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Introduction Two-Period Example Life-Cycle Model Discussion Insuring Consumption Using Income-Linked Assets Andreas Fuster and Paul Willen Harvard University and Federal Reserve Bank of Boston Conference on Household Finance and

  1. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion What we find 1 Usefulness of income-linked assets depends strongly on how they are implemented: � ILL generally more beneficial than IHI � Correlation with income shocks � Volatility Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 38

  2. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion What we find 1 Usefulness of income-linked assets depends strongly on how they are implemented: � ILL generally more beneficial than IHI � Correlation with income shocks � Volatility 2 The income-linked assets (in particular ILL) can produce non-negligible welfare gains ( > 1%) Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 38

  3. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion What we find 1 Usefulness of income-linked assets depends strongly on how they are implemented: � ILL generally more beneficial than IHI � Correlation with income shocks � Volatility 2 The income-linked assets (in particular ILL) can produce non-negligible welfare gains ( > 1%) 3 But difficult to reduce a large fraction of the welfare costs from labor income risk with the assets we consider Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 38

  4. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion “Asset Pricing” Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38

  5. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion “Asset Pricing” What would be... � E( r ) ? � σ ( r ) ? � corr ( r, income ) ? Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38

  6. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion “Asset Pricing” What would be... � E( r ) ? � σ ( r ) ? � corr ( r, income ) ? We remain relatively agnostic & try various assumptions Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38

  7. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion “Asset Pricing” What would be... � E( r ) ? � σ ( r ) ? � corr ( r, income ) ? We remain relatively agnostic & try various assumptions Baseline assumption for | corr ( r, income ) | : 0.5, based on CPS occupation-level income series (Davis et al. 2009) Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38

  8. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion “Asset Pricing” What would be... � E( r ) ? � σ ( r ) ? � corr ( r, income ) ? We remain relatively agnostic & try various assumptions Baseline assumption for | corr ( r, income ) | : 0.5, based on CPS occupation-level income series (Davis et al. 2009) Baseline assumption for E( r ) : “actuarial fairness” � E(˜ r IHI ) = r l (risk-free saving rate) � E(˜ r ILL ) = r b (risk-free borrowing rate) Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38

  9. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion “Asset Pricing” What would be... � E( r ) ? � σ ( r ) ? � corr ( r, income ) ? We remain relatively agnostic & try various assumptions Baseline assumption for | corr ( r, income ) | : 0.5, based on CPS occupation-level income series (Davis et al. 2009) Baseline assumption for E( r ) : “actuarial fairness” � E(˜ r IHI ) = r l (risk-free saving rate) � E(˜ r ILL ) = r b (risk-free borrowing rate) This assumes that risks are cross-sectional (not aggregate), and in that sense stacks the deck in favor of these assets Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38

  10. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion Related Literature Quantitative dynamic macro models that consider welfare costs of income shocks � Storesletten, Telmer, Yaron (2004), Heathcote, Storesletten, Violante (2008) Risk-sharing and partial insurance � Attanasio and Davis (1996), Krueger and Perri (2006), Blundell et al. (2008) Optimal portfolio choice over the life cycle � Cocco, Gomes, Maenhout (2005), Gomes and Michaelides (2005) � Our model builds on Davis, K¨ ubler, Willen (2006) � Close in spirit: De Jong, Driessen, Van Hemert (2008) on housing futures; Cocco and Gomes (2009) on longevity bonds Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 9 / 38

  11. Introduction Motivation Two-Period Example Overview Life-Cycle Model Related Literature Discussion Outline 1 Two-period example � Goal: provide intuition for what determines demand for and welfare gains from income-linked assets 2 Life-cycle model � Goal: show that intuition carries over; quantitatively assess use and usefulness of assets over life cycle 3 Discussion / Conclusion Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 10 / 38

  12. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Two-Period Example: Setup CRRA=2 investor lives for 2 periods Objective: max u ( c 1 ) + Eu ( c 2 ) Has some cash-on-hand in period 1 Receives stochastic income in period 2 with mean 8 � Y 2 ∈ { 5.4, 8, 10.6 } with p = { 1/6, 2/3, 1/6 } Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 11 / 38

  13. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Two-Period Example: Setup CRRA=2 investor lives for 2 periods Objective: max u ( c 1 ) + Eu ( c 2 ) Has some cash-on-hand in period 1 Receives stochastic income in period 2 with mean 8 � Y 2 ∈ { 5.4, 8, 10.6 } with p = { 1/6, 2/3, 1/6 } Benchmark: Investor can... � save at r l = 2% � invest in equity with E(˜ r e ) = 6% and σ (˜ r e ) = 16% � borrow at r b = 8% Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 11 / 38

  14. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Two-Period Example: Setup CRRA=2 investor lives for 2 periods Objective: max u ( c 1 ) + Eu ( c 2 ) Has some cash-on-hand in period 1 Receives stochastic income in period 2 with mean 8 � Y 2 ∈ { 5.4, 8, 10.6 } with p = { 1/6, 2/3, 1/6 } Benchmark: Investor can... � save at r l = 2% � invest in equity with E(˜ r e ) = 6% and σ (˜ r e ) = 16% � borrow at r b = 8% Constraints: b , l , e ≥ 0 No default in model, so positive lower bound on Y 2 important (otherwise no borrowing possible) Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 11 / 38

  15. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Benchmark E ( Y 2 ) = 8 , r l = 0 . 02 , r b = 0 . 08 , E(˜ r e ) = 0 . 06 , σ e = 0 . 16 Benchmark: Optimal Asset Holdings 4 3 2 � Equity 1 Asset holding 0 -1 � Borrowing -2 -3 -4 0 2 4 6 8 10 12 14 16 Cash-on-hand Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 12 / 38

  16. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Income-Hedging Instrument Now, investor can additionally invest in income-hedging instrument E(˜ r IHI ) = r l = 2% σ (˜ r IHI ) = 25% corr (˜ r IHI , Y 2 ) = –0.5 ⇒ IHI provides some insurance benefits, but not perfect insurance Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 13 / 38

  17. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare IHI: Optimal Asset Holdings Optimal IHI holdings nonlinear With Income-Hedging Instrument in cash-on-hand 4 � Over some range of 3 cash-on-hand, no IHI holdings � Relatively poor and relatively 2 � Income-hedging instrument � Equity rich investors find IHI most 1 ↓ Asset holding attractive 0 -1 � Borrowing -2 -3 -4 0 2 4 6 8 10 12 14 16 Cash-on-hand Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 14 / 38

  18. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare IHI: Optimal Asset Holdings Optimal IHI holdings nonlinear With Income-Hedging Instrument in cash-on-hand 4 � Over some range of 3 cash-on-hand, no IHI holdings � Relatively poor and relatively 2 � Income-hedging instrument � Equity rich investors find IHI most 1 Asset holding attractive 0 Compared with benchmark: -1 � Borrowing by poor investor � Borrowing -2 increases -3 � Equity holdings by rich decrease -4 0 2 4 6 8 10 12 14 16 Cash-on-hand Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 14 / 38

  19. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare What determines optimal asset holdings? How much a households holds of each asset depends on the risk-adjusted returns E Q ( ˜ R i ) of all assets � E Q ( ˜ R i ) higher if i pays off a lot in states of the world with high u � ( c ) Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 15 / 38

  20. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare What determines optimal asset holdings? How much a households holds of each asset depends on the risk-adjusted returns E Q ( ˜ R i ) of all assets � E Q ( ˜ R i ) higher if i pays off a lot in states of the world with high u � ( c ) For IHI, have that E Q ( ˜ R IHI ) > E( ˜ R IHI ) = R l Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 15 / 38

  21. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare What determines optimal asset holdings? How much a households holds of each asset depends on the risk-adjusted returns E Q ( ˜ R i ) of all assets � E Q ( ˜ R i ) higher if i pays off a lot in states of the world with high u � ( c ) For IHI, have that E Q ( ˜ R IHI ) > E( ˜ R IHI ) = R l So if household could borrow at R l , would always hold IHI Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 15 / 38

  22. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare What determines optimal asset holdings? How much a households holds of each asset depends on the risk-adjusted returns E Q ( ˜ R i ) of all assets � E Q ( ˜ R i ) higher if i pays off a lot in states of the world with high u � ( c ) For IHI, have that E Q ( ˜ R IHI ) > E( ˜ R IHI ) = R l So if household could borrow at R l , would always hold IHI However, for households who must borrow at higher rate, only buy IHI if E Q ( ˜ R IHI ) ≥ R b � What determines whether a household borrows? Expected future consumption growth ⇒ borrow if relatively poor today Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 15 / 38

  23. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare What determines optimal asset holdings? How much a households holds of each asset depends on the risk-adjusted returns E Q ( ˜ R i ) of all assets � E Q ( ˜ R i ) higher if i pays off a lot in states of the world with high u � ( c ) For IHI, have that E Q ( ˜ R IHI ) > E( ˜ R IHI ) = R l So if household could borrow at R l , would always hold IHI However, for households who must borrow at higher rate, only buy IHI if E Q ( ˜ R IHI ) ≥ R b � What determines whether a household borrows? Expected future consumption growth ⇒ borrow if relatively poor today And for households who save, IHI “competes” against equity ⇒ only buy IHI if E Q ( ˜ R IHI ) ≥ E Q ( ˜ R e ) Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 15 / 38

  24. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare IHI: Optimal Asset Holdings Optimal IHI holdings nonlinear With Income-Hedging Instrument in cash-on-hand 4 � Over some range of 3 cash-on-hand, no IHI holdings � Relatively poor and relatively 2 � Income-hedging instrument � Equity rich investors find IHI most 1 Asset holding attractive 0 Compared with benchmark: -1 � Borrowing by poor investor � Borrowing -2 increases -3 � Equity holdings by rich decrease -4 0 2 4 6 8 10 12 14 16 Cash-on-hand Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 16 / 38

  25. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Bottom line: Whether and how extensively investor will use income-linked asset will depend on Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 17 / 38

  26. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Bottom line: Whether and how extensively investor will use income-linked asset will depend on � his financial wealth Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 17 / 38

  27. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Bottom line: Whether and how extensively investor will use income-linked asset will depend on � his financial wealth � his life-cycle income profile Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 17 / 38

  28. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Bottom line: Whether and how extensively investor will use income-linked asset will depend on � his financial wealth � his life-cycle income profile � the risk-adjusted returns of other investment opportunities Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 17 / 38

  29. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Bottom line: Whether and how extensively investor will use income-linked asset will depend on � his financial wealth � his life-cycle income profile � the risk-adjusted returns of other investment opportunities The welfare gain from an income-linked asset will depend on its (opportunity) cost � High for IHI Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 17 / 38

  30. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Income-Linked Loan Now, instead, investor can additionally borrow through income-linked loan E(˜ r ILL ) = r b = 8% σ (˜ r ILL ) = 25% corr (˜ r ILL , Y 2 ) = +0.5 Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 18 / 38

  31. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare ILL: Optimal Asset Holdings Optimal ILL borrowing nonlinear With Income-Linked Loan in cash-on-hand 4 � Goes to zero as cash-on-hand 3 increases 2 Equity � 1 Asset holding 0 � Borrowing -1 � Income-linked loan -2 -3 -4 0 2 4 6 8 10 12 14 16 Cash-on-hand Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 19 / 38

  32. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare ILL: Optimal Asset Holdings Optimal ILL borrowing nonlinear With Income-Linked Loan in cash-on-hand 4 � Goes to zero as cash-on-hand 3 increases 2 Equity � Compared with benchmark: 1 Asset holding � ILL substitutes for unsecured 0 borrowing � Borrowing � Over some range, additional -1 � Income-linked loan borrowing & investment in -2 equity ( E Q ( ˜ R ILL ) = E Q ( ˜ -3 R e ) , even though E( ˜ R ILL ) > E( ˜ R e ) ) -4 0 2 4 6 8 10 12 14 16 Cash-on-hand Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 19 / 38

  33. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Welfare Gains over Benchmark Welfare measure: certainty-equivalent Welfare gains from the two assets consumption ¯ c s.th. 2.5 Gain in certainty-equivalent consumption (in %) u ( c 1 ) + Eu ( c 2 ) = 2 u (¯ c ) 2 1.5 � ILL 1 0.5 � IHI 0 0 2 4 6 8 10 12 14 16 Cash-on-hand Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 20 / 38

  34. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Welfare Gains over Benchmark Welfare measure: certainty-equivalent Welfare gains from the two assets consumption ¯ c s.th. 2.5 Gain in certainty-equivalent consumption (in %) u ( c 1 ) + Eu ( c 2 ) = 2 u (¯ c ) 2 ILL provides larger gains over wide range of cash-on-hand 1.5 � ILL 1 0.5 � IHI 0 0 2 4 6 8 10 12 14 16 Cash-on-hand Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 20 / 38

  35. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Welfare Gains over Benchmark Welfare measure: certainty-equivalent Welfare gains from the two assets consumption ¯ c s.th. 2.5 Gain in certainty-equivalent consumption (in %) u ( c 1 ) + Eu ( c 2 ) = 2 u (¯ c ) 2 ILL provides larger gains over wide range of cash-on-hand 1.5 � ILL Intuitively: lower (opportunity) 1 cost 0.5 � IHI 0 0 2 4 6 8 10 12 14 16 Cash-on-hand Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 20 / 38

  36. Introduction Benchmark Two-Period Example Income-Hedging Instrument Life-Cycle Model Income-Linked Loan Discussion Welfare Welfare Gains over Benchmark Welfare measure: certainty-equivalent Welfare gains from the two assets consumption ¯ c s.th. 2.5 Gain in certainty-equivalent consumption (in %) u ( c 1 ) + Eu ( c 2 ) = 2 u (¯ c ) 2 ILL provides larger gains over wide range of cash-on-hand 1.5 � ILL Intuitively: lower (opportunity) 1 cost Welfare gain small as compared 0.5 to having Y 2 =8 for sure � IHI � 9.21% for c-o-h=0 0 � 2.81% for c-o-h=5 0 2 4 6 8 10 12 14 16 � 1.40% for c-o-h=10 Cash-on-hand Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 20 / 38

  37. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  38. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  39. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Can trade three or four financial assets. Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  40. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Can trade three or four financial assets. � equity with stochastic net return ˜ r e , Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  41. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Can trade three or four financial assets. � equity with stochastic net return ˜ r e , � save at a net risk-free rate r l , Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  42. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Can trade three or four financial assets. � equity with stochastic net return ˜ r e , � save at a net risk-free rate r l , � engage in uncollateralized borrowing at the rate r b > r l Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  43. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Can trade three or four financial assets. � equity with stochastic net return ˜ r e , � save at a net risk-free rate r l , � engage in uncollateralized borrowing at the rate r b > r l and either Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  44. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Can trade three or four financial assets. � equity with stochastic net return ˜ r e , � save at a net risk-free rate r l , � engage in uncollateralized borrowing at the rate r b > r l and either � invest in income-hedging instrument with stochastic net return ˜ r IHI , or Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  45. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Can trade three or four financial assets. � equity with stochastic net return ˜ r e , � save at a net risk-free rate r l , � engage in uncollateralized borrowing at the rate r b > r l and either � invest in income-hedging instrument with stochastic net return ˜ r IHI , or � borrow through income-linked loans at the stochastic rate ˜ r ILL . Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  46. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Can trade three or four financial assets. � equity with stochastic net return ˜ r e , � save at a net risk-free rate r l , � engage in uncollateralized borrowing at the rate r b > r l and either � invest in income-hedging instrument with stochastic net return ˜ r IHI , or � borrow through income-linked loans at the stochastic rate ˜ r ILL . No explicit limit on borrowing; have to be able to repay with prob. 1 Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  47. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Can trade three or four financial assets. � equity with stochastic net return ˜ r e , � save at a net risk-free rate r l , � engage in uncollateralized borrowing at the rate r b > r l and either � invest in income-hedging instrument with stochastic net return ˜ r IHI , or � borrow through income-linked loans at the stochastic rate ˜ r ILL . No explicit limit on borrowing; have to be able to repay with prob. 1 Short-sale constraints: e t ≥ 0 , l t ≥ 0 , b t ≥ 0 , IHI t ≥ 0 , ILL t ≥ 0 Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  48. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Life-Cycle Model Households in partial equilibrium, live from 20 to 80, retire at 65 Receive stochastic income Y t during working life Can trade three or four financial assets. � equity with stochastic net return ˜ r e , � save at a net risk-free rate r l , � engage in uncollateralized borrowing at the rate r b > r l and either � invest in income-hedging instrument with stochastic net return ˜ r IHI , or � borrow through income-linked loans at the stochastic rate ˜ r ILL . No explicit limit on borrowing; have to be able to repay with prob. 1 Short-sale constraints: e t ≥ 0 , l t ≥ 0 , b t ≥ 0 , IHI t ≥ 0 , ILL t ≥ 0 Finite-horizon dynamic program, solved computationally Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38

  49. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Our Strategy Start with model that only features e, l, b � Calibrate to match wealth/income before retirement � Demonstrate that this model makes reasonable predictions regarding equity holdings and borrowing over the LC � Use this as benchmark model Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 22 / 38

  50. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Our Strategy Start with model that only features e, l, b � Calibrate to match wealth/income before retirement � Demonstrate that this model makes reasonable predictions regarding equity holdings and borrowing over the LC � Use this as benchmark model Then, add either income-hedging instrument or income-linked loan, with various assumptions about return process � Look at effect on asset holdings over the LC � Evaluate welfare gain from having access to new asset Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 22 / 38

  51. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income Process Income process as standard in consumption/pf choice literature (following Gourinchas-Parker 2002, Cocco et al. 2005): log ( Y it ) = ˜ y t = d t + ˜ η t + ˜ ε t Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38

  52. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income Process Income process as standard in consumption/pf choice literature (following Gourinchas-Parker 2002, Cocco et al. 2005): log ( Y it ) = ˜ y t = d t + ˜ η t + ˜ ε t Deterministic component d t Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38

  53. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income Process Income process as standard in consumption/pf choice literature (following Gourinchas-Parker 2002, Cocco et al. 2005): log ( Y it ) = ˜ y t = d t + ˜ η t + ˜ ε t Deterministic component d t Permanent (random walk) component u t ∼ N ( − σ 2 u / 2 , σ 2 η t = η t − 1 + ˜ ˜ u t , with ˜ u ) Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38

  54. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income Process Income process as standard in consumption/pf choice literature (following Gourinchas-Parker 2002, Cocco et al. 2005): log ( Y it ) = ˜ y t = d t + ˜ η t + ˜ ε t Deterministic component d t Permanent (random walk) component u t ∼ N ( − σ 2 u / 2 , σ 2 η t = η t − 1 + ˜ ˜ u t , with ˜ u ) Temporary component ε t ∼ N ( − σ 2 ε / 2 , σ 2 ˜ ε ) Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38

  55. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income Process Income process as standard in consumption/pf choice literature (following Gourinchas-Parker 2002, Cocco et al. 2005): log ( Y it ) = ˜ y t = d t + ˜ η t + ˜ ε t Deterministic component d t Permanent (random walk) component u t ∼ N ( − σ 2 u / 2 , σ 2 η t = η t − 1 + ˜ ˜ u t , with ˜ u ) Temporary component ε t ∼ N ( − σ 2 ε / 2 , σ 2 ˜ ε ) Shocks are effectively bounded; no zero-income temporary shock Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38

  56. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income Process Income process as standard in consumption/pf choice literature (following Gourinchas-Parker 2002, Cocco et al. 2005): log ( Y it ) = ˜ y t = d t + ˜ η t + ˜ ε t Deterministic component d t Permanent (random walk) component u t ∼ N ( − σ 2 u / 2 , σ 2 η t = η t − 1 + ˜ ˜ u t , with ˜ u ) Temporary component ε t ∼ N ( − σ 2 ε / 2 , σ 2 ˜ ε ) Shocks are effectively bounded; no zero-income temporary shock Retirement income: ˜ y t = log( λ ) + d t R + η t R , t > t R Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38

  57. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income Process Use parameters from Cocco et al. for HS grads: σ u = 0.103, σ ε = 0.272, λ = 0.682. Enter at 20, retire at 65, die at 80. Income over the Lifecycle (Thousands of 1992 USD) 45 40 35 30 ← Mean 25 20 15 10 20 30 40 50 60 70 80 age Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 24 / 38

  58. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income Process Use parameters from Cocco et al. for HS grads: σ u = 0.103, σ ε = 0.272, λ = 0.682. Enter at 20, retire at 65, die at 80. Income over the Lifecycle (Thousands of 1992 USD) 45 40 ← One realization 35 30 ← Mean 25 20 15 10 20 30 40 50 60 70 80 age Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 24 / 38

  59. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Other Parameters for Benchmark CRRA utility with curvature γ = 2 Taste-shifter s.th. consumption drops 10% at retirement Risk-free saving rate: r l = 0.02 Risk-free borrowing rate: r b = 0.08 (Davis et al. 2006) Equity returns: E(˜ r e ) = 0.06, σ e = 0.16 Discount factor: β = 0.936. Chosen to match W/Y = 2.6 of households with head aged 50 to 59 (Laibson et al. 2007) Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 25 / 38

  60. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Benchmark Results Investment and Borrowing over the LC (means) Borrowing & Stock Market Participation over the LC 80 100 70 90 80 60 ← Equity 70 Thousands of 1992 USD 50 % of households 60 40 50 30 40 � Equity 20 ← Borrowing 30 10 20 0 10 � Borrowing -10 0 20 30 40 50 60 70 80 20 30 40 50 60 70 80 age age Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 26 / 38

  61. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Benchmark Results Investment and Borrowing over the LC (means) Borrowing & Stock Market Participation over the LC 80 100 70 90 80 60 ← Equity 70 Thousands of 1992 USD 50 % of households 60 40 50 30 40 � Equity 20 ← Borrowing 30 10 20 0 10 � Borrowing -10 0 20 30 40 50 60 70 80 20 30 40 50 60 70 80 age age Successes: general pattern of borrowing and risky asset holdings (and participation) over the LC Failures: no bond holdings, and almost no borrowing late in life Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 26 / 38

  62. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income-Hedging Instrument Add IHI to benchmark setting. Parameters: r l = 0.02, r b = 0.08, E(˜ r e ) = 0.06, σ e = 0.16 Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 27 / 38

  63. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income-Hedging Instrument Add IHI to benchmark setting. Parameters: r l = 0.02, r b = 0.08, E(˜ r e ) = 0.06, σ e = 0.16 E(˜ r IHI ) = r l = 0.02 Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 27 / 38

  64. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income-Hedging Instrument Add IHI to benchmark setting. Parameters: r l = 0.02, r b = 0.08, E(˜ r e ) = 0.06, σ e = 0.16 E(˜ r IHI ) = r l = 0.02 corr (˜ r IHI , ˜ u ) = { –0.25,–0.5,–0.75,–1 } � Return negatively correlated with permanent shock to income Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 27 / 38

  65. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income-Hedging Instrument Add IHI to benchmark setting. Parameters: r l = 0.02, r b = 0.08, E(˜ r e ) = 0.06, σ e = 0.16 E(˜ r IHI ) = r l = 0.02 corr (˜ r IHI , ˜ u ) = { –0.25,–0.5,–0.75,–1 } � Return negatively correlated with permanent shock to income σ (˜ r IHI ) = { 0.3,0.5 } Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 27 / 38

  66. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income-Hedging Instrument Add IHI to benchmark setting. Parameters: r l = 0.02, r b = 0.08, E(˜ r e ) = 0.06, σ e = 0.16 E(˜ r IHI ) = r l = 0.02 corr (˜ r IHI , ˜ u ) = { –0.25,–0.5,–0.75,–1 } � Return negatively correlated with permanent shock to income σ (˜ r IHI ) = { 0.3,0.5 } LC profiles in Focus on welfare gains from introducing IHI (in paper, look at detail) Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 27 / 38

  67. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Welfare Gains from IHI 3 Gain in CE consumption (% over Benchmark) 2 1 σ = 0 . 5 � σ = 0 . 3 � 0 -0.25 -0.5 -0.75 -1 Correlation Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 28 / 38

  68. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Welfare Gains from IHI 3 Gain in CE consumption (% over Benchmark) 2 1 σ = 0 . 5 � σ = 0 . 3 � 0 -0.25 -0.5 -0.75 -1 Correlation Compare to gain from reducing permanent income shock variance by 25%: 3.5% gain from eliminating all income risk: 16.4% Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 28 / 38

  69. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans IHI: Conclusions Unless returns very highly correlated with income shock and very volatile, IHI not very useful Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 29 / 38

  70. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans IHI: Conclusions Unless returns very highly correlated with income shock and very volatile, IHI not very useful Too “expensive” for young households, who would benefit most from hedge Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 29 / 38

  71. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans IHI: Conclusions Unless returns very highly correlated with income shock and very volatile, IHI not very useful Too “expensive” for young households, who would benefit most from hedge Richer (older) households hold more of IHI, but at expense of equity Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 29 / 38

  72. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans IHI: Conclusions Unless returns very highly correlated with income shock and very volatile, IHI not very useful Too “expensive” for young households, who would benefit most from hedge Richer (older) households hold more of IHI, but at expense of equity u ) (and even in corr 2 ) Welfare gains convex in corr (˜ r IHI , ˜ Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 29 / 38

  73. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income-Linked Loans Now instead add ILL to benchmark setting. Parameters: r l = 0.02, r b =0.08, E(˜ r e ) = 0.06, σ e = 0.16 Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 30 / 38

  74. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income-Linked Loans Now instead add ILL to benchmark setting. Parameters: r l = 0.02, r b =0.08, E(˜ r e ) = 0.06, σ e = 0.16 E(˜ r ILL ) = r b = 0.08 Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 30 / 38

  75. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income-Linked Loans Now instead add ILL to benchmark setting. Parameters: r l = 0.02, r b =0.08, E(˜ r e ) = 0.06, σ e = 0.16 E(˜ r ILL ) = r b = 0.08 corr (˜ r ILL , ˜ u ) = { 0.25,0.5,0.75,1 } � Rate positively correlated with permanent shock to income Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 30 / 38

  76. Introduction Setup Two-Period Example Benchmark Results Life-Cycle Model Income-Hedging Instrument Discussion Income-Linked Loans Income-Linked Loans Now instead add ILL to benchmark setting. Parameters: r l = 0.02, r b =0.08, E(˜ r e ) = 0.06, σ e = 0.16 E(˜ r ILL ) = r b = 0.08 corr (˜ r ILL , ˜ u ) = { 0.25,0.5,0.75,1 } � Rate positively correlated with permanent shock to income σ (˜ r ILL ) = { 0.3,0.5 } LC profiles Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 30 / 38

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