Insuring Consumption Using Income-Linked Assets
Transcript of Insuring Consumption Using Income-Linked Assets
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
Insuring Consumption Using Income-Linked Assets
Andreas Fuster and Paul Willen
Harvard University and Federal Reserve Bank of Boston
Conference on Household Finance and MacroeconomicsBanco de Espana, Madrid
October 16, 2009
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 1 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Introduction
Human capital is the largest component of total household wealth formuch of life
It is also risky: income volatility is high (and supposedly has increasedover past decades)
Much evidence that this leads to consumption volatility, due toimperfect risk-sharing
Not too surprising: risk-sharing is generally difficult because of! informational asymmetries (moral hazard)! limited commitment
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 2 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Introduction
Yet, part of human capital risk is group-specific and cross-sectional
Such risk could be hedged through financial assets with payoffs linkedto group-level income indices
! and without requiring a risk premium for aggregate risk
Shiller (2003) and others have advocated the introduction of newfinancial assets to allow households to better insure against humancapital risk (among others)
Our goal is to evaluate the potential use and usefulness of such assetsfor households’ income risk management over the life cycle
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 3 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
Shiller proposes six types ofinsurance:
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
Shiller proposes six types ofinsurance:
1 Livelihood insurance
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
Shiller proposes six types ofinsurance:
1 Livelihood insurance
2 Home equity insurance
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
Shiller proposes six types ofinsurance:
1 Livelihood insurance
2 Home equity insurance
3 Macro markets
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
Shiller proposes six types ofinsurance:
1 Livelihood insurance
2 Home equity insurance
3 Macro markets
4 Income-linked loans
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
Shiller proposes six types ofinsurance:
1 Livelihood insurance
2 Home equity insurance
3 Macro markets
4 Income-linked loans
5 Inequality insurance
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
Shiller proposes six types ofinsurance:
1 Livelihood insurance
2 Home equity insurance
3 Macro markets
4 Income-linked loans
5 Inequality insurance
6 Intergenerational social security
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
1 Livelihood insurance
3 Macro markets
4 Income-linked loans
In this paper, we consider (anexample of) 1/3, and 4
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
“Imagining the social and economicachievement that could come from anew financial order is difficultbecause we have not seen such analternate world.”
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 5 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
“Imagining the social and economicachievement that could come from anew financial order is difficultbecause we have not seen such analternate world.”⇒ Need to use a model
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 5 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
“Imagining the social and economicachievement that could come from anew financial order is difficultbecause we have not seen such analternate world.”⇒ Need to use a model
“Making such [assets] more widelyavailable would entail work fromboth the private sector and thegovernment.”
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 5 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Motivation
“Imagining the social and economicachievement that could come from anew financial order is difficultbecause we have not seen such analternate world.”⇒ Need to use a model
“Making such [assets] more widelyavailable would entail work fromboth the private sector and thegovernment.”⇒ How large are the benefits? Is it
worth it?
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 5 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we do
Consider a life-cycle portfolio choice model with realistic borrowingand investment opportunities
! Key feature: borrowing rate > lending rate
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 6 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we do
Consider a life-cycle portfolio choice model with realistic borrowingand investment opportunities
! Key feature: borrowing rate > lending rate
Introduce new assets: income-hedging instrument or income-linkedloans
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 6 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we do
Consider a life-cycle portfolio choice model with realistic borrowingand investment opportunities
! Key feature: borrowing rate > lending rate
Introduce new assets: income-hedging instrument or income-linkedloans
! IHI: limited liability asset with returns negatively correlated withincome shock
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 6 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we do
Consider a life-cycle portfolio choice model with realistic borrowingand investment opportunities
! Key feature: borrowing rate > lending rate
Introduce new assets: income-hedging instrument or income-linkedloans
! IHI: limited liability asset with returns negatively correlated withincome shock
! ILL: loan with required repayment positively correlated with incomeshock
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 6 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we do
Consider a life-cycle portfolio choice model with realistic borrowingand investment opportunities
! Key feature: borrowing rate > lending rate
Introduce new assets: income-hedging instrument or income-linkedloans
! IHI: limited liability asset with returns negatively correlated withincome shock
! ILL: loan with required repayment positively correlated with incomeshock
Look at demand for these assets over the life cycle, and predictedwelfare gains that their availability would generate for households
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IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we find
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IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we find
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we find
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL generally more beneficial than IHI
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we find
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL generally more beneficial than IHI! Correlation with income shocks
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we find
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL generally more beneficial than IHI! Correlation with income shocks! Volatility
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we find
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL generally more beneficial than IHI! Correlation with income shocks! Volatility
2 The income-linked assets (in particular ILL) can producenon-negligible welfare gains (>1%)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
What we find
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL generally more beneficial than IHI! Correlation with income shocks! Volatility
2 The income-linked assets (in particular ILL) can producenon-negligible welfare gains (>1%)
3 But difficult to reduce a large fraction of the welfare costs from laborincome risk with the assets we consider
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
“Asset Pricing”
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
“Asset Pricing”
What would be...! E(r)?! σ(r)?! corr(r, income)?
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
“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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
“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 CPSoccupation-level income series (Davis et al. 2009)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
“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 CPSoccupation-level income series (Davis et al. 2009)
Baseline assumption for E(r): “actuarial fairness”! E(rIHI ) = rl (risk-free saving rate)! E(rILL) = rb (risk-free borrowing rate)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
“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 CPSoccupation-level income series (Davis et al. 2009)
Baseline assumption for E(r): “actuarial fairness”! E(rIHI ) = rl (risk-free saving rate)! E(rILL) = rb (risk-free borrowing rate)
This assumes that risks are cross-sectional (not aggregate), and inthat sense stacks the deck in favor of these assets
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
Related Literature
Quantitative dynamic macro models that consider welfare costs ofincome 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, Kubler, Willen (2006)! Close in spirit: De Jong, Driessen, Van Hemert (2008) on housing
futures; Cocco and Gomes (2009) on longevity bonds
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IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
MotivationOverviewRelated Literature
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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Two-Period Example: Setup
CRRA=2 investor lives for 2 periods
Objective: max u(c1) + Eu(c2)
Has some cash-on-hand in period 1
Receives stochastic income in period 2 with mean 8! Y2 ∈ {5.4, 8, 10.6} with p = {1/6, 2/3, 1/6}
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 11 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Two-Period Example: Setup
CRRA=2 investor lives for 2 periods
Objective: max u(c1) + Eu(c2)
Has some cash-on-hand in period 1
Receives stochastic income in period 2 with mean 8! Y2 ∈ {5.4, 8, 10.6} with p = {1/6, 2/3, 1/6}
Benchmark: Investor can...! save at rl = 2%! invest in equity with E(re) = 6% and σ(re) = 16%! borrow at rb = 8%
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 11 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Two-Period Example: Setup
CRRA=2 investor lives for 2 periods
Objective: max u(c1) + Eu(c2)
Has some cash-on-hand in period 1
Receives stochastic income in period 2 with mean 8! Y2 ∈ {5.4, 8, 10.6} with p = {1/6, 2/3, 1/6}
Benchmark: Investor can...! save at rl = 2%! invest in equity with E(re) = 6% and σ(re) = 16%! borrow at rb = 8%
Constraints: b, l, e ≥ 0
No default in model, so positive lower bound on Y2 important(otherwise no borrowing possible)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 11 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Benchmark
E(Y2) = 8, rl = 0.02, rb = 0.08, E(re) = 0.06, σe = 0.16
Cash-on-hand
Ass
etho
ldin
gBenchmark: Optimal Asset Holdings
↖ Borrowing
↖ Equity
0 2 4 6 8 10 12 14 16-4
-3
-2
-1
0
1
2
3
4
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 12 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Income-Hedging Instrument
Now, investor can additionally invest in income-hedging instrument
E(rIHI) = rl = 2%
σ(rIHI) = 25%
corr(rIHI , Y2) = –0.5
⇒ IHI provides some insurance benefits, but not perfect insurance
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 13 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
IHI: Optimal Asset Holdings
Cash-on-hand
Ass
etho
ldin
g
With Income-Hedging Instrument
↖ Borrowing
↖ Equity↙ Income-hedging instrument↓
0 2 4 6 8 10 12 14 16-4
-3
-2
-1
0
1
2
3
4
Optimal IHI holdings nonlinearin cash-on-hand
! Over some range ofcash-on-hand, no IHI holdings
! Relatively poor and relativelyrich investors find IHI mostattractive
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 14 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
IHI: Optimal Asset Holdings
Cash-on-hand
Ass
etho
ldin
g
With Income-Hedging Instrument
↖ Borrowing
↖ Equity↙ Income-hedging instrument
0 2 4 6 8 10 12 14 16-4
-3
-2
-1
0
1
2
3
4
Optimal IHI holdings nonlinearin cash-on-hand
! Over some range ofcash-on-hand, no IHI holdings
! Relatively poor and relativelyrich investors find IHI mostattractive
Compared with benchmark:! Borrowing by poor investor
increases! Equity holdings by rich
decrease
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 14 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
What determines optimal asset holdings?
How much a households holds of each asset depends on therisk-adjusted returns EQ(Ri) of all assets
! EQ(Ri) 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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
What determines optimal asset holdings?
How much a households holds of each asset depends on therisk-adjusted returns EQ(Ri) of all assets
! EQ(Ri) higher if i pays off a lot in states of the world with high u′(c)
For IHI, have that
EQ(RIHI) > E(RIHI) = Rl
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 15 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
What determines optimal asset holdings?
How much a households holds of each asset depends on therisk-adjusted returns EQ(Ri) of all assets
! EQ(Ri) higher if i pays off a lot in states of the world with high u′(c)
For IHI, have that
EQ(RIHI) > E(RIHI) = Rl
So if household could borrow at Rl, would always hold IHI
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 15 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
What determines optimal asset holdings?
How much a households holds of each asset depends on therisk-adjusted returns EQ(Ri) of all assets
! EQ(Ri) higher if i pays off a lot in states of the world with high u′(c)
For IHI, have that
EQ(RIHI) > E(RIHI) = Rl
So if household could borrow at Rl, would always hold IHI
However, for households who must borrow at higher rate, only buyIHI if EQ(RIHI) ≥ Rb
! What determines whether a household borrows? Expected futureconsumption growth ⇒ borrow if relatively poor today
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 15 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
What determines optimal asset holdings?
How much a households holds of each asset depends on therisk-adjusted returns EQ(Ri) of all assets
! EQ(Ri) higher if i pays off a lot in states of the world with high u′(c)
For IHI, have that
EQ(RIHI) > E(RIHI) = Rl
So if household could borrow at Rl, would always hold IHI
However, for households who must borrow at higher rate, only buyIHI if EQ(RIHI) ≥ Rb
! What determines whether a household borrows? Expected futureconsumption growth ⇒ borrow if relatively poor today
And for households who save, IHI “competes” against equity ⇒ onlybuy IHI if EQ(RIHI) ≥ EQ(Re)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 15 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
IHI: Optimal Asset Holdings
Cash-on-hand
Ass
etho
ldin
g
With Income-Hedging Instrument
↖ Borrowing
↖ Equity↙ Income-hedging instrument
0 2 4 6 8 10 12 14 16-4
-3
-2
-1
0
1
2
3
4
Optimal IHI holdings nonlinearin cash-on-hand
! Over some range ofcash-on-hand, no IHI holdings
! Relatively poor and relativelyrich investors find IHI mostattractive
Compared with benchmark:! Borrowing by poor investor
increases! Equity holdings by rich
decrease
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 16 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Bottom line:
Whether and how extensively investor will use income-linked asset willdepend on
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 17 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Bottom line:
Whether and how extensively investor will use income-linked asset willdepend on
! his financial wealth
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 17 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Bottom line:
Whether and how extensively investor will use income-linked asset willdepend on
! his financial wealth! his life-cycle income profile
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 17 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Bottom line:
Whether and how extensively investor will use income-linked asset willdepend 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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Bottom line:
Whether and how extensively investor will use income-linked asset willdepend 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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Income-Linked Loan
Now, instead, investor can additionally borrow through income-linked loan
E(rILL) = rb = 8%
σ(rILL) = 25%
corr(rILL, Y2) = +0.5
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 18 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
ILL: Optimal Asset Holdings
Cash-on-hand
Ass
etho
ldin
g
With Income-Linked Loan
↖ Borrowing
Equity↘
↖ Income-linked loan
0 2 4 6 8 10 12 14 16-4
-3
-2
-1
0
1
2
3
4
Optimal ILL borrowing nonlinearin cash-on-hand
! Goes to zero as cash-on-handincreases
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 19 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
ILL: Optimal Asset Holdings
Cash-on-hand
Ass
etho
ldin
g
With Income-Linked Loan
↖ Borrowing
Equity↘
↖ Income-linked loan
0 2 4 6 8 10 12 14 16-4
-3
-2
-1
0
1
2
3
4
Optimal ILL borrowing nonlinearin cash-on-hand
! Goes to zero as cash-on-handincreases
Compared with benchmark:! ILL substitutes for unsecured
borrowing! Over some range, additional
borrowing & investment inequity(EQ(RILL) = EQ(Re), eventhough E(RILL) > E(Re))
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 19 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Welfare Gains over Benchmark
Cash-on-hand
Gai
nin
cert
aint
y-eq
uiva
lent
cons
umpt
ion
(in
%)
Welfare gains from the two assets
↙ ILL
↙ IHI
0 2 4 6 8 10 12 14 16
0
0.5
1
1.5
2
2.5
Welfare measure:certainty-equivalentconsumption c s.th.u(c1) + Eu(c2) = 2u(c)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 20 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Welfare Gains over Benchmark
Cash-on-hand
Gai
nin
cert
aint
y-eq
uiva
lent
cons
umpt
ion
(in
%)
Welfare gains from the two assets
↙ ILL
↙ IHI
0 2 4 6 8 10 12 14 16
0
0.5
1
1.5
2
2.5
Welfare measure:certainty-equivalentconsumption c s.th.u(c1) + Eu(c2) = 2u(c)
ILL provides larger gains overwide range of cash-on-hand
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 20 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Welfare Gains over Benchmark
Cash-on-hand
Gai
nin
cert
aint
y-eq
uiva
lent
cons
umpt
ion
(in
%)
Welfare gains from the two assets
↙ ILL
↙ IHI
0 2 4 6 8 10 12 14 16
0
0.5
1
1.5
2
2.5
Welfare measure:certainty-equivalentconsumption c s.th.u(c1) + Eu(c2) = 2u(c)
ILL provides larger gains overwide range of cash-on-hand
Intuitively: lower (opportunity)cost
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 20 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
BenchmarkIncome-Hedging InstrumentIncome-Linked LoanWelfare
Welfare Gains over Benchmark
Cash-on-hand
Gai
nin
cert
aint
y-eq
uiva
lent
cons
umpt
ion
(in
%)
Welfare gains from the two assets
↙ ILL
↙ IHI
0 2 4 6 8 10 12 14 16
0
0.5
1
1.5
2
2.5
Welfare measure:certainty-equivalentconsumption c s.th.u(c1) + Eu(c2) = 2u(c)
ILL provides larger gains overwide range of cash-on-hand
Intuitively: lower (opportunity)cost
Welfare gain small as comparedto having Y2=8 for sure
! 9.21% for c-o-h=0! 2.81% for c-o-h=5! 1.40% for c-o-h=10
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 20 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working life
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working lifeCan trade three or four financial assets.
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working lifeCan trade three or four financial assets.
! equity with stochastic net return re,
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working lifeCan trade three or four financial assets.
! equity with stochastic net return re,! save at a net risk-free rate rl,
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working lifeCan trade three or four financial assets.
! equity with stochastic net return re,! save at a net risk-free rate rl,! engage in uncollateralized borrowing at the rate rb > rl
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working lifeCan trade three or four financial assets.
! equity with stochastic net return re,! save at a net risk-free rate rl,! engage in uncollateralized borrowing at the rate rb > rl
and either
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working lifeCan trade three or four financial assets.
! equity with stochastic net return re,! save at a net risk-free rate rl,! engage in uncollateralized borrowing at the rate rb > rl
and either! invest in income-hedging instrument with stochastic net return rIHI ,
or
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working lifeCan trade three or four financial assets.
! equity with stochastic net return re,! save at a net risk-free rate rl,! engage in uncollateralized borrowing at the rate rb > rl
and either! invest in income-hedging instrument with stochastic net return rIHI ,
or! borrow through income-linked loans at the stochastic rate rILL.
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working lifeCan trade three or four financial assets.
! equity with stochastic net return re,! save at a net risk-free rate rl,! engage in uncollateralized borrowing at the rate rb > rl
and either! invest in income-hedging instrument with stochastic net return rIHI ,
or! borrow through income-linked loans at the stochastic rate rILL.
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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working lifeCan trade three or four financial assets.
! equity with stochastic net return re,! save at a net risk-free rate rl,! engage in uncollateralized borrowing at the rate rb > rl
and either! invest in income-hedging instrument with stochastic net return rIHI ,
or! borrow through income-linked loans at the stochastic rate rILL.
No explicit limit on borrowing; have to be able to repay with prob. 1
Short-sale constraints:
et ≥ 0, lt ≥ 0, bt ≥ 0, IHIt ≥ 0, ILLt ≥ 0
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Life-Cycle Model
Households in partial equilibrium, live from 20 to 80, retire at 65
Receive stochastic income Yt during working lifeCan trade three or four financial assets.
! equity with stochastic net return re,! save at a net risk-free rate rl,! engage in uncollateralized borrowing at the rate rb > rl
and either! invest in income-hedging instrument with stochastic net return rIHI ,
or! borrow through income-linked loans at the stochastic rate rILL.
No explicit limit on borrowing; have to be able to repay with prob. 1
Short-sale constraints:
et ≥ 0, lt ≥ 0, bt ≥ 0, IHIt ≥ 0, ILLt ≥ 0
Finite-horizon dynamic program, solved computationally
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 21 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income Process
Income process as standard in consumption/pf choice literature(following Gourinchas-Parker 2002, Cocco et al. 2005):
log(Yit) = yt = dt + ηt + εt
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income Process
Income process as standard in consumption/pf choice literature(following Gourinchas-Parker 2002, Cocco et al. 2005):
log(Yit) = yt = dt + ηt + εt
Deterministic component dt
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income Process
Income process as standard in consumption/pf choice literature(following Gourinchas-Parker 2002, Cocco et al. 2005):
log(Yit) = yt = dt + ηt + εt
Deterministic component dt
Permanent (random walk) component
ηt = ηt−1 + ut, with ut ∼ N(−σ2u/2,σ2
u)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income Process
Income process as standard in consumption/pf choice literature(following Gourinchas-Parker 2002, Cocco et al. 2005):
log(Yit) = yt = dt + ηt + εt
Deterministic component dt
Permanent (random walk) component
ηt = ηt−1 + ut, with ut ∼ N(−σ2u/2,σ2
u)
Temporary component
εt ∼ N(−σ2ε/2,σ
2ε )
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income Process
Income process as standard in consumption/pf choice literature(following Gourinchas-Parker 2002, Cocco et al. 2005):
log(Yit) = yt = dt + ηt + εt
Deterministic component dt
Permanent (random walk) component
ηt = ηt−1 + ut, with ut ∼ N(−σ2u/2,σ2
u)
Temporary component
εt ∼ N(−σ2ε/2,σ
2ε )
Shocks are effectively bounded; no zero-income temporary shock
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income Process
Income process as standard in consumption/pf choice literature(following Gourinchas-Parker 2002, Cocco et al. 2005):
log(Yit) = yt = dt + ηt + εt
Deterministic component dt
Permanent (random walk) component
ηt = ηt−1 + ut, with ut ∼ N(−σ2u/2,σ2
u)
Temporary component
εt ∼ N(−σ2ε/2,σ
2ε )
Shocks are effectively bounded; no zero-income temporary shock
Retirement income: yt = log(λ) + dtR + ηtR , t > tR
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 23 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income ProcessUse parameters from Cocco et al. for HS grads: σu =0.103, σε =0.272,λ =0.682. Enter at 20, retire at 65, die at 80.
←Mean
Income over the Lifecycle (Thousands of 1992 USD)
age20 30 40 50 60 70 80
10
15
20
25
30
35
40
45
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 24 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income ProcessUse parameters from Cocco et al. for HS grads: σu =0.103, σε =0.272,λ =0.682. Enter at 20, retire at 65, die at 80.
←Mean
← One realization
Income over the Lifecycle (Thousands of 1992 USD)
age20 30 40 50 60 70 80
10
15
20
25
30
35
40
45
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 24 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-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: rl =0.02
Risk-free borrowing rate: rb =0.08 (Davis et al. 2006)
Equity returns: E(re) =0.06, σe =0.16
Discount factor: β =0.936. Chosen to match W/Y =2.6 ofhouseholds with head aged 50 to 59 (Laibson et al. 2007)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 25 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Benchmark Results
Investment and Borrowing over the LC (means)
Tho
usan
dsof
1992
USD
age
↖ Equity
↖ Borrowing
20 30 40 50 60 70 80
-10
0
10
20
30
40
50
60
70
80
Borrowing & Stock Market Participation over the LC
%of
hous
ehol
ds
age
← Borrowing
← Equity
20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
100
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 26 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Benchmark Results
Investment and Borrowing over the LC (means)
Tho
usan
dsof
1992
USD
age
↖ Equity
↖ Borrowing
20 30 40 50 60 70 80
-10
0
10
20
30
40
50
60
70
80
Borrowing & Stock Market Participation over the LC
%of
hous
ehol
ds
age
← Borrowing
← Equity
20 30 40 50 60 70 800
10
20
30
40
50
60
70
80
90
100
Successes: general pattern of borrowing and risky asset holdings (andparticipation) 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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income-Hedging Instrument
Add IHI to benchmark setting. Parameters:
rl =0.02, rb =0.08, E(re) =0.06, σe =0.16
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 27 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income-Hedging Instrument
Add IHI to benchmark setting. Parameters:
rl =0.02, rb =0.08, E(re) =0.06, σe =0.16
E(rIHI) = rl =0.02
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 27 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income-Hedging Instrument
Add IHI to benchmark setting. Parameters:
rl =0.02, rb =0.08, E(re) =0.06, σe =0.16
E(rIHI) = rl =0.02
corr(rIHI , 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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income-Hedging Instrument
Add IHI to benchmark setting. Parameters:
rl =0.02, rb =0.08, E(re) =0.06, σe =0.16
E(rIHI) = rl =0.02
corr(rIHI , u) = {–0.25,–0.5,–0.75,–1}! Return negatively correlated with permanent shock to income
σ(rIHI) = {0.3,0.5}
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 27 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income-Hedging Instrument
Add IHI to benchmark setting. Parameters:
rl =0.02, rb =0.08, E(re) =0.06, σe =0.16
E(rIHI) = rl =0.02
corr(rIHI , u) = {–0.25,–0.5,–0.75,–1}! Return negatively correlated with permanent shock to income
σ(rIHI) = {0.3,0.5}
Focus on welfare gains from introducing IHI (in paper, look at LC profiles indetail)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 27 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Welfare Gains from IHI
Correlation
Gai
nin
CE
cons
umpt
ion
(%ov
erB
ench
mar
k)
σ = 0.3 ↘
σ = 0.5 ↘
-0.25 -0.5 -0.75 -1
0
1
2
3
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 28 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Welfare Gains from IHI
Correlation
Gai
nin
CE
cons
umpt
ion
(%ov
erB
ench
mar
k)
σ = 0.3 ↘
σ = 0.5 ↘
-0.25 -0.5 -0.75 -1
0
1
2
3
Compare togain 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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
IHI: Conclusions
Unless returns very highly correlated with income shock and veryvolatile, IHI not very useful
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 29 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
IHI: Conclusions
Unless returns very highly correlated with income shock and veryvolatile, IHI not very useful
Too “expensive” for young households, who would benefit most fromhedge
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 29 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
IHI: Conclusions
Unless returns very highly correlated with income shock and veryvolatile, IHI not very useful
Too “expensive” for young households, who would benefit most fromhedge
Richer (older) households hold more of IHI, but at expense of equity
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 29 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
IHI: Conclusions
Unless returns very highly correlated with income shock and veryvolatile, IHI not very useful
Too “expensive” for young households, who would benefit most fromhedge
Richer (older) households hold more of IHI, but at expense of equity
Welfare gains convex in corr(rIHI , u) (and even in corr2)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 29 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income-Linked Loans
Now instead add ILL to benchmark setting. Parameters:
rl =0.02, rb=0.08, E(re) =0.06, σe =0.16
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 30 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income-Linked Loans
Now instead add ILL to benchmark setting. Parameters:
rl =0.02, rb=0.08, E(re) =0.06, σe =0.16
E(rILL) = rb =0.08
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 30 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income-Linked Loans
Now instead add ILL to benchmark setting. Parameters:
rl =0.02, rb=0.08, E(re) =0.06, σe =0.16
E(rILL) = rb =0.08
corr(rILL, 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
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Income-Linked Loans
Now instead add ILL to benchmark setting. Parameters:
rl =0.02, rb=0.08, E(re) =0.06, σe =0.16
E(rILL) = rb =0.08
corr(rILL, u) = {0.25,0.5,0.75,1}! Rate positively correlated with permanent shock to income
σ(rILL) = {0.3,0.5}
LC profiles
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 30 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
Welfare Gains from ILL
Correlation
Gai
nin
CE
cons
umpt
ion
(%ov
erB
ench
mar
k)
↖ σ = 0.3
σ = 0.5 ↘
0.25 0.5 0.75 1
0
1
2
3
4
5
6
7
8
9
10
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 31 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
ILL: Conclusions
ILL offer more potential for welfare gains than IHI
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 32 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
ILL: Conclusions
ILL offer more potential for welfare gains than IHI! Even for relatively moderate ρ, σ
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 32 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
ILL: Conclusions
ILL offer more potential for welfare gains than IHI! Even for relatively moderate ρ, σ
Young households (who would borrow anyway) would use itextensively and benefit from improved insurance
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 32 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
ILL: Conclusions
ILL offer more potential for welfare gains than IHI! Even for relatively moderate ρ, σ
Young households (who would borrow anyway) would use itextensively and benefit from improved insurance
They invest part of their ILL borrowing in high-return equity
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 32 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
SetupBenchmark ResultsIncome-Hedging InstrumentIncome-Linked Loans
ILL: Conclusions
ILL offer more potential for welfare gains than IHI! Even for relatively moderate ρ, σ
Young households (who would borrow anyway) would use itextensively and benefit from improved insurance
They invest part of their ILL borrowing in high-return equity
Yet, still far from hypothetical welfare gain achieved by reducingincome risk to zero
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 32 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Alternative Investment Option
Our model does not generate enough (any) riskfree asset holdings
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 33 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Alternative Investment Option
Our model does not generate enough (any) riskfree asset holdings
Consequence: may understate benefits from IHI; overstate benefitsfrom ILL
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 33 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Alternative Investment Option
Our model does not generate enough (any) riskfree asset holdings
Consequence: may understate benefits from IHI; overstate benefitsfrom ILL
Check: version of the model where agent can only invest in 50/50stocks-bonds fund (expected return (E(re) + rl)/2, st. dev. 0.5σe)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 33 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Alternative Investment Option
Our model does not generate enough (any) riskfree asset holdings
Consequence: may understate benefits from IHI; overstate benefitsfrom ILL
Check: version of the model where agent can only invest in 50/50stocks-bonds fund (expected return (E(re) + rl)/2, st. dev. 0.5σe)
Set β = 0.947 to match W/Y
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 33 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Alternative Investment Option
Our model does not generate enough (any) riskfree asset holdings
Consequence: may understate benefits from IHI; overstate benefitsfrom ILL
Check: version of the model where agent can only invest in 50/50stocks-bonds fund (expected return (E(re) + rl)/2, st. dev. 0.5σe)
Set β = 0.947 to match W/Y
Gain from IHI with ρ = –0.5 and σ = 0.5 is now 0.33% instead of0.04%
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 33 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Alternative Investment Option
Our model does not generate enough (any) riskfree asset holdings
Consequence: may understate benefits from IHI; overstate benefitsfrom ILL
Check: version of the model where agent can only invest in 50/50stocks-bonds fund (expected return (E(re) + rl)/2, st. dev. 0.5σe)
Set β = 0.947 to match W/Y
Gain from IHI with ρ = –0.5 and σ = 0.5 is now 0.33% instead of0.04%
Gain from ILL with ρ = +0.5 and σ = 0.5 is now 0.85% instead of1.36%
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 33 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Alternative Investment Option
Our model does not generate enough (any) riskfree asset holdings
Consequence: may understate benefits from IHI; overstate benefitsfrom ILL
Check: version of the model where agent can only invest in 50/50stocks-bonds fund (expected return (E(re) + rl)/2, st. dev. 0.5σe)
Set β = 0.947 to match W/Y
Gain from IHI with ρ = –0.5 and σ = 0.5 is now 0.33% instead of0.04%
Gain from ILL with ρ = +0.5 and σ = 0.5 is now 0.85% instead of1.36%
⇒ ILL still generates larger welfare gain than IHI, but differencesmaller
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 33 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Borrowing Cost
We assume an interest rate wedge between borrowing and lending of6%, based on Davis et al. (2006)
! Adjust for tax considerations and charge-offs ⇒ remaining wedge dueto transaction costs etc.
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 34 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Borrowing Cost
We assume an interest rate wedge between borrowing and lending of6%, based on Davis et al. (2006)
! Adjust for tax considerations and charge-offs ⇒ remaining wedge dueto transaction costs etc.
What happens if households had access to cheaper borrowing, at 5%?
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 34 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Borrowing Cost
We assume an interest rate wedge between borrowing and lending of6%, based on Davis et al. (2006)
! Adjust for tax considerations and charge-offs ⇒ remaining wedge dueto transaction costs etc.
What happens if households had access to cheaper borrowing, at 5%?
Welfare gain from baseline IHI: 0.8%
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 34 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Borrowing Cost
We assume an interest rate wedge between borrowing and lending of6%, based on Davis et al. (2006)
! Adjust for tax considerations and charge-offs ⇒ remaining wedge dueto transaction costs etc.
What happens if households had access to cheaper borrowing, at 5%?
Welfare gain from baseline IHI: 0.8%
Welfare gain from baseline ILL with E(rILL) = rb = 0.05: 3%
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 34 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Borrowing Cost
We assume an interest rate wedge between borrowing and lending of6%, based on Davis et al. (2006)
! Adjust for tax considerations and charge-offs ⇒ remaining wedge dueto transaction costs etc.
What happens if households had access to cheaper borrowing, at 5%?
Welfare gain from baseline IHI: 0.8%
Welfare gain from baseline ILL with E(rILL) = rb = 0.05: 3%
Welfare gain from baseline ILL but with E(rILL) = 0.08 > rb: 0.5%
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 34 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Borrowing Cost
We assume an interest rate wedge between borrowing and lending of6%, based on Davis et al. (2006)
! Adjust for tax considerations and charge-offs ⇒ remaining wedge dueto transaction costs etc.
What happens if households had access to cheaper borrowing, at 5%?
Welfare gain from baseline IHI: 0.8%
Welfare gain from baseline ILL with E(rILL) = rb = 0.05: 3%
Welfare gain from baseline ILL but with E(rILL) = 0.08 > rb: 0.5%
Thus, if households had access to borrowing at a cheaper rate thanwhat they would pay on the ILL, result that ILL generates larger gainsthan IHI may be reversed
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 34 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Preferences
With higher risk aversion, welfare gains increase
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 35 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Preferences
With higher risk aversion, welfare gains increase
Try γ = 3, β = 0.92 (and equity as earlier, not 50/50)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 35 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Preferences
With higher risk aversion, welfare gains increase
Try γ = 3, β = 0.92 (and equity as earlier, not 50/50)
Gain from IHI with ρ = –0.5 and σ = 0.5 is now 0.42%(γ = 2: 0.04%)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 35 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Preferences
With higher risk aversion, welfare gains increase
Try γ = 3, β = 0.92 (and equity as earlier, not 50/50)
Gain from IHI with ρ = –0.5 and σ = 0.5 is now 0.42%(γ = 2: 0.04%)
Gain from ILL with ρ = +0.5 and σ = 0.5 is now 2.42%(γ = 2: 1.36%)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 35 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Robustness – Preferences
With higher risk aversion, welfare gains increase
Try γ = 3, β = 0.92 (and equity as earlier, not 50/50)
Gain from IHI with ρ = –0.5 and σ = 0.5 is now 0.42%(γ = 2: 0.04%)
Gain from ILL with ρ = +0.5 and σ = 0.5 is now 2.42%(γ = 2: 1.36%)
⇒ Gains significantly larger with higher risk aversion (as is the welfarecost from life cycle income shocks in the benchmark: 24.5%)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 35 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 36 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL vs. IHI
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 36 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL vs. IHI! Correlation with income shocks
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 36 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL vs. IHI! Correlation with income shocks! Volatility
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 36 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL vs. IHI! Correlation with income shocks! Volatility
2 The income-linked assets (in particular ILL) can producenon-negligible welfare gains
! Baseline welfare gains with |ρ| =0.5, σ =0.5: IHI ≈ 0, ILL ≈ 1.4%(US$400/year)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 36 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL vs. IHI! Correlation with income shocks! Volatility
2 The income-linked assets (in particular ILL) can producenon-negligible welfare gains
! Baseline welfare gains with |ρ| =0.5, σ =0.5: IHI ≈ 0, ILL ≈ 1.4%(US$400/year)
! Attractiveness of alternative investment options matters for relativegains from ILL vs. IHI
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 36 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL vs. IHI! Correlation with income shocks! Volatility
2 The income-linked assets (in particular ILL) can producenon-negligible welfare gains
! Baseline welfare gains with |ρ| =0.5, σ =0.5: IHI ≈ 0, ILL ≈ 1.4%(US$400/year)
! Attractiveness of alternative investment options matters for relativegains from ILL vs. IHI
3 But difficult to reduce a large fraction of the welfare costs from laborincome risk with the assets we have considered
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 36 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL vs. IHI! Correlation with income shocks! Volatility
2 The income-linked assets (in particular ILL) can producenon-negligible welfare gains
! Baseline welfare gains with |ρ| =0.5, σ =0.5: IHI ≈ 0, ILL ≈ 1.4%(US$400/year)
! Attractiveness of alternative investment options matters for relativegains from ILL vs. IHI
3 But difficult to reduce a large fraction of the welfare costs from laborincome risk with the assets we have considered
! Unless they were highly correlated with shocks to permanent income...! or households had access to cheap borrowing
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 36 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
1 Usefulness of income-linked assets depends strongly on how they areimplemented:
! ILL vs. IHI! Correlation with income shocks! Volatility
2 The income-linked assets (in particular ILL) can producenon-negligible welfare gains
! Baseline welfare gains with |ρ| =0.5, σ =0.5: IHI ≈ 0, ILL ≈ 1.4%(US$400/year)
! Attractiveness of alternative investment options matters for relativegains from ILL vs. IHI
3 But difficult to reduce a large fraction of the welfare costs from laborincome risk with the assets we have considered
! Unless they were highly correlated with shocks to permanent income...! or households had access to cheap borrowing
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 36 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
Using a model with realistic portfolio constraints & opportunity costsis key to evaluating new assets
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 37 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
Using a model with realistic portfolio constraints & opportunity costsis key to evaluating new assets
If instead assumed rb = rl = E(rILA) =0.02, model would predict! IHI and ILL equivalent! σ does not matter! even with |ρ| =0.5, welfare gain > 4%
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 37 / 38
IntroductionTwo-Period Example
Life-Cycle ModelDiscussion
RobustnessConclusion
Discussion & Conclusion
THE END – THANK YOU!
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 38 / 38
Appendix: Welfare Measures
Through large number of simulations of stochastic variables, findex-ante lifetime expected utility U
Then, compute certainty equivalent consumption c: constant level ofconsumption such that lifetime utility equals U
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 1 / 8
Appendix: Welfare Measures
Through large number of simulations of stochastic variables, findex-ante lifetime expected utility U
Then, compute certainty equivalent consumption c: constant level ofconsumption such that lifetime utility equals U
For benchmark parameters, eliminating income shocks would raise cby 16.4%
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 1 / 8
Appendix: Welfare Measures
Through large number of simulations of stochastic variables, findex-ante lifetime expected utility U
Then, compute certainty equivalent consumption c: constant level ofconsumption such that lifetime utility equals U
For benchmark parameters, eliminating income shocks would raise cby 16.4%
Alternative measure to assess effect of new assets: coefficient ofpartial insurance against shocks (Kaplan and Violante, 2008):
φut = 1 −
cov(∆cit, uit)
var(uit)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 1 / 8
Appendix: Welfare Measures
Through large number of simulations of stochastic variables, findex-ante lifetime expected utility U
Then, compute certainty equivalent consumption c: constant level ofconsumption such that lifetime utility equals U
For benchmark parameters, eliminating income shocks would raise cby 16.4%
Alternative measure to assess effect of new assets: coefficient ofpartial insurance against shocks (Kaplan and Violante, 2008):
φut = 1 −
cov(∆cit, uit)
var(uit)The lower this coefficient, the more an income shock translates intoconsumption changes. φ = 1: perfect insurance.
In benchmark, φu = 0.08 and φε = 0.9: easy to insure againsttransitory shocks, hard to insure against permanent ones.
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 1 / 8
Partial Insurance Coefficients with IHI
ρ = −0.5 ↘↖ Benchmark
ρ = −0.75 ↘
ρ = −1 ↘
Insurance coefficients (against perm. shocks) during working life
age
Insu
ranc
eco
effici
ent
20 25 30 35 40 45 50 55 60 65
0
0.2
0.4
0.6
0.8
1
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 2 / 8
Partial Insurance Coefficients with ILL
ρ = 0.5 ↘
↖ Benchmark
ρ = 0.75 ↓
ρ = 1 ↓
Insurance coefficients (against perm. shocks) during working life
age
Insu
ranc
eco
effici
ent
20 25 30 35 40 45 50 55 60 65
0
0.2
0.4
0.6
0.8
1
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 3 / 8
Computational Solution
Closely follow Davis, Kubler, Willen (2006)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 8
Computational Solution
Closely follow Davis, Kubler, Willen (2006)
Finite-horizon dynamic program, solved by backward induction
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 8
Computational Solution
Closely follow Davis, Kubler, Willen (2006)
Finite-horizon dynamic program, solved by backward induction
Get rid of one state by exploiting scale-independence and dividingeverything by permanent income
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 8
Computational Solution
Closely follow Davis, Kubler, Willen (2006)
Finite-horizon dynamic program, solved by backward induction
Get rid of one state by exploiting scale-independence and dividingeverything by permanent income
Thus, state variables: normalized cash-on-hand (xt) and age (t)
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 8
Computational Solution
Closely follow Davis, Kubler, Willen (2006)
Finite-horizon dynamic program, solved by backward induction
Get rid of one state by exploiting scale-independence and dividingeverything by permanent income
Thus, state variables: normalized cash-on-hand (xt) and age (t)
3 or 4 asset holding decisions, with short-sale constraints on all ofthem
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 8
Computational Solution
Closely follow Davis, Kubler, Willen (2006)
Finite-horizon dynamic program, solved by backward induction
Get rid of one state by exploiting scale-independence and dividingeverything by permanent income
Thus, state variables: normalized cash-on-hand (xt) and age (t)
3 or 4 asset holding decisions, with short-sale constraints on all ofthem
Solve by policy-function iteration, as FOCs necessary and sufficient
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 4 / 8
Computational Solution
Use Garcia-Zangwill (1981) “trick” to transform Kuhn-Tuckerconditions into system of nonlinear equations. E.g. for et:
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 5 / 8
Computational Solution
Use Garcia-Zangwill (1981) “trick” to transform Kuhn-Tuckerconditions into system of nonlinear equations. E.g. for et:
u′(
ct︷ ︸︸ ︷
xt + bt − lt − et) − βE[(1 + re)u′(ct+1)] − µe,t = 0et ≥ 0, µe,t ≥ 0, etµe,t = 0
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 5 / 8
Computational Solution
Use Garcia-Zangwill (1981) “trick” to transform Kuhn-Tuckerconditions into system of nonlinear equations. E.g. for et:
u′(
ct︷ ︸︸ ︷
xt + bt − lt − et) − βE[(1 + re)u′(ct+1)] − µe,t = 0et ≥ 0, µe,t ≥ 0, etµe,t = 0
Define et = (max{0,λe})κ and µe,t = max({0,−λe})κ.
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 5 / 8
Computational Solution
Use Garcia-Zangwill (1981) “trick” to transform Kuhn-Tuckerconditions into system of nonlinear equations. E.g. for et:
u′(
ct︷ ︸︸ ︷
xt + bt − lt − et) − βE[(1 + re)u′(ct+1)] − µe,t = 0et ≥ 0, µe,t ≥ 0, etµe,t = 0
Define et = (max{0,λe})κ and µe,t = max({0,−λe})κ.
Then, (max{0,λe})κ ≥ 0, (max{0,−λe})κ ≥ 0, and(max{0,λe})κ · (max{0,−λe})κ = 0
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 5 / 8
Computational Solution
Use Garcia-Zangwill (1981) “trick” to transform Kuhn-Tuckerconditions into system of nonlinear equations. E.g. for et:
u′(
ct︷ ︸︸ ︷
xt + bt − lt − et) − βE[(1 + re)u′(ct+1)] − µe,t = 0et ≥ 0, µe,t ≥ 0, etµe,t = 0
Define et = (max{0,λe})κ and µe,t = max({0,−λe})κ.
Then, (max{0,λe})κ ≥ 0, (max{0,−λe})κ ≥ 0, and(max{0,λe})κ · (max{0,−λe})κ = 0
⇒ can solve equation that is differentiable of degree κ–1 for λe (andsimilarly for other assets and multipliers).
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 5 / 8
Computational Solution
Discretization: 2 nodes for income shocks, 3 for equity, 4 forincome-linked assets
! Results not sensitive to adding more nodes, as long as lowest incomeshock “not too small”
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 6 / 8
Computational Solution
Discretization: 2 nodes for income shocks, 3 for equity, 4 forincome-linked assets
! Results not sensitive to adding more nodes, as long as lowest incomeshock “not too small”
Set bounds of grid for cash-on-hand s.th. never move out of boundsin simulations
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 6 / 8
Computational Solution
Discretization: 2 nodes for income shocks, 3 for equity, 4 forincome-linked assets
! Results not sensitive to adding more nodes, as long as lowest incomeshock “not too small”
Set bounds of grid for cash-on-hand s.th. never move out of boundsin simulations
Denser grid at low values of cash-on-hand
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 6 / 8
Computational Solution
Discretization: 2 nodes for income shocks, 3 for equity, 4 forincome-linked assets
! Results not sensitive to adding more nodes, as long as lowest incomeshock “not too small”
Set bounds of grid for cash-on-hand s.th. never move out of boundsin simulations
Denser grid at low values of cash-on-hand
Solve in Matlab, using dogleg algorithm by H.B. Nielsen
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 6 / 8
Computational Solution
Discretization: 2 nodes for income shocks, 3 for equity, 4 forincome-linked assets
! Results not sensitive to adding more nodes, as long as lowest incomeshock “not too small”
Set bounds of grid for cash-on-hand s.th. never move out of boundsin simulations
Denser grid at low values of cash-on-hand
Solve in Matlab, using dogleg algorithm by H.B. Nielsen
Average consumption-equivalent EE error of order 10−6
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 6 / 8
Optimal Asset Holdings with IHI
Asset Holdings and Borrowing over the LC (means)
Tho
usan
dsof
1992
USD
age
↖ Equity
↖ Borrowing
No IHI
20 30 40 50 60 70 80-60
-40
-20
0
20
40
60
Back
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 8
Optimal Asset Holdings with IHI
Asset Holdings and Borrowing over the LC (means)
Tho
usan
dsof
1992
USD
age
↖ Equity
↖ Borrowing
↙ IHI
ρ = −0.5
20 30 40 50 60 70 80-60
-40
-20
0
20
40
60
Back
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 8
Optimal Asset Holdings with IHI
Asset Holdings and Borrowing over the LC (means)
Tho
usan
dsof
1992
USD
age
Equity ↘
↖ Borrowing
← IHI
ρ = −0.75
20 30 40 50 60 70 80-60
-40
-20
0
20
40
60
Back
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 8
Optimal Asset Holdings with IHI
Asset Holdings and Borrowing over the LC (means)
Tho
usan
dsof
1992
USD
age
Equity →
↖ Borrowing
← IHI
ρ = −1
20 30 40 50 60 70 80-60
-40
-20
0
20
40
60
Back
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 7 / 8
Optimal Asset Holdings with ILL
Asset Holdings and Borrowing over the LC (means)
Tho
usan
dsof
1992
USD
age
↖ Equity
↖ Borrowing
No ILL
20 30 40 50 60 70 80
-60
-40
-20
0
20
40
60
Back
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 8
Optimal Asset Holdings with ILL
Asset Holdings and Borrowing over the LC (means)
Tho
usan
dsof
1992
USD
age
↖ Equity
↖ Borrowing ↖ ILL
ρ = 0.5
20 30 40 50 60 70 80
-60
-40
-20
0
20
40
60
Back
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 8
Optimal Asset Holdings with ILL
Asset Holdings and Borrowing over the LC (means)
Tho
usan
dsof
1992
USD
age
↖ Equity
↖ Borrowing ← ILL
ρ = 0.75
20 30 40 50 60 70 80
-60
-40
-20
0
20
40
60
Back
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 8
Optimal Asset Holdings with ILL
Asset Holdings and Borrowing over the LC (means)
Tho
usan
dsof
1992
USD
age
← Equity
↙ Borrowing← ILL
ρ = 1
20 30 40 50 60 70 80
-60
-40
-20
0
20
40
60
Back
Andreas Fuster (Harvard) Income-Linked Assets October 16, 2009 8 / 8