1. Model
Solution
Under the Hood
Spending and Investment
for Shortfall Averse Endowments
Paolo Guasoni1,2
Gur Huberman3
Dan Ren4
Boston University1
Dublin City University2
Columbia Business School3
University of Dayton4
Financial and Actuarial Mathematics Seminar
University of Michigan
October 31st , 2013
2. Model
Solution
Under the Hood
Tobin (1974)
• "The trustees of an endowed institution are the guardians of the future
against the claims of the present.
• "Their task is to preserve equity among generations.
• "The trustees of an endowed university like my own assume the institution
to be immortal.
• "They want to know, therefore, the rate of consumption from endowment
which can be sustained indeﬁnitely.
• "Sustainable consumption is their conception of permanent endowment
income.
• "In formal terms, the trustees are supposed to have a zero subjective
rate of time preference."
3. Model
Solution
Under the Hood
Tobin (1974)
• "The trustees of an endowed institution are the guardians of the future
against the claims of the present.
• "Their task is to preserve equity among generations.
• "The trustees of an endowed university like my own assume the institution
to be immortal.
• "They want to know, therefore, the rate of consumption from endowment
which can be sustained indeﬁnitely.
• "Sustainable consumption is their conception of permanent endowment
income.
• "In formal terms, the trustees are supposed to have a zero subjective
rate of time preference."
4. Model
Solution
Under the Hood
Tobin (1974)
• "The trustees of an endowed institution are the guardians of the future
against the claims of the present.
• "Their task is to preserve equity among generations.
• "The trustees of an endowed university like my own assume the institution
to be immortal.
• "They want to know, therefore, the rate of consumption from endowment
which can be sustained indeﬁnitely.
• "Sustainable consumption is their conception of permanent endowment
income.
• "In formal terms, the trustees are supposed to have a zero subjective
rate of time preference."
5. Model
Solution
Under the Hood
Tobin (1974)
• "The trustees of an endowed institution are the guardians of the future
against the claims of the present.
• "Their task is to preserve equity among generations.
• "The trustees of an endowed university like my own assume the institution
to be immortal.
• "They want to know, therefore, the rate of consumption from endowment
which can be sustained indeﬁnitely.
• "Sustainable consumption is their conception of permanent endowment
income.
• "In formal terms, the trustees are supposed to have a zero subjective
rate of time preference."
6. Model
Solution
Under the Hood
Tobin (1974)
• "The trustees of an endowed institution are the guardians of the future
against the claims of the present.
• "Their task is to preserve equity among generations.
• "The trustees of an endowed university like my own assume the institution
to be immortal.
• "They want to know, therefore, the rate of consumption from endowment
which can be sustained indeﬁnitely.
• "Sustainable consumption is their conception of permanent endowment
income.
• "In formal terms, the trustees are supposed to have a zero subjective
rate of time preference."
7. Model
Solution
Under the Hood
Tobin (1974)
• "The trustees of an endowed institution are the guardians of the future
against the claims of the present.
• "Their task is to preserve equity among generations.
• "The trustees of an endowed university like my own assume the institution
to be immortal.
• "They want to know, therefore, the rate of consumption from endowment
which can be sustained indeﬁnitely.
• "Sustainable consumption is their conception of permanent endowment
income.
• "In formal terms, the trustees are supposed to have a zero subjective
rate of time preference."
8. Model
Solution
Under the Hood
Outline
• Motivation:
Endowment Management. Universities, sovereign funds, trust funds.
Retirement planning?
• Model:
Constant investment opportunities.
Constant Relative Risk and Shortfall Aversion.
• Result:
Optimal spending and investment.
9. Model
Solution
Under the Hood
Outline
• Motivation:
Endowment Management. Universities, sovereign funds, trust funds.
Retirement planning?
• Model:
Constant investment opportunities.
Constant Relative Risk and Shortfall Aversion.
• Result:
Optimal spending and investment.
10. Model
Solution
Under the Hood
Outline
• Motivation:
Endowment Management. Universities, sovereign funds, trust funds.
Retirement planning?
• Model:
Constant investment opportunities.
Constant Relative Risk and Shortfall Aversion.
• Result:
Optimal spending and investment.
11. Model
Solution
Under the Hood
Endowment Management
• How to invest? How much to withdraw?
• Major goal: keep spending level.
• Minor goal: increasing it.
• Increasing and then decreasing spending is worse than not increasing in
the ﬁrst place.
• How to capture this feature?
• Heuristic rule among endowments and private foundations:
spend a constant proportion of the moving average of assets.
12. Model
Solution
Under the Hood
Endowment Management
• How to invest? How much to withdraw?
• Major goal: keep spending level.
• Minor goal: increasing it.
• Increasing and then decreasing spending is worse than not increasing in
the ﬁrst place.
• How to capture this feature?
• Heuristic rule among endowments and private foundations:
spend a constant proportion of the moving average of assets.
13. Model
Solution
Under the Hood
Endowment Management
• How to invest? How much to withdraw?
• Major goal: keep spending level.
• Minor goal: increasing it.
• Increasing and then decreasing spending is worse than not increasing in
the ﬁrst place.
• How to capture this feature?
• Heuristic rule among endowments and private foundations:
spend a constant proportion of the moving average of assets.
14. Model
Solution
Under the Hood
Endowment Management
• How to invest? How much to withdraw?
• Major goal: keep spending level.
• Minor goal: increasing it.
• Increasing and then decreasing spending is worse than not increasing in
the ﬁrst place.
• How to capture this feature?
• Heuristic rule among endowments and private foundations:
spend a constant proportion of the moving average of assets.
15. Model
Solution
Under the Hood
Endowment Management
• How to invest? How much to withdraw?
• Major goal: keep spending level.
• Minor goal: increasing it.
• Increasing and then decreasing spending is worse than not increasing in
the ﬁrst place.
• How to capture this feature?
• Heuristic rule among endowments and private foundations:
spend a constant proportion of the moving average of assets.
16. Model
Solution
Under the Hood
Endowment Management
• How to invest? How much to withdraw?
• Major goal: keep spending level.
• Minor goal: increasing it.
• Increasing and then decreasing spending is worse than not increasing in
the ﬁrst place.
• How to capture this feature?
• Heuristic rule among endowments and private foundations:
spend a constant proportion of the moving average of assets.
17. Model
Solution
Under the Hood
Theory
• Classical Merton model: maximize
∞
e−βt
E
0
ct1−γ
dt
1−γ
over spending rate ct and wealth in stocks Ht .
• Optimal ct and Ht are constant fractions of current wealth Xt :
c
1
1
= β+ 1−
X
γ
γ
r+
µ2
2γσ 2
H
µ
=
X
γσ 2
with interest rate r , equity premium µ, and stocks volatility σ.
• Spending as volatile as wealth.
• Want stable spending? Hold nearly no stocks. But then ct ≈ rXt .
• Not easy with interest rates near zero. Not used by ﬁnancial planners.
18. Model
Solution
Under the Hood
Theory
• Classical Merton model: maximize
∞
e−βt
E
0
ct1−γ
dt
1−γ
over spending rate ct and wealth in stocks Ht .
• Optimal ct and Ht are constant fractions of current wealth Xt :
c
1
1
= β+ 1−
X
γ
γ
r+
µ2
2γσ 2
H
µ
=
X
γσ 2
with interest rate r , equity premium µ, and stocks volatility σ.
• Spending as volatile as wealth.
• Want stable spending? Hold nearly no stocks. But then ct ≈ rXt .
• Not easy with interest rates near zero. Not used by ﬁnancial planners.
19. Model
Solution
Under the Hood
Theory
• Classical Merton model: maximize
∞
e−βt
E
0
ct1−γ
dt
1−γ
over spending rate ct and wealth in stocks Ht .
• Optimal ct and Ht are constant fractions of current wealth Xt :
c
1
1
= β+ 1−
X
γ
γ
r+
µ2
2γσ 2
H
µ
=
X
γσ 2
with interest rate r , equity premium µ, and stocks volatility σ.
• Spending as volatile as wealth.
• Want stable spending? Hold nearly no stocks. But then ct ≈ rXt .
• Not easy with interest rates near zero. Not used by ﬁnancial planners.
20. Model
Solution
Under the Hood
Theory
• Classical Merton model: maximize
∞
e−βt
E
0
ct1−γ
dt
1−γ
over spending rate ct and wealth in stocks Ht .
• Optimal ct and Ht are constant fractions of current wealth Xt :
c
1
1
= β+ 1−
X
γ
γ
r+
µ2
2γσ 2
H
µ
=
X
γσ 2
with interest rate r , equity premium µ, and stocks volatility σ.
• Spending as volatile as wealth.
• Want stable spending? Hold nearly no stocks. But then ct ≈ rXt .
• Not easy with interest rates near zero. Not used by ﬁnancial planners.
21. Model
Solution
Under the Hood
Theory
• Classical Merton model: maximize
∞
e−βt
E
0
ct1−γ
dt
1−γ
over spending rate ct and wealth in stocks Ht .
• Optimal ct and Ht are constant fractions of current wealth Xt :
c
1
1
= β+ 1−
X
γ
γ
r+
µ2
2γσ 2
H
µ
=
X
γσ 2
with interest rate r , equity premium µ, and stocks volatility σ.
• Spending as volatile as wealth.
• Want stable spending? Hold nearly no stocks. But then ct ≈ rXt .
• Not easy with interest rates near zero. Not used by ﬁnancial planners.
22. Model
Solution
Under the Hood
The Four-Percent Rule
• Bengen (1994). Popular with ﬁnancial planners.
• When you retire, spend in the ﬁrst year 4% of your savings.
Keep same amount for future years. Invest 50% to 75% in stocks.
• Historically, savings will last at least 30 years.
• Inconsistent with theory.
Spending-wealth ratio variable, but portfolio weight ﬁxed.
Bankruptcy possible.
• Time-inconsistent.
With equal initial capital, consume more if retired in Dec 2007 than if
retired in Dec 2008. But savings will always be lower.
• How to embed preference for stable spending in objective?
23. Model
Solution
Under the Hood
The Four-Percent Rule
• Bengen (1994). Popular with ﬁnancial planners.
• When you retire, spend in the ﬁrst year 4% of your savings.
Keep same amount for future years. Invest 50% to 75% in stocks.
• Historically, savings will last at least 30 years.
• Inconsistent with theory.
Spending-wealth ratio variable, but portfolio weight ﬁxed.
Bankruptcy possible.
• Time-inconsistent.
With equal initial capital, consume more if retired in Dec 2007 than if
retired in Dec 2008. But savings will always be lower.
• How to embed preference for stable spending in objective?
24. Model
Solution
Under the Hood
The Four-Percent Rule
• Bengen (1994). Popular with ﬁnancial planners.
• When you retire, spend in the ﬁrst year 4% of your savings.
Keep same amount for future years. Invest 50% to 75% in stocks.
• Historically, savings will last at least 30 years.
• Inconsistent with theory.
Spending-wealth ratio variable, but portfolio weight ﬁxed.
Bankruptcy possible.
• Time-inconsistent.
With equal initial capital, consume more if retired in Dec 2007 than if
retired in Dec 2008. But savings will always be lower.
• How to embed preference for stable spending in objective?
25. Model
Solution
Under the Hood
The Four-Percent Rule
• Bengen (1994). Popular with ﬁnancial planners.
• When you retire, spend in the ﬁrst year 4% of your savings.
Keep same amount for future years. Invest 50% to 75% in stocks.
• Historically, savings will last at least 30 years.
• Inconsistent with theory.
Spending-wealth ratio variable, but portfolio weight ﬁxed.
Bankruptcy possible.
• Time-inconsistent.
With equal initial capital, consume more if retired in Dec 2007 than if
retired in Dec 2008. But savings will always be lower.
• How to embed preference for stable spending in objective?
26. Model
Solution
Under the Hood
The Four-Percent Rule
• Bengen (1994). Popular with ﬁnancial planners.
• When you retire, spend in the ﬁrst year 4% of your savings.
Keep same amount for future years. Invest 50% to 75% in stocks.
• Historically, savings will last at least 30 years.
• Inconsistent with theory.
Spending-wealth ratio variable, but portfolio weight ﬁxed.
Bankruptcy possible.
• Time-inconsistent.
With equal initial capital, consume more if retired in Dec 2007 than if
retired in Dec 2008. But savings will always be lower.
• How to embed preference for stable spending in objective?
27. Model
Solution
Under the Hood
The Four-Percent Rule
• Bengen (1994). Popular with ﬁnancial planners.
• When you retire, spend in the ﬁrst year 4% of your savings.
Keep same amount for future years. Invest 50% to 75% in stocks.
• Historically, savings will last at least 30 years.
• Inconsistent with theory.
Spending-wealth ratio variable, but portfolio weight ﬁxed.
Bankruptcy possible.
• Time-inconsistent.
With equal initial capital, consume more if retired in Dec 2007 than if
retired in Dec 2008. But savings will always be lower.
• How to embed preference for stable spending in objective?
28. Model
Solution
Under the Hood
Related Literature
• Consumption ratcheting: allow only increasing spending rates...
Dybvig (1995), Bayraktar and Young (2008), Riedel (2009).
• ...or force maximum drawdown on consumption from its peak.
Thillaisundaram (2012).
• Consume less than interest, or bankruptcy possible.
No solvency with zero interest.
• Habit formation: utility from consumption minus habit.
Sundaresan (1989), Constantinides (1990), Detemple and Zapatero
(1992), Campbell and Cochrane (1999), Detemple and Karatzas (2003).
∞
e−βt
max E
0
(ct − xt )1−γ
dt
1−γ
where dxt = (bct − axt )dt
• Inﬁnite marginal utility at habit level. Like zero consumption.
No solution if habit too high relative to wealth (x0 > X0 (r + a − b)).
• Habit transitory. Time heals.
29. Model
Solution
Under the Hood
Related Literature
• Consumption ratcheting: allow only increasing spending rates...
Dybvig (1995), Bayraktar and Young (2008), Riedel (2009).
• ...or force maximum drawdown on consumption from its peak.
Thillaisundaram (2012).
• Consume less than interest, or bankruptcy possible.
No solvency with zero interest.
• Habit formation: utility from consumption minus habit.
Sundaresan (1989), Constantinides (1990), Detemple and Zapatero
(1992), Campbell and Cochrane (1999), Detemple and Karatzas (2003).
∞
e−βt
max E
0
(ct − xt )1−γ
dt
1−γ
where dxt = (bct − axt )dt
• Inﬁnite marginal utility at habit level. Like zero consumption.
No solution if habit too high relative to wealth (x0 > X0 (r + a − b)).
• Habit transitory. Time heals.
30. Model
Solution
Under the Hood
Related Literature
• Consumption ratcheting: allow only increasing spending rates...
Dybvig (1995), Bayraktar and Young (2008), Riedel (2009).
• ...or force maximum drawdown on consumption from its peak.
Thillaisundaram (2012).
• Consume less than interest, or bankruptcy possible.
No solvency with zero interest.
• Habit formation: utility from consumption minus habit.
Sundaresan (1989), Constantinides (1990), Detemple and Zapatero
(1992), Campbell and Cochrane (1999), Detemple and Karatzas (2003).
∞
e−βt
max E
0
(ct − xt )1−γ
dt
1−γ
where dxt = (bct − axt )dt
• Inﬁnite marginal utility at habit level. Like zero consumption.
No solution if habit too high relative to wealth (x0 > X0 (r + a − b)).
• Habit transitory. Time heals.
31. Model
Solution
Under the Hood
Related Literature
• Consumption ratcheting: allow only increasing spending rates...
Dybvig (1995), Bayraktar and Young (2008), Riedel (2009).
• ...or force maximum drawdown on consumption from its peak.
Thillaisundaram (2012).
• Consume less than interest, or bankruptcy possible.
No solvency with zero interest.
• Habit formation: utility from consumption minus habit.
Sundaresan (1989), Constantinides (1990), Detemple and Zapatero
(1992), Campbell and Cochrane (1999), Detemple and Karatzas (2003).
∞
e−βt
max E
0
(ct − xt )1−γ
dt
1−γ
where dxt = (bct − axt )dt
• Inﬁnite marginal utility at habit level. Like zero consumption.
No solution if habit too high relative to wealth (x0 > X0 (r + a − b)).
• Habit transitory. Time heals.
32. Model
Solution
Under the Hood
Related Literature
• Consumption ratcheting: allow only increasing spending rates...
Dybvig (1995), Bayraktar and Young (2008), Riedel (2009).
• ...or force maximum drawdown on consumption from its peak.
Thillaisundaram (2012).
• Consume less than interest, or bankruptcy possible.
No solvency with zero interest.
• Habit formation: utility from consumption minus habit.
Sundaresan (1989), Constantinides (1990), Detemple and Zapatero
(1992), Campbell and Cochrane (1999), Detemple and Karatzas (2003).
∞
e−βt
max E
0
(ct − xt )1−γ
dt
1−γ
where dxt = (bct − axt )dt
• Inﬁnite marginal utility at habit level. Like zero consumption.
No solution if habit too high relative to wealth (x0 > X0 (r + a − b)).
• Habit transitory. Time heals.
33. Model
Solution
Under the Hood
Related Literature
• Consumption ratcheting: allow only increasing spending rates...
Dybvig (1995), Bayraktar and Young (2008), Riedel (2009).
• ...or force maximum drawdown on consumption from its peak.
Thillaisundaram (2012).
• Consume less than interest, or bankruptcy possible.
No solvency with zero interest.
• Habit formation: utility from consumption minus habit.
Sundaresan (1989), Constantinides (1990), Detemple and Zapatero
(1992), Campbell and Cochrane (1999), Detemple and Karatzas (2003).
∞
e−βt
max E
0
(ct − xt )1−γ
dt
1−γ
where dxt = (bct − axt )dt
• Inﬁnite marginal utility at habit level. Like zero consumption.
No solution if habit too high relative to wealth (x0 > X0 (r + a − b)).
• Habit transitory. Time heals.
34. Model
Solution
Under the Hood
This Paper
• Inputs
• Risky asset follows geometric Brownian motion. Constant interest rate.
• Constant relative risk aversion.
Constant relative shortfall aversion by new parameter α.
• Outputs
• Spending constant between endogenous boundaries, bliss and gloom.
Increases in small amounts at bliss.
Declines smoothly at gloom.
• Portfolio weight varies between high and low bounds.
• Features
• Solution solvent for any initial capital. Even with zero interest.
• Spending target permanent. You never forget.
35. Model
Solution
Under the Hood
This Paper
• Inputs
• Risky asset follows geometric Brownian motion. Constant interest rate.
• Constant relative risk aversion.
Constant relative shortfall aversion by new parameter α.
• Outputs
• Spending constant between endogenous boundaries, bliss and gloom.
Increases in small amounts at bliss.
Declines smoothly at gloom.
• Portfolio weight varies between high and low bounds.
• Features
• Solution solvent for any initial capital. Even with zero interest.
• Spending target permanent. You never forget.
36. Model
Solution
Under the Hood
This Paper
• Inputs
• Risky asset follows geometric Brownian motion. Constant interest rate.
• Constant relative risk aversion.
Constant relative shortfall aversion by new parameter α.
• Outputs
• Spending constant between endogenous boundaries, bliss and gloom.
Increases in small amounts at bliss.
Declines smoothly at gloom.
• Portfolio weight varies between high and low bounds.
• Features
• Solution solvent for any initial capital. Even with zero interest.
• Spending target permanent. You never forget.
37. Model
Solution
Under the Hood
This Paper
• Inputs
• Risky asset follows geometric Brownian motion. Constant interest rate.
• Constant relative risk aversion.
Constant relative shortfall aversion by new parameter α.
• Outputs
• Spending constant between endogenous boundaries, bliss and gloom.
Increases in small amounts at bliss.
Declines smoothly at gloom.
• Portfolio weight varies between high and low bounds.
• Features
• Solution solvent for any initial capital. Even with zero interest.
• Spending target permanent. You never forget.
38. Model
Solution
Under the Hood
This Paper
• Inputs
• Risky asset follows geometric Brownian motion. Constant interest rate.
• Constant relative risk aversion.
Constant relative shortfall aversion by new parameter α.
• Outputs
• Spending constant between endogenous boundaries, bliss and gloom.
Increases in small amounts at bliss.
Declines smoothly at gloom.
• Portfolio weight varies between high and low bounds.
• Features
• Solution solvent for any initial capital. Even with zero interest.
• Spending target permanent. You never forget.
39. Model
Solution
Under the Hood
This Paper
• Inputs
• Risky asset follows geometric Brownian motion. Constant interest rate.
• Constant relative risk aversion.
Constant relative shortfall aversion by new parameter α.
• Outputs
• Spending constant between endogenous boundaries, bliss and gloom.
Increases in small amounts at bliss.
Declines smoothly at gloom.
• Portfolio weight varies between high and low bounds.
• Features
• Solution solvent for any initial capital. Even with zero interest.
• Spending target permanent. You never forget.
40. Model
Solution
Under the Hood
This Paper
• Inputs
• Risky asset follows geometric Brownian motion. Constant interest rate.
• Constant relative risk aversion.
Constant relative shortfall aversion by new parameter α.
• Outputs
• Spending constant between endogenous boundaries, bliss and gloom.
Increases in small amounts at bliss.
Declines smoothly at gloom.
• Portfolio weight varies between high and low bounds.
• Features
• Solution solvent for any initial capital. Even with zero interest.
• Spending target permanent. You never forget.
41. Model
Solution
Under the Hood
This Paper
• Inputs
• Risky asset follows geometric Brownian motion. Constant interest rate.
• Constant relative risk aversion.
Constant relative shortfall aversion by new parameter α.
• Outputs
• Spending constant between endogenous boundaries, bliss and gloom.
Increases in small amounts at bliss.
Declines smoothly at gloom.
• Portfolio weight varies between high and low bounds.
• Features
• Solution solvent for any initial capital. Even with zero interest.
• Spending target permanent. You never forget.
42. Model
Solution
Under the Hood
This Paper
• Inputs
• Risky asset follows geometric Brownian motion. Constant interest rate.
• Constant relative risk aversion.
Constant relative shortfall aversion by new parameter α.
• Outputs
• Spending constant between endogenous boundaries, bliss and gloom.
Increases in small amounts at bliss.
Declines smoothly at gloom.
• Portfolio weight varies between high and low bounds.
• Features
• Solution solvent for any initial capital. Even with zero interest.
• Spending target permanent. You never forget.
43. Model
Solution
Under the Hood
Shortfall Aversion and Prospect Theory
• Kanheman and Tversky (1979,1991). Precursor: Markowitz (1952).
• Reference dependence:
utility is from gains and losses relative to reference level.
• Loss Aversion:
marginal utility lower for gains than for losses.
• Prospect theory focuses on wealth gambles.
• We bring these features to spending instead.
Shortfall aversion as losses aversion for spending.
44. Model
Solution
Under the Hood
Shortfall Aversion and Prospect Theory
• Kanheman and Tversky (1979,1991). Precursor: Markowitz (1952).
• Reference dependence:
utility is from gains and losses relative to reference level.
• Loss Aversion:
marginal utility lower for gains than for losses.
• Prospect theory focuses on wealth gambles.
• We bring these features to spending instead.
Shortfall aversion as losses aversion for spending.
45. Model
Solution
Under the Hood
Shortfall Aversion and Prospect Theory
• Kanheman and Tversky (1979,1991). Precursor: Markowitz (1952).
• Reference dependence:
utility is from gains and losses relative to reference level.
• Loss Aversion:
marginal utility lower for gains than for losses.
• Prospect theory focuses on wealth gambles.
• We bring these features to spending instead.
Shortfall aversion as losses aversion for spending.
46. Model
Solution
Under the Hood
Shortfall Aversion and Prospect Theory
• Kanheman and Tversky (1979,1991). Precursor: Markowitz (1952).
• Reference dependence:
utility is from gains and losses relative to reference level.
• Loss Aversion:
marginal utility lower for gains than for losses.
• Prospect theory focuses on wealth gambles.
• We bring these features to spending instead.
Shortfall aversion as losses aversion for spending.
47. Model
Solution
Under the Hood
Shortfall Aversion and Prospect Theory
• Kanheman and Tversky (1979,1991). Precursor: Markowitz (1952).
• Reference dependence:
utility is from gains and losses relative to reference level.
• Loss Aversion:
marginal utility lower for gains than for losses.
• Prospect theory focuses on wealth gambles.
• We bring these features to spending instead.
Shortfall aversion as losses aversion for spending.
48. Model
Solution
Under the Hood
Model
• Safe rate r . Risky asset price follows geometric Brownian motion:
dSt
= µdt + σdWt
St
for Brownian Motion (Wt )t≥0 with natural ﬁltration (Ft )t≥0 .
• Utility from consumption rate ct :
∞
E
0
1−γ
(ct /htα )
1−γ
dt
where
ht = sup cs
0≤s≤t
• Risk aversion γ > 1 to make the problem well posed.
• α ∈ [0, 1) relative loss aversion.
Strict monotonicity in c implies that α < 1 and strict concavity that α ≥ 0.
• If you could forget the past you would be happier. But you can’t.
• More is better today... but makes less worse tomorrow.
• No loss aversion with ct increasing... or decreasing.
49. Model
Solution
Under the Hood
Model
• Safe rate r . Risky asset price follows geometric Brownian motion:
dSt
= µdt + σdWt
St
for Brownian Motion (Wt )t≥0 with natural ﬁltration (Ft )t≥0 .
• Utility from consumption rate ct :
∞
E
0
1−γ
(ct /htα )
1−γ
dt
where
ht = sup cs
0≤s≤t
• Risk aversion γ > 1 to make the problem well posed.
• α ∈ [0, 1) relative loss aversion.
Strict monotonicity in c implies that α < 1 and strict concavity that α ≥ 0.
• If you could forget the past you would be happier. But you can’t.
• More is better today... but makes less worse tomorrow.
• No loss aversion with ct increasing... or decreasing.
50. Model
Solution
Under the Hood
Model
• Safe rate r . Risky asset price follows geometric Brownian motion:
dSt
= µdt + σdWt
St
for Brownian Motion (Wt )t≥0 with natural ﬁltration (Ft )t≥0 .
• Utility from consumption rate ct :
∞
E
0
1−γ
(ct /htα )
1−γ
dt
where
ht = sup cs
0≤s≤t
• Risk aversion γ > 1 to make the problem well posed.
• α ∈ [0, 1) relative loss aversion.
Strict monotonicity in c implies that α < 1 and strict concavity that α ≥ 0.
• If you could forget the past you would be happier. But you can’t.
• More is better today... but makes less worse tomorrow.
• No loss aversion with ct increasing... or decreasing.
51. Model
Solution
Under the Hood
Model
• Safe rate r . Risky asset price follows geometric Brownian motion:
dSt
= µdt + σdWt
St
for Brownian Motion (Wt )t≥0 with natural ﬁltration (Ft )t≥0 .
• Utility from consumption rate ct :
∞
E
0
1−γ
(ct /htα )
1−γ
dt
where
ht = sup cs
0≤s≤t
• Risk aversion γ > 1 to make the problem well posed.
• α ∈ [0, 1) relative loss aversion.
Strict monotonicity in c implies that α < 1 and strict concavity that α ≥ 0.
• If you could forget the past you would be happier. But you can’t.
• More is better today... but makes less worse tomorrow.
• No loss aversion with ct increasing... or decreasing.
52. Model
Solution
Under the Hood
Model
• Safe rate r . Risky asset price follows geometric Brownian motion:
dSt
= µdt + σdWt
St
for Brownian Motion (Wt )t≥0 with natural ﬁltration (Ft )t≥0 .
• Utility from consumption rate ct :
∞
E
0
1−γ
(ct /htα )
1−γ
dt
where
ht = sup cs
0≤s≤t
• Risk aversion γ > 1 to make the problem well posed.
• α ∈ [0, 1) relative loss aversion.
Strict monotonicity in c implies that α < 1 and strict concavity that α ≥ 0.
• If you could forget the past you would be happier. But you can’t.
• More is better today... but makes less worse tomorrow.
• No loss aversion with ct increasing... or decreasing.
53. Model
Solution
Under the Hood
Model
• Safe rate r . Risky asset price follows geometric Brownian motion:
dSt
= µdt + σdWt
St
for Brownian Motion (Wt )t≥0 with natural ﬁltration (Ft )t≥0 .
• Utility from consumption rate ct :
∞
E
0
1−γ
(ct /htα )
1−γ
dt
where
ht = sup cs
0≤s≤t
• Risk aversion γ > 1 to make the problem well posed.
• α ∈ [0, 1) relative loss aversion.
Strict monotonicity in c implies that α < 1 and strict concavity that α ≥ 0.
• If you could forget the past you would be happier. But you can’t.
• More is better today... but makes less worse tomorrow.
• No loss aversion with ct increasing... or decreasing.
54. Model
Solution
Under the Hood
Model
• Safe rate r . Risky asset price follows geometric Brownian motion:
dSt
= µdt + σdWt
St
for Brownian Motion (Wt )t≥0 with natural ﬁltration (Ft )t≥0 .
• Utility from consumption rate ct :
∞
E
0
1−γ
(ct /htα )
1−γ
dt
where
ht = sup cs
0≤s≤t
• Risk aversion γ > 1 to make the problem well posed.
• α ∈ [0, 1) relative loss aversion.
Strict monotonicity in c implies that α < 1 and strict concavity that α ≥ 0.
• If you could forget the past you would be happier. But you can’t.
• More is better today... but makes less worse tomorrow.
• No loss aversion with ct increasing... or decreasing.
55. Model
Solution
Under the Hood
Sliding Kink
• Because ct ≤ ht , utility effectively has a kink at ht :
U c, h
h
c
• Optimality: marginal utilities of spending and wealth equal.
• Spending ht optimal for marginal utility of wealth between left and right
derivative at kink.
• State variable: ratio ht /Xt between reference spending and wealth.
• Investor rich when ht /Xt low, and poor when ht /Xt high.
56. Model
Solution
Under the Hood
Sliding Kink
• Because ct ≤ ht , utility effectively has a kink at ht :
U c, h
h
c
• Optimality: marginal utilities of spending and wealth equal.
• Spending ht optimal for marginal utility of wealth between left and right
derivative at kink.
• State variable: ratio ht /Xt between reference spending and wealth.
• Investor rich when ht /Xt low, and poor when ht /Xt high.
57. Model
Solution
Under the Hood
Sliding Kink
• Because ct ≤ ht , utility effectively has a kink at ht :
U c, h
h
c
• Optimality: marginal utilities of spending and wealth equal.
• Spending ht optimal for marginal utility of wealth between left and right
derivative at kink.
• State variable: ratio ht /Xt between reference spending and wealth.
• Investor rich when ht /Xt low, and poor when ht /Xt high.
58. Model
Solution
Under the Hood
Sliding Kink
• Because ct ≤ ht , utility effectively has a kink at ht :
U c, h
h
c
• Optimality: marginal utilities of spending and wealth equal.
• Spending ht optimal for marginal utility of wealth between left and right
derivative at kink.
• State variable: ratio ht /Xt between reference spending and wealth.
• Investor rich when ht /Xt low, and poor when ht /Xt high.
59. Model
Solution
Under the Hood
Sliding Kink
• Because ct ≤ ht , utility effectively has a kink at ht :
U c, h
h
c
• Optimality: marginal utilities of spending and wealth equal.
• Spending ht optimal for marginal utility of wealth between left and right
derivative at kink.
• State variable: ratio ht /Xt between reference spending and wealth.
• Investor rich when ht /Xt low, and poor when ht /Xt high.
60. Model
Solution
Under the Hood
Control Argument
• Value function V (x, h) depends on wealth x = Xt and target h = ht .
• Set Jt =
t
0
U(cs , hs )ds + V (Xt , ht ). By Itô’s formula:
dJt = L(Xt , πt , ct , ht )dt + Vh (Xt , ht )dht + Vx (Xt , ht )Xt πt σ dWt
where drift L(Xt , πt , ct , ht ) equals
U(ct , ht ) + (Xt rt − ct + Xt πt µ)Vx (Xt , ht ) +
Vxx (Xt , ht ) 2
Xt πt Σπt
2
ˆ
• Jt supermartingale for any c, and martingale for optimizer c . Hence:
max(sup L(x, π, c, h), Vh (x, h)) = 0
π,c
• That is, either supπ,c L(x, π, c, h) = 0, or Vh (x, h) = 0.
V
• Optimal portfolio has usual expression π = − xVx Σ−1 µ.
ˆ
xx
• Free-boundaries:
When do you increase spending? When do you cut it below ht ?
61. Model
Solution
Under the Hood
Control Argument
• Value function V (x, h) depends on wealth x = Xt and target h = ht .
• Set Jt =
t
0
U(cs , hs )ds + V (Xt , ht ). By Itô’s formula:
dJt = L(Xt , πt , ct , ht )dt + Vh (Xt , ht )dht + Vx (Xt , ht )Xt πt σ dWt
where drift L(Xt , πt , ct , ht ) equals
U(ct , ht ) + (Xt rt − ct + Xt πt µ)Vx (Xt , ht ) +
Vxx (Xt , ht ) 2
Xt πt Σπt
2
ˆ
• Jt supermartingale for any c, and martingale for optimizer c . Hence:
max(sup L(x, π, c, h), Vh (x, h)) = 0
π,c
• That is, either supπ,c L(x, π, c, h) = 0, or Vh (x, h) = 0.
V
• Optimal portfolio has usual expression π = − xVx Σ−1 µ.
ˆ
xx
• Free-boundaries:
When do you increase spending? When do you cut it below ht ?
62. Model
Solution
Under the Hood
Control Argument
• Value function V (x, h) depends on wealth x = Xt and target h = ht .
• Set Jt =
t
0
U(cs , hs )ds + V (Xt , ht ). By Itô’s formula:
dJt = L(Xt , πt , ct , ht )dt + Vh (Xt , ht )dht + Vx (Xt , ht )Xt πt σ dWt
where drift L(Xt , πt , ct , ht ) equals
U(ct , ht ) + (Xt rt − ct + Xt πt µ)Vx (Xt , ht ) +
Vxx (Xt , ht ) 2
Xt πt Σπt
2
ˆ
• Jt supermartingale for any c, and martingale for optimizer c . Hence:
max(sup L(x, π, c, h), Vh (x, h)) = 0
π,c
• That is, either supπ,c L(x, π, c, h) = 0, or Vh (x, h) = 0.
V
• Optimal portfolio has usual expression π = − xVx Σ−1 µ.
ˆ
xx
• Free-boundaries:
When do you increase spending? When do you cut it below ht ?
63. Model
Solution
Under the Hood
Control Argument
• Value function V (x, h) depends on wealth x = Xt and target h = ht .
• Set Jt =
t
0
U(cs , hs )ds + V (Xt , ht ). By Itô’s formula:
dJt = L(Xt , πt , ct , ht )dt + Vh (Xt , ht )dht + Vx (Xt , ht )Xt πt σ dWt
where drift L(Xt , πt , ct , ht ) equals
U(ct , ht ) + (Xt rt − ct + Xt πt µ)Vx (Xt , ht ) +
Vxx (Xt , ht ) 2
Xt πt Σπt
2
ˆ
• Jt supermartingale for any c, and martingale for optimizer c . Hence:
max(sup L(x, π, c, h), Vh (x, h)) = 0
π,c
• That is, either supπ,c L(x, π, c, h) = 0, or Vh (x, h) = 0.
V
• Optimal portfolio has usual expression π = − xVx Σ−1 µ.
ˆ
xx
• Free-boundaries:
When do you increase spending? When do you cut it below ht ?
64. Model
Solution
Under the Hood
Control Argument
• Value function V (x, h) depends on wealth x = Xt and target h = ht .
• Set Jt =
t
0
U(cs , hs )ds + V (Xt , ht ). By Itô’s formula:
dJt = L(Xt , πt , ct , ht )dt + Vh (Xt , ht )dht + Vx (Xt , ht )Xt πt σ dWt
where drift L(Xt , πt , ct , ht ) equals
U(ct , ht ) + (Xt rt − ct + Xt πt µ)Vx (Xt , ht ) +
Vxx (Xt , ht ) 2
Xt πt Σπt
2
ˆ
• Jt supermartingale for any c, and martingale for optimizer c . Hence:
max(sup L(x, π, c, h), Vh (x, h)) = 0
π,c
• That is, either supπ,c L(x, π, c, h) = 0, or Vh (x, h) = 0.
V
• Optimal portfolio has usual expression π = − xVx Σ−1 µ.
ˆ
xx
• Free-boundaries:
When do you increase spending? When do you cut it below ht ?
65. Model
Solution
Under the Hood
Control Argument
• Value function V (x, h) depends on wealth x = Xt and target h = ht .
• Set Jt =
t
0
U(cs , hs )ds + V (Xt , ht ). By Itô’s formula:
dJt = L(Xt , πt , ct , ht )dt + Vh (Xt , ht )dht + Vx (Xt , ht )Xt πt σ dWt
where drift L(Xt , πt , ct , ht ) equals
U(ct , ht ) + (Xt rt − ct + Xt πt µ)Vx (Xt , ht ) +
Vxx (Xt , ht ) 2
Xt πt Σπt
2
ˆ
• Jt supermartingale for any c, and martingale for optimizer c . Hence:
max(sup L(x, π, c, h), Vh (x, h)) = 0
π,c
• That is, either supπ,c L(x, π, c, h) = 0, or Vh (x, h) = 0.
V
• Optimal portfolio has usual expression π = − xVx Σ−1 µ.
ˆ
xx
• Free-boundaries:
When do you increase spending? When do you cut it below ht ?
66. Model
Solution
Under the Hood
Duality
˜
• Setting U(y , h) = supc≥0 [U(c, h) − cy ], HJB equation becomes
V 2 (x, h)
˜
U(Vx , h) + xrVx (x, h) − x
µ Σ−1 µ = 0
2Vxx (x, h)
˜
• Nonlinear equation. Linear in dual V (y , h) = supx≥0 [V (x, h) − xy ]
µ Σ−1 µ 2 ˜
˜
˜
U(y , h) − ry Vy +
y Vyy = 0
2
• Plug U(c, h) =
(c/hα )1−γ
.
1−γ
Kink spawns two cases:
∗
h1−γ
− hy
˜
U(y , h) = 1 − γ
(yhα )1−1/γ
1−1/γ
where γ ∗ = α + (1 − α)γ.
∗
if (1 − α)h−γ ≤ y ≤ h−γ
if y > h−γ
∗
∗
67. Model
Solution
Under the Hood
Duality
˜
• Setting U(y , h) = supc≥0 [U(c, h) − cy ], HJB equation becomes
V 2 (x, h)
˜
U(Vx , h) + xrVx (x, h) − x
µ Σ−1 µ = 0
2Vxx (x, h)
˜
• Nonlinear equation. Linear in dual V (y , h) = supx≥0 [V (x, h) − xy ]
µ Σ−1 µ 2 ˜
˜
˜
U(y , h) − ry Vy +
y Vyy = 0
2
• Plug U(c, h) =
(c/hα )1−γ
.
1−γ
Kink spawns two cases:
∗
h1−γ
− hy
˜
U(y , h) = 1 − γ
(yhα )1−1/γ
1−1/γ
where γ ∗ = α + (1 − α)γ.
∗
if (1 − α)h−γ ≤ y ≤ h−γ
if y > h−γ
∗
∗
68. Model
Solution
Under the Hood
Duality
˜
• Setting U(y , h) = supc≥0 [U(c, h) − cy ], HJB equation becomes
V 2 (x, h)
˜
U(Vx , h) + xrVx (x, h) − x
µ Σ−1 µ = 0
2Vxx (x, h)
˜
• Nonlinear equation. Linear in dual V (y , h) = supx≥0 [V (x, h) − xy ]
µ Σ−1 µ 2 ˜
˜
˜
U(y , h) − ry Vy +
y Vyy = 0
2
• Plug U(c, h) =
(c/hα )1−γ
.
1−γ
Kink spawns two cases:
∗
h1−γ
− hy
˜
U(y , h) = 1 − γ
(yhα )1−1/γ
1−1/γ
where γ ∗ = α + (1 − α)γ.
∗
if (1 − α)h−γ ≤ y ≤ h−γ
if y > h−γ
∗
∗
69. Model
Solution
Under the Hood
Homogeneity
∗
∗
˜
• V (λx, λh) = λ1−γ V (x, h) implies that V (y , h) = h1−γ q(z), where
γ∗
z = yh .
• HJB equation reduces to:
µ Σ−1 µ 2
z q
2
(z) − rzq (z) =
z−
1
1−γ
1−1/γ
z
1−1/γ
1−α≤z ≤1
z>1
• Condition Vh (x, h) = 0 holds when desired spending must increase:
(1 − γ ∗ )q(z) + γ ∗ zq (z) = 0
for z ≤ 1 − α
• Agent “rich” for z ∈ [1 − α, 1].
Consumes at desired level, and increases it at bliss point 1 − α.
• Agent becomes “poor” at gloom point 1.
For z > 1, consumes below desired level.
70. Model
Solution
Under the Hood
Homogeneity
∗
∗
˜
• V (λx, λh) = λ1−γ V (x, h) implies that V (y , h) = h1−γ q(z), where
γ∗
z = yh .
• HJB equation reduces to:
µ Σ−1 µ 2
z q
2
(z) − rzq (z) =
z−
1
1−γ
1−1/γ
z
1−1/γ
1−α≤z ≤1
z>1
• Condition Vh (x, h) = 0 holds when desired spending must increase:
(1 − γ ∗ )q(z) + γ ∗ zq (z) = 0
for z ≤ 1 − α
• Agent “rich” for z ∈ [1 − α, 1].
Consumes at desired level, and increases it at bliss point 1 − α.
• Agent becomes “poor” at gloom point 1.
For z > 1, consumes below desired level.
71. Model
Solution
Under the Hood
Homogeneity
∗
∗
˜
• V (λx, λh) = λ1−γ V (x, h) implies that V (y , h) = h1−γ q(z), where
γ∗
z = yh .
• HJB equation reduces to:
µ Σ−1 µ 2
z q
2
(z) − rzq (z) =
z−
1
1−γ
1−1/γ
z
1−1/γ
1−α≤z ≤1
z>1
• Condition Vh (x, h) = 0 holds when desired spending must increase:
(1 − γ ∗ )q(z) + γ ∗ zq (z) = 0
for z ≤ 1 − α
• Agent “rich” for z ∈ [1 − α, 1].
Consumes at desired level, and increases it at bliss point 1 − α.
• Agent becomes “poor” at gloom point 1.
For z > 1, consumes below desired level.
72. Model
Solution
Under the Hood
Homogeneity
∗
∗
˜
• V (λx, λh) = λ1−γ V (x, h) implies that V (y , h) = h1−γ q(z), where
γ∗
z = yh .
• HJB equation reduces to:
µ Σ−1 µ 2
z q
2
(z) − rzq (z) =
z−
1
1−γ
1−1/γ
z
1−1/γ
1−α≤z ≤1
z>1
• Condition Vh (x, h) = 0 holds when desired spending must increase:
(1 − γ ∗ )q(z) + γ ∗ zq (z) = 0
for z ≤ 1 − α
• Agent “rich” for z ∈ [1 − α, 1].
Consumes at desired level, and increases it at bliss point 1 − α.
• Agent becomes “poor” at gloom point 1.
For z > 1, consumes below desired level.
73. Model
Solution
Under the Hood
Homogeneity
∗
∗
˜
• V (λx, λh) = λ1−γ V (x, h) implies that V (y , h) = h1−γ q(z), where
γ∗
z = yh .
• HJB equation reduces to:
µ Σ−1 µ 2
z q
2
(z) − rzq (z) =
z−
1
1−γ
1−1/γ
z
1−1/γ
1−α≤z ≤1
z>1
• Condition Vh (x, h) = 0 holds when desired spending must increase:
(1 − γ ∗ )q(z) + γ ∗ zq (z) = 0
for z ≤ 1 − α
• Agent “rich” for z ∈ [1 − α, 1].
Consumes at desired level, and increases it at bliss point 1 − α.
• Agent becomes “poor” at gloom point 1.
For z > 1, consumes below desired level.
74. Model
Solution
Under the Hood
Solving it
• For r = 0, q(z) has solution:
q(z) =
C21 + C22 z
C31 +
1+
µ
2r
Σ−1 µ
−
z
r
+
2 log z
(1−γ)(2r +µ Σ−1 µ)
γ
1−1/γ
(1−γ)δ0 z
if 0 < z ≤ 1,
if z > 1,
where δα deﬁned shortly.
• Neumann condition at z = 1 − α: (1 − γ ∗ )q(z) + γ ∗ zq (z) = 0.
• Value matching at z = 1: q(z−) = q(z+).
• Smooth-pasting at z = 1: q (z−) = q (z+).
• Fourth condition?
• Intuitively, it should be at z = ∞.
• Marginal utility z inﬁnite when wealth x is zero.
• Since q (z) = −x/h, the condition is limz→∞ q (z) = 0.
75. Model
Solution
Under the Hood
Solving it
• For r = 0, q(z) has solution:
q(z) =
C21 + C22 z
C31 +
1+
µ
2r
Σ−1 µ
−
z
r
+
2 log z
(1−γ)(2r +µ Σ−1 µ)
γ
1−1/γ
(1−γ)δ0 z
if 0 < z ≤ 1,
if z > 1,
where δα deﬁned shortly.
• Neumann condition at z = 1 − α: (1 − γ ∗ )q(z) + γ ∗ zq (z) = 0.
• Value matching at z = 1: q(z−) = q(z+).
• Smooth-pasting at z = 1: q (z−) = q (z+).
• Fourth condition?
• Intuitively, it should be at z = ∞.
• Marginal utility z inﬁnite when wealth x is zero.
• Since q (z) = −x/h, the condition is limz→∞ q (z) = 0.
76. Model
Solution
Under the Hood
Solving it
• For r = 0, q(z) has solution:
q(z) =
C21 + C22 z
C31 +
1+
µ
2r
Σ−1 µ
−
z
r
+
2 log z
(1−γ)(2r +µ Σ−1 µ)
γ
1−1/γ
(1−γ)δ0 z
if 0 < z ≤ 1,
if z > 1,
where δα deﬁned shortly.
• Neumann condition at z = 1 − α: (1 − γ ∗ )q(z) + γ ∗ zq (z) = 0.
• Value matching at z = 1: q(z−) = q(z+).
• Smooth-pasting at z = 1: q (z−) = q (z+).
• Fourth condition?
• Intuitively, it should be at z = ∞.
• Marginal utility z inﬁnite when wealth x is zero.
• Since q (z) = −x/h, the condition is limz→∞ q (z) = 0.
77. Model
Solution
Under the Hood
Solving it
• For r = 0, q(z) has solution:
q(z) =
C21 + C22 z
C31 +
1+
µ
2r
Σ−1 µ
−
z
r
+
2 log z
(1−γ)(2r +µ Σ−1 µ)
γ
1−1/γ
(1−γ)δ0 z
if 0 < z ≤ 1,
if z > 1,
where δα deﬁned shortly.
• Neumann condition at z = 1 − α: (1 − γ ∗ )q(z) + γ ∗ zq (z) = 0.
• Value matching at z = 1: q(z−) = q(z+).
• Smooth-pasting at z = 1: q (z−) = q (z+).
• Fourth condition?
• Intuitively, it should be at z = ∞.
• Marginal utility z inﬁnite when wealth x is zero.
• Since q (z) = −x/h, the condition is limz→∞ q (z) = 0.
78. Model
Solution
Under the Hood
Solving it
• For r = 0, q(z) has solution:
q(z) =
C21 + C22 z
C31 +
1+
µ
2r
Σ−1 µ
−
z
r
+
2 log z
(1−γ)(2r +µ Σ−1 µ)
γ
1−1/γ
(1−γ)δ0 z
if 0 < z ≤ 1,
if z > 1,
where δα deﬁned shortly.
• Neumann condition at z = 1 − α: (1 − γ ∗ )q(z) + γ ∗ zq (z) = 0.
• Value matching at z = 1: q(z−) = q(z+).
• Smooth-pasting at z = 1: q (z−) = q (z+).
• Fourth condition?
• Intuitively, it should be at z = ∞.
• Marginal utility z inﬁnite when wealth x is zero.
• Since q (z) = −x/h, the condition is limz→∞ q (z) = 0.
79. Model
Solution
Under the Hood
Solving it
• For r = 0, q(z) has solution:
q(z) =
C21 + C22 z
C31 +
1+
µ
2r
Σ−1 µ
−
z
r
+
2 log z
(1−γ)(2r +µ Σ−1 µ)
γ
1−1/γ
(1−γ)δ0 z
if 0 < z ≤ 1,
if z > 1,
where δα deﬁned shortly.
• Neumann condition at z = 1 − α: (1 − γ ∗ )q(z) + γ ∗ zq (z) = 0.
• Value matching at z = 1: q(z−) = q(z+).
• Smooth-pasting at z = 1: q (z−) = q (z+).
• Fourth condition?
• Intuitively, it should be at z = ∞.
• Marginal utility z inﬁnite when wealth x is zero.
• Since q (z) = −x/h, the condition is limz→∞ q (z) = 0.
80. Model
Solution
Under the Hood
Solving it
• For r = 0, q(z) has solution:
q(z) =
C21 + C22 z
C31 +
1+
µ
2r
Σ−1 µ
−
z
r
+
2 log z
(1−γ)(2r +µ Σ−1 µ)
γ
1−1/γ
(1−γ)δ0 z
if 0 < z ≤ 1,
if z > 1,
where δα deﬁned shortly.
• Neumann condition at z = 1 − α: (1 − γ ∗ )q(z) + γ ∗ zq (z) = 0.
• Value matching at z = 1: q(z−) = q(z+).
• Smooth-pasting at z = 1: q (z−) = q (z+).
• Fourth condition?
• Intuitively, it should be at z = ∞.
• Marginal utility z inﬁnite when wealth x is zero.
• Since q (z) = −x/h, the condition is limz→∞ q (z) = 0.
81. Model
Solution
Under the Hood
Solving it
• For r = 0, q(z) has solution:
q(z) =
C21 + C22 z
C31 +
1+
µ
2r
Σ−1 µ
−
z
r
+
2 log z
(1−γ)(2r +µ Σ−1 µ)
γ
1−1/γ
(1−γ)δ0 z
if 0 < z ≤ 1,
if z > 1,
where δα deﬁned shortly.
• Neumann condition at z = 1 − α: (1 − γ ∗ )q(z) + γ ∗ zq (z) = 0.
• Value matching at z = 1: q(z−) = q(z+).
• Smooth-pasting at z = 1: q (z−) = q (z+).
• Fourth condition?
• Intuitively, it should be at z = ∞.
• Marginal utility z inﬁnite when wealth x is zero.
• Since q (z) = −x/h, the condition is limz→∞ q (z) = 0.
82. Model
Solution
Under the Hood
Main Quantities
• Ratio between safe rate and half squared Sharpe ratio:
ρ=
2r
(µ/σ)2
In practice, ρ is small ≈ 8%.
• Gloom ratio g (as wealth/target). 1/g coincides with Merton ratio:
1
=
g
1−
1
γ
r+
µ2
2γσ 2
• Bliss ratio:
b=g
(γ − 1)(1 − α)ρ+1 + (γρ + 1)(α(γ − 1)(ρ + 1) − γ(ρ + 1) + 1)
(α − 1)γ 2 ρ(ρ + 1)
• For ρ small, limit:
g
≈
b
1−α+
α
γ2
−
1
γ (1
1−α
1
− γ )(1 − α) log(1 − α)
Ratio insensitive to market parameters, for typical investment
opportunities.
83. Model
Solution
Under the Hood
Main Quantities
• Ratio between safe rate and half squared Sharpe ratio:
ρ=
2r
(µ/σ)2
In practice, ρ is small ≈ 8%.
• Gloom ratio g (as wealth/target). 1/g coincides with Merton ratio:
1
=
g
1−
1
γ
r+
µ2
2γσ 2
• Bliss ratio:
b=g
(γ − 1)(1 − α)ρ+1 + (γρ + 1)(α(γ − 1)(ρ + 1) − γ(ρ + 1) + 1)
(α − 1)γ 2 ρ(ρ + 1)
• For ρ small, limit:
g
≈
b
1−α+
α
γ2
−
1
γ (1
1−α
1
− γ )(1 − α) log(1 − α)
Ratio insensitive to market parameters, for typical investment
opportunities.
84. Model
Solution
Under the Hood
Main Quantities
• Ratio between safe rate and half squared Sharpe ratio:
ρ=
2r
(µ/σ)2
In practice, ρ is small ≈ 8%.
• Gloom ratio g (as wealth/target). 1/g coincides with Merton ratio:
1
=
g
1−
1
γ
r+
µ2
2γσ 2
• Bliss ratio:
b=g
(γ − 1)(1 − α)ρ+1 + (γρ + 1)(α(γ − 1)(ρ + 1) − γ(ρ + 1) + 1)
(α − 1)γ 2 ρ(ρ + 1)
• For ρ small, limit:
g
≈
b
1−α+
α
γ2
−
1
γ (1
1−α
1
− γ )(1 − α) log(1 − α)
Ratio insensitive to market parameters, for typical investment
opportunities.
85. Model
Solution
Under the Hood
Main Quantities
• Ratio between safe rate and half squared Sharpe ratio:
ρ=
2r
(µ/σ)2
In practice, ρ is small ≈ 8%.
• Gloom ratio g (as wealth/target). 1/g coincides with Merton ratio:
1
=
g
1−
1
γ
r+
µ2
2γσ 2
• Bliss ratio:
b=g
(γ − 1)(1 − α)ρ+1 + (γρ + 1)(α(γ − 1)(ρ + 1) − γ(ρ + 1) + 1)
(α − 1)γ 2 ρ(ρ + 1)
• For ρ small, limit:
g
≈
b
1−α+
α
γ2
−
1
γ (1
1−α
1
− γ )(1 − α) log(1 − α)
Ratio insensitive to market parameters, for typical investment
opportunities.
86. Model
Solution
Under the Hood
Main Result
Theorem
For r = 0, the optimal spending policy is:
Xt /b if ht ≤ Xt /b
ˆ
ct = ht
if Xt /b ≤ ht ≤ Xt /g
Xt /g if ht ≥ Xt /g
(1)
The optimal weight of the risky asset is the Merton risky asset weight when the
wealth to target ratio is lower than the gloom point, Xt /ht ≤ g. Otherwise, i.e.,
when Xt /ht ≥ g the weight of the risky asset is
πt =
ˆ
ρ γρ + (γ − 1)z ρ+1 + 1
µ
(γρ + 1)(ρ + (γ − 1)(ρ + 1)z) − (γ − 1)z ρ+1 σ 2
(2)
where the variable z satisﬁes the equation:
(γρ + 1)(ρ + (γ − 1)(ρ + 1)z) − (γ − 1)z ρ+1
x
=
(γ − 1)(ρ + 1)rz(γρ + 1)
h
(3)
87. Model
Solution
Under the Hood
Bliss and Gloom – α
2.0
gloom
Habit Wealth Ratio
1.5
bliss
1.0
0.5
0.0
0.0
0.2
0.4
0.6
Loss Aversion
µ/σ = 0.4, γ = 2, r = 1%
0.8
1.0
88. Model
Solution
Under the Hood
Bliss and Gloom
• Gloom ratio independent of the shortfall aversion α, and its inverse equals
the Merton consumption rate.
• Bliss ratio increases as the shortfall aversion α increases.
• Within the target region, the optimal investment policy is independent of
the shortfall aversion α.
• At α = 0 the model degenerates to the Merton model and b = g, i.e., the
bliss and the gloom points coincide. At α = 1 the bliss point is inﬁnity, i.e.,
shortfall aversion is so strong that the solution calls for no spending
increases at all.
89. Model
Solution
Under the Hood
Bliss and Gloom
• Gloom ratio independent of the shortfall aversion α, and its inverse equals
the Merton consumption rate.
• Bliss ratio increases as the shortfall aversion α increases.
• Within the target region, the optimal investment policy is independent of
the shortfall aversion α.
• At α = 0 the model degenerates to the Merton model and b = g, i.e., the
bliss and the gloom points coincide. At α = 1 the bliss point is inﬁnity, i.e.,
shortfall aversion is so strong that the solution calls for no spending
increases at all.
90. Model
Solution
Under the Hood
Bliss and Gloom
• Gloom ratio independent of the shortfall aversion α, and its inverse equals
the Merton consumption rate.
• Bliss ratio increases as the shortfall aversion α increases.
• Within the target region, the optimal investment policy is independent of
the shortfall aversion α.
• At α = 0 the model degenerates to the Merton model and b = g, i.e., the
bliss and the gloom points coincide. At α = 1 the bliss point is inﬁnity, i.e.,
shortfall aversion is so strong that the solution calls for no spending
increases at all.
91. Model
Solution
Under the Hood
Bliss and Gloom
• Gloom ratio independent of the shortfall aversion α, and its inverse equals
the Merton consumption rate.
• Bliss ratio increases as the shortfall aversion α increases.
• Within the target region, the optimal investment policy is independent of
the shortfall aversion α.
• At α = 0 the model degenerates to the Merton model and b = g, i.e., the
bliss and the gloom points coincide. At α = 1 the bliss point is inﬁnity, i.e.,
shortfall aversion is so strong that the solution calls for no spending
increases at all.
92. Model
Solution
Under the Hood
Spending and Investment as Target/Wealth Varies
5
110
105
100
3
95
90
Portfolio Weight
Spending•Wealth
4
2
85
1
1.0
1.5 Bliss
2.0
Gloom 2.5
80
3.0
Wealth•Target
µ/σ = 0.4,
γ = 2, r = 1%
93. Model
Solution
Under the Hood
Spending and Investment as Wealth/Target Varies
5
110
100
90
3
80
70
Portfolio Weight
Spending•Wealth
4
2
60
1
40
Gloom 45
50
55
60Bliss
50
65
Wealth•Target
µ/
94. Model
Solution
Under the Hood
Investment – α
140
Portfolio
130
120
a = 0.5
a = 0.75
a = .25
110
100
a=0
90
20
40
60
Wealth•Target
80
100
95. Model
Solution
Under the Hood
Steady State
Theorem
• The long-run average time spent in the target zone is a fraction
1 − (1 − α)1+ρ of the total time. This fraction is approximately α because
reasonable values of ρ are close to zero.
• Starting from a point z0 ∈ [0, 1] in the target region, the expected time
before reaching gloom is
Ex,h [τgloom ] =
ρ
log(z0 ) −
(ρ + 1)r
ρ+1
(1 − α)−ρ−1 z0 − 1
ρ+1
In particular, starting from bliss (z0 = 1 − α) and for small ρ,
Ex,h [τgloom ] =
ρ
r
α
+ log(1 − α) + O(ρ2 )
1−α
96. Model
Solution
Under the Hood
Steady State
Theorem
• The long-run average time spent in the target zone is a fraction
1 − (1 − α)1+ρ of the total time. This fraction is approximately α because
reasonable values of ρ are close to zero.
• Starting from a point z0 ∈ [0, 1] in the target region, the expected time
before reaching gloom is
Ex,h [τgloom ] =
ρ
log(z0 ) −
(ρ + 1)r
ρ+1
(1 − α)−ρ−1 z0 − 1
ρ+1
In particular, starting from bliss (z0 = 1 − α) and for small ρ,
Ex,h [τgloom ] =
ρ
r
α
+ log(1 − α) + O(ρ2 )
1−α
97. Model
Solution
Under the Hood
Steady State
Theorem
• The long-run average time spent in the target zone is a fraction
1 − (1 − α)1+ρ of the total time. This fraction is approximately α because
reasonable values of ρ are close to zero.
• Starting from a point z0 ∈ [0, 1] in the target region, the expected time
before reaching gloom is
Ex,h [τgloom ] =
ρ
log(z0 ) −
(ρ + 1)r
ρ+1
(1 − α)−ρ−1 z0 − 1
ρ+1
In particular, starting from bliss (z0 = 1 − α) and for small ρ,
Ex,h [τgloom ] =
ρ
r
α
+ log(1 − α) + O(ρ2 )
1−α
98. Model
Solution
Under the Hood
Time Spent at Target – α
Portion of Constant Consumption
1.0
0.8
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
Loss Aversion
µ/σ = 0.35, γ = 2, r = 1%
0.8
1.0
99. Model
Solution
Under the Hood
Time from Bliss to Gloom – α
Expected Hitting Time at Gloom
120
100
80
60
40
20
0
0.0
0.2
0.4
0.6
Loss Aversion
µ/σ = 0.35, γ = 2, r = 1%
0.8
1.0
100. Model
Solution
Under the Hood
Under the Hood
• With
∞
0
U(ct , ht )dt, ﬁrst-order condition is:
Uc (ct , ht ) = yMt
where Mt = e−(r +µ
Σ−1 µ/2)t−µ Σ−1 Wt
is stochastic discount factor.
−1
• Candidate ct = I(yMt , ht ) with I(y , h) = Uc (y , h). But what is ht ?
• ht increases only at bliss. And at bliss Uc independent of past maximum:
−1/γ
ht = c0 ∨ y −1/γ
inf Ms
s≤t
101. Model
Solution
Under the Hood
Under the Hood
• With
∞
0
U(ct , ht )dt, ﬁrst-order condition is:
Uc (ct , ht ) = yMt
where Mt = e−(r +µ
Σ−1 µ/2)t−µ Σ−1 Wt
is stochastic discount factor.
−1
• Candidate ct = I(yMt , ht ) with I(y , h) = Uc (y , h). But what is ht ?
• ht increases only at bliss. And at bliss Uc independent of past maximum:
−1/γ
ht = c0 ∨ y −1/γ
inf Ms
s≤t
102. Model
Solution
Under the Hood
Under the Hood
• With
∞
0
U(ct , ht )dt, ﬁrst-order condition is:
Uc (ct , ht ) = yMt
where Mt = e−(r +µ
Σ−1 µ/2)t−µ Σ−1 Wt
is stochastic discount factor.
−1
• Candidate ct = I(yMt , ht ) with I(y , h) = Uc (y , h). But what is ht ?
• ht increases only at bliss. And at bliss Uc independent of past maximum:
−1/γ
ht = c0 ∨ y −1/γ
inf Ms
s≤t
103. Model
Solution
Under the Hood
Duality Bound
• For any spending plan ct :
∞
E
0
(ct /ht α )1−γ
x 1−γ
dt ≤
E
1−γ
1−γ
∞
0
1−1/γ
Z∗ t
˜
u
1−1/γ
Zt
Z∗t
γ
dt
γ
˜
˜
where Zt = Mt and Z∗t = infs≤t Zs , and U(y , h) = − h
1−1/γ u (yh ).
• Show that equality holds for candidate optimizer.
Lemma
˜
ˆ
For γ > 1, limT ↑∞ Ex,h V (yMT , ht (y )) = 0 for all y ≥ 0.
• Estimates on Brownian motion and its running maximum.
104. Model
Solution
Under the Hood
Duality Bound
• For any spending plan ct :
∞
E
0
(ct /ht α )1−γ
x 1−γ
dt ≤
E
1−γ
1−γ
∞
0
1−1/γ
Z∗ t
˜
u
1−1/γ
Zt
Z∗t
γ
dt
γ
˜
˜
where Zt = Mt and Z∗t = infs≤t Zs , and U(y , h) = − h
1−1/γ u (yh ).
• Show that equality holds for candidate optimizer.
Lemma
˜
ˆ
For γ > 1, limT ↑∞ Ex,h V (yMT , ht (y )) = 0 for all y ≥ 0.
• Estimates on Brownian motion and its running maximum.
105. Model
Solution
Under the Hood
Duality Bound
• For any spending plan ct :
∞
E
0
(ct /ht α )1−γ
x 1−γ
dt ≤
E
1−γ
1−γ
∞
0
1−1/γ
Z∗ t
˜
u
1−1/γ
Zt
Z∗t
γ
dt
γ
˜
˜
where Zt = Mt and Z∗t = infs≤t Zs , and U(y , h) = − h
1−1/γ u (yh ).
• Show that equality holds for candidate optimizer.
Lemma
˜
ˆ
For γ > 1, limT ↑∞ Ex,h V (yMT , ht (y )) = 0 for all y ≥ 0.
• Estimates on Brownian motion and its running maximum.
106. Model
Solution
Under the Hood
Duality Bound
• For any spending plan ct :
∞
E
0
(ct /ht α )1−γ
x 1−γ
dt ≤
E
1−γ
1−γ
∞
0
1−1/γ
Z∗ t
˜
u
1−1/γ
Zt
Z∗t
γ
dt
γ
˜
˜
where Zt = Mt and Z∗t = infs≤t Zs , and U(y , h) = − h
1−1/γ u (yh ).
• Show that equality holds for candidate optimizer.
Lemma
˜
ˆ
For γ > 1, limT ↑∞ Ex,h V (yMT , ht (y )) = 0 for all y ≥ 0.
• Estimates on Brownian motion and its running maximum.
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