Cash Flow Multipliers and Optimal Investment Decisions Holger Kraft - PowerPoint PPT Presentation

Cash Flow Multipliers and Optimal Investment Decisions Holger Kraft 1 Eduardo S. Schwartz 2 1 Goethe University Frankfurt 2 UCLA Anderson School Kraft, Schwartz Cash Flow Multipliers 1/51

Agenda Contributions 1 Model 2 Optimal Cash-Flow Multiplier 3 Panel Regressions 4 Robustness Checks 5 Value of the Option to Invest 6 Conclusion 7 Kraft, Schwartz Cash Flow Multipliers 2/51

Agenda Contributions 1 Model 2 Optimal Cash-Flow Multiplier 3 Panel Regressions 4 Robustness Checks 5 Value of the Option to Invest 6 Conclusion 7 Kraft, Schwartz Cash Flow Multipliers 3/51

Contributions We develop a theoretical discounted cash flow valuation model , determine the optimal investment policy and calculate the ratio of the current value of the firm and the current cash flow which we call the “cash flow multiplier”. The model provides a link between the cash flow multiplier and the optimal investment policy . Using a very extensive data set comprised of more than 16,500 firms over 38 years we examine the determinants of the cash flow multiplier. We include as explanatory variables macro and firm specific variables suggested by the theoretical model. We find strong support for the variables suggested by the model. Perhaps the most interesting aspect of the paper is the formulation of a parsimonious empirical asset pricing model . Kraft, Schwartz Cash Flow Multipliers 4/51

Related Literature Discounting of stochastic cash flows and stock valuation Ang and Liu (2001, 2004, and 2007) Conditional expected returns when there are growth options Theoretical: Berk, Green and Naik (1999) and Carlson, Fisher and Giammarino (2004) Empirical: Titman, Wei and Xie (2004), Anderson and Garcia (2005), and Li and Zhang (2009) Real options and competitive markets Real options: Brennan and Schwartz (1985) and McDonald and Siegel (1986) Competitive markets: Grenadier (2002) and Aguerrevere (2009) Multipliers in Accounting Boatsman and Baskin (1981), Alford (1992), Baker and Ruback (1999), Nissim and Thomas (2001), Liu, Nissim and Thomas (2002, 2007), Bhojraj and Ng (2007) Kraft, Schwartz Cash Flow Multipliers 5/51

Agenda Contributions 1 Model 2 Optimal Cash-Flow Multiplier 3 Panel Regressions 4 Robustness Checks 5 Value of the Option to Invest 6 Conclusion 7 Kraft, Schwartz Cash Flow Multipliers 6/51

Model We consider a firm with cash flow dynamics (before investment) dC = C [ µ ( π, X ) dt + σ ( π, X ) dW ] , C (0) = c , where X is a state process, π is the percentage of the firm’s cash flow reinvested, W is a Brownian motion. Important: Firm can control its cash flow stream by investing! Firm Value with Endogenous Investment The firm value is given by � � ∞ � � s e − 0 R ( X u ) du ( C s − I s ) ds V ( c , x ) = max E , π 0 where I = π C . Kraft, Schwartz Cash Flow Multipliers 7/51

Linearity and Cash Flow Multiplier Proposition: Linearity of Firm Value Firm value is linear in the cash flow, i.e. V ( c , x ) = f ( x ) c , where f ( x ) = V (1 , x ). The result leads to the following definition. Definition: Cash Flow Multiplier In our model, the function f is said to be the cash flow multiplier. Interpretation: This is the multiple by which the current cash flow must be multiplied to obtain the firm value. Kraft, Schwartz Cash Flow Multipliers 8/51

Examples: Specifications of Risk-adjusted Discount Rates Two-factor model Stochastic riskfree rate r , stochastic beta β R = r + βλ, where λ = λ + λ r r is the risk premium In this talk: One-factor model R = ϕ + ψ r where ϕ and ψ are constants. Possible interpretation: ϕ = βλ and ψ = 1 + βλ r Kraft, Schwartz Cash Flow Multipliers 9/51

Endogenous Expected Growth and Volatility Dividend-discount model: Cash flow multiplier beyond the control of the firm (exogenous). In contrast, we explicitly model the firm’s opportunity to change its risk-return tradeoff. More precisely, we allow the firm to control the expected growth rate and the volatility of the cash flow stream by its investment policy. Benchmark Specification Expected cash flow growth and volatility are given by the concave functions √ π + µ 2 π, √ π + σ 2 π, µ ( π, r ) = µ 0 ( r ) + µ 1 σ ( π ) = σ 0 + σ 1 where µ 0 ( r ) = µ 0 + � µ 0 r . Kraft, Schwartz Cash Flow Multipliers 10/51

Endogenous Expected Growth mu2=-0.03 mu2=-0.06 0.05 0.04 0.03 0.02 0.01 0 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -0.01 -0.02 -0.03 -0.04 Investment Proportion pi We assume µ 0 = − 0 . 03 and µ 1 = 0 . 1. For the upper curve, we have µ 2 = − 0 . 03 and for the lower one µ 2 = − 0 . 06. Kraft, Schwartz Cash Flow Multipliers 11/51

Agenda Contributions 1 Model 2 Optimal Cash-Flow Multiplier 3 Panel Regressions 4 Robustness Checks 5 Value of the Option to Invest 6 Conclusion 7 Kraft, Schwartz Cash Flow Multipliers 12/51

HJB and Optimal Investment Policy We assume the short rate to have Vasicek dynamics dr = ( θ − κ r ) dt + η dW r , where W r is a Brownian motion with d < W , W r > = ρ dt . The optimal cash flow multiplier satisfies the HJB-equation √ π + µ 2 π ) f + 1 − π − ( ϕ + ψ r ) f 0 = max π { ( µ 0 + � µ 0 r + µ 1 √ π + σ 2 π ) f r } . +( θ − κ r ) f r + 0 . 5 η 2 f rr + ρη ( σ 0 + σ 1 Optimal Investment Proportion The first-order condition yields the firm’s optimal investment � � 2 µ 1 f + ρησ 1 f r π ∗ = . 2(1 − µ 2 f − ρησ 2 f r ) Kraft, Schwartz Cash Flow Multipliers 13/51

Optimal Cash Flow Multiplier Substituting π ∗ into HJB equation yields ( µ 1 f + ρησ 1 f r ) 2 ϕ + � ψ r ) f +1+( θ + ρησ 0 − κ r ) f r +0 . 5 η 2 f rr + 0 = ( � 4(1 − µ 2 f − ρσ 2 η f r ) . Proposition: Optimal Cash Flow Multiplier The optimal cash flow multiplier has the stochastic representation � ∞ e A ( s ) − B ( s ) r ds f ( r ) = + O ( r ; f ) , � �� � 0 � �� � Growth Opport. CFM without Invest. where A and B are known deterministic functions and � ∞ � � ( µ 1 f ( r s ) + ρησ 1 f r ( r s )) 2 � s ϕ + � � 0 � ψ r u du O ( r ; f ) ≡ E e ds 4(1 − µ 2 f ( r s ) − ρσ 2 η f r ( r s )) 0 captures the firm’s growth opportunities. Kraft, Schwartz Cash Flow Multipliers 14/51 This is an implicit representation!

Special Case: Constant Interest We assume that the risk-adjusted discount rate is the sum of a constant short rate and spread, i.e. R = r + λ = const . Then the cash flow multiplier has the representation � ∞ � ∞ ( µ 1 f ) 2 e ( µ 0 − r − λ ) s ds + e ( µ 0 − r − λ ) s f = 4(1 − µ 2 f ) ds , 0 0 � �� � = O ( f ) which can be solved explicitly. Optimal Cash Flow Multiplier under Constant Interest If µ 2 1 / 4 − µ 2 ( µ 0 − r − λ ) < 0, then the optimal cash flow multiplier is uniquely given as the positive root of the quadratic equation � � f 2 + ( µ 0 − r − λ − µ 2 ) f + 1 . µ 2 0 = 1 / 4 − µ 2 ( µ 0 − r − λ ) Kraft, Schwartz Cash Flow Multipliers 15/51

Dependence on Pi The cash flow multiplier has the representation � ∞ � ∞ ( µ 1 f ) 2 e ( µ 0 − r − λ ) s ds + e ( µ 0 − r − λ ) s f = 4(1 − µ 2 f ) ds , 0 0 which can be rewritten f = f 0 + f 0 (1 − µ 2 f ) π ∗ . Solving for f and taking logarithms yields ln f 0 + ln(1 + π ∗ ) − ln(1 + f 0 µ 2 π ∗ ) ln f = ln f 0 + β 1 π ∗ + β 2 ( π ∗ ) 2 ≈ where β 1 is positive and β 2 is negative (diminishing marginal returns on capital). Kraft, Schwartz Cash Flow Multipliers 16/51

Special Case: Constant Interest and No Growth Options Recall from the last slide: � � f 2 + ( µ 0 − r − λ − µ 2 ) f + 1 . µ 2 0 = 1 / 4 − µ 2 ( µ 0 − r − λ ) If the firm has no control over the expected growth rate of its cash flow stream ( µ 1 = µ 2 = 0), then we obtain Cash Flow Multiplier of Gordon Growth Model 1 f = r + λ − µ 0 Kraft, Schwartz Cash Flow Multipliers 17/51

Numerical Example 1st Example. µ 0 = − 0 . 03, µ 1 = 0 . 1, µ 2 = − 0 . 03, R = 0 . 07 Cash flow multiplier with optimal investment: 13.06 Cash flow multiplier without investment: 10 Growth option O = 3 . 06 Opportunity to invest increases cash flow multiplier by 30% 2nd Example. µ 0 = − 0 . 05, . . . Cash flow multiplier with optimal investment: 9.91 Cash flow multiplier without investment: 8.33 Growth option O = 1 . 58 Opportunity to invest increases cash flow multiplier by 18% This patterns hold in general! Kraft, Schwartz Cash Flow Multipliers 18/51

Cash Flow Multiplier and Growth Options Net Present Value of Growth Opportunities If µ 2 1 / 4 − µ 2 ( µ 0 − r − λ ) < 0 and µ 0 − r − λ − µ 2 < 0 hold, then the optimal cash flow multiplier f , the option value O , and the ratio O / f are increasing in µ 0 . This result puts some of the classical results on real options into perspective. If the firm is forced to invest for instance because competitors do the same, then the option to invest loses (part of) its value. Hence, the cash flow multiplier decreases. Kraft, Schwartz Cash Flow Multipliers 19/51