBV is included for illustrative purposes and can be seen as a special case of the P/B model where the ratio by definition equals 1, making it an absolute valuation model rather than a relative valuation model.

### Robustness and estimator characteristics
- The arithmetic mean and weighted mean produce the highest estimates:
  - The arithmetic mean is upward biased because it is affected by skewness in the distribution of P/B ratios.
  - The weighted mean is upward biased because of the significant influence of a few large companies included in major stock market indexes.
- P/B models are considerably more robust than P/E models in terms of both estimation technique and the treatment of negative positions.
- Including or excluding negative direct investment equity positions makes only a small difference to aggregate estimates, even though almost 15% of the unlisted companies display negative OFBV:
  - Negative OFBV are relatively small compared to positive OFBV because companies with large negative OFBV will be forced to restore equity capital by authorities or creditors.

### Size, liquidity, and model choice
- Descriptive statistics indicate unlisted companies are generally smaller than listed companies.
- If size plays a role in valuation and is not taken into account, applying models calibrated on listed firms to unlisted equity will lead to biased results.
- The regression model on P/B ratios is the only model presented that includes size directly and also includes liquidity directly.
- For Denmark, the regression model on P/B ratios is preferred.

### Effect on Danish IIP: numerical results (DKK billion; percentage change in brackets)
- OFBV (official figures)
  - Direct investment equity: Assets 589, Liabilities 496, Net assets 93
  - All other financial instruments: Assets 2785, Liabilities 2895, Net assets -110
  - Total: Assets 3374, Liabilities 3391, Net assets -17
- Market value (P/B regression model approximations)
  - Direct investment equity: Assets 1439 (144%), Liabilities 1212 (144%), Net assets 227
  - All other financial instruments: Assets 2785 (0%), Liabilities 2895 (0%), Net assets -110
  - Total: Assets 4224 (25%), Liabilities 4107 (21%), Net assets 117
- Aggregate impacts and interpretation:
  - Denmark’s total external assets would increase by 25% while liabilities would increase by 21% compared to the official figures (which use OFBV for unlisted direct investment equity).
  - The asymmetric effect arises because Denmark has a positive net position in direct investment equity.
  - Denmark’s overall external financial position changes from a net liability position of DKK -17 billion to a net asset position of DKK 117 billion.
  - The DKK/EUR exchange rate was 7.4560 at end-2006.
- Caveat: the results use the crude assumption that the adjustment factor on assets is proportional to that on liabilities.

### Policy implications and recommendations (BPM6 context)
- BPM6 introduces seven recommended methods for valuing unlisted direct investment equity to achieve more reliable market value estimates and minimize bilateral asymmetry.
- All seven methods have strengths and weaknesses; no single method dominates across all companies.
- As an IIP compiler, it is essential to find the method or mix of methods that provides the most reliable estimates at the published aggregation level.
- Practical guidance for compilers:
  - Analyze the distribution of valuation multiples and compare to data for unlisted companies before choosing an estimation technique.
  - If multiples are robust, consider estimating a regression model with the multiple as dependent variable (deflated for scale differences), as done in the P/B regression model.
  - The regression approach allows direct inclusion of liquidity variables and can mitigate scale-related biases.
- Moving from OFBV to other market value approximations can have a significant impact on IIP figures and a country’s published net financial position; the net impact is likely to be large for countries with unbalanced direct investment equity positions or considerable differences between P/B ratios for inward and outward direct investment equity.

*Source: Calculations based on data from Bureau van Dijk's ODIN Database and official Danish IIP figures (excerpt from _wp09242 Section 4C and related text).*

### Annex  6  sums  up  the  practical  advice  for  the  compilers  given  throughout  this  paper.  It  also  provides  an

### Annex 6 — Practical advice for compilers and overview of relations between equity valuation theory, BPM6-recommended methods, and empirical estimation

### Key practical findings and recommendations
- The interpretation within each valuation method may need to be reduced over time to enhance cross-country consistency in direct investment data and increase international comparability.
- Studies similar to this one need to be carried out for other countries to provide precise guidelines on dealing with remaining methods; it may be necessary to use more than one valuation method or combinations depending on individual country circumstances.
- Estimating valuation models is a time-consuming process: Danish compilers would, in principle, need to estimate models for every single country in which Denmark has outward direct investment equity before revaluing the asset side of the Danish IIP.
- For practical reasons it is not realistic that every country can estimate country-specific models for every counterpart country on the outward side.
- There is an evident risk that bilateral asymmetries would occur due to the choice of estimation technique and input data.
- A proposed solution to i) lower the burden for IIP compilers, ii) reduce bilateral asymmetries, and iii) achieve better market value approximations is to establish a set-up allowing IIP compilers to share experiences and valuation models.
  - If every country developed models for valuing inward direct investment equity and shared them, much progress could be made on the three goals.
  - Workload could be eased if IIP compilers only have to develop models for valuing inward direct investment equity.
  - Bilateral asymmetries would ceteris paribus be reduced if everybody uses the same model to value direct investment enterprises resident in a given country.
  - The quality of estimates may be improved if valuation models are estimated by compilers with extensive knowledge about the specific country.

### Practical implementation issues to resolve for shared models
- Agreement on the detail level of industry breakdowns is important; using different breakdowns across countries is likely to cause problems.
- Treatment of negative direct investment equity positions matters if the valuation indicator can take on negative values; preferred valuation indicators rarely take on negative values and in such cases only small values.
- Treatment of special purpose entities (SPEs):
  - Valuation of SPEs may differ from regular companies; SPEs are often established for tax purposes and typically channel earnings through countries with favorable tax rates.
  - A specific concern is that the value of inward direct investment should equal the value of outward direct investment for pass-through companies; this may fail if parameter estimates vary across countries.
  - One solution is sharing information on valuation of pass-through companies between IIP compilers.

### IMF recommendation and implications
- To promote consistency in estimates of bilateral direct investment positions, the IMF recommends the use of OFBV as the valuation principle for unlisted equity in the CDIS; this symmetric valuation principle could help IIP compilers detect bilateral asymmetries.
- The recommendation to use OFBV aligns with this study’s conclusion that P/B ratios should be used to value unlisted direct investment equity.
- If P/B models are estimated for all countries, the CDIS will be able to provide the necessary input data to approximate market values for direct investment equity in all participating IMF member states based on a consistent valuation principle.

---

### Annex 1 — BPM6-recommended valuation methods (summarized)
- (A) Recent Transaction Price
  - Use transaction price when unlisted equity has been traded recently; BPM6 recommends using the transaction price to approximate market value for a maximum of one year without adjustments for changes in corporation’s position and general market conditions.
  - Formula provided for market value based on PTRS, TS, TRS (see source).
  - Advantages: easy to implement when available; transaction between independent parties equals market price at transaction time.
  - Drawbacks: market values change rapidly; transaction prices often not available for unlisted equity.
- (B) Net Asset Value
  - Fair value estimated as total assets at current value minus total liabilities (excluding equity) at market value; appraisals by management or independent auditors; intangibles should be covered but may be excluded per BD4; appraisals must have been conducted within the last year.
  - Advantages: appraisers close to the company may have superior knowledge; takes company-specific details into account.
  - Drawbacks: compilers do not know appraiser effort or consistency; potential for misreporting due to tax evasion, shareholder protection, etc.
- (C1) Present Value of Earnings
  - Absolute valuation: discount expected future earnings to present to obtain fundamental value; formula given using E and r (see source).
  - Advantages: theoretically sound; includes expected future earnings when they differ from past earnings.
  - Drawbacks: extremely time-consuming to forecast earnings at company level; fundamental value may differ from market value due to exuberance/bubbles; problematic for approximating market value equivalents.
- (C2) Price to Earnings (P/E)
  - Relative valuation: calculate P/E ratios for listed companies by industry group and apply to unlisted equity; formula referenced (see source).
  - Advantages: fairly easy to implement; uses market values rather than fundamentals; LOOP (Law of One Price) more likely to hold.
  - Drawbacks: does not capture individual company characteristics; assumes P/E ratios identical for listed and unlisted equity; accounting differences (transfer pricing) may bias results.
- (D) Price to Book Value (P/B)
  - Relative valuation using OFBV as book value measure; reuse formula from C2 with OFBV instead of earnings (see source).
  - Shares pros and cons with P/E method; uses a stock variable (book value) which is more stable than earnings.
- (E) Own Funds at Book Value (OFBV)
  - BPM6 definition: sum of (i) paid-up capital (excluding shares the enterprise holds in itself and including share premium accounts); (ii) all types of reserves identified as equity including investment grants when considered company reserves; (iii) cumulated reinvested earnings; (iv) holding gains or losses included in own funds.
  - Advantages: precise and easy to implement; promotes symmetric bilateral recording if all countries use same accounting standard and interpretation.
  - Drawbacks: IFRS may still exclude many intangibles; book values may underestimate market values; IFRS not fully implemented in most countries.
- (F) Apportioning Global Value
  - Prorate overall market value of a listed international group to entities using indicators such as sales, net income, assets, employment; formula given using IND (see source).
  - Advantages: based on actual market value of the group; straightforward once indicator data selected.
  - Drawbacks: difficult to determine how to prorate group value to entities; not general method for unlisted entities not part of a listed group.

---

### Annex 2 — Issues particularly related to valuation of unlisted equity

- Illiquidity Discounts
  - Trading costs components: the bid-ask spread, the price impact when buying or selling, opportunity costs, and commission (Damodaran, 2005a).
  - Illiquidity discounts are generally higher for unlisted equity than for traded listed equity.
  - BPM6 does not mention liquidity considerations explicitly among the seven recommended methods, but valuation principle is to estimate market value equivalents if market prices are not readily available; inference: illiquidity discounts should be taken into account if significant.
  - ESA 95 (paragraph 7.54) and BD4 (paragraphs 521 and 525) explicitly mention valuation with reference to listed equity and adjustment for differences in liquidity.
  - Two empirical approaches on whether marketability (listed vs unlisted dummy) should be included:
    - Include D_UNLISTED dummy plus liquidity variables (Stowe et al. (2002) argue for both an illiquidity discount and a marketability discount).
    - Treat liquidity as a continuum using liquidity variables only (Damodaran (2005a) argues against a separate dummy).
  - Multi-factor regression model illustration: equation A2.1 (see source) including optional D_UNLISTED.
  - Empirical literature finds significant illiquidity discounts but no consensus on size:
    - Koeplin, Sarin and Shapiro (2000) compare publicly traded companies to similar private acquisition targets and compute an average illiquidity discount of 20-30%.
    - Brennan, Chorida, and Schwartz (1998) find a negative relationship between trading volume and stock returns, implying investors pay a premium for liquidity.
    - Nguyen, Mishra, Prakash, and Ghosh (2007) find correlations between turnover ratios and expected returns, and find market capitalization and P/B ratios do not proxy for liquidity in their study.
  - Practical implication:
    - If liquidity is a continuum, set liquidity variable to zero for non-traded unlisted equity in models estimated on listed data.
    - If a marketability discount is believed relevant, add a dummy variable for non-listed status.
    - Transformations of liquidity variables (e.g., square root of trading volume) can be used as compromises; this study tests transformations and does not introduce an explicit marketability dummy in addition to the liquidity variable.

- The Value of Control
  - Control value defined as CV_i = V_i* − V_i (Equation A2.2 in source), where V_i* is value if optimally run and V_i is status quo value.
  - Investors may pay a premium for control for reasons including private benefits, synergy, strategic considerations.
  - Literature findings:
    - Basic studies looking at acquisition premiums conclude control value is approximately 25%, but Hitchner (2006) argues this includes synergy and strategic gains so the pure control premium is considerably smaller than 25%.
    - Zingales (1995) finds voting shares in the US trade at a premium of 5-10% over limited-voting shares.
    - Nenova (2003) finds control-block premiums vary significantly across countries: as high as 48% in countries with low minority investor protection and less than 1% in Denmark, Finland, and Sweden.
  - Implications for valuation models:
    - Absolute and forward-looking relative models can incorporate expected earnings improvements and probabilities of acquisition.
    - Relative valuation models estimated on listed companies may reflect control premiums; compilers could include largest investor's ownership share as an independent variable to capture such effects, though this may not capture all cases where control premium arises.
    - Macro approach applies same price to all shares within a share class; care needed when applying models to minority shareholders in countries with low minority investor protection.

- Treatment of Negative Positions
  - Listed equity has lower bound zero; negative market prices are not observed on exchanges.
  - Direct investment enterprises (including quasi-corporations, branches, units) can have non-equity liabilities exceed assets; estimated market value for money-losing sub-units could be negative if owner would pay a buyer to assume ownership.
  - BPM6 (paragraph 7.19) recognizes that direct investment enterprise non-equity liabilities may exceed assets and allows inclusion of negative equity positions under direct investment in the IIP.
  - Some countries revalue negative equity positions in limited liability enterprises to zero; differences in treatment can significantly impact IIP figures and be a source of asymmetry.
  - Recommendation: for symmetry reasons, countries should follow the statistical standard and accept negative values recorded under direct investment equity; alternatively, recommend valuation methods that are not too sensitive to treatment of negative positions.

*Source: Annexes and sections from the supplied IMF working paper content.*

### ANNEX 3: MODEL ASSUMPTIONS

### ANNEX 3: MODEL ASSUMPTIONS

### Central assumptions underpinning relative valuation methods
- The relative methods, price to earnings (Method C2) and price to book value (Method D), rely on four essential assumptions:
  - LOOP
  - Existence of comparables
  - Transferability of model
  - Projections outside data range
- If the four assumptions are not fulfilled, it would not be reasonable to use the relative valuation methods to estimate a model for the valuation of unlisted equity.
- LOOP is the theoretical basis of the relative models; empirical evidence (Levy Yeyati et al. (2006)) indicates price differences on financial markets are generally arbitraged away quickly. Capital controls can produce persistent price dispersions and would require IIP compilers to investigate their importance for asset pricing if models are built from data in regions with inter-regional capital controls.
- For the Nordic dataset used in this study, the LOOP assumption is likely to hold due to:
  - a large number of investors in the market
  - lack of capital restrictions in the region

### Assumption 2 — Existence of comparables and model validity checks
- BPM6 suggests single-factor models with either earnings or book value as the independent variable; industry classification may also be taken into account.
- When using a regression approach, the models’ coefficients of determination, R2, can be used to check validity:
  - If R2 is low, the model does not seem able to explain equity value variation and the model validity is questionable.
  - R2 is a measure for the value relevance of the factors in the model.
  - Note: R2 will be upwards biased in the presence of scale effects; R2 can only be used as an indicator of value relevance and analysis should include a plausibility test of the parameter estimates.
- For central-tendency multiple methods, the distribution of multiples can be analyzed; in principle, multiples for all companies should be identical for a correctly defined peer group.
- Even if independent variables do not explain all variation in market values, IIP compilers can still use a model for economy-wide aggregates provided there are no systematic differences between listed and unlisted equity (differences may even out in aggregation).

### Assumption 3 — Transferability from listed to unlisted companies (three practical implications)
- Implication 1: Listed and unlisted companies must face identical potential earnings and future risks.
  - If only listed companies secure lucrative contracts, models would overestimate unlisted equity market values.
  - For the Nordic countries, there are no indications that one group is favored in contract allocation; general industry risks are assumed identical for listed and unlisted firms.
- Implication 2: Financial leverage ratios must be similar between listed and unlisted companies if non-EV models are applied to unlisted companies.
  - Both types face the same optimization problem when deciding on leverage; firms will increase debt if expected return on assets exceeds borrowing interest rate.
  - Since tax rates do not differ between listed and unlisted companies in Denmark, leverage ratios are ceteris paribus expected to be identical.
- Implication 3: Accounting regulations must not lead to systematic differences in accounting data for listed vs. unlisted companies.
  - Misuse of transfer pricing or private benefits more likely in unlisted companies could create systematic bias; strong minority investor protection in the Nordic countries (especially Denmark, Finland, Sweden) reduces this risk (Nenova (2003)).
  - Danish tax regulations comply with OECD Transfer Pricing Guidelines; intra-group transactions must be documented on an arm’s-length basis.
  - EU rule (2002) required listed companies to prepare consolidated accounts based on IFRS beginning in 2005; unlisted companies may still use local GAAP. To avoid asymmetries, this study uses unconsolidated accounting data, which both listed and unlisted companies are required to follow under local GAAP.
  - Empirical evidence: Vedran et al. (2007) find only small (but significant) changes in net income and owners’ equity when transitioning from local GAAP to IFRS; Gjerde, Knivsflå, and Sættem (2007) report adjusted R2 of 80.5% for Norwegian GAAP and 79.2% for IFRS in an unadjusted price regression with book value and earnings.

- If systematic differences exist, they can be quantified and incorporated by adjusting parameter estimates. If differences are non-systematic, the model can still be applied but uncertainty of market value estimates increases.

### Assumption 4 — Projections outside the input-data range
- Relative models are estimated on traded liquid equity; it must be possible to project relationships (e.g., between market value and liquidity) outside the observed liquidity range to apply models to less-liquid unlisted equity.
- Unlisted companies tend to be smaller, but the dataset includes listed companies with negative OFBV and market capitalization as low as EUR 0.3 million, providing observations for relatively small firms and supporting transferability.

### Overall conclusion from Annex 3 (Nordic dataset)
- There are no obvious violations of the four assumptions for the Nordic dataset.
- Differences in reporting requirements and structural aspects between listed and unlisted companies are not expected to have a systematic effect on accounting figures that would hamper transferability of models estimated on listed companies to unlisted companies.
- Illiquidity discounts can be included directly in the models.
- The dataset includes relatively small companies, enabling size to be taken into account in estimations.
- Therefore, valuation models estimated on listed companies can be transferred to unlisted companies in this study without additional adjustments to parameter estimates, because structural differences between listed and unlisted companies are expected to be already reflected in the models.

*Source: ANNEX 3: MODEL ASSUMPTIONS (from the supplied IMF content unit)*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp09242.pdf_