The Money Value of a Man Mark Huggett Greg Kaplan 1 Common View: - PowerPoint PPT Presentation

The Money Value of a Man Mark Huggett Greg Kaplan 1

Common View: Most valuable asset that most people hold is their own human capital. Questions: What are the properties of the value of an individual’s human capital? What are the properties of the associated returns? 2

Value and Return Concepts: J � v j ≡ E j [ m j,k e k ] k = j +1 j +1 ≡ v j +1 + e j +1 R h v j What we do: (1) justify this notion of value and (2) analyze ( v j , R h j ) using US data . 3

Motivation: 1. Portfolio advice. 2. International portfolio diversification puzzle 3. Welfare Gains of moving to a smoother consumption plan 4. Estimating preference parameters 4

Preview of our Findings 1. Value - Far below the value of discounting future earnings at the risk- free rate - Stock component is smaller than the bond component - Large negative orthogonal component early in life 2. Returns - Mean returns very large early in life and decline with age - Small positive correlation with stock return 5

Related Literature: 1. Value of Human Capital - Farr (1853), Weisbrod (1961), Becker (1975), ... discount earnings at a deterministic rate. Not useful for analyzing returns. 2. Return to Human Capital - Campbell (1996), Baxter and Jermann (1997), ... use aggregate earnings data - Huggett and Kaplan (2011) put bounds on values/returns. - Mincerian Returns literature focuses on a different notion of a return. 3. Portfolio Allocation - Campbell et al (2001), Benzoni, Collin-Dufresne and Goldstein (2007), Lynch and Tan (2011). - focus is on understanding what impacts portfolio composition 6

Justification for the Notion of Value: Problem P1 max U ( c, n ) where c = ( c 1 , .., c J ) and n = ( n 1 , .., n J ) i ∈I a i i ∈I a i j R i 1 c j + � j +1 = e j + � j 2 e j = G j ( y j , n j , z j ), 0 ≤ n j ≤ 1 and a i J +1 ≥ 0 Important: U is concave Unimportant: Restrictions on G , number of assets, ... 7

Problem P2 max U ( c, n ) i ∈I a i i ∈I a i j R i 1 c j + � j +1 + s j +1 v j + p j n j = s j ( v j + d j ) + � j 2 0 ≤ n j ≤ 1 and a i J +1 ≥ 0 Personalized prices: ( p j , v j ) 8

Definitions: 1. m j,k ( z k ) = dU ( c ∗ ,n ∗ ) /dc k ( z k ) 1 P ( z k | z j ) - stochastic discount factor dU ( c ∗ ,n ∗ ) /dc j ( z j ) 2. v j ( z j ) = E [ � J k = j +1 m j,k d k | z j ] - value of human capital 3. d j ( z j ) = e ∗ j ( z j ) + p j ( z j ) n ∗ j ( z j ) - dividends 4. p j ( z j ) = dU ( c ∗ ,n ∗ ) /dn j ( z j ) dU ( c ∗ ,n ∗ ) /dc j ( z j ) - price of leisure 9

Theorem: If ( c ∗ , n ∗ , e ∗ , y ∗ , a ∗ ) solves P1 and c ∗ is strictly positive, then ( c ∗ , n ∗ , a ∗ , s ∗ ) solves P2, where s ∗ j = 1 , when the agent takes ( v j , d j , p j ) as given. Intuition : P2 is a concave program. Necessary conditions to P1 are sufficient for P2. Extensions: value earnings and leisure components separately or allow financials frictions 10

1 Naive Value: v naive = E [ � J (1+ r ) k − j d k | z j ] j k = j +1 Our Notion of Value: v j = E [ � J k = j +1 m j,k d k | z j ] When will the two notions differ? J J E [ m j,k | z j ] E [ d k | z j ] + cov ( m j,k , d k | z j ) � � v j = k = j +1 k = j +1 When: E [ m j,k | z j ] < 1 / (1 + r ) k − j agent is on a corner When: cov ( m j,k , d k | z j ) < 0 11

Simple Example max E [ � J j =1 β j − 1 u ( c j ) | z 1 ] subject to (1) c j + a j +1 ≤ a j (1 + r ) + e j , (2) c j ≥ 0 , a J +1 ≥ 0 c 1 − ρ u ( c ) = (1 − ρ ) - CRRA e j = � j k =1 z k and ln z k ∼ N ( µ, σ 2 ) i.i.d. (1 + r ) > 0 - one risk-free asset 12

β exp ( ρµ − ρ 2 σ 2 Theorem: If 1 + r = 1 2 ) and initial assets are zero, then ( c j , a j +1 ) = ( e j , 0) , ∀ j solves the decision problem. Further- more, at this solution: (i) the value of human capital is v j ( z j ) = f j e j , k = j +1 β k − j exp (( k − j )[( − ρ + 1) µ + ( − ρ + 1) 2 σ 2 f j = � J 2 ]) . (ii) the return to human capital satisfies: j +1 = ( 1+ f j +1 (a) R h ) z j +1 f j β exp ( µρ + σ 2 j +1 | z j ] = 1 (b) E [ R h 2 (1 − (1 − ρ ) 2 )) 13

Quantitative Analysis: Model Periods: age 20 − 65 Flat Earnings Profile: e 1 = 1 and µ = − σ 2 / 2 and σ ∈ [0 , . 3] Risk Aversion Parameter: ρ ∈ { 1 , 2 , 4 } β exp ( ρµ − ρ 2 σ 2 Interest Rate: 1 . 01 = 1 + r = 1 2 ) 14

40 45 40 35 ρ = 1 ρ = 2 35 30 ρ = 4 30 25 25 20 20 15 15 Naive value ρ = 1 10 10 ρ = 2 ρ = 4 5 5 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.05 0.1 0.15 0.2 0.25 0.3 σ : st dev earnings shocks σ : st dev earnings shocks (a) Value of human capital (b) Mean return human capital (%) 600 Marginal benefit ( ρ = 2) Total benefit ( ρ = 2) 500 400 300 200 100 0 0 0.05 0.1 0.15 0.2 0.25 0.3 σ : st dev earnings shocks (c) Benefit of moving to a smooth consumption plan (%) Figure 1: Human capital values and returns: simple example Notes: 42

Marginal Welfare Gains - Alvarez and Jermann (2004) U ((1 + Ω( α )) c ) = U ((1 − α ) c + αc smooth ) z j dU ( c ) � J dc j ( z j ) ( c smooth ( z j ) − c j ( z j )) � j =1 j Ω ′ (0) = z j dU ( c ) � J dc j ( z j ) c j ( z j ) � j =1 E [ � J j =1 m 1 ,j c smooth | z 1 ] j Ω ′ (0) = 1 ( z 1 ) R i ( z 1 ) − 1 v e 1 ( z 1 ) + e 1 ( z 1 ) + � i ∈I a i 15

Benchmark Model: U ( c 1 , ..., c J ) = W ( c 1 , F ( U ( c 2 , ..., c J )); j ) - Epstein-Zin utility W ( a, b ; j ) = [(1 − β ) a 1 − ρ + βψ j +1 b 1 − ρ ] 1 / (1 − ρ ) F ( x ) = ( E [ x 1 − α ]) 1 / (1 − α ) Two assets: bond and stock Exogenous earnings: e j = G j ( z j ) Extra restrictions: a s j +1 ≥ 0 and a s j +1 ≤ p ( a s j +1 + a b j +1 ) 16

Empirical Framework I: log e i,j,t = u 1 t + u 2 i,j,t i,j,t = α i + κ j + ζ i,j,t + ν i,j,t u 2 ζ i,j,t +1 = ρζ i,j,t + η i,j,t +1 and ζ i, 0 ,t = 0 α ∼ N (0 , σ 2 α ) , η ∼ N (0 , σ 2 η (∆ u 1 t )) , ν ∼ N (0 , σ 2 ν (∆ u 1 t )) 17

t , P t ) ′ where log R s Empirical Framework II: y t = ( u 1 t = ∆ P t p � y t = v + A i y t − i + ε t i =1 If y t ∼ I (1) can rewrite as VECM ( w t ≡ β ′ y t + µ ) p − 1 � � � � � � � � � � ∆ y t Γ i γ α ε t � = + ∆ y t − i + w t − 1 + β ′ γ β ′ Γ i 1 + β ′ α β ′ ε t w t i =1 � � � � � � � � � � ∆ y t Γ ∆ y t − 1 γ α ε t = + + β ′ γ β ′ Γ 1 + β ′ α β ′ ε t w t w t − 1 18

Data Sources: Idiosyncratic Earnings: - PSID 1967- 1996 (three samples: FULL, HS and COL) - Male heads age 22-60 w/ annual earnings > 1 , 000 Aggregate Component: - PSID 1967-96 and CPS 1967-2009 (three samples) Asset Returns: Ken French’s Data Archive: (1) stock return is NYSE/AMEX/NASDAQ return, (2) bond return is based on 6-month TBill 19

Table 1: Idiosyncratic Earnings Process Parameters Full College High School Sample Sub-sample Sub-sample ρ 0.957 0.959 0.902 σ 2 0.110 0.092 0.121 α av. σ 2 0.033 0.033 0.039 η σ 2 η ( L ) − σ 2 η ( H ) 0.020 0.012 0.025 av. σ 2 0.150 0.151 0.147 ε Linear time trend in σ 2 0.0011 0.0014 0.0013 η 20

Table 3: Steady-State Statistics for the Aggregate Process Full Sample Data No Cointegration With Cointegration logR b E � � 0.012 0.012 0.012 t E (log R s 0.041 0.045 0.047 t ) E � ∆ u 1 � -0.002 -0.004 -0.003 t ∆ u 1 � � sd 0.025 0.025 0.025 t sd (log R s t ): 0.187 0.187 0.188 ∆ u 1 t , log R s corr � � 0.184 0.177 0.172 t corr � ∆ u 1 t , ∆ u 1 � 0.425 0.441 0.420 t − 1 corr � log R s t , log R s � 0.057 0.055 0.039 t − 1 ∆ u 1 t log R s corr � � 0.372 0.398 0.390 t − 1 corr � log R s t , ∆ u 1 � -0.292 -0.270 -0.292 t − 1 21

Table 3: Steady-State Statistics for the Aggregate Process College Sub-sample Data No Cointegration With Cointegration logR b E � � 0.012 0.012 0.012 t E (log R s 0.041 0.040 0.045 t ) E � ∆ u 1 � 0.000 -0.001 -0.001 t ∆ u 1 � � sd 0.023 0.023 0.023 t sd (log R s t ): 0.187 0.187 0.186 ∆ u 1 t , log R s corr � � 0.248 0.251 0.243 t corr � ∆ u 1 t , ∆ u 1 � 0.346 0.341 0.342 t − 1 corr � log R s t , log R s � 0.057 0.084 0.050 t − 1 ∆ u 1 t log R s corr � � 0.377 0.387 0.367 t − 1 corr � log R s t , ∆ u 1 � -0.225 -0.235 -0.229 t − 1 22

Table 3: Steady-State Statistics for the Aggregate Process High School Sub-sample Data No Cointegration With Cointegration logR b E � � 0.012 0.012 0.012 t E (log R s 0.041 0.045 0.055 t ) E � ∆ u 1 � -0.007 -0.010 -0.008 t ∆ u 1 � � sd 0.030 0.030 0.030 t sd (log R s t ): 0.187 0.187 0.190 ∆ u 1 t , log R s corr � � 0.207 0.194 0.192 t corr � ∆ u 1 t , ∆ u 1 � 0.386 0.416 0.408 t − 1 corr � log R s t , log R s � 0.057 0.047 0.037 t − 1 ∆ u 1 t log R s corr � � 0.387 0.420 0.420 t − 1 corr � log R s t , ∆ u 1 � -0.289 -0.261 -0.277 t − 1 23