optimal investment to minimize the probability of drawdownoptimal investment to minimize the...
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Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Optimal investment to minimize the
probability of drawdown
Bahman Angoshtari
Joint work with E. Bayraktar and V. R. Young
Financial/Actuarial Mathematics Seminar
University of Michigan
November 2nd, 2016
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Drawdown
ddt = maxs∈[0,t]
Ws −Wt =: Mt −Wt
Max. drawdown
maxt∈[0,T ]
ddt
is a conservative measure for thepotential loss of a portfolio
Max. drawdown duration
maxt∈[0,T ]
{t− arg max
s∈[0,t]Ws
}is a conservative measure of howlong a loss lasts
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Drawdown
ddt = maxs∈[0,t]
Ws −Wt =: Mt −Wt
Max. drawdown
maxt∈[0,T ]
ddt
is a conservative measure for thepotential loss of a portfolio
Max. drawdown duration
maxt∈[0,T ]
{t− arg max
s∈[0,t]Ws
}is a conservative measure of howlong a loss lasts
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Drawdown
ddt = maxs∈[0,t]
Ws −Wt =: Mt −Wt
Max. drawdown
maxt∈[0,T ]
ddt
is a conservative measure for thepotential loss of a portfolio
Max. drawdown duration
maxt∈[0,T ]
{t− arg max
s∈[0,t]Ws
}is a conservative measure of howlong a loss lasts
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Related Literature
Drawdown constraint: Grossman and Zhou (1993), Cvitanic and Karatzas(1995), ... , Cherny and Ob loj (2013), Kardaras et al. (2014), ...
Goal-seeking problems: Typically to minimize probability of ruin or tomaximize probability of reaching a bequest
Dubins and Savage (1965, 1975), Pestien and Sudderth (1985), Karatzas (1997),Browne (1995, 1997, 1999a,b), Young (2004), Promislow and Young (2005),Bayraktar and Young (2007), Bauerle and Bayraktar (2014), Bayraktar andYoung (2016), ...
Maximizingdrift
volatility2, maximizes the prob. of reaching b before reaching a < b
Minimizing probability of drawdown (≥ a fraction of the running maximum):
Bauerle and Bayraktar (2014): the optimizer is the same as the ruin/bequestproblem if the maximized ratio is independent of the state variable
Chen et al. (2015), Angoshtari et al. (2016a), Angoshtari et al. (2016b)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Related Literature
Drawdown constraint: Grossman and Zhou (1993), Cvitanic and Karatzas(1995), ... , Cherny and Ob loj (2013), Kardaras et al. (2014), ...
Goal-seeking problems: Typically to minimize probability of ruin or tomaximize probability of reaching a bequest
Dubins and Savage (1965, 1975), Pestien and Sudderth (1985), Karatzas (1997),Browne (1995, 1997, 1999a,b), Young (2004), Promislow and Young (2005),Bayraktar and Young (2007), Bauerle and Bayraktar (2014), Bayraktar andYoung (2016), ...
Maximizingdrift
volatility2, maximizes the prob. of reaching b before reaching a < b
Minimizing probability of drawdown (≥ a fraction of the running maximum):
Bauerle and Bayraktar (2014): the optimizer is the same as the ruin/bequestproblem if the maximized ratio is independent of the state variable
Chen et al. (2015), Angoshtari et al. (2016a), Angoshtari et al. (2016b)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Related Literature
Drawdown constraint: Grossman and Zhou (1993), Cvitanic and Karatzas(1995), ... , Cherny and Ob loj (2013), Kardaras et al. (2014), ...
Goal-seeking problems: Typically to minimize probability of ruin or tomaximize probability of reaching a bequest
Dubins and Savage (1965, 1975), Pestien and Sudderth (1985), Karatzas (1997),Browne (1995, 1997, 1999a,b), Young (2004), Promislow and Young (2005),Bayraktar and Young (2007), Bauerle and Bayraktar (2014), Bayraktar andYoung (2016), ...
Maximizingdrift
volatility2, maximizes the prob. of reaching b before reaching a < b
Minimizing probability of drawdown (≥ a fraction of the running maximum):
Bauerle and Bayraktar (2014): the optimizer is the same as the ruin/bequestproblem if the maximized ratio is independent of the state variable
Chen et al. (2015), Angoshtari et al. (2016a), Angoshtari et al. (2016b)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement
A riskless with short-rate r > 0 and a stock dSt = µStdt+ σStdBt
Wt the value of the fund, and πt the amount invested in the stock
The fund pays out at at determinist rate c(Wt) ≥ 0, e.g. c(Wt) ≡ c orc(Wt) = c Wt
The budget constraint: dWt = [rWt + (µ− r)πt − c(Wt)] dt+ σπtdBt
We assume (πt) to be progressively measurable and∫ t0π2sds <∞ a.s.
The running maximum (or the “high-water-mark”):
Mt = max{M0, sup0≤s≤tWt
}W0 = w and M0 = m are given, where 0 < w ≤ m
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement
A riskless with short-rate r > 0 and a stock dSt = µStdt+ σStdBt
Wt the value of the fund, and πt the amount invested in the stock
The fund pays out at at determinist rate c(Wt) ≥ 0, e.g. c(Wt) ≡ c orc(Wt) = c Wt
The budget constraint: dWt = [rWt + (µ− r)πt − c(Wt)] dt+ σπtdBt
We assume (πt) to be progressively measurable and∫ t0π2sds <∞ a.s.
The running maximum (or the “high-water-mark”):
Mt = max{M0, sup0≤s≤tWt
}W0 = w and M0 = m are given, where 0 < w ≤ m
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement
A riskless with short-rate r > 0 and a stock dSt = µStdt+ σStdBt
Wt the value of the fund, and πt the amount invested in the stock
The fund pays out at at determinist rate c(Wt) ≥ 0, e.g. c(Wt) ≡ c orc(Wt) = c Wt
The budget constraint: dWt = [rWt + (µ− r)πt − c(Wt)] dt+ σπtdBt
We assume (πt) to be progressively measurable and∫ t0π2sds <∞ a.s.
The running maximum (or the “high-water-mark”):
Mt = max{M0, sup0≤s≤tWt
}W0 = w and M0 = m are given, where 0 < w ≤ m
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement
A riskless with short-rate r > 0 and a stock dSt = µStdt+ σStdBt
Wt the value of the fund, and πt the amount invested in the stock
The fund pays out at at determinist rate c(Wt) ≥ 0, e.g. c(Wt) ≡ c orc(Wt) = c Wt
The budget constraint: dWt = [rWt + (µ− r)πt − c(Wt)] dt+ σπtdBt
We assume (πt) to be progressively measurable and∫ t0π2sds <∞ a.s.
The running maximum (or the “high-water-mark”):
Mt = max{M0, sup0≤s≤tWt
}W0 = w and M0 = m are given, where 0 < w ≤ m
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement
A riskless with short-rate r > 0 and a stock dSt = µStdt+ σStdBt
Wt the value of the fund, and πt the amount invested in the stock
The fund pays out at at determinist rate c(Wt) ≥ 0, e.g. c(Wt) ≡ c orc(Wt) = c Wt
The budget constraint: dWt = [rWt + (µ− r)πt − c(Wt)] dt+ σπtdBt
We assume (πt) to be progressively measurable and∫ t0π2sds <∞ a.s.
The running maximum (or the “high-water-mark”):
Mt = max{M0, sup0≤s≤tWt
}
W0 = w and M0 = m are given, where 0 < w ≤ m
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement
A riskless with short-rate r > 0 and a stock dSt = µStdt+ σStdBt
Wt the value of the fund, and πt the amount invested in the stock
The fund pays out at at determinist rate c(Wt) ≥ 0, e.g. c(Wt) ≡ c orc(Wt) = c Wt
The budget constraint: dWt = [rWt + (µ− r)πt − c(Wt)] dt+ σπtdBt
We assume (πt) to be progressively measurable and∫ t0π2sds <∞ a.s.
The running maximum (or the “high-water-mark”):
Mt = max{M0, sup0≤s≤tWt
}W0 = w and M0 = m are given, where 0 < w ≤ m
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement, cont’
Given a constant 0 < α < 1, drawdownhappens if Wt ≤ α Mt
Define τα = inf{t ≥ 0 : Wt ≤ α Mt}
Minimum probability of (eternal)drawdown
φ(w,m) = infπ
Pw,m(τα <∞); 0 < w ≤ m
m
w
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement, cont’
Given a constant 0 < α < 1, drawdownhappens if Wt ≤ α Mt
Define τα = inf{t ≥ 0 : Wt ≤ α Mt}
Minimum probability of (eternal)drawdown
φ(w,m) = infπ
Pw,m(τα <∞); 0 < w ≤ m
m
w
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement, cont’
Given a constant 0 < α < 1, drawdownhappens if Wt ≤ α Mt
Define τα = inf{t ≥ 0 : Wt ≤ α Mt}
Minimum probability of (eternal)drawdown
φ(w,m) = infπ
Pw,m(τα <∞); 0 < w ≤ m
m
w
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement, cont’
Given a constant 0 < α < 1, drawdownhappens if Wt ≤ α Mt
Define τα = inf{t ≥ 0 : Wt ≤ α Mt}
Minimum probability of (eternal)drawdown
φ(w,m) = infπ
Pw,m(τα <∞); 0 < w ≤ m
m
w
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Problem statement, cont’
Given a constant 0 < α < 1, drawdownhappens if Wt ≤ α Mt
Define τα = inf{t ≥ 0 : Wt ≤ α Mt}
Minimum probability of (eternal)drawdown
φ(w,m) = infπ
Pw,m(τα <∞); 0 < w ≤ m
m
w
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Safe level ws
Assumption: c(w) is continuous,non-negative and non-decreasing
There is a 0 < ws ≤ +∞ such that{r w < c(w); ∀w < ws
r w > c(w); ∀w > ws
Example: c(w) ≡ c ⇒ ws =c
r
c(w) = c w, c > r ⇒ ws = +∞
If Wt ≥ ws, the consumption can be financed by investing risk-free
For ws ≤ w ≤ m, we have φ(w,m) = 0 and π∗ ≡ 0
m
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Safe level ws
Assumption: c(w) is continuous,non-negative and non-decreasing
There is a 0 < ws ≤ +∞ such that{r w < c(w); ∀w < ws
r w > c(w); ∀w > ws
Example: c(w) ≡ c ⇒ ws =c
r
c(w) = c w, c > r ⇒ ws = +∞
If Wt ≥ ws, the consumption can be financed by investing risk-free
For ws ≤ w ≤ m, we have φ(w,m) = 0 and π∗ ≡ 0
m
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Safe level ws
Assumption: c(w) is continuous,non-negative and non-decreasing
There is a 0 < ws ≤ +∞ such that{r w < c(w); ∀w < ws
r w > c(w); ∀w > ws
Example: c(w) ≡ c ⇒ ws =c
r
c(w) = c w, c > r ⇒ ws = +∞
If Wt ≥ ws, the consumption can be financed by investing risk-free
For ws ≤ w ≤ m, we have φ(w,m) = 0 and π∗ ≡ 0
m
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m ≥ ws and α m < w < ws
Drawdown is equivalent to hitting αm=⇒ minimizing probability of ruin
Bauerle and Bayraktar (2014)-The optimizer is obtain by maximizing
r w + (µ− r)π − c(w)
σ2π2
which yields π∗(w) =2(c(w)− r w
)µ− r
independent of m and α
The optimal wealth
dWt = (c(Wt)− rWt){dt+ 2σ
µ−r dBt}
and the min. prob. of ruin/drawdown is φ(w,m) = 1− g(w,m)g(ws,m)
where
g(w,m) =
∫ w
αm
exp
(−∫ y
αm
δ du
c(u)− ru
)dy, δ :=
1
2
(µ− rσ
)2
m
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m ≥ ws and α m < w < ws
Drawdown is equivalent to hitting αm=⇒ minimizing probability of ruin
Bauerle and Bayraktar (2014)-The optimizer is obtain by maximizing
r w + (µ− r)π − c(w)
σ2π2
which yields π∗(w) =2(c(w)− r w
)µ− r
independent of m and α
The optimal wealth
dWt = (c(Wt)− rWt){dt+ 2σ
µ−r dBt}
and the min. prob. of ruin/drawdown is φ(w,m) = 1− g(w,m)g(ws,m)
where
g(w,m) =
∫ w
αm
exp
(−∫ y
αm
δ du
c(u)− ru
)dy, δ :=
1
2
(µ− rσ
)2
m
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m ≥ ws and α m < w < ws
Drawdown is equivalent to hitting αm=⇒ minimizing probability of ruin
Bauerle and Bayraktar (2014)-The optimizer is obtain by maximizing
r w + (µ− r)π − c(w)
σ2π2
which yields π∗(w) =2(c(w)− r w
)µ− r
independent of m and α
The optimal wealth
dWt = (c(Wt)− rWt){dt+ 2σ
µ−r dBt}
and the min. prob. of ruin/drawdown is φ(w,m) = 1− g(w,m)g(ws,m)
where
g(w,m) =
∫ w
αm
exp
(−∫ y
αm
δ du
c(u)− ru
)dy, δ :=
1
2
(µ− rσ
)2
m
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m < ws and α m < w < ws
Drawdown may happen at a levelhigher than α m
The maximumdrift
volatility2 is(µ− r)2
4σ2(c(w)− r w
)not independent of w =⇒Bauerle and Bayraktar (2014) does not applyto the drawdown problem
Let Lπ f(w,m) =[r w + (µ− r)π − c(w)
]fw(w,m) + 1
2σ2π2fww(w,m)
and D = {(w,m) : 0 < α m ≤ w ≤ min(m,ws)}
m
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m < ws and α m < w < ws
Drawdown may happen at a levelhigher than α m
The maximumdrift
volatility2 is(µ− r)2
4σ2(c(w)− r w
)not independent of w =⇒Bauerle and Bayraktar (2014) does not applyto the drawdown problem
Let Lπ f(w,m) =[r w + (µ− r)π − c(w)
]fw(w,m) + 1
2σ2π2fww(w,m)
and D = {(w,m) : 0 < α m ≤ w ≤ min(m,ws)}
m
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m < ws and α m < w < ws
Drawdown may happen at a levelhigher than α m
The maximumdrift
volatility2 is(µ− r)2
4σ2(c(w)− r w
)not independent of w =⇒Bauerle and Bayraktar (2014) does not applyto the drawdown problem
Let Lπ f(w,m) =[r w + (µ− r)π − c(w)
]fw(w,m) + 1
2σ2π2fww(w,m)
and D = {(w,m) : 0 < α m ≤ w ≤ min(m,ws)}
m
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Theorem (Verification for the eternal problem)
If h : D→ R is bounded and continuous and satisfies:
(i) h(·,m) ∈ C2 is non-increasing and convex
(ii) h(w, ·) is continuously differentiable,except possibly at wswhere it has right and left derivative
(iii) hm(m,m) ≥ 0 if m < ws
(iv) h(α m,m) = 1
(v) h(ws,m) = 0 if m > ws
(vi) Lπ h ≥ 0 for all π
Then, h(w,m) ≤ φ(w,m) on Dm
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Theorem (Verification for the eternal problem)
If h : D→ R is bounded and continuous and satisfies:
(i) h(·,m) ∈ C2 is non-increasing and convex
(ii) h(w, ·) is continuously differentiable,except possibly at wswhere it has right and left derivative
(iii) hm(m,m) ≥ 0 if m < ws
(iv) h(α m,m) = 1
(v) h(ws,m) = 0 if m > ws
(vi) Lπ h ≥ 0 for all π
Then, h(w,m) ≤ φ(w,m) on D
Proof: φ(w,m) = infπ
Ew,m(1{τα<∞}), apply Ito’s formula to h(Wπτn ,M
πτn) ...
m
w
ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m < ws: HJB equation
For N ≤ ws and α m ≤ w ≤ m ≤ NsupπL hN =
(r w − c(w)
)hNw − δ
(hNw )2
hNww= 0;
hN (α m,m) = 1, hNm(m,m) = 0
hN (N,N) = 0
(BVP)
Proposition
The solution of (BVP) is hN (w,m) = 1− e−∫Nm f(y)dy g(w,m)
g(N,N)
where f(m) = α
[1
g(m,m)− δ
c(αm)− rαm
].
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m < ws: The optimal strategy
hws is the minimum probability of drawdown on α m ≤ w ≤ m ≤ wsExtra care if ws = +∞
The optimal strategy is π∗(w) = −µ− rσ2
hwswhwsww
=2(c(w)− r w
)µ− r
The same as the one for probability of ruin!
Note that for m < ws, we have π∗(m) > 0The optimal strategy allows for the running max to increase to ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m < ws: The optimal strategy
hws is the minimum probability of drawdown on α m ≤ w ≤ m ≤ wsExtra care if ws = +∞
The optimal strategy is π∗(w) = −µ− rσ2
hwswhwsww
=2(c(w)− r w
)µ− r
The same as the one for probability of ruin!
Note that for m < ws, we have π∗(m) > 0The optimal strategy allows for the running max to increase to ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m < ws: The optimal strategy
hws is the minimum probability of drawdown on α m ≤ w ≤ m ≤ wsExtra care if ws = +∞
The optimal strategy is π∗(w) = −µ− rσ2
hwswhwsww
=2(c(w)− r w
)µ− r
The same as the one for probability of ruin!
Note that for m < ws, we have π∗(m) > 0The optimal strategy allows for the running max to increase to ws
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The optimal strategy
TheoremThe optimal strategy is
π∗(Wt) =
0; Wt ≥ ws2(c(w)−r w
)µ−r ; α m < Wt < ws
the minimum probability of drawdown is
φ(w,m) =
1− g(w,m)
g(ws,m), if αm ≤ w ≤ ws,m ≥ ws,
1− e−∫wsm f(y)dy g(w,m)
g(ws,ws), if αm ≤ w ≤ m < ws,
The optimal strategy is:
independent of α and Mt
for w < ws, it is optimal to let Mt to increase up to ws
m
w
ws
m
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The optimal strategy
TheoremThe optimal strategy is
π∗(Wt) =
0; Wt ≥ ws2(c(w)−r w
)µ−r ; α m < Wt < ws
the minimum probability of drawdown is
φ(w,m) =
1− g(w,m)
g(ws,m), if αm ≤ w ≤ ws,m ≥ ws,
1− e−∫wsm f(y)dy g(w,m)
g(ws,ws), if αm ≤ w ≤ m < ws,
The optimal strategy is:
independent of α and Mt
for w < ws, it is optimal to let Mt to increase up to ws
m
w
ws
m
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The lifetime drawdown problem
The same market setting as before
Introduce time of death of the investor τd ∼ Exp(λ)
Minimum probability of lifetime drawdown
φ(w,m) = infπ
Pw,m(τα < τd); 0 < w ≤ m
Chen et al. (2015) considered the case of
proportional consumption
c(w) = κ w for κ > r =⇒ ws =∞
They showed that it is not optimal for Mt to increase
We consider constant consumption rate: c(w) = c > 0
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The lifetime drawdown problem
The same market setting as before
Introduce time of death of the investor τd ∼ Exp(λ)
Minimum probability of lifetime drawdown
φ(w,m) = infπ
Pw,m(τα < τd); 0 < w ≤ m
Chen et al. (2015) considered the case of
proportional consumption
c(w) = κ w for κ > r =⇒ ws =∞
They showed that it is not optimal for Mt to increase
We consider constant consumption rate: c(w) = c > 0
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The lifetime drawdown problem
The same market setting as before
Introduce time of death of the investor τd ∼ Exp(λ)
Minimum probability of lifetime drawdown
φ(w,m) = infπ
Pw,m(τα < τd); 0 < w ≤ m
Chen et al. (2015) considered the case of
proportional consumption
c(w) = κ w for κ > r =⇒ ws =∞
They showed that it is not optimal for Mt to increase
We consider constant consumption rate: c(w) = c > 0
m
w
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The lifetime drawdown problem
The same market setting as before
Introduce time of death of the investor τd ∼ Exp(λ)
Minimum probability of lifetime drawdown
φ(w,m) = infπ
Pw,m(τα < τd); 0 < w ≤ m
Chen et al. (2015) considered the case of
proportional consumption
c(w) = κ w for κ > r =⇒ ws =∞
They showed that it is not optimal for Mt to increase
We consider constant consumption rate: c(w) = c > 0
m
w
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The lifetime drawdown problem
The same market setting as before
Introduce time of death of the investor τd ∼ Exp(λ)
Minimum probability of lifetime drawdown
φ(w,m) = infπ
Pw,m(τα < τd); 0 < w ≤ m
Chen et al. (2015) considered the case of
proportional consumption
c(w) = κ w for κ > r =⇒ ws =∞
They showed that it is not optimal for Mt to increase
We consider constant consumption rate: c(w) = c > 0
m
w
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m > cr
The safe level is cr: c < r w ⇔ w > c
r
For Wt ≥ cr, we have π∗t = 0 and ϕ(w,m) = 0
For w < cr≤ m
Drawdown is equivalent to hitting αm=⇒ minimizing probability of lifetime ruin
Young (2004)- π∗t =µ− rσ2
1
γ − 1
( cr−Wπ
t
)independent of m and α
γ =1
2r
[(r + λ+ δ) +
√(r + λ+ δ)2 − 4rλ
]> 1, δ =
1
2
(µ− rσ
)2The optimal wealth: dWπ
t =(cr−Wπ
t
){(2δγ−1− r)dt+ µ−r
σ1
γ−1dBt
}the min. prob. of ruin/drawdown: φ(w,m) =
(c/r−wc/r−α m
)γ
m
w
cr
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m > cr
The safe level is cr: c < r w ⇔ w > c
r
For Wt ≥ cr, we have π∗t = 0 and ϕ(w,m) = 0
For w < cr≤ m
Drawdown is equivalent to hitting αm=⇒ minimizing probability of lifetime ruin
Young (2004)- π∗t =µ− rσ2
1
γ − 1
( cr−Wπ
t
)independent of m and α
γ =1
2r
[(r + λ+ δ) +
√(r + λ+ δ)2 − 4rλ
]> 1, δ =
1
2
(µ− rσ
)2
The optimal wealth: dWπt =
(cr−Wπ
t
){(2δγ−1− r)dt+ µ−r
σ1
γ−1dBt
}the min. prob. of ruin/drawdown: φ(w,m) =
(c/r−wc/r−α m
)γ
m
w
cr
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
m > cr
The safe level is cr: c < r w ⇔ w > c
r
For Wt ≥ cr, we have π∗t = 0 and ϕ(w,m) = 0
For w < cr≤ m
Drawdown is equivalent to hitting αm=⇒ minimizing probability of lifetime ruin
Young (2004)- π∗t =µ− rσ2
1
γ − 1
( cr−Wπ
t
)independent of m and α
γ =1
2r
[(r + λ+ δ) +
√(r + λ+ δ)2 − 4rλ
]> 1, δ =
1
2
(µ− rσ
)2The optimal wealth: dWπ
t =(cr−Wπ
t
){(2δγ−1− r)dt+ µ−r
σ1
γ−1dBt
}the min. prob. of ruin/drawdown: φ(w,m) =
(c/r−wc/r−α m
)γ
m
w
cr
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
0 < m ≤ cr : Main result
There exists a critical high-water-mark
m∗ ∈ (0,c
r) such that:
(i) For m ∈ (m∗, cr), the optimal strategy
lets M to increase above m
(ii) For m ∈ (0,m∗], the optimal strategy
keeps Mt = m
m
w
crm∗
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
0 < m ≤ cr : Main result
There exists a critical high-water-mark
m∗ ∈ (0,c
r) such that:
(i) For m ∈ (m∗, cr), the optimal strategy
lets M to increase above m
(ii) For m ∈ (0,m∗], the optimal strategy
keeps Mt = m
m
w
crm∗
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
0 < m ≤ cr : Main result
There exists a critical high-water-mark
m∗ ∈ (0,c
r) such that:
(i) For m ∈ (m∗, cr), the optimal strategy
lets M to increase above m
(ii) For m ∈ (0,m∗], the optimal strategy
keeps Mt = m
m
w
crm∗
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Theorem (Verification for the lifetime problem)
If h : D→ R is bounded and continuous and satisfies:
(i) h(·,m) ∈ C2 is non-increasing and convex
(ii) h(w, ·) is continuously differentiable,except possibly at finitely many pointswhere it has right and left derivative
(iii) hm(m,m) ≥ 0 if m < ws
(iv) h(α m,m) = 1
(v) h(ws,m) = 0 if m > ws
(vi) Lπ h− λ h ≥ 0 for all π
Then, h(w,m) ≤ φ(w,m) on D
m
w
cr
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Theorem (Verification for the lifetime problem)
If h : D→ R is bounded and continuous and satisfies:
(i) h(·,m) ∈ C2 is non-increasing and convex
(ii) h(w, ·) is continuously differentiable,except possibly at finitely many pointswhere it has right and left derivative
(iii) hm(m,m) ≥ 0 if m < ws
(iv) h(α m,m) = 1
(v) h(ws,m) = 0 if m > ws
(vi) Lπ h− λ h ≥ 0 for all π
Then, h(w,m) ≤ φ(w,m) on D
Proof: φ(w,m) = infπ
Ew,m
[∫ ∞0
1{τα<t}λ e−λ tdt
]= inf
πEw,m
[e−λ τα
]apply Ito’s formula to e−λ τnh(Wπ
τn ,Mπτn) ...
m
w
cr
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
For an arbitrary m0 ∈ (0, c/r),consider the BVP on m0 ≤ m ≤ c/r and α m ≤ w ≤ m
supπL h = (r w − c)hw − δ
h2w
hww= λ h
h(α m,m) = 1, hm(m,m) = 0
limm→c/r−
h(m,m) = 0
(BVP)
Legendre transform: φ(y,m) = minw{h(w,m) + w y}
δy2φyy − (r − λ)yφy − λφ+ cy = 0
φ(yαm(m),m) = 1 + αm yαm(m), φy(yαm(m),m) = αm
φy(ym(m),m) = m, φm(ym(m),m) = 0
limm→c/r−
φ (ym(m),m) =c
rym( cr
)lim
m→c/r−φy (ym(m),m) =
c
r
(FBP)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
For an arbitrary m0 ∈ (0, c/r),consider the BVP on m0 ≤ m ≤ c/r and α m ≤ w ≤ m
supπL h = (r w − c)hw − δ
h2w
hww= λ h
h(α m,m) = 1, hm(m,m) = 0
limm→c/r−
h(m,m) = 0
(BVP)
Legendre transform: φ(y,m) = minw{h(w,m) + w y}
δy2φyy − (r − λ)yφy − λφ+ cy = 0
φ(yαm(m),m) = 1 + αm yαm(m), φy(yαm(m),m) = αm
φy(ym(m),m) = m, φm(ym(m),m) = 0
limm→c/r−
φ (ym(m),m) =c
rym( cr
)lim
m→c/r−φy (ym(m),m) =
c
r
(FBP)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Ansatz: φ(y,m) = D1(m)yB1 +D2(m)yB2 + cry
B1 = 12δ
[(r − λ+ δ) +
√(r − λ+ δ)2 + 4λδ
]= γ
γ−1> 1
B2 = 12δ
[(r − λ+ δ)−
√(r − λ+ δ)2 + 4λδ
]< 0
Proposition (solution of FBP)
Assume that z′(m) =g1(z)(c/r −m) + g0(z)
h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)
z(c/r) = 0
(ODE)
has a solution on z : [m0, c/r]→ [0, 1]. Here, gi and hi are known functions.
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Proposition (solution of FBP, cont’)
Then, the solution of (FBP) for (y,m) ∈ [ym(m), yαm(m)]× [m0, c/r] is
φ(y,m) =c
ry −
[B2
B1 −B2+( cr− αm
) 1−B2
B1 −B2yαm(m)
](y
yαm(m)
)B1
+
[B1
B1 −B2−( cr− αm
) B1 − 1
B1 −B2yαm(m)
](y
yαm(m)
)B2
,
where the free boundaries are
1
yαm(m)
B1B2
B1 −B2
(z(m)B1−1 − z(m)B2−1
)=( cr−m
)−( cr− αm
)[B1(1−B2)
B1 −B2z(m)B1−1 +
B2(B1 − 1)
B1 −B2z(m)B2−1
]and ym = z(m)yαm(m)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
z′(m) =g1(z)(c/r −m) + g0(z)
h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)
z(c/r) = 0
(ODE)
A solution of (ODE) on [m0, c/r] yields φ(w,m) for m0 ≤ m ≤ c/r andαm ≤ w ≤ m
The “inverse” of z(m) satisfies an Able equation
Any chance for a closed form solution?
Functions gi and hi are too complicated...
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
z′(m) =g1(z)(c/r −m) + g0(z)
h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)
z(c/r) = 0
(ODE)
A solution of (ODE) on [m0, c/r] yields φ(w,m) for m0 ≤ m ≤ c/r andαm ≤ w ≤ m
The “inverse” of z(m) satisfies an Able equation
Any chance for a closed form solution?
Functions gi and hi are too complicated...
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The auxiliary functions
g0(z) = (1 − α)c
r(zB2 − z
B1 )[(B2 − 1)z
B1−1 − (B1 − 1)zB2−1
]g1(z) = (z
B2 − zB1 )[B1 − B2 + α(B2 − 1)z
B1−1
− α(B1 − 1)zB2−1
+ α(B1 − 1)(B2 − 1)(zB1−1 − z
B2−1)]
h0(z) = (1 − α)2
(c
r
)2
(B1 − B2)zB1+B2−2
[(B1 − 1)z
B2−1 − (B2 − 1)zB1−1
]h1(z) = (1 − α)
c
r
{[(B2 − 1)z
B1−1 − (B1 − 1)zB2−1
]×[
(B1 − 1)zB1−1 − (B2 − 1)z
B2−1 − α(B1 − B2)zB1+B2−2
]− (B1 − B2)z
B1+B2−2[B1 − B2 + α(B2 − 1)z
B1−1 − α(B1 − 1)zB2−1
]}h2(z) =
[(B1 − 1)z
B1−1 − (B2 − 1)zB2−1 − α(B1 − B2)z
B1+B2−2]×[
B1 − B2 + α(B2 − 1)zB1−1 − α(B1 − 1)z
B2−1]
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
z′(m) =g1(z)(c/r −m) + g0(z)
h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)=:
G(m, z)
H(m, z)
z(c/r) = 0
(ODE)
0 bm 5 10 15 20 c
rm
0
0.2
0.4
1=x(bm)
0.8
1
1.2
z
Sign of F (m; z) = G(m; z)=H(m; z)
m = 9(z)
z = 1x(m)
H(
1x(m)
,m)≡ 0 for a known
functions x(m)
G (ξ(z), z) ≡ 0 for a knownfunctions ξ(z)
(ξ(z), z) and(
1x(m)
,m)
intersect at(
1x(m)
, m)
where
m ∈ (0, c/r)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
z′(m) =g1(z)(c/r −m) + g0(z)
h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)=:
G(m, z)
H(m, z)
z(c/r) = 0
(ODE)
0 bm 5 10 15 20 c
rm
0
0.2
0.4
1=x(bm)
0.8
1
1.2
z
Sign of F (m; z) = G(m; z)=H(m; z)
m = 9(z)
z = 1x(m)
H(
1x(m)
,m)≡ 0 for a known
functions x(m)
G (ξ(z), z) ≡ 0 for a knownfunctions ξ(z)
(ξ(z), z) and(
1x(m)
,m)
intersect at(
1x(m)
, m)
where
m ∈ (0, c/r)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
z′(m) =g1(z)(c/r −m) + g0(z)
h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)=:
G(m, z)
H(m, z)
z(c/r) = 0
(ODE)
0 bm 5 10 15 20 c
rm
0
0.2
0.4
1=x(bm)
0.8
1
1.2
z
Sign of F (m; z) = G(m; z)=H(m; z)
m = 9(z)
z = 1x(m)
H(
1x(m)
,m)≡ 0 for a known
functions x(m)
G (ξ(z), z) ≡ 0 for a knownfunctions ξ(z)
(ξ(z), z) and(
1x(m)
,m)
intersect at(
1x(m)
, m)
where
m ∈ (0, c/r)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
z′(m) =g1(z)(c/r −m) + g0(z)
h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)=:
G(m, z)
H(m, z)
z(c/r) = 0
(ODE)
0 bm 5 10 15 20 c
rm
0
0.2
0.4
1=x(bm)
0.8
1
1.2
z
Integral Curves
H(
1x(m)
,m)≡ 0 for a known
functions x(m)
G (ξ(z), z) ≡ 0 for a knownfunctions ξ(z)
(ξ(z), z) and(
1x(m)
,m)
intersect at(
1x(m)
, m)
where
m ∈ (0, c/r)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
z′(m) =g1(z)(c/r −m) + g0(z)
h2(z)(c/r −m)2 + h1(z)(c/r −m) + h0(z)=:
G(m, z)
H(m, z)
z(c/r) = 0
(ODE)
0 m$ bm 5 10 15 20 c
rm
0
0.2
0.4
1=x(bm)
1=x(m$)
0.8
1
1.2
z
The solution of BVP (5.22)
H(
1x(m)
,m)≡ 0 for a known
functions x(m)
G (ξ(z), z) ≡ 0 for a knownfunctions ξ(z)
(ξ(z), z) and(
1x(m)
,m)
intersect at(
1x(m)
, m)
where
m ∈ (0, c/r)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Proposition (solution of ODE)
Assume that there exist solutions z(m) and z(m) of (ODE) in D0 suchthat z (respectively, z) satisfies the terminal condition z(m0) = z0 for(m0, z0) ∈ ∂D0 (respectively, (m0, z0) ∈ ∂+D0) and extends on the left to∂−D0\
{(m, 1/x(m)
)}. Let m∗ (respectively, m∗) be the value of m where z
(respectively, z) intercepts ∂−D0.
Then, there exists a unique solution z(m) of (ODE) in D0 satisfying theterminal condition z(c/r) = 0 and extending on the left to the boundary∂−D0 such that z(m∗) = 1/x(m∗) for some m∗ ∈ (m∗,m∗). In particular,z(m) is not defined on (0,m∗).
0 bm 5 10 15 20 c
rm
0
0.2
0.4
1=x(bm)
0.8
1
1.2
z
D0 and its boundaries
@!D0
@D0
@D0
@+D0
D0
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Proposition (solution of ODE)
Assume that there exist solutions z(m) and z(m) of (ODE) in D0 suchthat z (respectively, z) satisfies the terminal condition z(m0) = z0 for(m0, z0) ∈ ∂D0 (respectively, (m0, z0) ∈ ∂+D0) and extends on the left to∂−D0\
{(m, 1/x(m)
)}. Let m∗ (respectively, m∗) be the value of m where z
(respectively, z) intercepts ∂−D0.
Then, there exists a unique solution z(m) of (ODE) in D0 satisfying theterminal condition z(c/r) = 0 and extending on the left to the boundary∂−D0 such that z(m∗) = 1/x(m∗) for some m∗ ∈ (m∗,m∗). In particular,z(m) is not defined on (0,m∗).
0 bm 5 10 15 20 c
rm
0
0.2
0.4
1=x(bm)
0.8
1
1.2
z
Integral Curves
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Proposition (solution of ODE)
Assume that there exist solutions z(m) and z(m) of (ODE) in D0 suchthat z (respectively, z) satisfies the terminal condition z(m0) = z0 for(m0, z0) ∈ ∂D0 (respectively, (m0, z0) ∈ ∂+D0) and extends on the left to∂−D0\
{(m, 1/x(m)
)}. Let m∗ (respectively, m∗) be the value of m where z
(respectively, z) intercepts ∂−D0.
Then, there exists a unique solution z(m) of (ODE) in D0 satisfying theterminal condition z(c/r) = 0 and extending on the left to the boundary∂−D0 such that z(m∗) = 1/x(m∗) for some m∗ ∈ (m∗,m∗). In particular,z(m) is not defined on (0,m∗).
0 m$ bm 5 10 15 20 c
rm
0
0.2
0.4
1=x(bm)
1=x(m$)
0.8
1
1.2
z
The solution of BVP (5.22)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Proposition (Minimum probability of DD for m∗ ≤ m ≤ cr)
Assume that z(m) is the solution of (ODE) on [m∗, c/r]
Then, for αm ≤ w ≤ m and m∗ ≤ m ≤ c/r
φ(w,m) =B1 − 1
B1 −B2
[B2 +
( cr− αm
)(1−B2) yαm(m)
]( y(w)
yαm(m)
)B1
+1−B2
B1 −B2
[B1 −
( cr− αm
)(B1 − 1) yαm(m)
]( y(w)
yαm(m)
)B2
where y(w) ∈ [ym(m), yαm(m)] uniquely solves
c
r− w =
B1
B1 −B2
[B2
yαm(m)+( cr− αm
)(1−B2)
](y(w)
yαm(m)
)B1−1
− B2
B1 −B2
[B1
yαm(m)−( cr− αm
)(B1 − 1)
](y(w)
yαm(m)
)B2−1
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
For αm ≤ w ≤ m and m∗ ≤ m ≤ c/r
π∗(w,m) =µ− rσ2
B1(B1 − 1)
B1 −B2
[B2
yαm(m)+( cr− αm
)(1−B2)
](y
yαm(m)
)B1−1
+µ− rσ2
B2(1−B2)
B1 −B2
[B1
yαm(m)−( cr− αm
)(B1 − 1)
](y
yαm(m)
)B2−1
0 m$ 5 10 15 20 c
rm
0
0.5
1
?(m
;m)
0 m$ 5 10 15 20 c
rm
0
10
20
?m(m
;m)
#10!3
0 m$ 5 10 15 20 c
rm
-0.1
-0.05
0
?w(m
;m)
0 m$ 5 10 15 20 c
rm
0
5000
10000
?w
w(m!0;
m)
0 m$ 5 10 15 20 c
rm
0
0.05
0.1
?w
w(m
;m)
0 m$ 5 10 15 20 c
rm
0
5
10
:$(m
;m)
m
w
crm∗
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
0 < m < m∗: The restricted problem
No solution for (ODE) on 0 < m < m∗
However, it seems that
π∗(m,m) = 0 for 0 < m ≤ m∗
Consider the restricted problem where
M is not allowed to increasesupπL h = (r w − c)hw − δ
h2w
hww= λ h
h(α m,m) = 1, limw→m−
hw(w,m)
hww(w,m)= 0
The dual FBP corresponds to an optimal controller-stopper problem
This is where we got the ansatz ...
The dual problem reduces to an ODE =⇒ its solution is1
x(m)!
0 m$ bm 5 10 15 20 c
rm
0
0.2
0.4
1=x(bm)
1=x(m$)
0.8
1
1.2
z
The solution of BVP (5.22)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
0 < m < m∗: The restricted problem
No solution for (ODE) on 0 < m < m∗
However, it seems that
π∗(m,m) = 0 for 0 < m ≤ m∗
Consider the restricted problem where
M is not allowed to increasesupπL h = (r w − c)hw − δ
h2w
hww= λ h
h(α m,m) = 1, limw→m−
hw(w,m)
hww(w,m)= 0
The dual FBP corresponds to an optimal controller-stopper problem
This is where we got the ansatz ...
The dual problem reduces to an ODE =⇒ its solution is1
x(m)!
m
w
crm∗
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
0 < m < m∗: The restricted problem
No solution for (ODE) on 0 < m < m∗
However, it seems that
π∗(m,m) = 0 for 0 < m ≤ m∗
Consider the restricted problem where
M is not allowed to increasesupπL h = (r w − c)hw − δ
h2w
hww= λ h
h(α m,m) = 1, limw→m−
hw(w,m)
hww(w,m)= 0
The dual FBP corresponds to an optimal controller-stopper problem
This is where we got the ansatz ...
The dual problem reduces to an ODE =⇒ its solution is1
x(m)!
m
w
crm∗
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
0 < m < m∗: The restricted problem
No solution for (ODE) on 0 < m < m∗
However, it seems that
π∗(m,m) = 0 for 0 < m ≤ m∗
Consider the restricted problem where
M is not allowed to increasesupπL h = (r w − c)hw − δ
h2w
hww= λ h
h(α m,m) = 1, limw→m−
hw(w,m)
hww(w,m)= 0
The dual FBP corresponds to an optimal controller-stopper problem
This is where we got the ansatz ...
The dual problem reduces to an ODE =⇒ its solution is1
x(m)!
m
w
crm∗
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
0 < m < m∗: The restricted problem
No solution for (ODE) on 0 < m < m∗
However, it seems that
π∗(m,m) = 0 for 0 < m ≤ m∗
Consider the restricted problem where
M is not allowed to increasesupπL h = (r w − c)hw − δ
h2w
hww= λ h
h(α m,m) = 1, limw→m−
hw(w,m)
hww(w,m)= 0
The dual FBP corresponds to an optimal controller-stopper problem
This is where we got the ansatz ...
The dual problem reduces to an ODE =⇒ its solution is1
x(m)!
m
w
crm∗
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The solution for 0 < m ≤ cr
The equations for “φ(w,m)” and π∗(w,m) in the restricted problem are thesame as the one for m ∈ [m∗, c/r], expect that we replace z(m) with 1
x(m)!
Let us glue the solutions of the two problems
η(m) =
{1/x(m), 0 ≤ m ≤ m∗
z(m), m∗ ≤ m ≤ c/r
The “free boundary”:
1
yαm(m)
B1B2
B1 −B2
(η(m)B1−1 − η(m)B2−1
)=( cr−m
)−( cr− αm
)[B1(1−B2)
B1 −B2η(m)B1−1 +
B2(B1 − 1)
B1 −B2η(m)B2−1
]
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The solution for 0 < m ≤ cr
The equations for “φ(w,m)” and π∗(w,m) in the restricted problem are thesame as the one for m ∈ [m∗, c/r], expect that we replace z(m) with 1
x(m)!
Let us glue the solutions of the two problems
η(m) =
{1/x(m), 0 ≤ m ≤ m∗
z(m), m∗ ≤ m ≤ c/r
The “free boundary”:
1
yαm(m)
B1B2
B1 −B2
(η(m)B1−1 − η(m)B2−1
)=( cr−m
)−( cr− αm
)[B1(1−B2)
B1 −B2η(m)B1−1 +
B2(B1 − 1)
B1 −B2η(m)B2−1
]
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The solution for 0 < m ≤ cr
The equations for “φ(w,m)” and π∗(w,m) in the restricted problem are thesame as the one for m ∈ [m∗, c/r], expect that we replace z(m) with 1
x(m)!
Let us glue the solutions of the two problems
η(m) =
{1/x(m), 0 ≤ m ≤ m∗
z(m), m∗ ≤ m ≤ c/r
The “free boundary”:
1
yαm(m)
B1B2
B1 −B2
(η(m)B1−1 − η(m)B2−1
)=( cr−m
)−( cr− αm
)[B1(1−B2)
B1 −B2η(m)B1−1 +
B2(B1 − 1)
B1 −B2η(m)B2−1
]
0 m$ bm 5 10 15 20 c
rm
0
0.2
0.4
1=x(bm)
1=x(m$)
0.8
1
1.2
z
The solution of BVP (5.22)
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
The solution for 0 < m ≤ cr
The equations for “φ(w,m)” and π∗(w,m) in the restricted problem are thesame as the one for m ∈ [m∗, c/r], expect that we replace z(m) with 1
x(m)!
Let us glue the solutions of the two problems
η(m) =
{1/x(m), 0 ≤ m ≤ m∗
z(m), m∗ ≤ m ≤ c/r
The “free boundary”:
1
yαm(m)
B1B2
B1 −B2
(η(m)B1−1 − η(m)B2−1
)=( cr−m
)−( cr− αm
)[B1(1−B2)
B1 −B2η(m)B1−1 +
B2(B1 − 1)
B1 −B2η(m)B2−1
]
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Theorem (Minimum probability of DD for 0 < m ≤ cr)
Assume that z(m) is the solution of (ODE) on [m∗, c/r]
and define η and yαm. Then, for αm ≤ w ≤ m and 0 < m ≤ c/r
φ(w,m) =B1 − 1
B1 −B2
[B2 +
( cr− αm
)(1−B2) yαm(m)
]( y(w)
yαm(m)
)B1
+1−B2
B1 −B2
[B1 −
( cr− αm
)(B1 − 1) yαm(m)
]( y(w)
yαm(m)
)B2
where y(w) ∈ [ym(m), yαm(m)] uniquely solves
c
r− w =
B1
B1 −B2
[B2
yαm(m)+( cr− αm
)(1−B2)
](y(w)
yαm(m)
)B1−1
− B2
B1 −B2
[B1
yαm(m)−( cr− αm
)(B1 − 1)
](y(w)
yαm(m)
)B2−1
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
For αm ≤ w ≤ m and 0 < m ≤ c/r
π∗(w,m) =µ− rσ2
B1(B1 − 1)
B1 −B2
[B2
yαm(m)+( cr− αm
)(1−B2)
](y
yαm(m)
)B1−1
+µ− rσ2
B2(1−B2)
B1 −B2
[B1
yαm(m)−( cr− αm
)(B1 − 1)
](y
yαm(m)
)B2−1
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
For αm ≤ w ≤ m and 0 < m ≤ c/r
π∗(w,m) =µ− rσ2
B1(B1 − 1)
B1 −B2
[B2
yαm(m)+( cr− αm
)(1−B2)
](y
yαm(m)
)B1−1
+µ− rσ2
B2(1−B2)
B1 −B2
[B1
yαm(m)−( cr− αm
)(B1 − 1)
](y
yαm(m)
)B2−1
0 m$ 5 10 15 20 c
rm
0
0.5
1
?(m
;m)
0 m$ 5 10 15 20 c
rm
0
10
20
?m(m
;m)
#10!3
0 m$ 5 10 15 20 c
rm
-0.1
-0.05
0
?w(m
;m)
0 m$ 5 10 15 20 c
rm
0
5000
10000
?w
w(m!0;
m)
0 m$ 5 10 15 20 c
rm
0
0.05
0.1
?w
w(m
;m)
0 m$ 5 10 15 20 c
rm
0
5
10
:$(m
;m)
m
w
crm∗
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
For αm ≤ w ≤ m and 0 < m ≤ c/r
π∗(w,m) =µ− rσ2
B1(B1 − 1)
B1 −B2
[B2
yαm(m)+( cr− αm
)(1−B2)
](y
yαm(m)
)B1−1
+µ− rσ2
B2(1−B2)
B1 −B2
[B1
yαm(m)−( cr− αm
)(B1 − 1)
](y
yαm(m)
)B2−1
0
0
5
5
10
:$(w
;m)
10
15
w
15
20
m
c
r0m$5101520c
r
m
w
crm∗
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
0
0.2
0
0.4
0.6
5
0.8
?(w
;m)
1
10
w
15
20
m
c
r0m$5101520c
r
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
0
0
0.2
0.4
5
0.6
?(w
;m)
0.8
10
1
w
15
20 0
m
510m$c
r20c
r
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Conclusion
We determined the optimal strategy to minimize the probability ofdrawdown in two scenarios
For the infinite time horizon problem, the optimal strategy is thesame as the one for infinite time ruin problem
For the lifetime problem with constant consumption, there is atrade-off in allowing the high-water-mark to increase
0 < m ≤ m∗: increasing high-water-mark level makes DD moreprobable (0 < m ≤ m∗)m∗ < m < c/r: letting wealth increase helps fund theconsumption, thus reducing the probability of DD
Will this behavior exist for other types of consumption wherews <∞?
Adding trade-off between risk and return?
Motivation & Lit. The Eternal Problem The Lifetime Problem Conclusion References
Thank you for your attention!
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