Stochastic Taxation and Pricing of CMBS REITs Robert H. Edelstein The University of California at Berkeley, Haas School of Business Konstantin Magin The University of California at Berkeley, Center for Risk Management Research May 11, 2014 () May 12, 2014 1 / 24
SUMMARY OF FINDINGS • Major Innovation: Introduction of Stochastic Taxation • After-tax Risk Premium resolves a substantial part of the Equity Premium Puzzle • Coefficient of Relative Risk Aversion: 7 . 43 − 10 . 59 () May 12, 2014 2 / 24
PRESENTATION STRATEGY • Review CCAPM • Outline the Equity Risk Premium Puzzle • Introducing Stochastic Taxation into the Analysis • Determining the Coefficient of Risk Aversion () May 12, 2014 3 / 24
SUMMARY OF CCAPM Theorem (Lucas Tree-Model (1978)): Assume • Preferences: ∞ b T U i ( c i ) = u i ( c it ) + E [ ∑ i u i ( c it + T )] ∀ i ∈ I . T = 1 u � i ( · ) > 0 , u i ” ( · ) < 0 ∀ i ∈ I . • Budget Constraint: n n ∑ z ikt + T ( p kt + T + d kt + T ) = c it + T + ∑ z ikt + T + 1 p kt + T ∀ i ∈ I , k = 1 k = 1 ∀ T = 0 , ..., ∞ . • Supply of Assets: z ikt + T = z kt + T > 0 ∀ k = 1 , ..., n ∀ T = 0 , ..., ∞ . ∑ i ∈ I () May 12, 2014 4 / 24
Then • Pricing Equation: � b i u � � i ( c it + 1 ) p kt = E i ( c it ) ( p kt + 1 + d kt + 1 ) ∀ k = 1 , ..., n . u � • Efficient Market Hypothesis: � ∞ � b T i u � i ( c it + T ) p kt = E ∑ d kt + T ∀ k = 1 , ..., n . u � i ( c it ) T = 1 • Euler Equation: � b i u � � i ( c it + 1 ) E R kt + 1 = 1 ∀ k = 1 , ..., n , u � i ( c it ) � b i u � � i ( c it + 1 ) E R f = 1 . u � i ( c it ) () May 12, 2014 5 / 24
COROLLARY 1: Assume • Lucas (1978) CCAPM • u � i ( c it + 1 ) = λ i · R mt + 1 Then • CAPM: E [ R kt + 1 − R f ] = β k · E [ R mt + 1 − R f ] ∀ k = 1 , ..., n . COROLLARY 2: Assume • Lucas (1978) CCAPM • Identical Agents Then • Efficient Market Hypothesis: n b T u � ( ∑ d kt + T ) ∞ . p kt = E k = 1 d kt + T ∑ n u � ( ∑ d kt ) T = 1 k = 1 () May 12, 2014 6 / 24
COROLLARY 3: Assume • Lucas (1978) CCAPM • Identical Agents • CRRA: u ( c ) = c 1 − a 1 − a Then • Efficient Market Hypothesis: − a n ∑ d kt + T ∞ . b T k = 1 p kt = E ∑ d kt + T n ∑ d kt T = 1 k = 1 () May 12, 2014 7 / 24
COROLLARY 4: Assume • Lucas (1978) CCAPM • Identical Agents • CRRA: u ( c ) = c 1 − a 1 − a • ln ( c t + T ) ∼ N ( µ c , σ c ) ∀ T = 1 , ..., ∞ • n = 1 Then 1 − a n ∑ d kt + T ∞ · d kt , b T k = 1 p kt = E ∑ n ∑ d kt T = 1 k = 1 d kt + T p kt + T = c t + T p kt + T = constant ∀ T = 1 , ..., ∞ . () May 12, 2014 8 / 24
THEOREM (RUBINSTEIN (1976)): Assume • Lucas (1978) CCAPM • Identical Agents • CRRA: u ( c ) = c 1 − a 1 − a • ln ( c t + T ) ∼ N ( µ c , σ c ) ∀ T = 1 , ..., ∞ • ln ( R kt + T ) ∼ N ( µ k , σ k ) ∀ T = 1 , ..., ∞ • ρ ln ( R kt + T ) , ln ( c t + T ) � 0 ∀ T = 1 , ..., ∞ Then ln E [ R kt + 1 ] − ln R f = a · cov [ ln R kt + 1 , ln ( c t + 1 c t )] . and • Black-Scholes-Rubinstein Formula: √ 1 S Call ( p kt , S , T , σ k , D , rf ) = T p kt N ( Z ks + T σ k ) − ( 1 + rf ) T N ( Z ks ) , ( 1 + D ) ln Pkt ( 1 + D ) T + ln R T 1 S + ln √ f − 1 Z ks = √ T σ k . 2 T σ k () May 12, 2014 9 / 24
EQUITY PREMIUM PUZZLE • The coefficient of relative risk aversion: � � − u �� ( c ) c rr ( c ) = . u � ( c ) LEMMA: Assume • u ( c ) = c 1 − a 1 − a Then • u � ( c ) = c − a • u �� ( c ) = − a · c − a − 1 � � − − a · c − a − 1 · c • rr ( c ) = = a c − a • Equity Premium Puzzle for β = 1 Portfolio, (Mehra and Prescott (1985) and Mehra (2003)): ln ( E [ R mt + 1 ]) − ln ( R f ) �� = 0 . 07 − 0 . 01 a = � � Ct + 1 = 47 . 6 . 0 . 00125 COV ln ( R mt + 1 ) , ln Ct () May 12, 2014 10 / 24
CALCULATING TAX YIELD FOR S&P 500 Components of tax yield: • Dividend tax • Short-term capital gains tax • Long-term capital gains tax Tax yield for the S&P 500 (Sialm (2008)): TY t + 1 = τ d t + 1 d mt + 1 + τ SCG t + 1 SCG mt + 1 + τ LCG t + 1 LCG mt + 1 = p mt mt + 1 · d mt + 1 t + 1 · SCG mt + 1 t + 1 · LCG mt + 1 = τ d p mt + τ SCG + τ LCG = p mt p mt = τ d mt + 1 · 0 . 045 + τ SCG t + 1 · 0 . 001 + τ LCG t + 1 · 0 . 018 , where p mt is the price per share of the market portfolio of risky assets, d mt is the dividend paid per share of the market portfolio of risky assets, R mt + 1 = 1 + r mt + 1 is the gross rate of return on the market portfolio of risky assets, τ d t + 1 is the dividend tax, () May 12, 2014 11 / 24
τ SCG t + 1 is the tax on short-term capital gains, τ LCG t + 1 is the tax on long-term capital gains, SCG t + 1 are realized short-term capital gains, LCG t + 1 are realized long-term capital gains, and TY t + 1 is the tax yield. • The dividend yield for the market portfolio of risky assets: d mt + 1 = 0 . 045 p mt • The realized short-term capital gains yield for the market portfolio of risky assets: SCG mt + 1 = 0 . 001 p mt • The realized long-term capital gains yield for the market portfolio of risky assets: LCG mt + 1 = 0 . 018 . p mt • Tax yield for the S&P 500 (Sialm (2008)): TY t + 1 = τ d mt + 1 · 0 . 045 + τ SCG t + 1 · 0 . 001 + τ LCG t + 1 · 0 . 018 . () May 12, 2014 12 / 24
• The tax τ t + 1 imposed on the wealth of the S&P 500 stockholders (Magin(2014)): τ t + 1 = τ d t + 1 d mt + 1 + τ SCG t + 1 SCG t + 1 + τ LCG t + 1 LCG t + 1 = p mt + 1 + d mt + 1 = τ d t + 1 d mt + 1 + τ SCG t + 1 SCG t + 1 + τ LCG t + 1 LCG t + 1 p mt TY t + 1 · = R mt + 1 , p mt p mt + 1 + d mt + 1 � �� � � �� � Tax Yield , TY t + 1 1 / R mt + 1 • Estimate for the tax τ t + 1 imposed on the wealth of the S&P 500 stockholders for 1913-2007: � � τ d t + 1 · 0 . 045 + τ SCG t + 1 · 0 . 001 + τ LCG 1 τ t + 1 = t + 1 · 0 . 018 · R mt + 1 . � �� � Tax Yield , TY t + 1 () May 12, 2014 13 / 24
CALCULATING TAX YIELD FOR CMBS REITs • About 20 % of all stock shares are held in taxable accounts. • Stock dividends are on average taxed at the ordinary income tax rate of about 20 % . • The average effective dividend tax rate estimate: τ d t + 1 = 0 . 2 · 0 . 2 = 0 . 04 . • REITs distribute at least 90 % of taxable income to shareholders in the form of dividends. • REITs dividend distributions constitute a significant portion of the overall before-tax return from REITs. • REITs dividends are ostensibly taxed as ordinary income. () May 12, 2014 14 / 24
• Expect that the typical investor in REITs may be subject to below average ordinary income tax rates. • Many tax exempt institutional investors may be attracted to REITs. • The average dividend tax rate appropriate for the S&P, in general, may not be appropriate for REITs investors. • The average effective dividend tax rate estimate: τ d cmbs reits t + 1 = 1 2 · 0 . 2 · 0 . 2 = 0 . 02 . • The average dividend yield for CMBS REITs is more than twice that of the average dividend yield for S&P 500 stocks: 0 . 123 vs. 0 . 45. () May 12, 2014 15 / 24
() May 12, 2014 16 / 24
TABLE 1: TAX YIELD PARAMETERS S&P 500 Equity REITs CMBS REITs d kt + 1 0.045 0.080 0.123 p kt SCG kt + 1 0.001 0.001 0.001 p kt LCG kt + 1 0.018 0.018 0.018 � � � � p kt τ d τ d 0 . 25 · τ d mt + 1 , τ d 0 . 25 · τ d mt + 1 , τ d t + 1 mt + 1 mt + 1 mt + 1 τ SCG τ SCG τ SCG τ SCG t + 1 mt + 1 mt + 1 mt + 1 τ LCG τ LCG τ LCG τ LCG t + 1 mt + 1 mt + 1 mt + 1 () May 12, 2014 17 / 24
• Tax Yield for CMBS REITs: TY cmbs reits t + 1 = τ d cmbs reits t + 1 d cmbs reits t + 1 + τ SCG t + 1 SCG cmbs reits t + 1 + τ LCG t + 1 LCG cmbs reits t + 1 = = p cmbs reits t = cmbs reits t + 1 · d cmbs reits t + 1 t + 1 · SCG cmbs reits t + 1 t + 1 · LCG cmbs reits t + 1 τ d p cmbs reits t + τ SCG + τ LCG = p cmbs reits t p cmbs reits t = 0 . 02 · 0 . 123 + τ SCG t + 1 · 0 . 001 + τ LCG t + 1 · 0 . 018 . • The dividend yield for CMBS REITs: d cmbs reits t + 1 = 0 . 123 p cmbs reits t • The realized short-term capital gains yield for CMBS REITs: SCG cmbs reits t + 1 = 0 . 001 p cmbs reits t • The realized long-term capital gains yield for CMBS REITs: LCG cmbs reits t + 1 = 0 . 018 p cmbs reits t () May 12, 2014 18 / 24
• Tax Yield for CMBS REITs: TY cmbs reits t + 1 = 0 . 02 · 0 . 123 + τ SCG t + 1 · 0 . 001 + τ LCG t + 1 · 0 . 018 . • The mean tax yield for shareholders of CMBS REITs: E [ TY cmbs reits t + 1 ] = 0 . 0061 . () May 12, 2014 19 / 24
() May 12, 2014 20 / 24
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